~$Binder1

1110 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 43, NO. 8, AUGUST 1998

Hence the feedback interconnection of two stable subsystems withbounding functions'1 and'2 is stable since the internal signals canbe bounded for any bounded exogenous input.

V. CONCLUSION

A boundedness result for nonlinear unity-feedback interconnectionswith stable subsystems is introduced in Theorem 1. Each subsystemis associated with a nondecreasing bounding function. The result re-duces to translating two curves that denote the boundaries of feasibleregions and seeking bounded intersections. No further analysis orconstruction of functions other than the original pair of boundingfunctions is required. The application of this result has a simpletwo-dimensional graphical interpretation.

REFERENCES

[1] C. A. Desoer and M. Vidyasagar,Feedback Systems: Input–OutputProperties. New York: Academic Press, 1975.

[2] C. A. Desoer and C. A. Lin, “Nonlinear unity feedback systems andQ-parametrization,”Analysis and Optimization of Systems, Lecture Notesin Control and Information Sciences, A. Bensoussan and J. L. Lions,Eds. Berlin: Springer Verlag, vol. 62, 1984.

[3] D. J. Hill, “A generalization of the small-gain theorem for nonlinearfeedback systems,”Automatica, vol. 27, no. 6, pp. 1043–1045, 1991.

[4] A. R. Teel, “On graphs, conic relations, and input–output stability ofnonlinear feedback systems,”IEEE Trans. Automat. Contr., vol. 41, pp.702–709, May 1996.

[5] M. Vidyasagar,Nonlinear Systems Analysis, 2nd ed. Englewood Cliffs,NJ: Prentice Hall, 1993.

A New Necessary and Sufficient Condition for StaticOutput Feedback Stabilizability and Commentson “Stabilization via Static Output Feedback”

Yong-Yan Cao, You-Xian Sun, and Wei-Jie Mao

Abstract—In this paper, a counterexample of the above-mentionedpaper1 is reported. It is pointed out that one of the conditions for alinear system to be stabilizable via static output feedback is not correct.A modified necessary and sufficient condition for this problem is alsopresented.

Index Terms—Stabilization, static output feedback.

I. INTRODUCTION

Stabilization of linear systems by static output feedback is aproblem that is practically important and theoretically appealing.Recently, Trofino-Neto and Kucera presented two necessary andsufficient conditions for the existence of a stabilizing static outputfeedback gain matrix,1 but one of them is incorrect.

Manuscript received April 26, 1996; revised September 17, 1996. Thiswork was supported by the National Natural Science Foundation of Chinaunder Grant 69604007.

The authors are with the Institute of Industrial Process Control andNational Laboratory of Industrial Control Technology of Zhejiang University,Hangzhou, 310027, China (e-mail: [email protected]).

Publisher Item Identifier S 0018-9286(98)05800-0.1A. Trofino-Neto and V. Kucera,IEEE Trans. Automat. Contr., vol. 38, pp.

764–765, 1993.

Consider the system described by the equations

_x = Ax +Bu; y = Cx (1)

where x is the state vector,u is the control vector,y is themeasurement vector, andA; B; and C are constant matrices. LetEi andEk denote the orthogonal projection matrices on theImCT

and KerC, respectively, i.e.,

Ei = C+C; Ek = I Ei (2)

and the Hamiltonian matrix be defined as

H(S) =ABR1S BR1BT

Q STR1S (ABR1S)T: (3)

Theorem 3.1 of the paper is stated as follows.Let Ei andEk be the projection matrices defined in (2) andH(s)

the Hamiltonian matrix in (3). Then, the following statements areequivalent.

i) System (1) is stabilizable via static output feedback.ii) There exist matricesQ > 0; R > 0; andL of compatible

dimensions such that the following algebraic matrix equation:

ATP +PAEi(PB+L

T )R1(BTP +L)Ei+Q = 0 (4)

has a unique solutionP > 0 andK = R1(L + BTP )Ei

is stabilizing.iii) The pair (A;B) is stabilizable (by state feedback) and there

exist P > 0; Q > 0; R > 0; and a matrixL in (4) suchthat the Hamiltonian matrix in (3) has no pure imaginaryeigenvalues forS = LEi BTPEk.

First, let us see a counterexample. Consider the system (1) with

A =0 11 0

; B =10; C = [1 a]

which is not stable since its two eigenvalues are 1 and1. It is notdifficult to find that this system cannot be stabilized via static outputfeedback whena > 0, while it can be whena < 0. Let a = 0:5 and

R = 1; L = [0:2 1:7]; Q =5:96 4:884:88 4:04

:

The algebraic matrix equation (4) has a unique solution

P =1:8 1:71:7 1:8

> 0

and thenS = [0:2 0:9]. So the output feedback gain isF = 1:6.But the eigenvalues of the closed-loop systemABFC are2.362and 0.762. Obviously, it is not stable. So the statements i) and iii) inTheorem 3.1 of the paper are not equivalent.

II. A CORRECTED STABILIZABILITY CONDITION

In this section, we give a new necessary and sufficient conditionfor the existence of a stabilizing static output feedback gain matrix.The following lemma is well known [1], [2].

Lemma 1: Let the linear time-invariant system (1) be given. Then,the following statements are equivalent.

1) System (1) is stabilizable via state feedback.2) There exist matricesQ > 0 and R > 0 of compatible

dimensions such that the following algebraic Riccati equation(ARE):

PA +ATP PBR

1BTP +Q = 0

has a unique solutionP > 0.

0018–9286/98$10.00 1998 IEEE

C:\work2011\shamov\yulia7\00704983.pdf 1

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 43, NO. 8, AUGUST 1998 1111

3) There exist matricesP > 0 and R > 0 of compatibledimensions satisfying the following algebraic Riccati inequality(ARI):

PA+ ATP PBR

1BTP < 0:

Now we consider theLQ control problem with the cross term,which we take as the basis for our development. Namely, we considerthe cost functions

v(x0) =1

0

(xTQx+ 2uTSx+ uTRu)dt (5)

whereQ > 0; R > 0; and S are constant weighting matrices ofcompatible dimensions satisfying

Q STR1S > 0: (6)

The solution for theLQ control problem associated with (1), (5),and (6) is [1]

u = Kx; K = R1(BT

P + S);

PA +ATP (BT

P + S)TR1(BTP + S) +Q = 0: (7)

In the paper, the authors ignored the inequality constraint (6).Lemma 2: Let (1) be given. Then, the following statements are

equivalent.

1) System (1) is stabilizable via state feedback.2) There exist matricesQ > 0; R > 0; and S of compatible

dimensions such that the ARE (7) with the cross term satisfyinginequality (6) has a solutionP > 0.

3) There exist matricesP > 0; R > 0; and S of compatibledimensions satisfying the following modified ARI with thecross term:

PA+ ATP (BT

P + S)TR1(BTP + S) + S

TR1S < 0:

(8)

Proof: From Lemma 1, we know that if(A;B) is stabilizable,then there exist matricesR > 0 andQ1 > 0 such that ARE

PA +ATP PBR

1BTP +Q1 = 0

has a unique solutionP > 0. BecauseQ1 > 0, we can always findsomeS such that

Q1 + PBR1S + S

TR1BTP > 0: (9)

For example, letS = S0; S0 = [Im 0nm], where

jj <min(Q1)

max PBR1S0 + ST0 R1BTP

:

Define

Q = Q1 + PBR1S + S

TR1BTP + S

TR1S

thenQ > 0; Q STR1S > 0; and ARE (4) hold. Therefore, 1)implies 2). Note that the ARE (4) can be rewritten as

P (ABR1S) + (ABR

1S)TP

PBR1BTP +Q S

TR1S = 0:

From Lemma 1, we know 2) implies that 1) holds. The rest of theproof is omitted because the equivalence of 2) and 3) is obvious.

Theorem 1: Let Ei andEk be the projection matrices defined in(2). Then, the following statements are equivalent.

1) System (1) is stabilizable via static output feedback.2) There exist matricesQ > 0; R > 0; and L of compatible

dimensions such that the constrained algebraic matrix equation

ATP + PA Ei(PB + L

T )R1(BTP + L)Ei +Q = 0

Q STR1S > 0; S = LEi B

TPEk

has a unique solutionP > 0.3) There exist matricesP > 0; R > 0 and L of compatible

dimensions satisfying the following modified ARI with suitablecross term

ATP + PA Ei(PB + L

T )R1(BTP + L)Ei

+ STR1S < 0: (10)

ThenF = R1(L+BTP )C+ is a stabilizing output feedbackgain matrix.

Proof: From the former part of the proof1 and Lemma 2, thenecessity is obvious and there is only a need to prove the sufficiency.It is not difficult to find that the ARE (4) can be written as

P (ABR1S) + (ABR

1S)TP

PBR1BTP +Q S

TR1S = 0:

From [1], the state feedbackK = R1(S + BTP ) = R1(L +BTP )C+C = R1(L+BTP )Ei = FC stabilizes (1).

In fact, the second condition of statement ii) of Theorem 3.1 in thepaper has been modified to inequality constraint (6).

Let us continue the above example witha = 1. It is not difficultto find that we can find (4) and (6) will hold if

L = [0 1:5]; R = 1; S = [0 1:5]

P =2 0:50:5 1

> 0; Q =3 11 3

> 0:

So F = 2 is a stabilizing output feedback gain. In fact, botheigenvalues of the closed-loop systemA BFC are1 and1.

III. CONCLUDING REMARKS

The main point of this paper is to correct the result of theabove-mentioned paper. A necessary and sufficient condition forstabilizability of a linear time-invariant plants via static outputfeedback is given using the forms of ARE and ARI.

We notice that the ARI (10) is a quadratic matrix inequality.Because of the negative sign in theEiMTR1MEi; (M =BTP+L) term, (10) cannot be simplified to a linear matrix inequality(LMI). But we may use an iterative LMI algorithm to solve it sincethis ARI will be reduced to an LMI problem, which may be solvednumerically very efficiently [2], ifM andR are previously fixed.This is also one of our research interests.

REFERENCES

[1] B. D. O. Anderson and J. B. Moore,Linear Optimal Control. Engle-wood Cliffs, NJ: Prentice-Hall, 1971.

[2] B. Boyd, L. E. Ghaoui, E. Feron, and V. Balakrishnan,Linear MatrixInequalities in System and Control Theory. Philadelphia, PA: SIAM,1994.

[3] G. Gu, “On the existence of linear optimal control with output feed-back,” SIAM J. Contr. Optimiz., pp. 711–719, 1990.


IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 49, NO. 6, JUNE 2004 995

Optimization of Static Output Feedback UsingSubstitutive LMI Formulation

Atsushi Fujimori

Abstract—This note proposes a new design tool for optimizing staticoutput feedback using a linear matrix inequality (LMI) formula called sub-stitutive LMI. A matrix inequality derived from static output feedback isnot usually linear. Adding a positive definite term including auxiliary vari-ables, the matrix inequality is transformed into an LMI with respect to thepositive definite matrix and the static output feedback gain. An iterativecalculation algorithm is given to solve the substitutive LMI. In this note,designs of the static output feedback gain are shown in the frame of HandH syntheses. A numerical example is shown to demonstrate the effec-tiveness of the proposed technique.

Index Terms—H andH syntheses, iterative calculation, linear matrixinequality (LMI), static output feedback.

I. INTRODUCTION

Static output feedback design is one of open problems in controlcommunity. A number of papers treating static output feedback designhave been published. Unlike state feedback design, a static output feed-back gain which stabilizes the closed-loop system is not always foundeven if the plant is controllable and/or observable [1]–[4]. Recently, de-sign methods of static output feedback using linear matrix inequality(LMI) have been proposed by many researchers [5]–[10]. Crusius andTrofino [5] presented LMI conditions for static output feedback and ex-tended them to the cases of discrete-time systems, decentralized outputfeedback, H1 control, and robust control systems with polytopic un-certainties. Benton and Smith [7] presented another LMI formula, inwhich any iterative calculation is not required. Cao et al. [8] discusseda simultaneous stabilization by static output feedback. Ghaoui et al.[9] and Iwasaki [10] developed iterative algorithms for designing staticoutput feedback gains and fixed-order controllers.

This note proposes a new design tool for optimizing static outputfeedback using an LMI formula called substitutive LMI. A matrixinequality derived from static output feedback is not usually linear.Adding a positive–definite term including auxiliary variables, thematrix inequality is transformed into an LMI with respect to thepositive definite matrix and the feedback gain. An iterative calculationalgorithm is given to solve the substitutive LMI. The iterative calcu-lation presented in this note is related to LQ control problems by Caoet al. [8] and Shimomura and Fujii [11]. This note refines the LMIsderived in [8] and [11] and extends them to the designs of static outputfeedback in the frame of H1 and H2 syntheses. A numerical exampleis shown to demonstrate the effectiveness of the proposed technique.

In this note, the n n unit matrix is written as In, and an n n

positive definite matrix is denoted as P 2 Rnn > 0.

II. STATIC OUTPUT FEEDBACK BY SUBSTITUTIVE LMI

This section proposes the substitutive LMI for static output feedback.Consider a generalized plant (1)

_x(t) = Ax(t) +B1w(t) +B2u(t)

z(t) = C1x(t) +D11w(t) +D12u(t)

y(t) = C2x(t) +D21w(t)

(1)

Manuscript received June 17, 2003; revised February 21, 2003 and October21, 2003. Recommended by Associate Editor A. Bemporad.

The author is with the Department of Mechanical Engineering, Shizuoka Uni-versity, Hamamatsu 432-8561, Japan (e-mail: [email protected]).

Digital Object Identifier 10.1109/TAC.2004.829633

where x 2 Rn is the state, w 2 Rr is the external signal, u 2 Rm isthe input, z 2 Rq is the controlled output and y 2 Rp is the measuredoutput. The following assumption is holds on the generalized plant (1):

A1) (A;B2) is stabilizable and (C2; A) is detectable.The objective in this note is to design a static output feedback law

given by

u(t) = Fy(t) (2)

which stabilizes the closed-loop system and satisfies specified controlperformances such as H1, H2 norm constraints, and so on.

A. Substitutive LMI Based on Quadratic Stability

In this section, static output feedback design using the substitutiveLMI is explained in the frame of the quadratic stability, where D21 =0. Substituting (2) into (1), the closed-loop system is stable only if thereexists a Lyapunov function V (x)

= xTPx, P 2 Rnn > 0 such that

_V < 0 (3)

holds. Expressing (3) in terms of a matrix inequality, we have

(P; FC2)= P (AB2FC2) + (AB2FC2)

TP < 0: (4)

Instead of (4), consider a sufficient matrix inequality

(P; FC2) + T < 0

=

LR1BT2 P

M FC2

=

R 0

0 R(5)

where L 2 Rmn and M 2 Rmn are introduced to solve (5) as anLMI and are called substitutive variables in this note.R 2 Rmm > 0is a positive–definite weighting matrix and is given according to eachcontrol specification. Expanding (5), we have

(P;L)++ FC2R1BT2 P

T

R FC2R1BT2 P <0

(P;L)=P (AB2L)+(AB2L)

TP <0

=LTRL+MT

RMMTRFC2(FC2)

TRM: (6)

Using the Schur complement [12], (6) is transformed into

(P;L) + FC2 R1BT2 P

T

FC2 R1BT2 P R1

< 0: (7)

If L and M are fixed, (7) is an LMI with respect to P and F and canbe solved by an LMI solver. In this note, (7) is referred as substitutiveLMI. Similar techniques were used in [8] and [11] which were basedon LQ control problems. The substitutive LMI proposed in this noteis an extended formula to a general control problem. If the followingrelations hold:

LR1BT2 P =0 (8)

M FC2 =0 (9)

then (5) coincides with (4). Taking into consideration this in the calcu-lation algorithm shown in the next section, L and M will be given soas to satisfy (8) and (9) as much as possible.

0018-9286/04$20.00 © 2004 IEEE


996 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 49, NO. 6, JUNE 2004

The dual form of (7) can be also derived. Defining Q= P1, (4) is

equivalent to the following matrix inequality:

(Q;B2F )= (AB2FC2)Q+Q(AB2FC2)

T< 0: (10)

Similar to (5), a sufficient matrix inequality for (10) is given by

(Q;B2F ) + T< 0

= K QCT

2 W1

N B2F =

W 0

0 W(11)

where K 2 Rnp and N 2 Rnp are the substitutive variables andare introduced to solve (11) as an LMI. W 2 Rpp > 0 is a pos-itive–definite weighting matrix. Expanding (11) and using the Schurcomplement, (11) is transformed into

(Q;K)+ B2FQCT2 W

1

B2FQCT2 W

1 TW1

<0

(Q;K)=(AKC2)Q+Q(AKC2)

T<0

=KWK

T+NWNTNW (B2F )

TB2FWNT: (12)

In this note, (7) is called standard form, while (12) is called dual form.

B. Iterative Calculation Algorithm of Substitutive LMI

This section shows an iterative calculation algorithm for solving thesubstitutive LMIs (7) and (12). A basic idea of the algorithm is as fol-lows. If the substitutive variables L,M ,K andN are fixed, (7) is thenan LMI with respect to P and F , while (12) is an LMI with respectto Q and F . Therefore, after L,M , K and N are given appropriately,(7) or (12) can be solved by a conventional LMI solver [13]. Thesemanipulations are iterated until the solutions are converged. The itera-tive calculation algorithm is given as follows, where the superscript kmeans the number of iteration and the equations in the round bracketsindicate the calculation of the dual form.

Step 1) Give a positive–definite weighting matrixR > 0(W > 0).Set k = 1. Select an initial value L(1)(K(1)) such thatAB2L

(1)(AK(1)C2) is stable.M (1)(N (1)) is givenas

M(1) = L

(1)N

(1) = K(1)

: (13)

Step 2) Solve LMI (7) [LMI (12)] with respect toP (k) > 0(Q(k) >

0) and F (k).Step 3) L(k+1) and M (k+1) (K(k+1) and N (k+1)) are given by

L(k+1)=R1BT

2 P(k)

K(k+1)=Q(k)

CT2 W

1 (14)

M(k+1)=F (k)

C2 N(k+1)=B2F

(k): (15)

Step 4) If it is judged that the solution is converged, stop the calcu-lation. Then, the static output feedback gain is obtained asF (k). Otherwise set k ! k + 1 and go to Step 2).

Some comments should be given on the aforementioned algorithm.From assumption A1), an initial value L(1) (K(1)) which stabilizesAB2L

(1) (AK(1)C2) is always found because L(1) (K(1)) cor-responds to the state feedback (full-order observer) gain. However, it isnot guaranteed whetherP (k) > 0 (Q(k) > 0) andF (k) satisfying LMI(7) [LMI (12)] are always found at Step 2) or not. It depends on L(1)

(K(1)) and R(W ). If LMI (7) [LMI (12)] could not be solved duringsome iteration, it is needed to change L(1) (K(1)) and/or R(W ).

At Step 4), the following norms of the difference between the (k)thand (k 1)th with respect to P (k) (Q(k)) and F (k):

(k)P

= P

(k) P (k1)(k)Q

= Q

(k) Q(k1)

(k)F

= F

(k) F (k1) (16)

may be indexes for the judging the convergence of the iterative calcu-lation. In H1 and H2 syntheses shown in the following section, H1or H2 norm of the closed-loop system can be used for this. Strictlyspeaking, the convergence of the proposed calculation is not guaran-teed. However, once a feasible solution is found, P (k), F (k), L(k), andM (k) are adjusted to reduce the sufficiency of the substitutive LMI (7);that is, to satisfy (8) and (9). This implicitly indicates the convergenceof the calculation.

III. SUBSTITUTIVE LMI FORMULAS FOR H1 AND H2 SYNTHESES

This section presents the substitutive LMIs for designing staticoutput feedback gains in H1 and H2 control syntheses.

A. H1 Synthesis

The generalized plant for H1 synthesis is given by (1). UsingDoyle’s notation [14], the transfer function of the closed-loop systemTzw(s) is written as

Tzw(s) =AB2FC2 B1 B2FD21

C1 D12FC2 D11 D12FD21

=

Acl Bcl

Ccl Dcl

: (17)

The objective of H1 synthesis is to stabilize the closed-loop systemand satisfy H1 norm constraint

kTzw(s)k1< for > 0 (18)

where is an upper bound of H1 norm and is given in advance.1) Standard Form: (18) can be represented as a matrix inequality

[14]

1=

PAcl + ATclP PBcl CTcl

BTclP Ir DT

cl

Ccl Dcl Iq

< 0: (19)

Consider the following matrix inequality which is given by adding apositive–semidefinite term to (19):

1+T11<0

1=

LR1BT2 P UFD21 0

MFC2 0 0

=

R 0

0 R(20)

L 2 Rmn, M 2 Rmn, and U 2 Rmr are the substitutive vari-ables. R 2 Rmm > 0 is a positive–definite weighting matrix. Ex-panding (20) and using the Schur complement, (20) is transformed into(21)

(P;L)+1

(B1B2U)TP Ir

C1 D12FC2 D11D12FD21 Iq

FC2R1BT

2 P 0 0 R1

L UFD21 0 0 R1

<0

1=MT

RMMTRFC2(FC2)

TRM (21)

where indicates the elements that are readily inferred by symmetry.



2) Dual Form: (18) can be represented as another matrix inequality[14]

1=

AclQ+QATcl QCT

cl Bcl

CclQ Iq Dcl

BTcl DT

cl Ir

< 0: (22)

Consider the following matrix inequality which is given by adding apositive–semidefinite term to (22)

1+1T1<0

1=

KQCT2 W

1 NB2F

V D12F 0

0 0

=

W 0

0 W(23)

K 2 Rnp,N 2 Rnp, and V 2 Rqp are the substitutive variables.W 2 Rpp > 0 is a positive–definite weighting matrix. Expanding(23) and using the Schur complement, (23) is transformed into (24), asshown at the bottom of the page. InH1 synthesis, another substitutivevariable U or V is introduced. At Step 3) of the iterative calculation,U (k+1) (V (k+1)) is given by

U(k+1) = F

(k)D21 V

(k+1) = D12F(k)

: (25)

B. H2 Synthesis

The generalized plant for H2 synthesis is given by (1), where theclosed-loop system Tzw(s) must be strictly proper (Dcl = 0). There-fore, the following assumptions have to hold.

A2) D11 = 0.A3) Either of the following is imposed on D12 or D21.

a) D21 = 0 and rankD12 = m.b) D12 = 0 and rankD21 = p.

The objective ofH2 synthesis is to stabilize the closed-loop system andsatisfy H2 norm constraint:

kTzw(s)k2 < ; for > 0 (26)

where is an upper bound of H2 norm and is given in advance. InH2 synthesis, the standard or the dual form is derived according toassumption A3).

1) Standard Form: Suppose that assumptions A1), A2), and A3a)are held. The closed-loop system Tzw(s) is given by

Tzw(s) =Acl B1

Ccl 0(27)

Equation (26) can be then represented as the following matrix inequal-ities [14]

2= PAcl + A

TclP + C

TclCcl < 0 (28a)

P PB1

BT1 P X

> 0 (28b)

trX < 2: (28c)

Equations (28b) and (28c) are LMIs with respect to P andX 2 Rrr > 0, while (28a) is a BMI with respect to P and F . Then,the substitutive LMI is applied to only (28a) as follows:

2 + LR1BT2 P

T

R LR1BT2 P < 0 (29)

where

R= D

T12D12 > 0: (30)

Expanding (29) and using the Schur complement, (29) is transformedinto

(P;L)+2 FC2R1BT

2 PT

FC2R1BT

2 P R1<0

2=CT

1 C1+LTRLCT

1 D12FC2(FC2)TD

T12C1: (31)

LMIs (28b), (28c), and (31) are simultaneously solved by the iterativecalculation.

2) Dual Form: Suppose that assumptions A1), A2), and A3b) areheld. The closed-loop system Tzw(s) is given by

Tzw(s) =Acl Bcl

C1 0: (32)

Equation (26) can be then represented as the following matrix inequal-ities [14]:

2= AclQ +QA

Tcl +BclB

Tcl < 0 (33a)

Q QCT1

C1Q Y> 0 (33b)

trY < 2: (33c)

Equations (33b) and (33c) are LMIs with respect to Q andY 2 Rqq > 0, while (33a) is a BMI with respect to Q and F . Then,The substitutive LMI is applied to only (33a) as follows.

2 + K QCT2 W

1W K QC

T2 W

1T

< 0 (34)

(Q;K) + 1

(C1 V C2)Q Iq

(B1 B2FD21)T (D11 D12FD21)

T Ir

(B2F QC2W1)

T0 0 W1

KT (V D12F )T 0 0 W1

< 0

1= NWN

T NW (B2F )T B2FWN

T: (24)


998 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 49, NO. 6, JUNE 2004

where

W= D21D

T

21 > 0: (35)

Expanding (34) and using the Schur complement, (34) is transformedinto

(Q;K)+2 B2FQCT

2 W1

B2KQCT

2 W1 T

W1<0

2=B1B

T

1 +KWKTB1(B2FD21)

TB2FD21BT

1 : (36)

That is, (33b), (33c), and (36) are simultaneously solved by the iterativecalculation.R and W given by (30) and (35) are positive definite matrices to

save the substitutive variables. Even ifR andW were not given by (30)and (35), the substitutive LMIs can be derived with another substitutivevariablesM and N such as the quadratic stability and H1 synthesis.

IV. NUMERICAL EXAMPLE

To demonstrate the proposed design method of static output feed-back by the substitutive LMI, a numerical example is shown in thissection. The state-space data of (1) were given by the longitudinal mo-tion of a helicopter [10], [15]

A =

0:0366 0:0271 0:0188 0:4555

0:0482 1:01 0:0024 4:0208

0:1002 0:3681 0:707 1:42

0 0 1 0

B1 =

1 0 0

0 0 0

0 0 0

0 0 0

B2 =

0:4422 0:1761

3:5446 7:5922

5:52 4:49

0 0

C1 =0 1 0 0

0 0 0 1C2 =

1 0 0 0

0 1 0 0

D11 =0:5 0 0

0 1 0D12 =

1 0

0 1

D21 =0 0:1 0

0 0 0:1: (37)

The size of the variable vectors in (1) was that n = 4, r = 3, m = 2,q = 2 and p = 2. The eigenvalues of A was 0.2325, 2.0727, and0:27579 j2:5758; that is, the controlled plant was unstable. Usingthis data, the static output feedback gain F was designed by H1 andH2 syntheses.

A. H1 Synthesis

The static output feedback gain F with H1 norm constraint wasdesigned by the standard and the dual forms. R in (21) was given byI2. The initial value L(1) was obtained by an optimal regulator whoseweights for the state and the input were, respectively, given by I4 andI2

L(1) =

0:9266 0:01474 0:9622 1:387

0:02248 0:8448 0:1886 0:7135:

Fig. 1 shows plots of the norms of the difference k(k)Pk and k(k)

Fk

with respect to the number of iteration, where the maximum singularvalue was used as the norm. The both norms almost converged untilthe tenth iteration. Fig. 2 shows a plot of H1 norm of the closed-loop

Fig. 1. Convergence evaluation by norms of difference and . Hsynthesis, standard form.

Fig. 2. H norm of closed-loop system.H synthesis, standard form.

system kTzwk1

with respect to the iteration. kTzwk1

finally con-verged to 1.183. The static output feedback gain was obtained as

F =0:02970 4:004

0:3892 5:244:

The eigenvalues of the closed-loop system was 55.03, 0.3138 and0:2345 j1:340.

Fig. 3 shows a plot of kTzwk1

, where the dual form ofH1 synthesiswas used. W in (24) was given by I2. Similar to Fig. 2, the iterativecalculation successfully converged and kTzwk

1at the twentieth was

1.207. The static output feedback gain was obtained as

F =0:05341 1:159

0:01364 1:551:

The eigenvalues of the closed-loop system was 17.10, 0.1307 and0:2191 j1:322.



Fig. 3. H norm of closed-loop system.H synthesis, dual form.

The H1 norm of the closed-loop system achieved by the standardform was almost the same as that by the dual form, but the static outputfeedback gains were different in this numerical example. should begiven as a larger value than that of kTzwk

1finally achieved. Other-

wise, P (k) > 0 and F (k) which satisfied (21) were not found by anLMI solver [13]. In this numerical example, in (21) and (24) wasgiven by 2 and 1.5, respectively.

B. H2 Synthesis

The static output feedback gain F withH2 norm constraint was nextdesigned for the same helicopter model (37), where we reset D11 = 0

and D12 = 0. Then, assumptions A2) and A3a) held. Applying thestandard form of H2 synthesis, H2 norm of the closed-loop systemkTzwk2 converged to 0.5747. The feedback gain was obtained as

F =2:7599 11:846

4:4460 19:370:

The eigenvalues of the closed-loop system was190.49,0.1404, and0:3029 j1:322.

Moreover, the dual form of H2 synthesis was examined, where wereset D11 = 0 and D21 = 0 in (37). kTzwk2 converged to 2.879. Thefeedback gain was obtained as

F =68:800 99:475

27:864 19:141

The eigenvalues of the closed-loop system was216.77, 26.92, and0:3366 j2:104.

V. CONCLUDING REMARKS

This note has proposed a new design tool for optimizing static outputfeedback using an LMI formula called substitutive LMI. A matrix in-equality for static output feedback is not usually linear. Adding a pos-itive definite term including auxiliary variables, the matrix inequalitywas transformed into an LMI with respect to the positive definite matrixand the static output feedback gain. An iterative calculation algorithm

was given to solve the substitutive LMI. For the sake of the page lim-itation, this note presented formulas of H1 and H2 syntheses by thesubstitutive LMI. The substitutive LMIs for the pole placement con-straints and the impulse response can be also derived.

In addition to the numerical example shown in this note, I appliedthe substitutive LMI to several numerical examples in [5], [7], [8], and[16]. The static output feedback gains for the fourth-order plant [5]and the third-order plants [7] and [8] were obtained by the substitutiveLMI, but the gain for the eighth-order plant [16] could not be found.It was depended on the initial and updated values of L and K in theiterative calculation. One of future subjects to research is to improvethe solvability of the substitutive LMI with respect to the initial values.

REFERENCES

[1] A. Trofino and V. Kucera, “Stabilization via static output feedback,”IEEE Trans. Automat. Contr., vol. 38, pp. 764–765, May 1993.

[2] V. Kucera and C. E. Souza, “A necessary and sufficient condition foroutput feedback stabilizability,” Automatica, vol. 31, pp. 1357–1359,Sept. 1995.

[3] V. L. Syrmos, C. T. Abdallah, P. Dorato, and K. Grigoriadis, “Staticoutput feedback—A survey,” Automatica, vol. 33, pp. 125–137, Feb.1997.

[4] Y.-Y. Cao, Y.-X. Sun, and W.-J. Mao, “A new necessary and sufficientcondition for output feedback stabilizability and comments on stabiliza-tion via static output feedback,” IEEE Trans. Automat. Contr., vol. 43,pp. 1110–1111, Aug. 1998.

[5] C. A. R. Crusius and A. Trofino, “Sufficient LMI conditions for outputfeedback control problems,” IEEE Trans. Automat. Contr., vol. 44, pp.1053–1057, May 1999.

[6] J. C. Geromel, C. C. de Souza, and R. E. Skelton, “LMI numerical so-lution for output feedback stabilization,” in Proc. Amer. Control Conf.,1994, pp. 40–44.

[7] R. E. Benton and D. Smith, “Static output feedback stabilization withprescribed degree of stability,” IEEE Trans. Automat. Contr., vol. 43,pp. 1493–1496, Oct. 1998.

[8] Y.-Y. Cao, Y.-X. Sun, and J. Lam, “Simultaneous stabilization via staticoutput feedback and state feedback,” IEEE Trans. Automat. Contr., vol.44, pp. 1277–1282, June 1999.

[9] L. E. Ghaoui, F. Oustry, and M. AitRami, “A cone complementaritylinearization algorithm for static output-feedback and related problems,”IEEE Trans. Automat. Contr., vol. 42, pp. 1171–1176, Aug. 1997.

[10] T. Iwasaki, “The dual iteration for fixed-order control,” IEEE Trans. Au-tomat. Contr., vol. 44, pp. 783–788, Apr. 1999.

[11] T. Shimomura and T. Fujii, “An iteration method for mixedH andHcontrol design with uncommon LMI solutions,” in Proc. Amer. ControlConf., 1999, pp. 3292–3296.

[12] S. Boyd, L. E. Ghaoui, E. Feron, and V. Balakrishnan, Linear MatrixInequalities in System and Control Theory. Philadelphia, PA: SIAM,1994, vol. 15, SIAM Studies in Applied Mathematics.

[13] P. Gahinet, A. Nemirovski, A. J. Laub, and M. Chilali, LMI ControlToolbox. Natick, MA: The Math Works Inc., 1995.

[14] C. Scherer, P. Gahinet, and M. Chilali, “Multiobjective output-feedbackcontrol via LMI optimization,” IEEE Trans. Automat. Contr., vol. 42,pp. 896–911, July 1997.

[15] W. E. Schmitendorf, “A design methodology for robust stabilizing con-trollers,” AIAA J. Guid., Control, Dyna., vol. 10, no. 2, 1987.

[16] X. A. Wang, “Grassmannian central projection and output feedback poleassignment of linear systems,” IEEE Trans. Automat. Contr., vol. 41, pp.786–794, June 1996.



IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 49, NO. 12, DECEMBER 2004 2271

[17] S. Boyd, L. E. Ghaoui, E. Feron, and V. Balakrishnan, Linear Matrix In-equality in System and Control Theory. Philadelphia, PA: SIAM, 1994.

[18] J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differ-ential Equations (Applied Mathematical Sciences, Vol. 99). NewYork:Springer-Verlag, 1993.

[19] L. E. Els’golts’ and S. B. Norkin, Introduction to the Theory and Ap-plication of Differential Equations With Deviating Arguments (Mathe-matics in Science and Engineering: Vol. 105). New York: Academic,1973.

[20] L. Xie, “Output feedbackH control of systems with parameter uncer-tainty,” Int. J. Control, vol. 63, no. 4, pp. 741–750, 1996.

A Convergent Algorithm for Computing StabilizingStatic Output Feedback Gains

Jen-te Yu

Abstract—We revisit the approach by Cao et al. that uses a fixed-struc-ture control law to find stabilizing static output feedback gains for lineartime-invariant systems. By performing singular value decomposition on theoutput matrix, together with similarity transformations, we present a newstabilization method. Unlike their results that involve a difficult modifiedRiccati equation whose solution is coupled with other two intermediate ma-trices that are difficult to find, we obtain Lyapunov equations. We presenta convergent algorithm to solve the new design equations for the gains. Wewill show that our new approach, like theirs, is a dual optimal output feed-back linear quadratic regulator theory. Numerical examples are given toillustrate the effectiveness of the algorithm and validate the new method.

Index Terms—Dual linear quadratic regulator (LQR), Lyapunov equa-tion, modified Riccati equation, static output feedback.

I. INTRODUCTION

Consider the following problem. Given a continuous linear time-in-variant system

_x = Ax +Bu y = Cx E [x(0)x(0)T ] = X > 0

x 2 <n; u 2 <

my 2 <

r (1)

where x(t) is the state, u(t) is the input, y(t) is the output, and E[ ]is the expected value. Find a static output feedback gain F such thatu = Fy stabilizes the given system; see [1], and [5]–[11]. In thisnote, we revisit the approach by Cao et al. [2] that uses a fixed-structurecontrol law to find the stabilizing output feedback gains; see also [3]and [4]. The main drawbacks of the aforementioned approach are thatits solution involves a difficult modified Riccati equation (or inequality[2]), together with two other matrices (coupled with the solution of themodified Riccati equation) that are difficult to find. To overcome thesedifficulties we develop a new stabilization method. In this note K de-notes the full state feedback gain, and F stands for the output feedbackgain. The superscript + denotes the Moore–Penrose inverse. A normwith subscript Fro denotes the Frobenius norm. The note is outlinedas follows. In Section II, we briefly review the results of [2]–[4], andpresent a new stabilization method. In Section III, we present a simpleiterative algorithm to solve the new design equations for the gains that

Manuscript received December 16, 2003; revised March 14, 2004. Recom-mended by Associate Editor Hong Wang.

The author is at 895 St. Charles Dr. #5, Thousand Oaks, CA 91360 USA(e-mail: [email protected]).


is guaranteed to converge. We show in Section IV that, like the existingapproach, the new approach is also a dual optimal output feedback LQRtheory. Numerical examples are given in Section V to illustrate the ef-fectiveness of the algorithm and the newmethod. We conclude the notein Section VI.

II. FIXED-STRUCTURE CONTROL LAW

A. Review of Existing Results

Equations (4)–(7) are the design equations from [2]; see also [3] and[4]. Note that K = FC .

min1

0

(xTQx+ 2uTNx+ uTRu)dt

QNTR1N 0; R > 0 (2)

Ei = C+C Ek = I Ei (3)

ATP + PA Ei(B

TP + L)TR1(BT

P + L)Ei +Q = 0 (4)

N = LEi BPEk (5)

F = R1(BT

P + L)C+ (6)

K = R1(BT

P +N): (7)

It is not clear howL andN can be generated by a systematic method,as they are all coupled with P—the solution of the modified Riccatiequation that is difficult to solve. In addition,N has to satisfy a matrixinequality constraint that involves the other two weighting matrices Qand R, as given in (2).

B. New Perspective on the Same Control Law

Consider again F = KC+, where K is stabilizing, and performsingular value decomposition on the output matrixC . In the following,U and V are unitary matrices, and S contains singular values ofC . Wepartition V into two parts.

C = USVT

UTU = I V

TV = I (8)

ABFC = ABKC+C = ABKV S

+UTUSV

T

= V (V TAV V

TBKV S

+S)V T

: (9)

Let us define the terms in the transformed domain and investigate theclosed-loop matrix A BFC

A = VTAV B = V

TB K = KV = [K1 K2] (10)

K1 = KV1 K2 = KV2 V1 2 <nr

V2 2 <n(nr)

ABFC = V A B[K1 K2]Ir 0

0 0VT: (11)

One shall see that K2 would be lost. To avoid this information loss,we may impose a constraint: K2 = 0, which is KV2 = 0. As V isunitary, we have V1V T

1 + V2VT

2 = I and the following:

KC+C = KV1V

T

1 = K I V2VT

2 = K KV2VT

2 : (12)

C. New Set of Design Equations

Theorem 1: Suppose the given system (1) is static output feedbackstabilizable and detectable. There exists a static stabilizing output feed-back gain F = KC+, if there exists a static stabilizing state feedbackgain K that satisfies

(ABK)Y + Y (ABK)T +X = 0 Y = YT> 0 (13)

(ABK)TP + P (ABK) +KTRK +Q = 0; P = P

T> 0

(14)

0018-9286/04$20.00 © 2004 IEEE


2272 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 49, NO. 12, DECEMBER 2004

K R1BTP I V2 V T2 Y

1V21

V T2 Y

1 = 0: (15)

Proof: Wemay formulate the static output feedback stabilizationproblem as follows:

_x = Ax +Bu y = Cx u = Kx E [x(0)x(0)T ] = X

minE1

0

(xTQx+ uTRu)dt ; Q 0 R > 0

subject toKV2 = 0:

Define J as the cost to be minimized, and we get

J = E1

0

xT (KTRK +Q)xdt :

Suppose there exists a matrix P = P T > 0 so that

d(xTPx)=dt = xT (KTRK +Q)x (16)

then

J = E1

0

[d(xTPx)=dt] dt

= E [xT (0)Px(0) xT (1)Px(1)]:

For an asymptotically stable system

J = tracefE [xT (0)Px(0)]g = trace(PX)

xT (ABK)TPx + xTP (ABK)x+ xT (Q+KTRK)x = 0:

(17)

The previous expression must hold for any x. As a result

(ABK)TP + P (ABK) +KTRK +Q = 0: (18)

Define the Hamiltonian H as

H = trace fPX + [(ABK)TP + P (ABK)

+KTRK +Q]Y g+ 2 trace V T2 K

TM : (19)

Matrices Y andM are Lagrange multipliers. The stationary conditionsare as follows:

@H

@P= 0

@H

@Y= 0

@H

@K= 0

@H

@M= 0: (20)

From which, we get

(ABK)Y + Y (ABK)T +X = 0

(ABK)TP + P (ABK) +KTRK +Q = 0

RKY BTPY +MV T2 = 0 (21)

KV2 = 0: (22)

The first two stationary equations are exactly (13) and (14). From (21),we obtain

K = R1 BTP MV T2 Y

1 : (23)

Combining (22) and (23), we obtain

R1 BTP MV T2 Y

1 V2 = 0:

From which, we get

M = BTPV2 V T2 Y

1V21

Substitution ofM into (23) yields (15).

III. A CONVERGENT ALGORITHM

We now present an iterative algorithm to solve the design equationsfor the gain. In the sequel, subscript i denotes the index of iteration,and " stands for the preset tolerance.

1) Initialization: Solve a standard Riccati equation and obtain theKalman gain as the initial gain.

2) ith iteration: Solve the Lyapunov equations for Yi and Pi, re-spectively, in

(ABKi)Yi + Yi(ABKi)T +X = 0

(ABKi)TPi + Pi(ABKi) +KT

i RKi +Q = 0:

Evaluate the gain increment

Ki = R1BTPi I V2 V T2 Y

1i V2

1

V T2 Y

1i Ki:

Update the gain by Ki+1 = Ki + iKi where 0 < i < 2,and i is chosen so that A BKi+1 is asymptotically stable.

Set i = i + 1.3) If kKiV2k < ", stop the procedure, and let F = KiC

+. Other-wise, go to 2).

Since Ki+1 = Ki + iKi, we get the following:

Ki+1V2 = KiV2 + iKiV2 = (1 i)KiV2:

As a result

kKi+1V2k = j1 ij kKiV2k (24)

0 < i < 2) limi!1

kKiV2k = 0: (25)

In order to get a successful implementation, it is important that wemaintain the closed-loop asymptotic stability for all steps during thesolution procedure. We can achieve this by the following theorem.

Theorem 2: The following conditions guarantee the asymptotic sta-bility of A BKi+1

0 < i <1

2kX1=2BKiYiX1=2k2 (26)

0 < i <1

2kX1=2BKiYiX1=2kFro (27)

Proof: SupposeM1 is asymptotically stable and satisfies the fol-lowing Lyapunov equation:

M1W +WMT1 + Z = 0 Z = ZT > 0: (28)

For a vector v that is compatible to a matrixG, we have vTGTGv 0. Take

G = (1=p2)Z1=2

p2Z1=2W MT

2 : (29)

We get

(1=2)vTZv + 2vT (M2)WZ1W MT2 v

vT W MT2 + (M2)W v: (30)

Suppose

Z > 4(M2)WZ1W MT2 : (31)

We then get

(1=2)vTZv + (1=2)vTZv > vT W MT2 + (M2)W v

) vTZv > vT W MT2 + (M2)W v

) vT M1W +WMT1 v > vT W MT

2 + (M2)W v

) 0 > vT [W (M1 + M2)T + (M1 + M2)W ]v:



This implies thatM1 + M2 is asymptotically stable.We used the sufficient condition

42 WMT2

T

Z1

WMT2 < Z:

Equivalently

42 Z1=2

WMT2 Z

1=2T

Z1=2

WMT2 Z

1=2< I

42 Z1=2

WMT2 Z

1=22

2

< 1

<1

2kZ1=2M2WZ1=2k2: (32)

RecallABKi+1 = ABKiiBKi. Compare (13) with (28),and letM1 = ABKi;W = Yi; Z = X;M2 = BKi , then (26)follows directly from (32). As a result of k k2 k kFro, we obtain(27). From (24). we can see that faster convergence can be achieved,if we choose i close to 1, while taking into account the asymptoticstability condition (26) or (27). Typically, it is assumed that X = I ,which greatly simplifies (26) and (27).

IV. DUAL OPTIMAL OUTPUT FEEDBACK LQR THEORY

We now show that our approach, like the previous one, is a dual op-timal output feedback LQR theory. The optimal output feedback LQRgain F can be obtained by solving three coupled matrix equations

(ABFC)Y + Y (ABFC)T +X = 0

(ABFC)TP + P (ABFC) + (FC)TR(FC) +Q = 0

F = R1B

TPY C

T (CYCT )1

where Y = Y T > 0; P = P T > 0.As K = KC+C = FC , we only need to show that the gains are

identical. We will be using the following equality:

I V2 VT2 Y

1V2

1

VT2 Y

1V1V

T1 Y V1 = Y V1: (33)

Equivalently

I V2 VT2 Y

1V2

1

VT2 Y

1V1 = Y V1 V

T1 Y V1

1

: (34)

Before we proceed, let us derive the aforementioned equality. Con-sider the left-hand side of (33)

I V2 VT2 Y

1V2

1

VT2 Y

1V1V

T1 Y V1

= I V2 VT2 Y

1V2

1

VT2 Y

1I V2V

T2 Y V1

= I V2 VT2 Y

1V2

1

VT2 Y

1 V2VT2

+V2 VT2 Y

1V2

1

VT2 Y

1V2V

T2 Y V1

= I V2 VT2 Y

1V2

1

VT2 Y

1Y V1

= Y V1:

We used V T2 V1 = 0 in the previous derivation, which results from

the fact that V is unitary, and the off-diagonal elements of V TV = I

are zero, i.e., V T2 V1 = 0.

By comparison, it is obvious that we need to show that the followingequality holds:

I V2 VT2 Y

1V2

1

VT2 Y

1C+ = Y C

T (CYCT )1: (35)

Matrix S consists of two parts: S = [S1 0], where S1 is diagonaland nonsingular. As a result

C = US1VT1 C

T = V1S1UT

C+ = V1S

11 U

T:

The right-hand side of (35) now becomes

Y CT (CY CT )1 = Y V1S1U

TUS1V

T1 Y V1S1U

T1

= Y V1 VT1 Y V1

1

S11 U

T:

The left-hand side of (35) becomes

I V2 VT2 Y

1V2

1

VT2 Y

1C+

= I V2 VT2 Y

1V2

1

VT2 Y

1V1S

11 U

T:

Now, we need to show

Y V1 VT1 Y V1

1

= I V2 VT2 Y

1V2

1

VT2 Y

1V1:

The conclusion follows directly from (34).

V. ILLUSTRATIVE NUMERICAL EXAMPLES

We give three examples (see Figs. 1–6) to illustrate the effective-ness of the new method. Example 2, taken from [7], is not static outputfeedback stabilizable. Example 3 is not static output feedback stabi-lizable for any h 2 < (defined below). We set " = 105, except forthe third example, where " = 104. The gainK0 stands for the initialgain (the Kalman gain), K is the measure of error from the optimalgain. Eigenvalues are denoted by ’s. It is interesting to note that theclosed-loop eigenvalues of Examples 2 and 3 fall on the imaginary axis.The best the algorithm can do for these two examples is not allowingthe closed-loop eigenvalues to go to the right complex plane.

Example 1:

A =

4 2 8 5 1 8 4

9 7 6 3 2 2 6

7 3 7 5 2 10 1

6 3 8 1 2 3 7

0 5 6 3 4 6 1

2 8 4 6 9 2 4

5 8 3 1 9 6 3

B =

3:9 0:5

2:0 0:5

0:1 1:0

2:5 0:5

1:0 1:0

2:5 2:0

1:0 0:05

C =3 6 5 2 1 7 5

1 4 7 1 6 5 3

Q = I7 = X R = I2

(A) = f18:4797;11:0100 j5:4496;4:6867 j9:7361

4:8710;5:0381g

K0 =6:5953 2:1799 18:0050 7:7135

1:7234 5:0666 12:3486 0:6713

2:7930 14:7524 0:2250

10:2665 6:0697 13:1018

(ABK0) = f18:4033;11:6814 j6:2995

4:7938 j9:5050;6:5629;5:2828g

K =4:4999 18:5063 33:6949 4:9221

1:9301 5:9048 5:6425 0:4170


2274 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 49, NO. 12, DECEMBER 2004

Fig. 1. Trajectory of Frobenius norm ofKV —Example 1.

28:6039 24:3573 13:8375

5:8308 2:9928 4:5799

K = 101 0:014 0:020 0:051 0:015

0:129 0:280 0:116 0:067

0:025 0:056 0:018

0:029 0:217 0:230

F = KC+ =0:0844 4:7532

0:3026 1:0222

(ABFC) = f16:8986 j33:7676;1:7898 j 12:1583

1:9086 j2:9663;3:4386g:

Example 2:

A =

0 0 1 0

0 0 0 1

1 1 0 0

1 1 0 0

B =

0

0

1

0

CT =

0

1

0

0

(A) = f0; 0; 0 j1:4142g Q = I4 = X R = 1

K0 = [1:7212 0:3070 2:1077 1:1365]

(ABK0) = f0:6801 j0:4961;0:3738 j1:3624g

K = [0 0:2651 0 0] F = KC+ = 0:2651

(ABFC) = f0 j0:3778;0 j1:3628g:

Example 3:

A =0 1

1 0B =

1

0CT =

1=h

h

(A) = f1:0; 1:0g

Q = I2 = X R = 1 K0 = [2:4142 2:4142]

(ABK0) = f1:4142;1:0g h = 500

K = [0:0001 1:5992] F = KC+ = 0:0032

(ABFC) = f0 j0:7741g:

VI. CONCLUSION

We gave a new perspective on an existing fixed-structure control law,and proposed a new method for computing stabilizing static outputfeedback gains. The new method only involves Lyapunov equations,

Fig. 2. Trajectory of scaling factor —Example 1.

Fig. 3. Trajectory of gain K—Example 1.

Fig. 4. Trajectory of K—Example 1.

and does not have the technical difficulties residing in the previous ap-proach. We presented an iterative algorithm to solve the new design





equations for the output feedback gains, which is guaranteed to con-verge. We showed that the new approach, like the previous one, is adual optimal output feedback LQR theory. Numerical examples clearlyillustrated the effectiveness of the algorithm, and validated the newmethod. Based on the presented framework, our future work will beto come up with a convergent algorithm that generates optimal outputfeedback gains in the sense of LQR.

ACKNOWLEDGMENT

The author would like to thank the reviewers for their valuable com-ments and suggestions.

REFERENCES

[1] V. Syrmos, C. Abdallah, and P. Dorato, “Static output feedback: Asurvey,” in Proc. 33rd IEEE Conf. Decision and Control, 1994, pp.837–842.

[2] Y. Cao, Y. Sun, and W. Mao, “A new necessary and sufficient conditionfor static output feedback stabilizability and comments on “stabilizationvia static output feedback”,” IEEE Trans. Automat. Contr., vol. 43, pp.1110–1111, Sept. 1998.

[3] A. Trofino-Neto and V. Kucera, “Stabilization via static output feed-back,” IEEE Trans. Automat. Contr., vol. 38, pp. 764–765, June 1993.

[4] A. Pimpalkhare and B. Bandyopadhyay, “Comments on ‘Stabilizationvia static output feedback’,” IEEE Trans. Automat. Contr., vol. Sept.,pp. 1148–1148, 1994.

[5] J. Geromel, C. de Souza, and R. Skelton, “Static output feedback con-trollers: Stability and convexity,” IEEE Trans. Automat. Contr., vol. 43,pp. 120–125, Jan. 1998.

[6] , “LMI numerical solution for output feedback stabilizations,” inProc. Amer. Control Conf., vol. 1, June 1994, pp. 40–44.

[7] M. Mesbahi, “A semi-definite programming solution of the least orderdynamic output feedback synthesis problem,” in Proc. Conf. Decisionand Control, Dec. 1999, pp. 1851–1856.

[8] F. Leibfritz. Computational design of stabilizing static output feedbackcontrollers. [Online]. Available: http://www.mathematik.uni-trier.de/~leibfritz/

[9] M. de Oliveira and J. Geromel, “Numerical comparison of output feed-back design methods,” in Proc. Amer. Control Conf., vol. 1, June 1997,pp. 72–76.

[10] F. Leibfritz. Trust region methods for solving the optimal output feed-back design problem. [Online]. Available: http://www.mathematik.uni-trier.de/~leibfritz/

[11] L. Ghaoui, F. Oustry, and M. AitRami, “A cone complementarity lin-earization algorithm for static output-feedback and related problems,”IEEE Trans. Automat. Contr., vol. 42, pp. 1171–1176, Aug. 1997.

Stability of a Riccati Equation Arising in RecursiveParameter Estimation Under Lack of Excitation

Alexander Medvedev

Abstract—Stability properties of the Riccati equation in a recently sug-gested antiwindup algorithm for recursive parameter estimation are ana-lyzed. Convergence of the resulting dynamic system is implied by that ofa linear time-varying difference matrix equation. By means of convergingmatrix products theory, the linear mapping associated with the system isshown to be a paracontraction with respect to a certain norm. Therefore,measured in that norm, the solution to the matrix equation will not di-verge notwithstanding excitation properties of the data. Thus the purposeof anti-windup is achieved.

Index Terms—Lyapunov methods, Riccati equations, recursive estima-tion, stability.

I. PRELIMINARIES

Consider the following difference Riccati equation typical to param-eter estimation of regression models:

P (t) = P (t 1)P (t 1)'(t)'T (t)P (t 1)

r(t) + 'T (t)P (t 1)'(t)+Q(t) (1)

where P ( ) 2 Rnn; Q( ) 2 Rnn;Q( ) = QT ( ); Q( ) 0; P (0) = P T (0); P (0) 0; '( ) 2 Rn is a regressor vector, r(t)is a positive scalar, and t 2 f1; 2; . . . ;1g.This equation is typically associated with parameter estimation in

the regressor model

y(t) = 'T (t) + e(t) (2)

Manuscript received September 18, 2003; revised March 31, 2004 andJuly 30, 2004. Recommended by Associate Editor A. Garulli. This workwas supported in part by the Royal Swedish Academy of Sciences andJernkontoret—the Swedish Steel Producers’ Association.

The author is with the Department of Information Technology, Uppsala Uni-versity, SE-751 05 Uppsala, Sweden (e-mail: [email protected]).


0018-9286/04$20.00 © 2004 IEEE



1678 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 51, NO. 10, OCTOBER 2006

the value of K . For the Van der Pol system, the optimization problem(8) remains feasible even if K ! 1, i.e., K1 = 0. In this case,the solution of problem (8) provides a slightly poor estimate of theattracting set, = 6:4912. However, the stability region is the wholeR2, so that the limit cycle is globally stable.

V. CONCLUSION

This note presents a systematic procedure for estimating the at-tracting set of a class of nonlinear systems called the extended Luriesystem. The extended Lurie system consists of systems that can bewritten like the Lurie problem, but with just one nonlinearity thatviolates the sector condition.

The procedure was applied to two classical nonlinear systems: theforced Duffing equation and the Van der Pol system. The numericalresults show that the proposed procedure provides good estimates ofthe attracting sets. Moreover, for the Van der Pol system the attractingset was proved to be the whole R2. The calculus of matrix P , usingthe LMI problems in (7) and (8), tends to improve the stability regionestimate since the negative term to be minimized is on the denominatorof . However, that improvement is bounded since the norm of matrixP is on the numerator of .

APPENDIX

Theorem 3: Let h : [a; b]D RRn ! Rp be a C1 function

and W (t; x) = [xT h(t; x)T ]

T x

h(t;x), where

T is symmetric negative definite. Then, W (t; x) xT [1T ]x8(t; x) 2 [a; b] D, where [ 1T ] is symmetric negativedefinite.

REFERENCES

[1] S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan, Linear MatrixInequalities in System and Control Theory. Philadelphia, PA: SIAM,1994.

[2] M. Dellnitz and A. Hohmann, “A subdivision algorithm for the compu-tation of unstable manifolds and global attractors,” Numerische Math-ematik, vol. 75, pp. 293–317, 1997.

[3] J.-Y. Dieudelot and P. Borne, “An estimation of the stability domainof a fuzzy controled pendulum using overlapping attractors and vectornorms,” in Proc. IEEE Int. Conf. Systems, Man, and Cybernetics, 2001,vol. 4, pp. 2245–2249.

[4] M. F. Gameiro and H. M. Rodrigues, “Applications of robust synchro-nization to communication systems,” Appl. Anal., vol. 79, pp. 21–45,2001.

[5] W. M. Haddad and V. Kapila, “Absolute stability criteria for multipleslope-restricted monotonic nonlinearities,” IEEE Trans. Autom. Con-trol, vol. 40, no. 2, pp. 361–365, Feb. 1995.

[6] S. Heidari and C. L. Nikias, “Characterizing chaotic attractors usingfourth-order off-diagonal cumulant slices,” in Proc. 27th AsilomarConf. Signal, Systems, and Computers, 1993, vol. 1, pp. 466–470.

[7] R. A. Horn and C. R. Johnson, Matrix Analysis. New York: Cam-bridge Univ. Press, 1996.

[8] H. K. Khalil, Nonlinear Systems. Upper Saddle River, NJ: Prentice-Hall, 1992.

[9] J. LaSalle and S. Lefschetz, Stability by Liapunov’s Direct Method WithApplications. New York: Academic, 1991.

[10] R. K. Miller, A. N. Michel, and G. S. Krens, “Stability analysis oflimit cycles in nonlinear feedback systems using describing functions:improved results,” IEEE Trans. Circuit Syst., vol. CAS-31, no. 6, pp.561–567, Jun. 1984.

[11] H. M. Rodrigues, L. F. C. Alberto, and N. G. Bretas, “On the invarianceprinciple: generalizations and applications to synchronization,” IEEETrans. Circuits Syst. I, vol. 47, no. 5, pp. 730–739, May 2000.

[12] J. Argyris, G. Faust, and M. Haase, An Exploration of Chaos. Ams-terdan, The Netherlands: North-Holland, 1994.

An Improved ILMI Method for Static Output FeedbackControl With Application to Multivariable PID Control

Yong He and Qing-Guo Wang

Abstract—An improved iterative linear matrix inequality (ILMI) algo-rithm for static output feedback (SOF) stabilization problem without in-troducing any additional variables is proposed in this note. The proposedILMI algorithm is also extended to solve the SOF controller designproblem. They are applied to the multivariable PID controllers. Numericalexamples show that the proposed algorithms yield better results and fasterconvergence than the existing ones.

Index Terms— control, linear matrix inequality (LMI), multivari-able PID control, stabilization, static output feedback (SOF).

I. INTRODUCTION

The static output feedback (SOF) plays a very important role in con-trol theory and applications. Recently, it has attracted considerable at-tention (see, e.g., [1]–[7] and the references therein). Yet, it is still leftwith some open problems. Unlike the state feedback case, a SOF gainwhich stabilizes the system is not easy to find. Linear matrix inequality(LMI) [8] is one of the most effective and efficient tools in controller de-sign and a great deal of LMI-based design methods of SOF design havebeen proposed over the last decade [9]–[19]. Among these methods,an iterative linear matrix inequality (ILMI) method was proposed byCao et al. [13] and later employed to solve some multivariable PIDcontroller design problems [20], [21]. In this context, a new additionalvariable was introduced such that the stability condition becomes a suf-ficient one. The iterative algorithm in [13] tried to find a sequence ofthe additional variables such that the sufficient condition is close to thenecessary and sufficient one. The similar idea is used in the so-calledsubstitutive LMI method by [19]. Both of them set the additional matrixvariables at the current step with the matrices derived in the precedingstep. With additional variables, the dimensions of the LMIs becomehigher. We observe that it is possible that the matrices derived in thepreceding step can be used in the next one directly without introducingthe additional variables and the dimensions of the LMIs need not be in-creased. In addition, we can develop some efficient ways to get suitableinitial values in the iterative procedure, which the existing approacheshave not been thought of.

In this note, a new ILMI algorithm is proposed for SOF stabilizationproblem without introducing any additional variables, and assisted witha separate ILMI algorithm to find good initial variables. The algorithmsfor SOF stabilization are also extended to solve the SOF H1 controlproblem. They are applied to multivariable PID control. Numerical ex-amples show the effectiveness and an improvement of the algorithmsover the existing methods.

Manuscript received April 3, 2005; revised January 12, 2006 and April 23,2006. Recommended by Associate Editor M. Kothare. The work with Y. Hewas supported in part by the National Science Foundation of China under Grant60574014 and in part by the Doctor Subject Foundation of China under Grant20050533015.

Y. He is with the Department of Electrical and Computer Engineering, Na-tional University of Singapore, Singapore 119260, Singapore. He is also withthe School of Information Science and Engineering, Central South University,Changsha 410083, China.

Q.-G. Wang is with the Department of Electrical and Computer Engineering,National University of Singapore, Singapore 119260, Singapore (e-mail:[email protected]).


0018-9286/$20.00 © 2006 IEEE


IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 51, NO. 10, OCTOBER 2006 1679

II. SOF STABILIZATION

Consider the following system:

_x = Ax(t) +B1!(t) +B2u(t)

z(t) = C1x(t) +D11!(t) +D12u(t)

y(t) = C2x(t) +D21!(t)

(1)

where x(t) 2 Rn is the state vector, !(t) 2 Rr is the external signal,u(t) 2 Rm is the controlled input, z 2 Rq is the controlled output, andy(t) 2 Rp is the measured output. A, B1, B2, C1, C2, D11, D12, andD21 are constant matrices with appropriate dimensions. The followingassumption is made on (1).

Assumption 1: (A;B2) is stabilizable and (C2; A) is detectable.The SOF stabilization problem is to find a SOF controller

u(t) = Fy(t) (2)

where F 2 Rmp such that the closed-loop system with !(t) = 0given by

_x = (A+B2FC2)x(t) (3)

is stable. As we all know, the closed-loop system (3) is stable if andonly if there exists a P = P T > 0 such that

P (A+B2FC2) + (A+B2FC2)TP < 0: (4)

Condition (4) is a BMI which is not a convex optimal problem. AnILMI method was proposed in [13], where a new variableX was intro-duced such that the stability condition becomes a sufficient one whenX 6= P . The algorithm presented in [13] tried to find some X closeto P by using an iterative method and the iterative procedure carriesbetween P and X . On the other hand, a substitutive LMI formulationwas proposed in [19], where some new variables such asL andM wereintroduced such that the stability condition also becomes sufficient onewhen L 6= R1B2P or M 6= FC2. The algorithm presented in [19]also tried to find some L and M close to R1B2P and FC2, respec-tively, by using iterative method. It is clear that so introduced variablesare given with the information on P and F in the last step in the it-erative procedure, that is, when deriving P , they employed the P ob-tained in the last step to express these additional variables. In fact, theinformation on P can be used directly without introducing additionalvariables. For example, when we derive a P in this step, we can em-ploy it to derive the F in the next step. In the contrary, the F is alsoused to derive P , and so on. Therefore, these variables in the iterativeprocedures in [13] and [19] are unnecessary and the iteration can becarried out between P and F directly. Base on this idea, we propose anew algorithm as follows.

As mentioned in [13], if

P (A+B2FC2) + (A+B2FC2)TP P < 0 (5)

holds, the closed-loop system matrix A + B2FC2 has its eigenvaluesin the strict left-hand side of the line =2 in the complex s-plane. If a 0 satisfying (5) can be found, the SOF stabilization problem issolved.

The key point in our algorithm is to find an initial P . The P whichsatisfies (4) cannot be derived using LMI due to unknownF . By settingV1 = PB2F , (4) becomes

PA +ATP + V1C2 + CT2 V

T1 < 0: (6)

However, (6) ignores B2 so that this P does not take into account allthe information. On the other hand, (4) is transformed to the followinginequality by pro- and postmultiply L = P1:

(A+B2FC2)L+ L(A+B2FC2)T < 0: (7)

By setting V2 = FC2L, (7) becomes

AL+ LAT +B2V2 + V T2 BT

2 < 0: (8)

According to the idea of the cone complementary linearizationmethod [11], L = P1 yields PL = I , which is relaxed with thefollowing LMI:

P I

I L 0 (9)

and the linearized version of trace(PL) is minimized. Then, an iterativealgorithm to find an initial P is stated as follows.

Algorithm 1

Step 1) Set i = 1 and P0 = I and L0 = I .Step 2) Derive a Pi and Li by solving the following optimization

problem for Pi, Li, V1 and V2:OP1: Minimize trace(Pi Li1 + Li Pi1) subject tothe following LMI constraints:

PiA +ATPi + V1C2 + CT2 V

T1 < 0 (10)

ALi + LiAT +B2V2 + V T

2 BT2 < 0 (11)

Pi I

I Li 0: (12)

Step 3) If trace(PiLi) n < "1, a prescribed tolerance, an initialP = Pi is found, stop.

Step 4) If the difference of two iterations satisfiestrace(PiLi) trace(Pi1Li1) < "2, a prescribedtolerance, the initial P may not be found, stop.

Step 5) Set i = i + 1, goto Step 2).

If an initial P can not be found by Algorithm 1, the SOF controlproblem for system (1) with !(t) = 0 may not have solutions. On theother hand, after an initial P is found, an ILMI algorithm that stabilizessystem (1) with !(t) = 0 using SOF is stated as follows.

Algorithm 2

Step 1) Set i = 1 and P1 = P as obtained from Algorithm 1.Step 2) Solve the following optimization problem for F with given

Pi:OP1: Minimize i subject to the following LMI constraint:

Pi(A+B2FC2) + (A+B2FC2)TPi iPi < 0: (13)



TABLE ISOF RESULTS (EXAMPLE 1)

Step 3) If i 0, F is a stabilizating SOF gain, stop.Step 4) Set i = i + 1. Solve the following optimization problem

for Pi with given F :OP2: Minimize i subject to the aforementioned LMIconstraint (13).

Step 5) If i 0, F is a stabilization SOF gain, stop.Step 6) Solve the following optimization problem for Pi with

given F and i:OP3: Minimize trace(Pi) subject to the previous LMIconstraint (13).

Step 7) If kPi Pi1k=kPik < , a prescribed tolerance, gotoStep 8), else set i = i+ 1 and Pi = Pi1, then goto Step2).

Step 8) The system may not be stabilizable via SOF, stop.

Remark 1: The discussions on the iterative procedure and conver-gence of Algorithm 2 follow those in [13].

Remark 2: If C2 = I , the SOF stabilization problem reduces to astate feedback problem. In fact, the initial P derived in Algorithm 1 isalso the solution of state feedback stabilization problem since the statefeedback gain F can be derived by F = V2L

1 = V2P . On the otherhand, the F can also be obtained directly by OP1 in Algorithm 2 whenthe previous initial P is given. Thus, the state feedback stabilizationgain without conservativeness can also be derived using our Algorithms1 and 2.

Remark 3: The algorithm presented in our note is different fromthose in [11]. The stopping criterion in [11] is given in terms of "eofwhich depends on selected and , where and should be suffi-ciently small. However, it is difficult to determine how small the valuesshould be as they depend on a particular situation under consideration.If inappropriate and are selected, the algorithm may not be conver-gent. In our procedure, Algorithm 1 finds an initial P for a given "1,where the selection of "1 is not crucial as P obtained is not the finalsolution but will be elaborated in Algorithm 2. With the initial P , Al-gorithm 2 produces a static output feedback gain matrix F which canguarantee the stability of closed-loop system (3). The stopping con-dition is < 0, where need not be specified a prior. Overall, ourprocedure is easier to use.

Example 1: [13] Consider the SOF stabilization problem of system(1) with !(t) = 0 and the following parameter matrices:

A =0 1

1 0B2 =

1

0C2 = [1 ]: (14)

The open-loop system is unstable since the eigenvalues are 1 and 1.The calculation results for SOF stabilization using the method in [13]and our algorithms are listed in Table I. It is noted that the iterationnumber in form of l+k in the table means that l is the iteration numberof Algorithm 1 to find an initial P and k is the iteration number ofAlgorithm 2 to find a SOF gain. It can be seen that our convergencespeed is greatly faster than that in [13].

Example 2: [7] Consider the SOF stabilization problem of system(1) with !(t) = 0 and the following parameter matrices:

A =

4 2 8 5 1 8 4

9 7 6 3 2 2 6

7 3 7 5 2 10 1

6 3 8 1 2 3 7

0 5 6 3 4 6 1

2 8 4 6 9 2 4

5 8 3 1 9 6 3

B2 =3:9 2 0:1 2:5 1 2:5 1

0:5 0:5 1 0:5 1 2 0:05

T

C2 =3 6 5 2 1 7 5

1 4 7 1 6 5 3: (15)

After more than 250 iterations, the algorithm in [7] converges. Algo-rithm 1 yields an initial P with only four iterations. Then, a SOF gain

F is found as0:8871 4:9310

0:6576 0:9869using Algorithm 2 with two itera-

tions. In this case, = 0:7481 and the eigenvalues of the closed-loopsystem are 7:8846 j36:0334, 0:6354 j12:2411, 0.3742,4:9779 j6:3825. The convergence speed is faster than that in [7].

III. H1 SYNTHESIS

An ILMI algorithm presented in the preceding section is now em-ployed to solve the SOFH1 control problem. The objective of the SOFH1 synthesis is to to find a SOF controller (2) such that the transferfunction of the closed-loop system satisfies H1 norm constraint

kTz!(s)k1

< ; for > 0: (16)

(16) can be represented as a matrix inequality [13]

PAcl + AT

clP PBcl CT

cl

BT

clP I DT

cl

Ccl Dcl I

< 0 (17)

where

Acl =A+B2FC2

Bcl =B1 +B2FD21

Ccl =C1 +D12FC2

Dcl =D11 +D12FD21:

Similarly to Algorithm 2, we propose an algorithm to obtain the so-lution of matrix inequality (17) for a given > 0. It relies on, like asAlgorithm 1, an algorithm for finding an initial P for SOF H1 controlproblem.



TABLE IISOF AND PID-CONTROLLER AND THEIR PERFORMANCES (EXAMPLE 4)

Algorithm 3

Step 1) Set i = 1 and P0 = I and L0 = I .Step 2) Derive a Pi and Li by solving the following optimization

problem for Pi, Li, V1, and V2:OP1: Minimize trace(Pi Li1 + Li Pi1) subject tothe LMI constraints (18) and (19), as shown at the bottomof the page, and (20)

Pi I

I Li

0: (20)

Step 3) If trace(PiLi) n < "1, a prescribed tolerance, an initialP = Pi is found, stop.

Step 4) If the difference of two iterations satisfiestrace(PiLi) trace(Pi1Li1) < "2, a prescribedtolerance, the initial P may not be found, stop.

Step 5) Set i = i + 1, goto Step 2).

After an initial P is found, an ILMI algorithm for SOF H1 controlproblem for system (1) is stated as follows.

Algorithm 4

Step 1) Set i = 1 and P = P1 as obtained from Algorithm 3.

PiA+ ATPi + V1C2 + CT

2 VT

1 PiB1 + V1D21 CT

1 + CT

2 FTDT

12

BT

1 Pi +DT

2 VT

1 I DT

11 +DT

21FTDT

12

C1 +D12FC2 D11 +D12FD21 I

< 0 (18)

ALi + LiAT +B2V2 + V T

2 BT

2 B1 +B2FD21 CT

1 Li + V T

2 DT

12

BT

1 +DT

21FTBT

2 I DT

11 +DT

21FTDT

12

LiC1 +D12V2 D11 +D12FD21 I

< 0 (19)



Step 2) Solve the following optimization problem for F with givenPi:OP1: Minimize i subject to the LMI constraint (21), asshown at the bottom of the page, where

11 = PiA+ ATPi + PiB2FC2 + CT2 F

TBT2 Pi Pi:

Step 3) If i 0, F is a stabilization SOF H1 control gain for , stop.

Step 4) Set i = i + 1. Solve the following optimization problemfor Pi with given F :OP2: Minimize i subject to the above LMI constraint(21).

Step 5) If i 0, F is a stabilizating SOF H1 control gain for , stop.

Step 6) Solve the following optimization problem for Pi withgiven F and i:OP3: Minimize trace(Pi) subject to the above LMIconstraint (21).

Step 7) If kPi Pi1k=kPik < , a prescribed tolerance, gotoStep 8), else set i = i+ 1 and Pi = Pi1, then goto Step2).

Step 8) It may not be decided by this algorithm whether SOF H1control problem is solvable, stop.

Example 3: [19] Consider system (1) with the following parametermatrices:

A =

0:0366 0:0271 0:0188 0:4555

0:0482 1:01 0:0024 4:0208

0:1002 0:3681 0:707 1:42

0 0 1 0

B1 =

1 0 0

0 0 0

0 0 0

0 0 0

B2 =

0:4422 0:1761

3:5446 7:5922

5:52 4:49

0 0

C1 =0 1 0 0

0 0 0 1C2 =

1 0 0 0

0 1 0 0

D11 =0:5 0 0

0 1 0D12 =

1 0

0 1

D21 =0 0:1 0

0 0 0:1:

The SOFH1 norm obtained in [19] is 1.183. For = 1:144, the initialmatrix P is obtained as

P1 =

0:7130 0:4054 0:3283 1:1658

0:4054 2:2769 1:4029 1:2100

0:3283 1:4029 1:5941 1:0533

1:1658 1:2100 1:0533 3:2305

by Algorithm 3 after 24 iterations. Then the SOF H1 norm convergesto 1.144 after two iterations using Algorithm 4. The resulting SOF gainis

F =0:0976 3:8054

0:4191 4:6958:

In this case, = 8:2776 107 and the eigenvalues of the closed-loop system are 50.1599, 0.2781, 0:2431 j1:3534.

IV. PID CONTROL

Motivated by its popularity in industry, let us consider now the fol-lowing PID controller:

u(t) = F1y(t) + F2

t

0

y()d + F3 _y(t) (22)

instead of SOF controller (2), where F1; F2; F3 2 Rmp are gainmatrices to be designed. Without lose of generality, D21 is set as zero.

It is noted that a new method is proposed in [20] to transform thisPID controller design problem to a SOF control problem. In order tousing the method proposed in [20], the following assumption is needed.

Assumption 2: The matrix I F3C2B2 is invertible.

Denote x(t) =x(t)

t

0y()d

and y(t) =

C2x(t)t

0y()d

C2Ax(t)

. The com-

posite system with (1) and (22) is described by

_x = Ax(t) + B1!(t) + B2u(t)

z(t) = C1x(t) + D11!(t) + D12u(t)

y(t) = C2x(t)

(23)

where

A =A 0

C 0B1 =

B1

0B2 =

B2

0

C21 = [C2 0] C22 = [0 I] C23 = [C2A 0]

C2 = CT21

CT22

CT23

TC1 = [C1 0]

D11 =D11D12 = D12:

The controller reduces to

u(t) = F y(t) (24)

where

Fi =(I F3C2B2)1Fi; i = 1; 2; 3

F = [ F1 F2 F3]:

11 PiB1 + PiB2FD21 CT1 + CT

2 FTDT

12

BT1 Pi +DT

21FTBT

2 Pi I DT11 +DT

21FTDT

12

C1 +D12FC2 D11 +D12FD21 I

< 0 (21)



A =

0:0266 36:6170 18:8970 32:0900 3:2509 0:7626

0:0001 1:8997 0:9831 0:0007 0:1708 0:0050

0:0123 11:7200 2:6316 0:0009 31:6040 22:3960

0 0 1:0000 0 0 0

0 0 0 0 30:0000 0

0 0 0 0 0 30:0000

B1 =

0

0

0

0

30

0

B2 =

0 0

0 0

0 0

0 0

30 0

0 30

C1 = [0 1 0 0 0 0] C2 =0 1 0 0 0 0

0 0 0 1 0 0D11 = [0] D12 = [1 1] D21 =

0

0: (25)

Once the composite matrix F = [ F1 F2 F3] is found, the original PIDcontroller gain can be recovered from

F3 = F3(I + C2B2F3)1

F2 =(I F3C2B2) F2

F1 =(I F3C2B2) F1:

As stated in [20], the invertibility of matrix I + C2B2F3 is guaran-

teed by the following Lemma.Lemma 1: Matrix I +C2B2

F3 is invertible if and only if Assump-tion 2 holds, where F3 and F3 are related to each other by

F3 = (I F3C2B2)1F3; or F3 = F3(I + C2B2

F3)1:

Under Assumption 2, one easily see that Algorithms 1 and 2 canbe employed to derive the stabiliziable PID control gains, F1, F2, andF3. Similarly, Algorithms 3 and 4 can be used to derive the PID H1controller for a given performance > 0.

Example 4: [20] One of the state space realization of the aircraftcontroller system model is as system (1) with the following parametersin (25), as shown at the top of the page.

The calculation results using our algorithms and the method in [13]and [20] for SOF control and PID control are listed in Table II. It canbe seen that the convergence speed is faster than that in [13], [20].

On the other hand, Remark 3 is demonstrated in this example. Forexample, when "1 in Algorithm 3, which is similar to those in [11],is set as 103, two iterations obtain an initial P . However, based onthis P , F is not a stabilization H1 control gain for given = 1:001.On the contrary, the corresponding F in Table II is derived after 14iterations by using Algorithm 4.

V. CONCLUSION

New ILMI algorithms for SOF stabilization andH1 control are pro-posed in this note, which avoid introduction of the additional variables,leading to lower dimensions of the LMIs than [13] and [19]. In partic-ular, an algorithm to derive an initial value is also given. The algo-rithms are also applied to design the multivariable PID controller. Nu-merical examples show that the proposed algorithms produces betterresult and/or faster convergence than the existing ones.

REFERENCES

[1] D. S. Bernstein, “Some open problems in matrix theory arising in linearsystems and control,” Linear Alg. Appl., vol. 162–164, pp. 409–432,1992.

[2] A. Trofino and V. Kucera, “Stabilization via static output feedback,”IEEE Trans. Autom. Control, vol. 38, no. 5, pp. 764–765, May 1993.

[3] V. Kucera and C. E. Souza, “A necessary and sufficient condition foroutput feedback stabilizability,” Automatica, vol. 31, pp. 1357–1359,Sep. 1995.

[4] M. Fu and Z. Q. Luo, “Computational complexity of a problem arisingin fixed order output feedback design,” Syst. Control Lett., vol. 30, pp.209–215, 1997.

[5] V. L. Syrmos, C. T. Abdallah, P. Dorato, and K. Grigoriadis, “Staticoutput feedback—A survey,” Automatica, vol. 33, pp. 125–137, 1997.

[6] H. Gao, J. Lam, C. Wang, and Y. Wang, “Delay-dependent output-feed-back stabilisation of discrete-time systems with time-varying statedelay,” Proc. Inst. Elect. Eng. Control Theory Appl., vol. 151, pp.691–698, 2004.

[7] J. T. Yu, “A convergent algorithm for computing stabilizing staticoutput feedback gains,” IEEE Trans. Autom. Control, vol. 49, no. 12,pp. 2271–2275, Dec. 2004.

[8] S. Boyd, L. E. Ghaoui, E. Feron, and V. Balakrishnan, Linear MatrixInequalities in System and Control Theory. Philadelphia, PA: SIAM,1994, vol. 15, SIAM Studies in Applied Mathematics.

[9] J. C. Geromel, C. C. de Souza, and R. E. Skelton, “LMI numerical so-lution foroutput feedback stabilization,” in Proc. Amer. Control Conf.,1994, pp. 40–44.

[10] T. Iwasaki and R. E. Skelton, “The XY-centring algorithm for the dualLMI problem: a new approach to fixed-order control design,” Int. J.Control, vol. 62, pp. 1257–1272, 1995.

[11] L. E. Ghaoui, F. Oustry, and M. AitRami, “A cone complementaritylinearization algorithm for static output-feedback and related prob-lems,” IEEE Trans. Autom. Control, vol. 42, no. 8, pp. 1171–1176,Aug. 1997.

[12] R. E. Benton and D. Smith, “Static output feedback stabilization withprescribed degree of stability,” IEEE Trans. Autom. Control, vol. 43,no. 10, pp. 1493–1496, Oct. 1998.

[13] Y. Y. Cao, J. Lam, and Y. X. Sun, “Static output feedback stabilization:An ILMI approach,” Automatica, vol. 34, pp. 1641–1645, 1998.

[14] Y. Y. Cao, Y. X. Sun, and W. J. Mao, “A new necessary and sufficientcondition foroutput feedback stabilizability and comments on stabiliza-tion via static output feedback,” IEEE Trans. Autom. Control, vol. 43,no. 8, pp. 1110–1111, Aug. 1998.

[15] Y. Y. Cao, Y. X. Sun, and J. Lam, “Simultaneous stabilization via staticoutput feedback and state feedback,” IEEE Trans. Autom. Control, vol.44, no. 6, pp. 1277–1282, Jun. 1999.

[16] C. A. R. Crusius and A. Trofino, “Sufficient LMI conditions for outputfeedback control problems,” IEEE Trans. Autom. Control, vol. 44, no.5, pp. 1053–1057, May 1999.

[17] F. Leibfritz, “An LMI-based algorithm for designing suboptimal staticoutput feedback controllers,” SIAM J. Control Optim., vol.

39, pp. 1711–1735, 2001.[18] M. Rotunno and R. A. de Callafon, “A Bundle method for solving the

fixed order control problem,” in Proc. 41st IEEE Conf. Decision andControl, Las Vegas, NV, Dec. 2002, pp. 3156–3161.

[19] A. Fujimori, “Optimization of static output feedback using substitu-tive LMI formulation,” IEEE Trans. Autom. Control, vol. 49, no. 6, pp.995–999, Jun. 2004.

[20] F. Zheng, Q. G. Wang, and T. H. Lee, “On the design of multivariablePID controllers via LMI approach,” Automatica, vol. 38, pp. 517–526,2002.

[21] C. Lin, Q. G. Wang, and T. H. Lee, “An improvement on multivariablePID controller design via iterative LMI approach,” Automatica, vol. 40,pp. 519–525, 2004.


IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 16, NO. 3, JUNE 2008 615

Control Synthesis of Singularly PerturbedFuzzy Systems

Guang-Hong Yang, Senior Member, IEEE, and Jiuxiang Dong

Abstract—This paper considers the problem of designing stabi-lizing and controllers for nonlinear singularly perturbed sys-tems described by Takagi–Sugeno fuzzy models with the consider-ation of the bound of singular perturbation parameter . For thesynthesis problem of simultaneously designing the bound of andstabilizing or controllers, linear matrix inequalities (LMI)-based methods are presented. For evaluating the upper bound ofsubject to stability or a prescribed performance bound con-straint for the resulting closed-loop system, sufficient conditionsare developed, respectively. For the stabilizing and controlsynthesis without the consideration of improving the bound of ,new design methods are also given in terms of solutions to a setof LMIs. Examples are given to illustrate the efficiency of the pro-posed methods.

Index Terms— performance, linear matrix inequalities(LMIs), nonlinear control systems, singularly perturbed systems,stabilizing control, state feedback control, Takagi–Sugeno (T-S)fuzzy models.

I. INTRODUCTION

I N CONTROL engineering applications, it is well known thatthe multiple time-scale systems or known as singularly per-

turbed systems often raise serious numerical problems. For thepurpose of avoiding the difficulties linked with the stiffness ofthe equations involved in the design, the singular perturbationdesign method has been developed [1], where singular pertur-bation with a small parameter, say , is exploited to determinethe degree of separation between “slow” and “fast” modes ofthe system, and the so-called reduction technique is proposed tohandle these systems. For the stabilization and control oflinear singularly perturbed systems, many important advanceshave been achieved, see [1]–[6] and the references therein. Inparticular, the fundamental results are given in [1], [4], and [5].

Manuscript received April 19, 2006; revised January 11, 2007. This work wassupported in part by the Program for New Century Excellent Talents in Univer-sity (NCET-04-0283), by the Funds for Creative Research Groups of China (No.60521003), by the Program for Changjiang Scholars and Innovative ResearchTeam in University (No. IRT0421), by the State Key Program of National Nat-ural Science of China under Grant 60534010, by the Funds of National Scienceof China under Grant 60674021, and by the Funds of the Ph.D. program ofMOE, China, under Grant 20060145019.

G.-H. Yang is with the College of Information Science and Engineering,Northeastern University, Shenyang 110004, China, and also with theKey Laboratory of Integrated Automation of Process Industry, North-eastern University, Ministry of Education, Shenyang 110004, China (e-mail:[email protected]; [email protected]).

J. Dong is with the College of Information Science and Engineering, North-eastern University, Shenyang 110004, China (e-mail: [email protected]; [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TFUZZ.2007.905911

In recent years, there have been some attempts to address thecontrol problem for nonlinear singularly perturbed sys-

tems. In [7] and [8], the control for a class of singularlyperturbed systems with nonlinearity in the slow variables is ex-amined. A local state feedback control problem for affinenonlinear singularly perturbed systems is studied in [9]. How-ever, the control design for general nonlinear singularlyperturbed systems still remains as an open research subject.

An important approach to the synthesis problems for non-linear systems is to model the considered system as Takagi andSugeno (T-S) fuzzy systems [10], which are locally linear time-invariant systems connected by IF-THEN rules. In [11] and [12],it has shown that the T-S fuzzy systems can approximate anycontinuous functions at any preciseness, which shows that theT-S fuzzy models can approximate a wide class of nonlinearsystems. As a result, the conventional linear system theory canbe applied to analysis and synthesis of the class of nonlinearcontrol systems. In recent years, the T-S fuzzy control systemshave been studied extensively, and many significant advanceshave been achieved (see [13]–[15] and the references therein).For nonlinear singularly perturbed systems, some control syn-thesis problems have been studied [16]–[19]. In [16] and [18],design methods for the stabilization and control of non-linear singularly perturbed systems via state feedback are givenin terms of solutions of linear matrix inequalities (LMIs) [20],respectively. A robust state feedback control design is presentedin [17]. An LMI-based method of designing output feed-back controllers for uncertain fuzzy singularly perturbed sys-tems is presented in [19].

For the effective applications of the design methods for sin-gularly perturbed control systems, the accurate knowledge ofthe stability bound of a singularly perturbed system (i.e., thesystem is stable for ) is very important. The charac-terization and computation of the stability bound have attractedconsiderable efforts for the past over two decades [21]–[28]. Ingeneral, there are two classes of methods to characterize andcompute the stability bounds, one is based on frequency domaintransfer functions and another is based on state space models.Both of the two methods can provide the exact bounds as shownin [22], [25], and [26]. However, the issue of how to improvethe bound in controller designs has not been addressed in theliterature, which undoubtedly is very important for the applica-tions of singularly perturbed system theory.

This paper is concerned with the problem of designing sta-bilizing and controllers for nonlinear singularly perturbedsystems described by T-S fuzzy models with the considera-tion of the bound of singular perturbation parameter . TwoLMI-based methods are presented for simultaneously designing

1063-6706/$25.00 © 2007 IEEE


616 IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 16, NO. 3, JUNE 2008

the bound of and stabilizing or controllers for a fuzzysingularly perturbed system, respectively. For the issue of com-puting the bound of singularly perturbed parameter , sufficientconditions are derived for evaluating the upper bound ofsubject to stability or a prescribed performance bound con-straint for the resulting closed-loop system for , re-spectively, where the upper bound can be obtained by solvinga generalized eigenvalue problem (GEVP) [20]. For the problemof designing stabilizing and controllers without the consid-eration of improving the bound of , design methods are alsogiven in terms of solutions to LMIs. The paper is organized asfollows. In Section II, the system description, the consideredproblems, and preliminary lemmas are presented. Section IIIconsiders the problem of designing stabilizing controllers andthe bound of , and the case for controller design is studiedin Section IV. Section V gives examples to illustrate the effec-tiveness of the new proposed methods. Finally, Section VI con-cludes this paper.

Notation: For a matrix , is defined as the largestsingular value of . For a square matrix , is definedas

The symbol within a matrix represents the symmetric entries

......

. . ....

II. SYSTEM DESCRIPTION AND PROBLEM STATEMENT

A. System Description

The class of nonlinear singularly perturbed systems underconsideration is described by the following fuzzy system model:

IF is and is is

THEN

(1)

where are fuzzy sets, are the premisevariables, and are the state vectors,

is the control input, is the disturbance,is the controlled output, the matrices , , ,

, , , , , , , and are of appropriatedimensions, is the number of IF-THEN rules, and is asmall constant.

Denote

is the grade of membership of in , whereit is assumed that

Let

then

(2)

is said to be normalized membership functions. TheT-S fuzzy model of (1) is inferred as follows:

(3)

The system (3) can be rewritten as follows:

(4)

where

In this paper, the concept of parallel distributed compensation(PDC) is used to design fuzzy controller, i.e., the designed fuzzycontroller shares the same fuzzy sets with the fuzzy model in


YANG AND DONG: CONTROL SYNTHESIS OF SINGULARLY PERTURBED FUZZY SYSTEMS 617

the premise parts (more details can be found in [13]). For thefuzzy model (1), the following state feedback controller [13] isadopted:

IF is and is is

THEN (5)

Because the controller rules are same as plant rules, we obtainthe state feedback controller as follows:

(6)

Combining (6) and (4), then the resulted closed-loop system isgiven as follows:

(7)

B. Problem Statement

In this paper, the following problems will be addressed.Controller Design With Consideration of Bound of :(i) Find , and an as big as possible

such that the closed-loop system (7) with isasymptotically stable for any and allsatisfying (2).

(ii) Let be a given constant. Findand an as big as possible such that the closed-loopsystem (7) is asymptotically stable and with an -normless than or equal to for any and allsatisfying (2).

Evaluation of Bound of :(iii) Let be given. Find an as

big as possible such that the closed-loop system (7) withis asymptotically stable for any and

all satisfying (2).(iv) Let and be given. Find an

as big as possible such that the closed-loop system (7) isasymptotically stable and with an -norm less than orequal to for any and all satisfying(2).

Remark 1: Problems (i) and (ii) are concerned with simul-taneously designing and finding the upperbound of with guaranteeing the stability and performanceof the closed-loop system (7), respectively. Problems (iii) and(iv) are related to the problem of finding the upper bound of

subject to that the closed-loop system (7) is asymptoticallystable or with the constraint of an performance bound when

are given. Moreover, the controller designproblems without the consideration of bound of will also bestudied in Sections III and IV, respectively.

C. Preliminaries

The following preliminaries will be used in the sequel.For the fuzzy control system (7), let be a constant.

If (7) is asymptotically stable, and for any(the space of square integrable functions) and , thefollowing inequality holds:

then the system (7) is said to be with an -norm less than orequal to [29], [30].

Denote

then the closed-loop system (7) can be rewritten as follows:

(8)

The following lemma from [29] gives a sufficient conditionfor the system (8) to be with an -norm less than or equal to

.Lemma 1: [29] Consider the system (8). If there exists a pos-

itive definite matrix such that

holds. Then the system (8) is asymptotically stable and with an-norm less than or equal to .

Lemma 2: If there exist symmetric matrices , ,, , and matrices , ,

such that the following LMIs hold:

(9)

(10)

(11)

where

then

...... (12)



Proof:

......

......

. . ....

From (9) to (11), it follows (12).Lemma 3: [16] If there exist matrices and , ,

with

where and are symmetric matrices, satisfying the fol-lowing LMIs:

(13)

(14)

then there exists a scaler such that, for , thestate feedback controller (6) with

renders the singularly perturbed fuzzy system (7) asymptoti-cally stable.

Lemma 4: [18] For given , if there exist matricesand , , with

where and are symmetric matrices, satisfying the fol-lowing LMIs, shown in (15) and (16) at the bottom of this page,then there exists a scaler such that, for , thestate feedback controller (6) with

renders the singularly perturbed fuzzy system (7) with annorm less than .

III. STABILITY BOUND AND STABILIZATION

In this section, a method of simultaneously designing theupper bound of and stabilizing controller gains is derivedwhere the upper bound of singularly perturbed parameter canbe improved, which provides a solution to Problem (i). Forsolving Problem (iii), a method of computing the upper boundof singularly perturbed parameter subject to the stability ofthe closed-loop system is presented. Moreover, a technique fordesigning stabilizing controllers for singularly perturbed fuzzysystems without consideration of improving the bound of sin-gularly perturbed parameter is also presented in Section III-C.

A. Design of Stability Bound of and Stabilizing Controllers

The following theorem presents a method of simultaneouslydesigning the upper bound of and stabilizing controller gains.

Theorem 1: If there exist matrices , , , , ,, , , , and positive scalars ,

, , , with

where , , , , , are symmetricmatrices, satisfying the following LMIs:

(17)

(18)

(19)

(20)

(21)

(22)

(23)

(24)

(25)

(26)

where

(15)He

(16)



and denote

(27)

where

then for , the state feedback controller (6) with


Proof: See Appendix A.Remark 2: Theorem 1 presents sufficient conditions under

which an upper bound of singularly perturbed parameterand stabilizing controller gains can be obtained. From (27) and

, , can be minimized by solving the followingoptimization problem:

minimize

subject to (17)–(26) (28)

where , are positive weighting constants to bechosen. However, the optimization problem cannot be solveddirectly due to the term . Consider

From (25) and (26), we have

then it follows that

Combining it with , , (28) can be minimizedby solving the optimization problem

minimize

subject to (17)–(26) (29)

where , are positive weighting constants tobe chosen. Since the constraints (17)–(26) are of LMIs, theoptimization problem can be effectively solved via LMI Con-trol Toolbox [31]. Regarding the issue of how to choose theweighting scalars , generally, one canchoose bigger for rendering smaller . Itshould be pointed out that the upper bound obtained bysolving the previous optimization problem may be conservative.After obtaining stabilizing controller gains, a less conservativebound of can be obtained by Theorem 2 in Section III-B.

B. Computation of Stability Bound of

In this subsection, assume that the controller has been de-signed. The following theorem gives a technique to estimate theupper bound of singularly perturbed parameter subject to thestability of the closed-loop system.

Theorem 2: If there exist matrices , , , ,, and a constant with

where , , , are symmetric matrices,satisfying (9)–(11), and the following LMIs:

(30)

(31)

(32)

(33)

where , , , .Then, for , the singularly perturbed closed-loopfuzzy system (7) is asymptotically stable, where

(34)

(35)

Proof: From condition (30) and (31), we have

(36)

By , and (36), it follows that

for (37)



Pre- and post multiplying (37) by

and its transpose, then the following inequality holds:

for

Consider the following Lyapunov function:

then

......

(38)

where , are given by (34) and (35).

On the other hand, from (32) and (33), we have

for (39)

Applying (9)–(11), (39) and Lemma 2, it follows

......

combined with (38), we have

for

Thus, for , the closed-loop singularly perturbedfuzzy system (7) is asymptotically stable.

Remark 3: By Theorem 2, an upper bound of can be ob-tained by solving inequalities (9)–(11), (30)–(33) for . The op-timization problem

Minimize subject to (9), (10), (11), (30)–(33)

is a generalized eigenvalue problem (GEVP) [20], which canbe effectively solved using LMI Control Toolbox [31]. Theproblem of computing the bound of was considered in [17],where a method of finding an interval so that the systemis stable for was derived. However, to search forsmall is related to -dependent computation, which cannotavoid the difficulties linked with the stiffness of the equationsinvolved in the design.

C. Stabilizing Controller Design Without Considering StabilityBound

In this section, a technique for designing stabilizing con-trollers for singularly perturbed fuzzy systems without con-sideration of improving the bound of singularly perturbedparameter , is given as follows.

Theorem 3: If there exist matrices , , , ,, , with

where , , , are symmetric matrices,satisfying the following LMIs:

(40)

(41)

(42)

(43)

(44)



where

then there exists a scaler such that, for , thestate feedback controller (6) with


Proof: By

and , , we choose , then by matrixinvertible formula, it follows that

Since and , there exists a scaler , suchthat for , which implies

Choose Lyapunov function

then

(45)On the other hand, substitute

(46)

for in (44) and pre- and post multiplying (44) byand , then we can obtain

(47)

where

Now, pre- and post multiplying (41)–(43) by and , then itfollows that

(48)

(49)

(50)

From (47), we have that there exist a scalar such that

for

(51)

where ,. By (48)–(50), (51) and Lemma 2, it follows

that

i.e.,

combined with (45), then we have

for

which implies that the closed-loop singularly perturbed fuzzysystem (7) is asymptotically stable, and from (46), we obtain

Thus, the proof is complete.Remark 4: Theorem 3 presents a method for designing state

feedback stabilizing controllers for singularly perturbed fuzzysystems. However, the upper bound of singularly perturbed pa-rameter is not addressed in the design.

The following theorem shows that the new method given inTheorem 3 is less conservative than that given in Lemma 3 [16].

Theorem 4: If the condition in Lemma 3 holds, then the con-dition in Theorem 3 holds.

Proof: Assume that the condition in Lemma 3 holds, thenchoose ,

, , . Then it follows thatconditions (40)–(43) hold from (13) and (14). Moreover, from(14) and the chosen , the following inequalities hold:

......

. . ....



which implies that (44) holds. Therefore, the condition of The-orem 3 holds. Thus, the proof is complete.

IV. BOUND OF AND CONTROL

In this section, the results in Section III are extended to thecase that the closed-loop system is required to be with anperformance bound. Solutions to Problem (ii) and (iv) formu-lated in Section II-B are, respectively, given in Section IV-A andIV-B. Moreover, Section IV-C also presented a new method fordesigning controllers without consideration of improvingthe bound of singularly perturbed parameter .

A. Design for Bound of and Performance

In this subsection, a method is given for designing con-trollers with consideration of improving the bound of singularlyperturbed parameter .

Theorem 5: For given , if there exist matrices , ,, , , , , , , and

positive scalar variables , , , , , with

where , , , , , are symmetricmatrices, satisfying the following LMIs:

(52)

(53)

(54)

(55)

(56)

(57)

(58)

(59)

(60)

(61)

where , are the same as in Theorem 3, and

Denote

(62)

where

(63)

then for , the singularly perturbed closed-loopfuzzy system (8) via state feedback controller

where

is asymptotically stable and with an norm less than .Proof: See Appendix A.

Remark 5: If the conditions of Theorem 5 hold, thenis an upper bound of singularly perturbed parameter , and

The problem of minimizing can be reduced to solving thefollowing optimization problem with LMI constraints:

Minimize

subject to (52)–(61)

where , , , , and are positive weighting constants tobe chosen.

B. Computation of Upper Bound of

In this subsection, we assume that the state feedback gains aregiven. Then the following theorem gives a method to estimatethe upper bound of singularly perturbed parameter subject tothe closed-loop system with an performance bound.

Theorem 6: For a given , if there exists , , ,, , and a constant with

where , , , are symmetric matricessatisfying (9)–(11) and the following LMIs:

(64)

(65)

(66)

(67)



where

where

then, for , the singularly perturbed closed-loopfuzzy system (8) is asymptotically stable and with an normless than .

Proof: From (64) to (67), it follows that, for any

(68)

and

By (9)–(11), and Lemma 2, which further implies that

...... (69)

On the other hand

......

Thus, from Lemma 1 and (69), the conclusion follows.Remark 6: When the state feedback gains are given, The-

orem 6 gives a method to estimate the upper bound of singularlyperturbed parameter subject to that the closed-loop systemis required to be with a prescribed performance bound.An upper bound of can be obtained by solving inequalities(9)–(11) and (64)–(67) for . The optimization problem

Minimize subject to (9), (10), (11), (64)–(67)

is also a generalized eigenvalue problem (GEVP) [20], whichcan be effectively solved using LMI Control Toolbox [31].

C. Controller Design

In this subsection, a new method is given for designingcontrollers, but without consideration of improving the boundof singularly perturbed parameter .

Theorem 7: For given , if there exist matrices , ,, , , , with

where , , , and are symmetric ma-trices satisfying the following LMIs:

where

then there exists a scaler such that, for ,the singularly perturbed fuzzy system (7) via the state feedbackcontroller (6) with

is asymptotically stable and with an norm less than .Proof: It is similar to Theorem 3, and omitted here.

The following theorem shows that the new method given inTheorem 7 is less conservative than that given in Lemma 4 [18].

Theorem 8: If the condition of Lemma 4 holds, then the con-dition of Theorem 7 holds.

Proof: It is similar to Theorem 4 and omitted here.

V. EXAMPLE

We consider an inverted pendulum controlled by a motor viaa gear train. It can be described by the following state equa-tions [32]:



TABLE IUPPER BOUNDS OF VIA THEOREM 6

where , ,, is the control input, is the disturbance input,is the motor torque constant, is the back emf constant,

and is the gear ratio. The parameters for the plant are given as9.8 m/s , 1 m, 1 kg, , 0.1 Nm/A,

0.1 Vs/rad, 1 and mH. Note that theinductance represents the small “parasitic” parameter in thesystem. Then, we get

Choose the membership functions of the fuzzy sets as

This fuzzy model exactly represents the dynamics of the non-linear mechanical system under . A T-S fuzzymodel can be obtained as follows

Plant Rule 1:

IF is

THEN

Plant Rule 2:

IF is

THEN

where

Lemma 4, Theorems 5 and 7 are applicable for designing fuzzycontrollers for the system.

By using Lemma 4 and Theorem 7, the obtained optimalperformance indexes are and , re-spectively, and the corresponding controller gains are given asfollows:

(70)

(71)

Now we apply Theorem 5 to design a fuzzy controller forthe system. Choose weighting scalars as

and , then the controller gains are given as follows:

(72)

By using Theorem 6, the upper bounds of subject to guaran-teeing performance bound can be estimated, and shownin Table I.

From Table I, it is easy to see that the new proposed designgiven by Theorem 5 gives a considerable improvement of upperbounds of . The upper bounds of given by Lemma 4 andTheorem 7 are very small.

Assume the initial states are zero, and the disturbance inputsignal is shown in Fig. 1. The simulation results of theoutput with the controller gains (70)–(72) are given inFigs. 2–4, respectively. The simulations for the square root ofratio of the regulated output energy to the disturbance inputnoise energy are depicted in Figs. 5–7. From the simulationresults, it can be seen that the fuzzy controller (72) guaranteesgood performance of the resulting closed-loop system,while the controllers (70) and (71) give poor system responses.



Fig. 1. Disturbance input w(t) used during simulation.

Fig. 2. Trajectory of z(t) via controller (70).



Fig. 5. z (t)z(t)dt= w (t)w(t)dt via controller (70).





VI. CONCLUSION

In this paper, we have studied the problem of designing sta-bilizing and controllers for nonlinear singularly perturbedsystems described by T-S fuzzy models with the considerationof improving the bound of singular perturbation parameter .The main contribution is as follows. For the synthesis problemof simultaneously designing the bound of and stabilizing or

controllers, LMI-based methods are presented. For evalu-ating the upper bound of subject to stability or a prescribed

performance bound constraint for the resulting closed-loopsystem, LMI-based sufficient conditions are developed, respec-tively. For the stabilizing and control synthesis without theconsideration of improving the bound of , new design methodsare also given in terms of solutions to a set of LMIs. The exam-ples have shown the efficiency of the proposed methods.

APPENDIX APROOF OF THEOREM 1

Proof: It consists of three parts. In Parts 1 and 2, we provethat (78) and (92) hold, respectively. In Part 3, the proof is com-pleted by using (78) and (92).

Part 1: First, from conditions (25) and (26), we have

(73)

From (27) and (73), it follows

(74)

then

(75)

Pre- and postmultiplying (75) by and its transpose andapplying the Schur complement, we obtain

which further implies that

for (76)

Let , then can be expressed as follows:

So (76) is equivalent to

(77)

Pre- and postmultiplying (77) by

and its transpose, then the following inequality holds:

(78)

Part 2: From (25), we can obtain

(79)

From (27), we have

(80)

By (17)–(19), (23), and Lemma 2, it follows that

i.e.,

(81)

Applying the Schur complement to (81), we have

i.e.,

(82)



Notice that

(83)

where are the same as in (34). Applying (82) to (83), itfollows that

(84)

By (80) and (84), we have

which implies that

(85)

Pre- and postmultiplying (24) by and its trans-pose, it follows that

...... (86)

Multiplying (86) by , and summing them, itfollows

......

Combining it with (79), we can obtain

......

Applying Lemma 2 to the previous inequality, then

which implies that

(87)

(88)

From (85) and (88), we have

(89)

Substituting into (89), then we have

(90)

Pre- and postmultiplying (90) by , then we can obtain

(91)

where are same as in (35). From (87), then

Combining it and (91), that yields

for (92)

Part 3: Choose Lyapunov function

Then, by (78) and (92), it follows that and

......

for

for . Thus, the proof is complete.



APPENDIX BPROOF OF THEOREM 5

Proof: From (58), (62), and Part 1 of the proof of Theorem3, we have

for

where

. By (62), it follows that

(93)

On the other hand, from (60), we have

(94)Substituting , , then (94) becomes

(95)

where

For brief expression, we also denote

Applying the Schur complement to (95), we obtain

which implies that

(96)

From (63), it follows that

(97)

Combining (96), (97), and (93), we have

which further implies that

for (98)

where

On the other hand, from (59), we have

(99)

Thus, from (98) and (99), it follows:

(100)

By (61) and Lemma 2, we can obtain (101) shown at the bottomof the page. Applying the Schur complement to (101), it followsthat

(102)

where

(101)



and

By (99), (100), and (102), we have

(103)

Pre- and postmultiplying (103) by and , then we obtain

i.e.,

Then, by Lemma 1, the proof is completed.

REFERENCES

[1] P. V. Kokotovic, J. O’Reilly, and H. K. Khalil, Singular PerturbationMethods in Control: Analysis and Design. Orlando, FL: AcademicPress, 1986.

[2] E. Fridman, “A descriptor system approach to nonlinear singularlyperturbed optimal control problem,” Automatica, vol. 37, no. 4, pp.543–549, 2001.

[3] E. Fridman, “Effects of small delays on stability of singularly perturbedsystems,” Automatica, vol. 38, no. 5, pp. 897–902, 2002.

[4] Z. Pan and T. Basar, “H -optimal control for singularly perturbedsystems. Part I: Perfect state measurements,” Automatica, vol. 29, no.2, pp. 401–423, 1993.

[5] Z. Pan and T. Basar, “H -optimal control for singularly perturbed sys-tems. II. Imperfect state measurements,” IEEE Trans. Autom. Control,vol. 39, no. 2, pp. 280–299, Feb. 1994.

[6] P. Shi and V. Dragan, “AsymptoticH control of singularly perturbedsystems with parametric uncertainties,” IEEE Trans. Autom. Control,vol. 44, no. 9, pp. 1738–1742, Sep. 1999.

[7] Z. Pan and T. Ba, “Time-scale separation and robust controller de-sign for uncertain nonlinear singularly perturbed systems under perfectstate measurements,” Int. J. Robust Nonlinear Control, vol. 6, no. 7, pp.585–608, 1996.

[8] H. D. Tuan and S. Hosoe, “On linear robust controllers for a class ofnonlinear singular perturbed systems,” Automatica, vol. 35, no. 4, pp.735–739, 1999.

[9] E. Fridman, “State-feedback H control of nonlinear singularly per-turbed systems,” Int. J. Robust Nonlinear Control, vol. 11, no. 12, pp.1115–1125, 2001.

[10] T. Takagi and M. Sugeno, “Fuzzy identification of systems and its ap-plications to modeling and control,” IEEE Trans. Syst., Man. Cybern.,vol. 15, no. 1, pp. 116–132, Jan. 1985.

[11] S. G. Cao, N. W. Rees, and G. Feng, “Analysis and design for a classof complex control systems part I: Fuzzy modelling and identification,”Automatica, vol. 33, no. 6, pp. 1017–1028, 1997.

[12] G. Feng, S. G. Cao, and N. W. Rees, “An approach to H controlof a class of nonlinear systems,” Automatica, vol. 32, no. 10, pp.1469–1474, 1996.

[13] K. Tanaka and H. O. Wang, Fuzzy Control Systems Design and Anal-ysis: A Linear Matrix Inequality Approach. New York: Wiley, 2001.

[14] X.-J. Ma, Z.-Q. Sun, and Y.-Y. He, “Analysis and design of fuzzy con-troller and fuzzy observer,” IEEE Trans. Fuzzy Syst., vol. 6, no. 1, pp.41–51, Feb. 1998.

[15] K. Tanaka, T. Hori, and H. O. Wang, “A multiple Lyapunov functionapproach to stabilization of fuzzy control systems,” IEEE Trans. FuzzySyst., vol. 11, no. 4, pp. 582–589, Aug. 2003.

[16] H. Liu, F. Sun, and Z. Sun, “Stability analysis and synthesis of fuzzysingularly perturbed systems,” IEEE Trans. Fuzzy Syst., vol. 13, no. 5,pp. 273–284, Oct. 2005.

[17] T. H. S. Li and K.-J. Lin, “Stabilization of singularly perturbed fuzzysystems,” IEEE Trans. Fuzzy Syst., vol. 12, no. 5, pp. 579–595, Oct.2004.

[18] W. Assawinchaichote and S. K. Nguang, “H fuzzy control designfor nonlinear singularly perturbed systems with pole placement con-straints: An LMI approach,” IEEE Trans. Syst., Man, Cybern. B, Cy-bern., vol. 34, no. 1, pp. 579–588, Jan. 2004.

[19] P. Shi, “H output feedback control design for uncertain fuzzy singu-larly perturbed systems: An LMI approach,” Automatica, vol. 40, no.12, pp. 2147–2152, Dec. 2004.

[20] S. Boyd, L. E. Ghaoui, E. Feron, and V. Balakrishnan, Linear matrixinequalities in system and control theory. Philadelphia, PA: Societyfor Industrial and Applied Mathematics (SIAM), 1994.

[21] N. R. Sandell, “Robust stability of systems with application to singularperturbations,” Automatica, vol. 15, no. 4, pp. 467–470, 1979.

[22] W. Feng, “Characterization and computation for the bound in lineartime-invariant singularly perturbed systems,” Syst. Control Lett., vol.11, no. 3, pp. 195–202, 1988.

[23] B.-S. Chen and C.-L. Lin, “On the stability bounds of singularly per-turbed systems,” IEEE Trans. Automat. Control, vol. 35, no. 11, pp.1265–1270, Nov. 1990.

[24] Z. H. Shao and M. E. Sawan, “Robust stability of singularly perturbedsystems,” Int. J. Control, vol. 58, no. 6, pp. 1469–1476, 1993.

[25] S. Sen and K. B. Datta, “Stability bounds of singularity perturbed sys-tems,” IEEE Trans. Autom. Control, vol. 38, no. 2, pp. 302–304, Feb.1993.

[26] D. Mustafa and T. N. Davidson, “Block bialternate sum and associatedstability formulae,” Automatica, vol. 31, no. 9, pp. 1263–1274, 1995.

[27] L. Cao and H. M. Schwartz, “Complementary results on the stabilitybounds of singularly perturbed systems,” IEEE Trans. Autom. Control,vol. 49, no. 11, pp. 2017–2021, Nov. 2004.

[28] Z. H. Shao, “Robust stability of two-time-scale systems with nonlinearuncertainties,” IEEE Trans. Autom. Control, vol. 49, no. 2, pp. 258–261,Feb. 2004.

[29] A. Isidori, “H control via measurement feedback for affine nonlinearsystems,” Int. J. Robust Nonlinear Control, vol. 4, pp. 553–574, 1994.

[30] K. Zhou, J. C. Doyle, and K. Glover, Robust and Optimal Control.Upper Saddle River, NJ, USA: Prentice-Hall, 1996.

[31] P. Gahinet, A. Nemirovski, A. J. Laub, and M. Chilali, LMI ControlToolbox. Natick, MA: The MathWorks, Inc., 1995.

[32] S. H. Z. Maccarley, “State-feedback control of non-linear systems,” Int.J. Control, vol. 43, pp. 1497–1514, 1986.

Guang-Hong Yang (SM’03) received the B.S.and M.S. degrees in mathematics from NortheastUniversity of Technology, Shenyang, China, in 1983and 1986, respectively, and the Ph.D. degree incontrol engineering from Northeastern University,Shenyang, China (formerly, Northeast University ofTechnology), in 1994.

He is currently a Professor with the Collegeof Information Science and Engineering, North-eastern University. From 1986 to 1995, he wasa Lecturer/Associate Professor with Northeastern

University. In 1996, he was as a Postdoctoral Fellow with the NanyangTechnological University, Singapore. From 2001 to 2005, he was a ResearchScientist/Senior Research Scientist with the National University of Singapore,Singapore. His current research interests include fault tolerant control, faultdetection and isolation, nonfragile control systems design, and robust control.

Dr. Yang is an Associate Editor for the International Journal of Control, Au-tomation, and Systems (IJCAS), and an Associate Editor on the Conference Ed-itorial Board of the IEEE Control Systems Society.

Jiuxiang Dong received the B.S. degree in mathe-matics and applied mathematics and the M.S. degreein applied mathematics from Liaoning NormalUniversity, Dalian, China, in 2001 and 2004, respec-tively. He is currently pursing the Ph.D. degree atNortheastern University, Shenyang, China.

His research interests include fuzzy control, robustcontrol, and fault-tolerant control.



104 IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 17, NO. 1, FEBRUARY 2009

Multiobjective Control for T–S FuzzySingularly Perturbed Systems

Chunyu Yang and Qingling Zhang

Abstract—This paper investigates the problem of multiobjec-tive control for a class of Takagi–Sugeno (T–S) fuzzy singularlyperturbed systems. Based on a linear matrix inequality (LMI) ap-proach, a state feedback controller that depends on the singularperturbation parameter ε is developed such that: 1) the H∞ per-formance of the resulting closed-loop system is less than or equal tosome prescribed value; 2) the closed-loop poles of each local systemare within a prespecified LMI stability region; and 3) for a givenupper bound ε for the singular perturbation parameter ε, both 1)and 2) are guaranteed for all ε ∈ (0, ε]. It is shown that the ε-dependent controller is well defined for any ε ∈ (0, ε], and can bereduced to an ε-independent one if ε is sufficiently small. Finally, apractical example is given to show the feasibility and effectivenessof the obtained method.

Index Terms—H∞ performance, linear matrix inequality(LMI), pole placement, singular perturbation bound, Takagi–Sugeno (T–S) fuzzy singularly perturbed systems.

I. INTRODUCTION

IN MODERN control systems, such as economic models,motor control systems, convection–diffusion systems, power

systems, and magnetic-ball suspension systems, small parame-ters are often involved, which can lead to high dimensionalityand ill-conditioned numerical issues in the system analysis andcontroller design. To deal with these problems, the theory of sin-gular perturbation has been developed in the past four decades(see [1]–[8] and the references therein).

Stability bound problem in a singularly perturbed system(SPS) has attracted much attention (see, e.g., [9]–[14]). Thisproblem is usually described as characterizing and computingan upper bound ε0 for the singular perturbation parameter ε,such that the stability of the SPS is ensured for all ε ∈ (0, ε0)or (0, ε0 ]. The stability bound problem of linear SPSs has beensolved perfectly, and some frequency- and time-domain meth-ods have been derived to provide the largest upper bound forε (see, e.g., [4], [5], and [13]). For nonlinear SPSs, the stabil-ity bound problem is a complex and challenging topic. Somemethods to estimate the upper bound for ε have been obtainedin [8], [10]–[12], and [14]. Stabilization bound problem has alsobeen considered (see [15]–[18]). The aim is to design a controllerto enlarge the stability bound. The authors in [15]–[17] have

Manuscript received December 19, 2007; revised May 30, 2008; First pub-lished September 9, 2008; current version published February 4, 2009. Thiswork was supported by the Natural Science Foundation of China under Grant60574011.

The authors are with the Institute of Systems Science, Northeastern Uni-versity, Shenyang 110004, China, and also with the Key Laboratory ofIntegrated Automation of Process Industry, Northeastern University, Min-istry of Education, Shenyang 110004, China (e-mail: [email protected];[email protected]).

Digital Object Identifier 10.1109/TFUZZ.2008.2005404

discussed the stabilization bound problem for linear SPSs andpresented some valuable results. Recently, the robust stabiliza-tion problem of a class of nonlinear SPSs has been consideredin [18], where a state feedback controller is designed by solvingtwo independent Lyapunov equations, and the stability boundof the closed-loop system is then computed by using the exist-ing methods given in [14]. In this two-step design procedure,stability bound is not regarded as one of design objects. To thebest of our knowledge, stability bound synthesis of nonlinearSPSs remains as an open area. Further, it is also a significantproblem to provide a bound for ε, such that the stability andother performances of the SPS are guaranteed.

In the past several decades, nonlinear control approachesbased on the Takagi–Sugeno (T–S) model have been extensivelystudied and successfully applied [19]–[21]. The main reasonsare as follows: 1) T–S model has been shown to be a universalapproximator for a wide class of nonlinear systems [22], [23]and 2) the design is usually formulated in the linear matrixinequality (LMI) framework that can be efficiently solved bythe existing tools [24]. Recently, many researchers have fo-cused on the analysis and design of T–S fuzzy SPSs. Stabilityanalysis and stabilization problems for both continuous- anddiscrete-time T–S fuzzy SPSs were investigated in [25], andsome LMI-based approaches were derived. H∞ control for T–Sfuzzy SPSs was considered in [26] and [27], and LMI-baseddesign methods were obtained. To get a satisfactory transientbehavior, H∞ control for T–S fuzzy SPSs with pole placementconstraints was considered in [28] and [29]. The obtained designmethods were expressed in terms of ε-independent LMIs. Usingthe results in [25]–[29], the stability and H∞ performance of theresulting closed-loop systems are only ensured for sufficientlysmall singular perturbation parameter ε. Since no reliable upperbound for ε can be determined, the effectiveness of the obtainedcontrollers has to be validated by trial and error. Such a problemhas been recognized by several researchers. Based on the sepa-ration of states into slow and fast ones, a composite fuzzy H∞controller was constructed in [30] and [31], and the upper boundfor the singular perturbation parameter ε is then determined. Inthese results, the controller design procedures are not concernedwith the bound of the singular perturbation parameter ε. More-over, since these approaches involve the separation of states intoslow and fast ones, they cannot be applied to nonstandard SPSs.H∞ control for T–S fuzzy SPSs with the consideration of en-larging the bound of ε was first studied in [32], where stabilizingand H∞ controllers with the consideration of maximizing thebound of ε were designed. However, in the design procedures,there are five weighting constants required to be chosen subjec-tively, which will result in conservatism and inconvenience in

1063-6706/$25.00 © 2009 IEEE


YANG AND ZHANG: MULTIOBJECTIVE CONTROL FOR T–S FUZZY SINGULARLY PERTURBED SYSTEMS 105

applications. In practice, the singular perturbation parameter εis usually very small, and a rough bound for the parameter ε isoften known. Thus, it is significant to regard a predefined boundfor ε as one of design objects.

This paper considers the problem of multiobjective controlfor T–S fuzzy SPSs. The problem consists of H∞ control, poleplacement, and singular perturbation bound design. Specifically,given an H∞ performance bound γ > 0, an LMI stability regionD , and an upper bound ε for the singular perturbation parameterε, this paper will construct an ε-dependent state feedback con-troller, such that ∀ε ∈ (0, ε], the L2-gain of the mapping fromthe exogenous input noise to the regulated output is less than orequal to γ, and the poles of each subsystem are all within theLMI stability region D . Two subproblems of the multiobjectivecontrol are discussed, and the main problem is then solved. Anε-dependent state feedback controller is designed by solving aset of ε-independent LMIs. It is shown that the controller is welldefined ∀ε ∈ (0, ε]. If ε is sufficiently small, the controller canbe reduced to an ε-independent one. At last, an inverted pendu-lum controlled by a dc motor via a gear train is used to illustratethe obtained approach.

The rest of this paper is organized as follows. In Section II,the problems under consideration are defined. The main resultsare given in Section III. The multiobjective control problemis reduced to the feasibility of a set of ε-independent LMIs. InSection IV, an illustrative example is given to show the effective-ness and advantage of the obtained method. Section V concludesthis paper and proposes a new topic for our future work.

II. PROBLEM FORMULATION

Consider a T–S fuzzy SPS, in which the ith rule is formulatedas follows:

Plant rule i :

IF v1(t) is Mi1 , v2(t) is Mi2 , . . . , vϑ (t) is Miϑ

THEN

E(ε)x(t) = Aix(t) + B1iw(t) + B2iu(t)

z(t) = Cix(t) + Diu(t)

for i = 1, 2, . . . , r (1)

where E(ε) = diagI, εI, ε > 0 is the singular pertur-bation parameter, Mij (i = 1, 2, . . . , r, j = 1, 2, . . . , ϑ) arefuzzy sets, r is the number of fuzzy rules, v(t) =[ v1(t) v1(t) · · · vϑ (t) ]T is the premise vector that may de-pend on states in many cases, ϑ is the number of premise vari-ables, x(t) ∈ n is the state, u(t) ∈ s is the input, w(t) ∈ p

is the disturbance that belongs to L2 [0,∞), z(t) ∈ s is thecontrolled output, and Ai,B1i , B2i , Ci,Di are constant matri-ces with appropriate dimensions.

Denote

wi(v(t)) =ϑ∏

k=1

Mik (vk (t)), i = 1, 2, . . . , r

where Mik (vk (t) is the grade of membership of vk (t) in Mik .

It is assumed in this paper that

wi(v(t)) ≥ 0,

r∑i=1

wi(v(t)) > 0, i = 1, 2, . . . , r ∀t ≥ 0.

Let

µi(v(t)) =wi(v(t))∑ri=1 wi(v(t))

, i = 1, 2, . . . , r.

Then

µi(v(t)) ≥ 0,

r∑i=1

µi(v(t)) = 1, i = 1, 2, . . . , r ∀t ≥ 0.

(2)For the convenience of notations, we denote µi =

µi(v(t)), i = 1, 2, . . . , r.Then, the T–S fuzzy model (1) is inferred as follows:

E(ε)x(t) =r∑

i=1

µi [Aix(t) + B1iw(t) + B2iu(t)]

z(t) =r∑

i=1

µi [Cix(t) + Diu(t)]. (3)

Throughout the paper, it is assumed that the singular perturba-tion parameter ε is available for feedback. Based on the conceptof parallel distributed compensation (PDC), the state feedbackfuzzy controller is described by

Controller rule i :

IF v1(t) is Mi1 , v2(t) is Mi2 , . . . , vϑ (t) is Miϑ

THEN

u(t) = Ki(ε)x(t)

for i = 1, 2, . . . , r. (4)

Because the controller rules are same as the plant rules, thestate feedback controller is given as follows:

u(t) =r∑

i=1

µiKi(ε)x(t). (5)

Remark 2.1: In many SPSs, the singular perturbation parame-ter ε can be measured. In these cases, ε is available for feedback,which has attracted much attention. For example, ε-dependentcontrollers were designed for T–S fuzzy SPSs in [27] and [29].Since ε is usually very small, an ε-dependent controller may beill-conditioning as ε tends to zero. Thus, it is a key task to ensurethe obtained controller to be well defined. This problem will bediscussed later.

Substituting (5) into (3) yields the closed-loop system

E(ε)x(t) =r∑

i=1

r∑j=1

µiµj [(Ai + B2iKj (ε))x(t) + B1iw(t)]

z(t) =r∑

i=1

r∑j=1

µiµj [(Ci + DiKj (ε))x(t)]. (6)

The following definitions that can be found in [28] will beused in this paper.



Definition 2.1: Given γ > 0, a system of the form (3) is saidto be with an H∞-norm less than or equal to γ if∫ Tf

0zT (t)z(t) dt ≤ γ2

∫ Tf

0wT (t)w(t) dt (7)

holds for x(0) = 0, where Tf is the terminal time of control andx(0) denotes the initial condition of system (3).

Definition 2.2: A subset D of the complex plane C is calledan LMI region if there exist a symmetric matrix L ∈ Rd×d anda matrix M ∈ Rd×d such that

D = z = x + jy ∈ C : fD(z) < 0 (8)

where the characteristic function fD(z) is given as follows:

fD(z) = L + Mz + MT z. (9)

In this paper, multiobjective control for T–S fuzzy SPSs isformulated as follows:

Problem 1: Given an H∞ performance bound γ, an LMIstability region D , and an upper bound ε for the singular per-turbation parameter ε, design a state feedback controller of theform (5), such that for all ε ∈ (0, ε], the closed-loop system (6)is with an H∞-norm less than or equal to γ, and the closed-looppoles of each local system are all within the LMI stability regionD .

Remark 2.2: Synthesis problems for T–S fuzzy SPSs havebeen investigated by many researchers (see [26]–[32]). Amongthese papers, only H∞ control was addressed in [26], [27], [30],and [31]. H∞ control with pole placement constraints was inves-tigated in [28] and [29]. H∞ control and singular perturbationbound design were simultaneously considered in [32]. This pa-per will consider a more general case. Problem 1 consists ofH∞ control, pole placement, and singular perturbation bounddesign.

To solve Problem 1, the following subproblems will bestudied.

Problem 2: Given an H∞ performance bound γ and an upperbound ε for the singular perturbation parameter ε, design a statefeedback controller of the form (5), such that for all ε ∈ (0, ε],the closed-loop system (6) is asymptotically stable and with anH∞-norm less than or equal to γ.

Problem 3: Given an LMI stability region D and an upperbound ε for the singular perturbation parameter ε, design a statefeedback controller of the form (5), such that for all ε ∈ (0, ε],the closed-loop poles of each local system are all within theLMI stability region D .

III. MAIN RESULTS

In this section, the main results will be derived by using thebasic lemmas given in the Appendix. For solving Problem 1,Problems 2 and 3 are considered. The following theorem pro-vides a solution to Problem 2.

Theorem 3.1: Given an H∞ performance bound γ and an up-per bound ε for the singular perturbation parameter ε, if there ex-ist matrices Zk (k = 1, 2, . . . , 5) with Zk = ZT

k (k = 1, 2, 3, 4),Fi(i = 1, 2, . . . , r) and Yij with Yij = Y T

ji (i, j = 1, 2, . . . , r)

satisfying the following LMIs

Z1 > 0 (10)[Z1 + εZ3 εZT

5

εZ5 εZ2

]> 0 (11)

[Z1 + εZ3 εZT

5

εZ5 εZ2 + ε2Z4

]> 0 (12)

Ψii1 < 0, i = 1, 2, . . . , r (13)

Ψii1 + εΨi

2 < 0, i = 1, 2, . . . , r (14)

Ψij1 + Ψj i

1 < 0, 1 ≤ i < j ≤ r (15)

Ψij1 + Ψj i

1 + ε(Ψi2 + Ψj

2) < 0, 1 ≤ i < j ≤ r (16)

Ψk3 < 0, k = 1, 2, . . . , r (17)

Ψk3 + εΨk

4 < 0, k = 1, 2, . . . , r (18)

where

U1 =[

Z1 0Z5 Z2

]U2 =

[Z3 ZT

50 Z4

]

Ψij1 = UT

1 ATi + FT

j BT2i + AiU1 + B2iFj

+1γ2 B1iB

T1j − Yij , i, j = 1, 2, . . . , r

Ψi2 = UT

2 ATi + AiU2 , i = 1, 2, . . . , r

Ψk3 =

Y11 · · · Y1r

.... . .

...

Yr1 · · · Yrr

C1U1 + D1Fk · · · CrU1 + DrFk

UT1 CT

1 + FTk DT

1

...

UT1 CT

r + FTk DT

r

−I

, k = 1, 2, . . . , r

Ψk4 =

0 · · · 0 UT2 CT

1

.... . .

......

0 · · · 0 UT2 CT

r

C1U2 · · · CrU2 0

, k = 1, 2, . . . , r

then, for any ε ∈ (0, ε], the closed-loop system (6) with Ki(ε) =Fi(U1 + εU2)−1(i = 1, 2, . . . , r) is asymptotically stable andwith an H∞-norm less than or equal to γ.

Proof: Suppose that LMIs (10)–(18) are feasible.From LMIs (13)–(18), it follows that

Ψii1 + εΨi

2 < 0 ∀ε ∈ (0, ε], i = 1, 2, . . . , r

(19)

Ψij1 + Ψj i

1 + ε(Ψi2 + Ψj

2) < 0 ∀ε ∈ (0, ε], 1 ≤ i < j ≤ r

(20)



and

Ψk3 + εΨk

4 < 0 ∀ε ∈ (0, ε], k = 1, 2, . . . , r. (21)

Substituting Z(ε) = U1 + εU2 into inequalities (19)–(21)shows that

Ψiia (ε) < 0 ∀ε ∈ (0, ε], i = 1, 2, . . . , r (22)

Ψija (ε) + Ψj i

a (ε) < 0 ∀ε ∈ (0, ε], 1 ≤ i < j ≤ r (23)

and

Y11 · · · Y1r

.... . .

...

Yr1 · · · Yrr

C1Z(ε) + D1Fk · · · CrZ(ε) + DrFk

ZT (ε)CT1 + FT

k DT1

...

ZT (ε)CTr + FT

k DTr

−I

< 0

∀ε ∈ (0, ε], k = 1, 2, . . . , r, (24)

where Ψija (ε) = ZT (ε)AT

i + FTj BT

2i + AiZ(ε) + B2iFj +1/γ2(B1iB

T1j − Yij ), i, j = 1, 2, . . . , r.

Using Lemma 1.3, LMIs (10)–(12) imply

E(ε)Z(ε) = ZT (ε)E(ε) > 0 ∀ε ∈ (0, ε]

which indicates that Z(ε) is nonsingular for any ε ∈ (0, ε].Pre- and postmultiplying (22) and (23) by Z−T (ε) and its

transpose, respectively, yield

Ψiib (ε) < 0 ∀ε ∈ (0, ε], i = 1, 2, . . . , r (25)

and

Ψijb (ε) + Ψj i

b (ε) < 0 ∀ε ∈ (0, ε], 1 ≤ i < j ≤ r(26)

where Ψijb = (Ai + B2iKj (ε))T P (ε) + PT (ε)(Ai + B2iKj

(ε)) + 1/γ2(PT (ε)B1iBT1jP (ε) − Xij (ε)), Kj (ε) = FjZ

−1

(ε), P (ε) = Z−1(ε), and Xij (ε) = Z−T (ε)YijZ−1(ε).

Pre- and postmultiplying (24) by diagZ−T (ε), . . . ,Z−T (ε), I and its transpose, respectively, yield

X11(ε) · · · X1r (ε) CT1 + KT

k (ε)DT1

.... . .

......

Xr1(ε) · · · Xrr (ε) CTr + KT

k (ε)DTr

C1 + D1Kk (ε) · · · Cr +DrKk (ε) −I

< 0

∀ε ∈ (0, ε], k = 1, 2, . . . , r. (27)

From (2) and (27), it follows that

0 >

r∑k=1

µk

X11(ε) · · · X1r (ε)...

. . ....

Xr1(ε) · · · Xrr (ε)

C1 + D1Kk (ε) · · · Cr + DrKk (ε)

CT1 + KT

k (ε)DT1

...

CTr + KT

k (ε)DTr

−I

=

X11(ε) · · ·...

. . .

Xr1(ε) · · ·∑rk=1 µk (C1 + D1Kk (ε)) · · ·

X1r (ε)...

Xrr (ε)∑rk=1 µk (Cr + DrKk (ε))∑r

k=1 µk (CT1 + KT

k (ε)DT1 )

...∑rk=1 µk (CT

r + KTk (ε)DT

r )

−I

.

Applying the Shur complement to the previous inequality, onegets

∑r

k=1 µk (CT1 + KT

k (ε)DT1 )

...∑rk=1 µk (CT

r + KTk (ε)DT

r )

×

∑rk=1 µk (CT

1 + KTk (ε)DT

1 )...∑r

k=1 µk (CTr + KT

k (ε)DTr )

T

+

X11(ε) · · · X1r (ε)...

. . ....

Xr1(ε) · · · Xrr (ε)

< 0. (28)

Then, for all x ∈ n , it holds that

0 ≥

µ1x

...

µrx

T

X11(ε) · · · X1r (ε)...

. . ....

Xr1(ε) · · · Xrr (ε)

µ1x

...

µrx



+

µ1x

...

µrx

T

∑rk=1 µk (CT

1 + KTk (ε)DT

1 )...∑r

k=1 µk (CTr + KT

k (ε)DTr )

×

∑rk=1 µk (CT

1 + KTk (ε)DT

1 )...∑r

k=1 µk (CTr + KT

k (ε)DTr )

T

µ1x

...

µrx

=r∑

i=1

r∑j=1

µiµjxT Xij (ε)x + zT z (29)

which indicatesr∑

i=1

r∑j=1

µiµjxT Xij (ε)x ≤ −zT z. (30)

Furthermore, it can be seen thatr∑

i=1

r∑j=1

µiµjxT Xij (ε)x < 0 ∀x = 0. (31)

Using Lemma 1.3, LMIs (10)–(12) imply

E(ε)Z(ε) = ZT (ε)E(ε) > 0 ∀ε ∈ (0, ε]

which shows that

E(ε)P (ε) = PT (ε)E(ε) > 0 ∀ε ∈ (0, ε].

Choose a Lyapunov function as

V (x(t)) = xT (t)E(ε)P (ε)x(t). (32)

Then

V (x(t)) = xT (t)E(ε)P (ε)x(t) + xT (t)PT (ε)E(ε)x(t)

=r∑

i=1

r∑j=1

µiµj [(Ai + B2iKj )x(t) + B1iw(t)]T

× P (ε)x(t) + xT (t)PT (ε)r∑

i=1

r∑j=1

µiµj

× [(Ai + B2iKj )x(t) + B1iw(t)]

=r∑

i=1

r∑j=1

µiµjxT (t)

× [(Ai + B2iKj )T P (ε)

+ PT (ε)(Ai + B2iKj ) +1γ2 PT (ε)B1i

× BT1jP (ε)]x(t) −

r∑i=1

r∑j=1

µiµjxT (t)

1γ2 PT (ε)

× B1iBT1jP (ε)x(t)

+r∑

i=1

µi [wT (t)BT1iP (ε)x(t)

+ xT (t)PT (ε)B1iw(t)]

=r∑

i=1

µ2i x

T (t)[(Ai + B2iKi)T P (ε)

+ PT (ε)(Ai + B2iKi)

+1γ2 PT (ε)B1iB

T1iP (ε)]x(t)

+r∑

i=1

r∑i<j

µiµjxT (t)

× [(Ai + B2iKj + Aj

+ B2jKi)T P (ε) + PT (ε)(Ai + B2iKj + Aj

+ B2jKi) +1γ2 PT (ε)B1iB

T1jP (ε)

+1γ2 PT (ε)B1jB

T1iP (ε)]x(t) −

r∑i=1

r∑j=1

µiµj

× 1γ2 xT (t)PT (ε)B1iB

T1jP (ε)x(t)

+r∑

i=1


+ xT (t)PT (ε)B1iw(t)]. (33)

From inequalities (25), (26), (30), and (33), it follows that

V (x(t)) ≤r∑

i=1

µ2i x

T (t)Xii(ε)x(t)

+r∑

i=1

r∑i<j

µiµjxT (t)(Xij (ε) + Xji(ε))x(t)

−r∑

i=1

r∑j=1

µiµjxT (t)

1γ2 PT (ε)B1iB

T1j

× P (ε)x(t) +r∑

i=1


+ xT (t)PT (ε)B1iw(t)]

=r∑

i=1

r∑j=1

µiµjxT (t)Xij (ε)x(t) + γ2wT (t)w(t)

−(

γw −r∑

i=1

µi1γ

BT1iP (ε)x(t)

)T

×(

γw −r∑

i=1

µi1γ

BT1iP (ε)x(t)

)

≤r∑

i=1

r∑j=1

µiµjxT (t)Xij (ε)x(t) + γ2wT (t)w(t)

≤ −zT (t)z(t) + γ2wT (t)w(t) (34)

which shows

V (x(t)) ≤ −zT (t)z(t) + γ2wT (t)w(t). (35)



Integrating both sides of (35) from 0 to Tf , with x(0) = 0,yields

∫ Tf

0zT (t)z(t) dt ≤ γ2

∫ Tf

0wT (t)w(t) dt.

Inequalities (31) and (34) indicate that V (x(t)) < 0 ∀x = 0when w(t) ≡ 0. This implies that the closed-loop system withw(t) ≡ 0 is asymptotically stable. This completes the proof.

Remark 3.1: LMIs (10) and (11) indicate that Z1 > 0and Z2 > 0. As a result, the matrix U1 = [ Z 1 0

Z 5 Z 2] is non-

singular. In addition, the proof of Theorem 3.1 has shownthat Z(ε) = U1 + εU2 is nonsingular for all ε ∈ (0, ε]. Then,Ki(ε) = Fi(U1 + εU2)−1 is always well defined for all ε ∈(0, ε] and limε→0+ Ki(ε) = FiU

−11 . Thus, if ε is sufficiently

small, controller (5) can be reduced to an ε-independent one.This remark is also suitable for Theorems 3.2 and 3.3.

Remark 3.2: H∞ control for T–S fuzzy SPSs with the con-sideration of singular perturbation bound has been studiedin [30]–[32]. The design approaches given by [30] and [31]involve the separation of states into slow and fast ones, whichlimits the applications of these methods to standard SPSs. AnH∞ controller with the consideration of maximizing the boundfor the singular perturbation parameter ε was designed in [32]by an LMI-based algorithm. To perform this algorithm, one hasfive weighting constants to choose subjectively, which may re-sults in conservatism and inconvenience in practice. Though thispaper has not proved that the design method described by Theo-rem 3.1 is less conservative than that given by [32], the examplein Section IV will show the advantage of Theorem 3.1. Thisadvantage may result from: 1) the information on the singularperturbation parameter ε used in the fuzzy controller (5) and 2)Lyapunov function (32) reduced to the one used by [32] if weset Z3 = 0 and Z4 = 0. Thus, the Lyapunov function, based onwhich Theorem 3.1 is derived, is more general than the one usedby [32].

Remark 3.3: The proof of Theorem 3.1 is partially motivatedby that of [21, Th. 2].

Now, a controller of the form of (5) will be constructed suchthat, for any admissible singular perturbation parameter ε, theresulting closed-loop system (6) with w(t) ≡ 0 is D-stable.

Theorem 3.2: Given an LMI stability region D and an upperbound ε for the singular perturbation parameter ε, if there ex-ist matrices Zk (k = 1, 2, . . . , 5) with Zk = ZT

k (k = 1, 2, 3, 4),Qij with Qij = QT

ji(i, j = 1, 2, . . . , r), and Fi(i = 1, 2, . . . , r)satisfying LMIs (10)–(12) and

Sii1 < 0, i = 1, 2, . . . , r (36)

Sii1 + εSi

2 < 0, i = 1, 2, . . . , r (37)

Sii1 + εSi

2 + ε2S3 < 0, i = 1, 2, . . . , r (38)

Sij1 + Sji

1 < 0, 1 ≤ i < j ≤ r (39)

Sij1 + Sji

1 + ε(Si2 + Sj

2 ) < 0, 1 ≤ i < j ≤ r (40)

Sij1 +Sji

1 +ε(Si2+Sj

2 )+2ε2S3 < 0, 1 ≤ i < j ≤ r (41)

Q11 · · · Q1r

.... . .

...Qr1 · · · Qrr

< 0 (42)

where

V1 =[

Z1 00 0

]V2 =

[Z3 ZT

5Z5 Z2

]V3 =

[0 00 Z4

]

U1 =[

Z1 0Z5 Z2

]U2 =

[Z3 ZT

50 Z4

]

Sij1 = L ⊗ V1 + M ⊗ (AiU1) + MT ⊗ (AiU1)T

+ M ⊗ (B2iFj ) + MT ⊗ (B2iFj )T − Qij

Si2 = L ⊗ V2 + M ⊗ (AiU2) + MT ⊗ (AiU2)T

S3 = L ⊗ V3

and ⊗ denotes the Kronecker product of the matrices; then forany ε ∈ (0, ε], the poles of each subsystem of system (6) withKi(ε) = Fi(U1 + εU2)−1 (i = 1, 2, . . . , r) are all within thegiven LMI region D .

Proof: Suppose LMIs (10)–(12) and (36)–(42) are feasible.Let Z(ε) = U1 + εU2 and Φij = Sij

1 + εSi2 + ε2S3 ,

i, j = 1, 2, . . . , r; then

E(ε)Z(ε) = V1 + εV2 + ε2V3

and

Φij = L ⊗ V1 + M ⊗ (AiU1) + MT ⊗ (AiU1)T

+ M ⊗ (B2iFj ) + MT ⊗ (B2iFj )T

+ ε(L ⊗ V2 + M ⊗ (AiU2) + MT ⊗ (AiU2)T )

+ ε2L ⊗ V3 − Qij

= L ⊗ (V1 + εV2 + ε2V3)

+ M ⊗ [Ai(U1 + εU2) + B2iFj ]

+ MT ⊗ [Ai(U1 + εU2) + B2iFj ]T − Qij

= L ⊗ [E(ε)Z(ε)] + M ⊗ [(Ai + B2iKj (ε))Z(ε)]

+ MT ⊗ [(Ai + B2iKj (ε))Z(ε)]T − Qij . (43)

Using Lemma 1.2, LMIs (36)–(38) imply that

Φii < 0 ∀ε ∈ (0, ε], i = 1, 2, · · · , r. (44)

Similarly, using Lemma 1.2, LMIs (39)–(41) show that

Φij + Φj i < 0 ∀ε ∈ (0, ε], 1 ≤ i < j ≤ r. (45)

Denote

FD = L ⊗ X(ε) + M

⊗ r∑

i=1

r∑j=1

µiµjE−1(ε)(Ai +B2iKj (ε))X(ε)

+ MT

⊗ r∑

i=1

r∑j=1

µiµjE−1(ε) (Ai + B2iKj (ε))X(ε)

T

(46)

where X(ε) is a matrix with appropriate dimensions.



By computation, it holds that

[I ⊗ E(ε)] × FD × [I ⊗ E(ε)]

=r∑

i=1

r∑j=1

µiµj

× L ⊗ [E(ε)X(ε)E(ε)]

+ M ⊗ [(Ai + B2iKj (ε))X(ε)E(ε)]

+ MT ⊗ [E(ε)X(ε)(Ai + B2iKj (ε))T ]. (47)

Let X(ε) = Z(ε)E−1(ε). Using Lemma 1.3, LMIs (10)–(12)imply that

E(ε)Z(ε) = ZT (ε)E(ε) > 0 ∀ε ∈ (0, ε]

which shows

X(ε) = XT (ε) > 0 ∀ε ∈ (0, ε]. (48)

Substituting X(ε) = Z(ε)E−1(ε) into (47) gives

[I ⊗ E(ε)] × FD × [I ⊗ E(ε)]

=r∑

i=1

r∑j=1

µiµjL ⊗ [E(ε)Z(ε)]

+ M ⊗ [(Ai + B2iKj (ε))Z(ε)]

+ MT ⊗ [ZT (ε)(Ai + B2iKj (ε))T ]

=r∑

i=1

µ2i L ⊗ [E(ε)Z(ε)]

+ M ⊗ [(Ai + B2iKi(ε))Z(ε)]

+ MT ⊗ [ZT (ε)(Ai + B2iKi(ε))T ]

+r∑

i=1

r∑i<j

µiµj2L ⊗ [E(ε)Z(ε)]

+ M ⊗ [(Ai + B2iKj (ε))Z(ε)]

+ MT ⊗ [ZT (ε)(Ai + B2iKj (ε))T ]

+ M ⊗ [(Aj + B2jKi(ε))Z(ε)]

+ MT ⊗ [ZT (ε)(Aj + B2jKi(ε))T .(49)

Inequalities (44), (45), and (49) imply that

[I ⊗ E(ε)] × FD × [I ⊗ E(ε)]

<r∑

i=1

r∑j=1

µiµjQij

=

µ1

...

µr

T

Q11 · · · Q1r

.... . .

...

Qr1 · · · Qrr

µ1

...

µr

. (50)

Inequalities (42) and (50) indicate that [I ⊗ E(ε)] × FD ×[I ⊗ E(ε)] < 0 ∀ε ∈ (0, ε], which shows that

FD < 0 ∀ε ∈ (0, ε]. (51)

From (48), (51), and Lemma 1.1, it follows that the closed-loop system (6) with w(t) ≡ 0 is D-stable, i.e., the poles of eachsubsystem of system (6) with Ki(ε) = Fi(U1 + εU2)−1(i =1, 2, . . . , r) are all within the given LMI region D . This com-pletes the proof.

Theorems 3.1 and 3.2 provide solutions to Problems 2 and 3,respectively. By Theorems 3.1 and 3.2, Problem 1 can be solvedby the following theorem.

Theorem 3.3: Given an H∞ performance bound γ, an LMIstability region D , and an upper bound ε for the singu-lar perturbation parameter ε, if there exist matrices Zk (k =1, 2, . . . , 5) with Zk = ZT

k (k = 1, 2, 3, 4), Yij with Yij =Y T

ji (i, j = 1, 2, . . . , r), Qij with Qij = QTji(i, j = 1, 2, . . . , r),

and Fi (i = 1, 2, . . . , r) satisfying LMIs (10)–(18) and (36)–(42), then for any ε ∈ (0, ε], the closed-loop system (6) withKi = Fi(U1 + εU2)−1 (i = 1, 2, . . . , r) is with an H∞-normless than or equal to γ, and the poles of each subsystem ofsystem (6) are within the given LMI region D .

Remark 3.4: H∞ control for T–S fuzzy SPSs with pole place-ment constraints via state and output feedback controllers hasbeen studied in [28] and [29], respectively. Using these meth-ods, the obtained controllers are only valid for the case that ε issufficiently small and the upper bounds for ε have to be deter-mined by trial and error. While in Theorem 3.3, the prescribedupper bound ε is one of design objects, which is very significantin practice.

Remark 3.5: Besides the approaches based on the T–S fuzzymodel [25]–[32], there have been various design methods fornonlinear SPSs in particular forms. Stabilization controller de-sign methods were proposed in [33]–[36] by using slidingmode control, feedback linearization, integral control, and gainscheduling, respectively. However, these methods can only beapplied to standard nonlinear SPSs since they are based ondecomposing the original system into two reduced-order sub-systems in different time scales. H∞ control for nonlinear SPSswas investigated in [37]–[39]. In particular, a class of SPSs be-ing nonlinear only on the slow variables was examined in [37]and [38]. A local state feedback H∞ control problem for anaffine nonlinear SPS was addressed in [39]. Each of the afore-mentioned designs presents a formula to compute the bound forthe perturbation parameter ε or shows the existence of the upperbound. In this paper, a given upper bound ε for ε is one of thedesign objects, which represents an advantage of this paper overliterature [33]–[39]. Furthermore, the obtained method can beapplied to a wide class of nonlinear SPSs including nonstandardones, and the LMI-based design procedure is easy to perform.

IV. EXAMPLE

This section considers an inverted pendulum controlled by adc motor via a gear train whose physical model can be foundin [28] and [40]. The system can be described by the followingstate equations [28], [40]:

x1(t) = x2(t) + 0.1w(t)

x2(t) =g

lsin x1(t) +

NKm

ml2x3(t)



TABLE IUPPER BOUNDS FOR ε

La x3(t) = −KbNx2(t) − Rax3(t) + u(t) + w(t)

z(t) = 0.1x1(t) + 0.1u(t) (52)

where x1(t) = θp(t), x2(t) = θp(t), x3(t) = Ia(t), u(t) is thecontrol input, w(t) is the disturbance input, km is themotor torque constant, Kb is the back emf constant, andN is the gear ratio. The parameters for the plant aregiven as g = 9.8 m/s2 , l = 1 m,m = 1 kg, N = 10,Km =0.1 N · m/A,Kb = 0.1 V · s/rad, Ra = 1 Ω, and La = ε mH.Note that the inductance La represents the small parameter inthe system.

Substituting the parameters into (52) gives

x1(t) = x2(t) + 0.1w(t)

x2(t) = 9.8 sin x1(t) + x3(t)

εx3(t) = −x2(t) − x3(t) + u(t) + w(t)

z(t) = 0.1x1(t) + 0.1u(t). (53)

As in [32], the membership functions of the fuzzy sets arechosen as follows:

M1(x1(t)) = 1 − |x1(t)|π

M2(x1(t)) =|x1(t)|

π.

Then, the following T–S fuzzy model can exactly represent thedynamics of nonlinear SPS (53) under −π ≤ x1(t) ≤ π

Plant rule 1 :

IF x1(t) is M1(x1(t)), THEN

E(ε)x(t) = A1x(t) + B11w(t) + B21u(t)

z(t) = C1x(t) + D1u(t)

Plant rule 2 :


E(ε)x(t) = A2x(t) + B12w(t) + B22u(t)

z(t) = C2x(t) + D2u(t) (54)

where

E(ε) =

1 0 0

0 1 00 0 ε

A1 =

0 1 0

9.8 0 10 −1 −1

A2 =

0 1 0

0 0 10 −1 −1

B11 =

0.1

01

B12 =

0.1

01

B21 =

0

01

B22 =

0

01

C1 = [ 0.1 0 0 ] C2 = [ 0.1 0 0 ]

D1 = 0.1 D2 = 0.1.

Fig. 1. LMI conic sector region of pole location.

Fig. 2. Simulation for

√∫ t

0zT (s)z(s) ds/

∫ t

0wT (s)w(s) ds with ε =

0.02, x(0) = 0, and w(t) = sin(10πt).

Fig. 3. Simulation for

√∫ t

0zT (s)z(s) ds/

∫ t

0wT (s)w(s) ds with ε =

0.2, x(0) = 0, and w(t) = sin(10πt).



Fig. 4. Pole map of each local system with ε = 0.02. (a) i = 1 and j = 1: poles at −2.4789,−64.7421 + 10.4377 i, and −64.7421 − 10.4377 i. (b) i = 2and j = 2: poles at −100.5430,−2.7396, and −43.3492. (c) i = 1 and j = 2: poles at −100.4619,−2.3880, and −43.7819. (d) i = 2 and j = 1: poles at−2.7997,−64.5817 + 9.9423 i, and −64.5817 − 9.9423 i.

The fuzzy controller is described as follows:

Plant rule 1 :


u(t) = K1(ε)x(t)

Plant rule 2 :


u(t) = K2(ε)x(t) (55)

where K1(ε) and K2(ε) are controller gains to be determined.The following two cases are considered to illustrate the design

method developed in this paper. Case I is used to show theadvantage of the obtained method over the one given by [32].Case II is utilized to show the applicability of the proposedcontroller design method.

A. Case I: D is the Open Left-Half Plane

In this case, Problem 1 is equivalent to Problem 2. ByTheorem 3.1, the upper bounds for ε subject to guaranteeingvarious H∞ performance bounds are shown in Table I. Theproblem of H∞ control for system (54) with the considerationof singular perturbation bound design was studied in [32] andthe upper bounds for ε were given (see Table I). It can be seen

that the upper bounds obtained by Theorem 3.1 are bigger thanthose given by [32].

B. Case II: D Is a Conic Sector Region

In this section, the closed-loop poles of each local system willbe placed within an LMI conic sector region with θ = 45 (seeFig. 1). Note that the LMI conic sector region can be describedby (8) with

L = 0 M =[

sin(θ) cos(θ)−cos(θ) sin(θ)

].

Solving the LMIs in Theorem 3.3 with ε = 0.3 and γ = 0.5gives

U1 =

1.8335 −4.7635 0

−4.7635 12.8429 0

0.1042 −36.7990 65.7377

U2 =

−0.9586 3.4179 0.1042

3.4179 8.2240 −36.7990

0 0 2.6154

F1 =[−3.7414 −9.0002 −40.9112

]F2 =

[8.1129 −31.0419 −58.4247

].



Fig. 5. Pole map of each local system with ε = 0.2. (a) i = 1 and j = 1: poles at −69.7388,−1.6183, and −7.3209. (b) i = 2 and j = 2: poles at−75.2666,−4.3815 + 1.6563 i, and −4.3815 − 1.6563 i. (c) i = 1 and j = 2: poles at −75.2495,−1.5142, and −7.2658. (d) i = 2 and j = 1: poles at−69.7593,−4.4594 + 1.7353 i, and −4.4594 − 1.7353 i.

Then, the fuzzy controller is given by

u(t) =2∑

i=1

µi(t)Ki(ε)x(t) (56)

where µ1(t) = M1(x1(t)), µ2(t) = M2(x1(t)),K1(ε) = F1(U1 + εU2)−1 , and K2(ε) = F2(U1 + εU2)−1 .

Apply controller (56) to system (54) and assume x(0) = 0and w(t) = sin(10πt). The simulation results corresponding toε = 0.02 and ε = 0.2 are shown in Figs. 2–5. It is easy to seethat, in either case, the H∞-norm is less than the prescribedvalue 0.5, and the closed-loop poles of each local system arewithin the LMI conic sector region with θ = 45.

V. CONCLUSION AND REMARKS

This paper has investigated multiobjective control for a classof T–S fuzzy singularly perturbed systems. An LMI-based ap-proach to design an ε-dependent state feedback controller hasbeen proposed. By virtue of this method, the obtained controllercan ensure that, for any singular perturbation parameter in thegiven bound, the prescribed H∞ performance bound is satisfied,and the closed-loop poles of each subsystem are all within thepredefined LMI stability region.

It should be noted that the approach presented in this pa-per requires that the perturbation parameter ε is available forfeedback. Consequently, a significant and challenging problem

emerges when the singular perturbation parameter ε is subject touncertainties. We will consider this problem in our future work.In addition, Lemmas 1.2 and 1.3 are important in the derivationof our main results. We expect that they are also useful in otherresearch areas of singularly perturbed systems.

APPENDIX

PRELIMINARY LEMMAS

Lemma 1.1: Given a dynamic system x(t) = Ax(t) and anLMI region D , the system is D-stable, i.e., Λ(A) ∈ D if thereexists a matrix X ∈ Rn×n with X = XT > 0 such that

L ⊗ X + M ⊗ (AX) + MT ⊗ (AX)T < 0

where Λ(A) is the set of eigenvalues of A and ⊗ denotes theKronecker product of the matrices.

Lemma 1.2: For a positive scalar ε and symmetric matricesS1 , S2 , and S3 with appropriate dimensions, inequality

S1 + εS2 + ε2S3 > 0 (57)

holds for all ε ∈ (0, ε], if

S1 ≥ 0 (58)

S1 + εS2 > 0 (59)

S1 + εS2 + ε2S3 > 0. (60)



Proof: Given λ1 ∈ [0, 1), multiplying inequalities (59) and(60) by λ1 and 1 − λ1 , respectively, yield

λ1S1 + λ1εS2 ≥ 0 (61)

and

(1 − λ1)S1 + (1 − λ1)εS2 + (1 − λ1)ε2S3 > 0. (62)

Adding (61) to (62) shows that

S1 + εS2 + (1 − λ1)ε2S3 > 0 (63)

holds for any λ1 ∈ [0, 1).Similarly, inequalities (58) and (63) imply that

S1 + (1 − λ2)εS2 + (1 − λ2)(1 − λ1)ε2S3 > 0 (64)

holds for any λ2 ∈ [0, 1).For any ε, δ ∈ (0, ε], there exist λ1 , λ2 ∈ [0, 1), such that δ =

(1 − λ1)ε, ε = (1 − λ2)ε. Then, it follows from (64) that

S1 + εS2 + εδS3 > 0 (65)

holds for any ε, δ ∈ (0, ε]. Thus, inequality (57) holds for any ε ∈ (0, ε].Lemma 1.3: If there exist matrices Zi (i = 1, 2, . . . , 5) with

Zi = ZTi (i = 1, 2, 3, 4) satisfying the LMIs (10)–(12), then

E(ε)Z(ε) = ZT (ε)E(ε) > 0 ∀ε ∈ (0, ε] (66)

where

Z(ε) =[

Z1 + εZ3 εZT5

Z5 Z2 + εZ4

]. (67)

Proof: Denote

S1 =

[Z1 0

0 0

]S2 =

[Z3 ZT

5

Z5 Z2

]S3 =

[0 0

0 Z4

]

then

E(ε)Z(ε) =

[Z1 + εZ3 εZT

5

εZ5 εZ2 + ε2Z4

]

= S1 + εS2 + ε2S3

= ZT (ε)E(ε). (68)

From inequalities (10)–(12), it follows that

S1 ≥ 0 (69)

S1 + εS2 > 0 (70)

S1 + εS2 + ε2S3 > 0. (71)

Then, using Lemma 1.2, one gets

S1 + εS2 + ε2S3 > 0, ε ∈ (0, ε]. (72)

As a result, (68) and (72) implies (66). This completes theproof.

ACKNOWLEDGMENT

The authors would like to thank the Editor and the anonymousreviewers for their valuable comments.

REFERENCES

[1] E. H. Abed, “A new parameter estimate in singular perturbations,” Syst.Control Lett., vol. 6, no. 3, pp. 193–198, 1985.

[2] H. K. Khalil, “Stability analysis of nonlinear multiparameter singularlyperturbed systems,” IEEE Trans. Autom. Control, vol. 32, no. 3, pp. 260–263, Mar. 1987.

[3] B. S. Chen and C. L. Lin, “On the stability bounds of singularly perturbedsystems,” IEEE Trans. Autom. Control, vol. 35, no. 11, pp. 1265–1270,Nov. 1990.

[4] W. Feng, “Characterization and computation for the bound ε∗in lineartime-invariant singularly perturbed systems,” Syst. Control Lett., vol. 11,no. 3, pp. 195–202, 1988.

[5] S. Sen and K. B. Datta, “Stability bounds of singularly perturbed systems,”IEEE Trans. Autom. Control, vol. 38, no. 2, pp. 302–304, Feb. 1993.

[6] L. Saydy, “New stability/performance results for singularly perturbedsystems,” Automatica, vol. 32, no. 6, pp. 807–818, 1996.

[7] S. J. Chen and J. L. Lin, “Maximal stability bounds of singularly perturbedsystems,” J. Franklin Inst., vol. 336, no. 8, pp. 1209–1218, 1999.

[8] H. D. Tuan and S. Hosoe, “Multivariable circle criteria for multiparametersingularly perturbed systems,” IEEE Trans. Autom. Control, vol. 45, no. 4,pp. 720–725, Apr. 2000.

[9] N. R. Sandell Jr., “Robust stability of systems with application to singularperturbations,” Automatica, vol. 15, no. 4, pp. 467–470, 1979.

[10] H. K. Khalil, “Asymptotic stability of nonlinear multiparameter singularlyperturbed systems,” Automatica, vol. 17, no. 6, pp. 797–804, 1981.

[11] A. Saberi and H. Khalil, “Quadratic-type lyapunov functions for singularlyperturbed systems,” IEEE Trans. Autom. Control, vol. 29, no. 6, pp. 542–550, Jun. 1984.

[12] K. Khorasani and M. A. Pai, “Asymptotic stability of nonlinear singularlyperturbed systems using higher order corrections,” Automatica, vol. 21,no. 6, pp. 717–727, 1985.



[15] T. H. S. Li and J. H. Li, “Stabilization bound of discrete two-time-scalesystems,” Syst. Control Lett., vol. 18, no. 6, pp. 479–489, 1992.

[16] W. Q. Liu, M. Paskota, and V. Sreeram, K. L. Teo, “Improvement onstability bounds for singularly perturbed systems via state feedback,” Int.J. Syst. Sci., vol. 28, no. 6, pp. 571–578, 1997.

[17] J. S. Chiou, F. C. Kung, and T. H. S. Li, “An infinite ε-bound stabilizationdesign for a class of singularly perturbed systems,” IEEE Trans. CircuitsSyst. I, Fundam Theory Appl., vol. 46, no. 12, pp. 1507–1510, Dec. 1999.

[18] Z. H. Shao and M. E. Sawan, “Stabilisation of uncertain singularly per-turbed systems,” Proc. Inst. Electr. Eng. Control Theory Appl., vol. 153,no. 1, pp. 99–103, 2006.

[19] T. Takagi and M. Sugeno, “Fuzzy identification of systems and its applica-tions to modeling and control,” IEEE Trans. Syst., Man, Cybern., vol. 15,no. 1, pp. 116–132, Jan./Feb. 1985.

[20] B. S. Chen, C. S. Tseng, and H. J. Uang, “Mixed H2 /H∞ fuzzy out-put feedback control design for nonlinear dynamic systems: An LMIapproach,” IEEE Trans. Fuzzy Syst., vol. 8, no. 3, pp. 249–265, Jun.2000.

[21] X. Liu and Q. Zhang, “Approaches to quadratic stability conditions andH∞control designs for T–S fuzzy systems,” IEEE Trans. Fuzzy Syst.,vol. 11, no. 6, pp. 830–839, Dec. 2003.

[22] S. G. Cao, N. W. Rees, and G. Feng, “Analysis and design for a classof complex control systems part I: Fuzzy modelling and identification,”Automatica, vol. 33, no. 6, pp. 1017–1028, 1997.

[23] S. G. Cao, N. W. Rees, and G. Feng, “Analysis and design for a class ofcomplex control systems part II: Fuzzy controller design,” Automatica,vol. 33, no. 6, pp. 1029–1039, 1997.

[24] S. Boyd, L. E. Ghaoui, E. Feron, and V. Balakrishnan, Linear MatrixInequalities in System and Control Theory. Philadelphia, PA: SIAM,1994.

[25] H. Liu, F. Sun, and Z. Sun, “Stability analysis and synthesis of fuzzysingularly perturbed systems,” IEEE Trans. Fuzzy Syst., vol. 13, no. 2,pp. 273–284, Apr. 2005.

[26] H. Liu, F. Sun, and Y. Hu, “H∞ control for fuzzy singularly perturbedsystems,” Fuzzy Sets Syst., vol. 155, no. 2, pp. 272–291, 2005.

[27] W. Assawinchaichote, S. K. Nguang, and P. Shi, “H∞ output feedbackcontrol design for uncertain fuzzy singularly perturbed systems: An LMIapproach,” Automatica, vol. 40, no. 12, pp. 2147–2152, 2004.



[28] W. Assawinchaichote and S. K. Nguang, “H∞ fuzzy control design fornonlinear singularly perturbed systems with pole placement constraints:An LMI approach,” IEEE Trans. Syst., Man, Cybern. B, Cybern., vol. 34,no. 1, pp. 579–588, Feb. 2004.

[29] W. Assawinchaichote and S. K. Nguang, “Fuzzy H∞ output feedbackcontrol design for singularly perturbed systems with pole placement con-straints: An LMI approach,” IEEE Trans. Fuzzy Syst., vol. 14, no. 3,pp. 361–371, Jun. 2006.

[30] T. H. S. Li and K. J. Lin, “Stabilization of singularly perturbed fuzzysystems,” IEEE Trans. Fuzzy Syst., vol. 12, no. 5, pp. 579–595, Oct.2004.

[31] T. H. S. Li and K. J. Lin, “Composite fuzzy control of nonlinear singularlyperturbed systems,” IEEE Trans. Fuzzy Syst., vol. 15, no. 2, pp. 176–187,Apr. 2007.

[32] G. H. Yang and J. X. Dong, “Control synthesis of singularly perturbedfuzzy systems,” IEEE Trans. Fuzzy Syst., vol. 16, no. 3, pp. 615–629, Jun.2008.

[33] J. Alvarez-Gallegos and G. Silva-Navarro, “Two-time scale sliding-modecontrol for a class of nonlinear systems,” Int. J. Robust Nonlinear Control,vol. 7, no. 9, pp. 865–879, 1997.

[34] H. L. Choi, Y. S. Shin, and J. T. Lim, “Control of nonlinear singularlyperturbed systems using feedback linearisation,” Proc. Inst. Electr. Eng.Control Theory Appl., vol. 152, no. 1, pp. 91–94, 2005.

[35] J. Wang, J. Wang, and H. Li, “Nonlinear PI control of a class of nonlinearsingularly perturbed systems,” Proc. Inst. Electr. Eng. Control TheoryAppl., vol. 152, no. 5, pp. 560–566, 2005.

[36] J. W. Son and J. T. Lim, “Robust stability of nonlinear singularly perturbedsystem with uncertainties,” Proc. Inst. Electr. Eng. Control Theory Appl.,vol. 153, no. 1, pp. 104–110, 2006.

[37] Z. Pan and T. Basar, “Time-scale separation and robust controller designfor uncertain nonlinear singularly perturbed systems under perfect statemeasurements,” Int. J. Robust Nonlinear Control, vol. 6, no. 7, pp. 585–608, 1996.

[38] H. D. Tuan and S. Hosoe, “On linear robust H∞ controllers for a classof nonlinear singular perturbed systems,” Automatica, vol. 35, no. 4,pp. 735–739, 1999.

[39] E. Fridman, “State-feedback H∞ control of nonlinear singularly perturbedsystems,” Int. J. Robust Nonlinear Control, vol. 11, no. 12, pp. 1115–1125,2001.

[40] S. H. Zak and C. A. Maccarley, “State-feedback control of non-linearsystems,” Int. J. Control, vol. 43, no. 5, pp. 1497–1514, 1986.

Chunyu Yang received the B.Sc. degree in 2002from the Department of Mathematics, NortheasternUniversity, Shenyang, China, where he is currentlyworking toward the Ph.D. degree.

His current research interests include descriptorsystems, robust control, and fuzzy control.

Qingling Zhang received the B.Sc. and M. Sc. de-grees from the Department of Mathematics, North-eastern University, Shenyang, China, in 1982 and1986, respectively, and the Ph.D. degree from theDepartment of Automatic Control, Northeastern Uni-versity, in 1995.

During 1997, he was a Postdoctoral Fellow inthe Department of Automatic Control, NorthwesternPolytechnical University, Xian, China. Since 1997,he has been a Professor at Northeastern University.He is also a member of the University Teaching Ad-

visory Committee of the National Ministry of Education. He visited Hong KongUniversity, Sydney University, Western Australia University, Niigata University,Pohan University of Science and Technology, Seoul University, Alberta Uni-versity, Lakehead University, and Wisor University as a Research Associate,Research Fellow, Senior Research Fellow, and Visiting Professor, respectively.He has authored or coauthored six books and more than 230 papers about con-trol theory and applications.

Prof. Zhang received 14 prizes from central and local governments for hisresearch. He has also received the Golden Scholarship from Australia in 2000.


IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART B: CYBERNETICS, VOL. 38, NO. 5, OCTOBER 2008 1371

H∞ Filtering for Fuzzy Singularly Perturbed SystemsGuang-Hong Yang, Senior Member, IEEE, and Jiuxiang Dong

Abstract—This paper considers the problem of designing H∞filters for fuzzy singularly perturbed systems with the considera-tion of improving the bound of singular-perturbation parameter ε.First, a linear-matrix-inequality (LMI)-based approach is pre-sented for simultaneously designing the bound of the singularlyperturbed parameter ε, and H∞ filters for a fuzzy singularlyperturbed system. When the bound of singularly perturbed pa-rameter ε is not under consideration, the result reduces to anLMI-based design method for H∞ filtering of fuzzy singularlyperturbed systems. Furthermore, a method is given for evaluatingthe upper bound of singularly perturbed parameter subject to theconstraint that the considered system is to be with a prescribedH∞ performance bound, and the upper bound can be obtainedby solving a generalized eigenvalue problem. Finally, numericalexamples are given to illustrate the effectiveness of the proposedmethods.

Index Terms—Fuzzy filtering, H∞ filtering, linear matrix in-equalities (LMIs), singularly perturbed systems, Takagi–Sugeno(T-S) fuzzy models.

I. INTRODUCTION

IN THE PAST 30 years, singularly perturbed systems havebeen intensively studied, where singular perturbation with a

small parameter, for example ε, is exploited to determine the de-gree of separation between “slow” and “fast” modes of the sys-tem. For the purpose of avoiding the difficulties linked with thestiffness of the equations involved in the design, the singular-perturbation design techniques have been developed [1].For the problem of filter design for linear singularly perturbedsystems, many significant advances have been achieved (see[2]–[4] and the reference therein). In particular, the decompo-sition solution of H∞ filter gain for linear singularly perturbedsystems is given in [3], and a method of designing reduced-order H∞ optimal filters for systems with slow and fast modesis proposed in [4]. However, the problem of designing H∞filters for nonlinear singularly perturbed systems still remainsas open research subject.

Manuscript received June 8, 2007; revised January 31, 2008 and April 26,2008. This work was supported in part by the Program for New CenturyExcellent Talents in University under Grant NCET-04-0283, by the Funds forCreative Research Groups of China under Grant 60521003, by the Programfor Changjiang Scholars and Innovative Research Team in University underGrant IRT0421, by the State Key Program of National Natural Science ofChina under Grant 60534010, by the Funds of National Science of Chinaunder Grant 60674021, by the 111 Project under Grant B08015, and by theFunds of Ph.D. program of the Ministry of Education, China under Grant20060145019. This paper was recommended by Associate Editor W. J. Wang.

The authors are with the Key Laboratory of Integrated Automation ofProcess Industry (Ministry of Education), and College of Information Scienceand Engineering, Northeastern University, Shenyang 110004, China (e-mail:[email protected]; [email protected]; [email protected]; [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TSMCB.2008.927277

An important approach to address the synthesis problemsfor nonlinear systems is to model the considered system asTakagi and Sugeno (T-S) fuzzy systems [5], which are locallylinear time-invariant systems connected by IF–THEN rules. Asa result, the conventional linear-system theory can be applied toanalysis and synthesis of the class of nonlinear control systems.In recent years, the T-S fuzzy control systems have been studiedextensively [6]–[10]. For the nonlinear-filtering problem viaT-S fuzzy models, some numerically effective design methodshave been obtained [11]–[14]. In [11], an H∞ fuzzy-filteringdesign for nonlinear discrete-time systems is proposed, andChen et al. [12] present a method of designing mixed H2/H∞filters for equalization/detection of nonlinear communicationsystems using fuzzy interpolation and linear-matrix-inequality(LMI) techniques [15]. The H∞ fuzzy-filtering problem withD stability constraints is studied in [13], and Feng [14] givesdesign methods for robust H∞ fuzzy filtering of nonlinearsystems. For nonlinear singularly perturbed systems, the earliermethods will lead to the ill-conditioning resulting from the in-teraction of slow and fast dynamic modes. For the fuzzy controland filtering problems of nonlinear singularly perturbed sys-tems, some progresses have been achieved [16]–[18], [20], [21].In [16] and [17], design methods for the stabilization andH∞ control of nonlinear singularly perturbed systems via statefeedback are given in terms of solutions of LMIs, respec-tively. LMI-based methods for designing robust controllers anddetermining allowable perturbation bounds are given in [18].In [19], a method for determining allowable perturbationbounds is presented via some algebra inequalities. A solution tothe nonlinear H∞ fuzzy-filtering problem with pole-placementconstraints is presented in [20]; a method based on Riccattiequation for multiparameter singularly perturbed systems ispresented in [21]. The aforementioned results are based onLMIs or Riccatti equations; the advantage of the existingmethods is that some performance indexes can be optimized.However, the upper bound of the singular-perturbation param-eter ε is not addressed at the stage of designing filters orcontrollers. In this paper, a filter will be designed with the con-sideration to improve the upper bound of singular-perturbationparameter ε.

It is well known that the accurate knowledge of the stabilitybound ε∗ of a singularly perturbed system (i.e., the system isstable for ε ∈ [0, ε∗]) is very important for applications. Thecharacterization and computation of the stability bound haveattracted considerable efforts for over two decades in the past([22]–[24] and the reference therein). In general, there are twoclasses of methods to characterize and compute the stabilitybounds: One is based on frequency-domain transfer functions,and another is based on state-space models. Both methods canprovide the exact bounds as shown in [22]–[24]. However,

1083-4419/$25.00 © 2008 IEEE


1372 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART B: CYBERNETICS, VOL. 38, NO. 5, OCTOBER 2008

the issue of how to improve the bound ε∗ by controller orfilter design has not been addressed in the literature, whichundoubtedly is very important for the applications of singularlyperturbed system theory. Furthermore, for nonlinear systems,no result has been presented on how to evaluate the upper boundof singularly perturbed parameter subject to the constraint thatthe considered system is to be with a prescribed performancebound.

This paper is concerned with the problem of designingH∞ filters for fuzzy singularly perturbed systems with theconsideration of improving the bound of singular-perturbationparameter ε. First, an LMI-based approach is presented forsimultaneously designing the bound of ε and H∞ filter for afuzzy singularly perturbed system. When the bound of singu-larly perturbed parameter ε is not under consideration, the resultreduces to a new LMI-based design method for H∞ filtering offuzzy singularly perturbed systems, which can provide less orat least the same conservative results than existing method [20].Furthermore, a method is given for evaluating the upper boundε∗ of singularly perturbed parameter ε subject to the consideredsystem with a designed filter to be with a prescribed perfor-mance bound for ε ∈ (0, ε∗], where the upper bound ε∗ can beobtained by solving a generalized eigenvalue problem (GEVP)[15]. Although a method of finding an interval (0 ε∗] so thatthe system is stable and satisfies an H∞ performance index forε ∈ (0 ε∗] is given in [19], the H∞ performance index in [19] isdependent on Lyapunov matrices and different from the generaldefinition for H∞ performance index. In this paper, a generalH∞ performance index (independent of Lyapunov matrices) isconsidered. Moreover, a convex method of finding an interval[ε, ε] is proposed in [18]; however, the search for small ε isrelated to ε-dependent computation, which cannot avoid thedifficulties linked with the stiffness of the equations. Therefore,a convex method of finding an interval (0, ε] in this paper ismore effective.

This paper is organized as follows. In Section II, the systemdescription and some preliminaries are presented. Section IIIconsiders the H∞ filter-design problem with the considerationof improving the bound of singular-perturbation parameter ε,and the estimation of the upper bound of singular-perturbationparameter ε with satisfying H∞ performance requirement.Section IV gives examples to illustrate the effectiveness ofthe new proposed methods. Finally, Section V concludes thispaper.

II. PROBLEM STATEMENT, NOTATION,AND PRELIMINARIES

In this section, we first give the notations used in this paper.

A. Notation

We use the following standard notations. For a matrix M ,‖M‖ is defined as the largest singular value of M . For a squarematrix E, He(E) is defined as

He(E) = E + ET

and E−T denotes (E−1)T if E is nonsingular. The symbol ∗within a matrix represents the symmetric entries

[Hij ]r×r =:

⎡⎢⎢⎣

H11 H12 · · · H1r

HT12 H22 · · · H2r

......

. . ....

HT1r HT

2r · · · Hrr

⎤⎥⎥⎦ .

B. System Description

The nonlinear singularly perturbed system under considera-tion is described by the following fuzzy-system model:

Plant Rule i :

IF v1(t) is Mi1 and v2(t) is Mi2, . . . , vp(t) is Mip

THEN Eεx(t) = Aix(t) + B1iw(t)

z(t) = C1ix(t)

y(t) = C2ix(t) + D21iw(t) (1)

where

Eε =[

In1×n1 00 εIn2×n2

]

x(t) ∈ Rn, n = n1 + n2 is the state vector, z(t) ∈ Rnz is thecontrolled output vector, y(t) ∈ Rny is the measurable outputvector, and w(t) ∈ Rnw is the exterior disturbance vector. r isthe number of IF–THEN rules, v(t) ∈ Rp×1 are the premisevariables, ε > 0 is the singular-perturbation parameter, and Mij

are the fuzzy sets. By using the fuzzy-inference method witha singleton fuzzifier, product inference, and center averagedefuzzifiers, the final T-S fuzzy model is obtained as

Eεx(t) =

r∑i=1

wi (v(t)) (Aix(t) + B1iw(t))

r∑i=1

wi (v(t))

z(t) =

r∑i=1

wi (v(t)) (C1ix(t))

r∑i=1

wi (v(t))

y(t) =

r∑i=1

wi (v(t)) (C2ix(t) + D21iw(t))

r∑i=1

wi (v(t))

where

wi (v(t)) =p∏

j=1

Mij (vj(t)) .

Mij(vj(t)) is the grade of membership of vj(t) in Mij , whereit is assumed that

r∑i=1

wi (v(t)) > 0, wi (v(t)) ≥ 0; i = 1, 2, . . . , r.


YANG AND DONG: H∞ FILTERING FOR FUZZY SINGULARLY PERTURBED SYSTEMS 1373

Denote

αi(t) =wi (v(t))

r∑i=1

wi (v(t))

then

0 ≤ αi (v(t)) ≤ 1r∑

i=1

αi (v(t)) = 1.

αi(v(t)) (which is also denoted as αi), 1 ≤ i ≤ r are said tobe normalized membership functions. Therefore, the singularlyperturbed fuzzy system (1) can be rewritten as follows:

Eεx(t) =r∑

i=1

αiAix(t) +r∑

i=1

αiB1iw(t)

z(t) =r∑

i=1

αiC1ix(t)

y(t) =r∑

i=1

αiC2ix(t) +r∑

i=1

αiD21iw(t)

i.e.,

Eεx(t) = A(α)x(t) + B1(α)w(t)

z(t) = C1(α)x(t)

y(t) = C2(α)x(t) + D21(α)w(t) (2)

where

A(α) =r∑

i=1

αiAi

B1(α) =r∑

i=1

αiB1i

C1(α) =r∑

i=1

αiC1i

C2(α) =r∑

i=1

αiC2i

D21(α) =r∑

i=1

αiD21i. (3)

In this paper, we use the following filter, which is from [20],for the H∞ filtering for fuzzy singularly perturbed systems:

Eεξ(t) =r∑

i=1

r∑j=1

αiαjAFijξ(t) +r∑

i=1

αiBFiy(t)

zF (t) =r∑

i=1

αiCFiξ(t) (4)

which can be rewritten as the follows:

Eεξ(t) =AF (α)ξ(t) + BF (α)y(t)

zF (t) =CF (α)ξ(t) (5)

where

AF (α) =r∑

i=1

r∑j=1

αi(t)αj(t)AFij

BF (α) =r∑

i=1

αi(t)BFi

CF (α) =r∑

i=1

αi(t)CFi.

Let

ze(t) = z(t) − zF (t) xe(t) =[x(t)ξ(t)

].

Augmenting the system (2) with the states of the filter (5), thenthe following filtering-error systems can be obtained:

xe(t) =Aeε(α)xe(t) + Beε(α)w(t)

ze(t) =Ce(α)xe(t) (6)

where

Aeε(α) =[

E−1ε A(α) 0

E−1ε BF (α)C2(α) E−1

ε AF (α)

]

Beε(α) =[

E−1ε B1(α)

E−1ε BF (α)D21(α)

]

Ce(α) = [C1(α) −CF (α) ] .

For the filter-error system (6), let γ > 0 be a constant. If (6) isasymptotically stable, and for any w(t) ∈ L2[0,∞) (the spaceof square-integrable functions) and x(0) = 0, the followinginequality holds:

∞∫0

zTe (t)ze(t)dt ≤ γ2

∞∫0

wT(t)w(t)dt

then the system (6) is said to be with an H∞-norm or L2-gainless than or equal to γ.

In this paper, the upper bound of the singularly perturbedparameter ε is the main topic to be addressed. A method willbe given for designing a filter with simultaneously improv-ing the upper bound of the singularly perturbed parameterε and satisfying H∞ performance requirement. Furthermore,an LMI-based method is proposed to estimate the upper boundof singular-perturbation parameter ε of the singularly per-turbed fuzzy system (6) satisfying H∞ performance-boundrequirement.

Remark 1: It should be pointed out that the existing filter-design method [20] is only considering the H∞ performancerequirement, which might result in a very small stability boundof ε so that the designed filter may not be applicable in practice.



C. Preliminary Lemmas

The following lemma, essentially from [26], gives a suffi-cient condition for the system (6) to be with an H∞-norm lessthan or equal to γ.

Lemma 1: For a given scalar γ > 0, if there exists matricesPε = PT

ε , satisfying the following matrix inequality:

Me1(α) = PεAeε(α) + ATeε(α)Pε

+PεBeε(α)BTeε(α)Pε +

1γ2

CTe (α)Ce(α) < 0 (7)

then the H∞-norm of the system (6) from exogenous input tothe filter error is less than or equal to γ.

Lemma 2: For a given scalar γ > 0, if there exist matricesSε = ST

ε and Nε = NTε , AF (α), BF (α), and CF (α), satisfy-

ing the following matrix inequalities:

Nε < 0 Sε > 0 (8)

Me3(α)= Me3a(α)+MTe3a(α)

+[

SεE−1ε B1(α)

(Sε−Nε)E−1ε B1(α)+NεE

−1ε BF (α)D21(α)

]

×[

SεE−1ε B1(α)


−1ε BF (α)D21(α)

]T

+1γ2

[ C1(α)−CF (α) C1(α) ]T

×[ C1(α)−CF (α) C1(α) ] < 0 (9)

where Me3a(α) is given in (10), shown at the bottom of thepage. Then, the system (6) is asymptotically stable, and theH∞-norm of the system (6) from exogenous input to the filtererror is less than or equal to γ.

Proof: See Appendix. Remark 2: Lemma 2 presents a sufficient condition for the

system (6) to be with an H∞ performance bound, and theconditions can be converted into LMIs by using appropriatevariable transformations. However, the resulting design condi-tions are ε-dependent and may become ill-conditioned whenε is sufficiently small, which always is the case for singularlyperturbed systems.

Lemma 3: If there exist symmetric matrices Jjii, 1 ≤ i =

j ≤ r and matrices J lij , 1 ≤ i < j ≤ r, 1 ≤ l ≤ r, such that the

following LMIs hold:

Jjii + J i

ij +(J i

ij

)T> 0, 1 ≤ i < j ≤ r (11)

J ijj + Jj

ij +(Jj

ij

)T

> 0, 1 ≤ i < j ≤ r (12)

He(J l

ij + Jjil + J i

jl

)> 0, 1 ≤ i < j < l ≤ r (13)

then

⎡⎣ α1I

...αrI

⎤⎦

Tr∑

l=1

αlJl

⎡⎣α1I

...αrI

⎤⎦ ≥ 0 (14)

where

J l =[J l

ij

]r×r

J iii = 0, for 1 ≤ l ≤ r.

Proof:

⎡⎣ α1I

...αrI

⎤⎦

Tr∑

l=1

αlJl

⎡⎣ α1I

...αrI

⎤⎦

=r∑

l=1

αl

⎡⎣α1I

...αrI

⎤⎦

T⎡⎢⎣

J l11 · · · J l

1r...

. . . · · ·(J l

1r

)T · · · J lrr

⎤⎥⎦

⎡⎣α1I

...αrI

⎤⎦

=r∑

i=1

r∑j=i+1

α2i αj

(J i

ij +(J i

ij

)T + Jjii

)

+r∑

i=1

r∑j=i+1

αiα2j

(J i

jj + Jjij +

(Jj

ij

)T)

+r∑

i=1

r∑j=i+1

r∑l=j+1

αiαjαlHe(J l

ij + Jjil + J i

jl

).

From (11)–(13), it follows (14). The following result from [20] gives a sufficient condition

for H∞ filtering of fuzzy singularly perturbed systems via thefilter (4).

Lemma 4: For a given γ > 0 [20], if there exist matrices X ,Y , AF0ij , BF0i, CF0i, 1 ≤ i, and j ≤ r, with

X =[

X11 0X21 X22

]Y =

[Y11 0Y21 Y22

]

where X11, X22, Y11, and Y22 are symmetric matrices, satisfy-ing the following LMIs:

⎡⎢⎣

X11 0 I 00 X22 0 II 0 Y11 00 I 0 Y22

⎤⎥⎦ > 0 (15)

Ψii < 0, 1 ≤ i ≤ r (16a)

Ψij + Ψji < 0, 1 ≤ i < j ≤ r (16b)

Me3a(α)=[

SεE−1ε A(α) SεE

−1ε A(α)

(Sε−Nε)E−1ε A(α)+NεE

−1ε AF (α)+NεE

−1ε BF (α)C2(α) (Sε−Nε)E−1

ε A(α)+NεE−1ε BF (α)C2(α)

](10)



where Ψij is solved as shown at the bottom of the page. Then,there exists an ε∗ > 0 such that the H∞-norm of the system(6) with

AFij = N−T[AF0ij − Y TAiX − BF0jC2iX

](MT)−1

BFi = N−TBF0i

CFi = CF0i(MT)−1 (17)

is less than or equal to γ for ε ∈ (0, ε∗], where M =[M11 M12

0 M22

]and N =

[N11 0N21 N22

]satisfy NMT =I−Y X .

III. MAIN RESULTS

A. Filter Design With Improving the Upper Bound of ε

The following theorem is concerned with the problem ofdesigning a filter with the consideration of improving the upperbound of singular-perturbation parameter ε and satisfying H∞performance requirement.

Theorem 1: If there exist matrices S, Sa, N , Na, Λ, AFij ,BFi, CFi, 1 ≤ i, j ≤ r, J l

ij , 1 ≤ l ≤ r, 1 ≤ i < j ≤ r, J lii, and

1 ≤ i = l ≤ r and scalar variable ηi, βi, and 1 ≤ i ≤ 6, with

S =[

S11 0S21 S22

]N =

[N11 0N21 N22

]

Sa =[

0 ST21

0 0

]Na =

[0 NT

21

0 0

]

where S11, S22, N11, N22, Λ, J lii, and 1 ≤ i = l ≤ r are

symmetric matrices, satisfying the following LMIs:

Jjii + J i

ij +(J i

ij

)T> 0, 1 ≤ i < j ≤ r (18)

J ijj + Jj

ij +(Jj

ij

)T

> 0, 1 ≤ i < j ≤ r (19)

He(J l

ij + Jjil + J i

jl

)> 0, 1 ≤ i < j < l ≤ r (20)[

−η1I II −S11

]< 0

[−η2I I

I −S22

]< 0

[−η3I ST

21

S21 −I

]< 0 (21)[

−η4I II N11

]< 0

[−η5I I

I N22

]< 0[

−η6I NT21

N21 −I

]< 0 (22)[

−Λ II −β4I

]< 0 (23)⎡

⎣−β26I 0 STB1i

∗ −β26I (ST − NT)B1i + BFjD21i

∗ ∗ −I

⎤⎦ < 0 (24)

[−β2

2I (BFjD21i)T

BFjD21i −I

]< 0 (25)⎡

⎣ −β23I 0 ST

a B1i

0 −β23I

(ST

a − NTa

)B1i

BT1iSa BT

1i(Sa − Na) −I

⎤⎦ < 0 (26)

⎡⎣ −β2

5I 0 ATFij + CT

2iBTFj

0 −β25I CT

2iBTFj

AFij + BFjC2i BFjC2i −I

⎤⎦ < 0 (27)

[Φij ]r×r +[J l

ij

]r×r

< 0, l = 1, . . . , r (28)

Me5aii + MTe5aii < β1I (29)

where

Φij = Hij +

⎡⎣Λ 0 0

0 0 00 0 0

⎤⎦

Me5aii =[

STa Ai ST

a Ai

(Sa − Na)TAi (Sa − Na)TAi

]

Me4aij =[

STAi STAi

AFij +BFjC2i+(S−N)TAi (S−N)TAi+BFjC2i

]

where Hij is given in (30), shown at the bottom of the page,and denote

λ∗ = maxη1η2η3, η4η5η6, β4β7 (31)

Ψij =

⎡⎢⎢⎢⎣

AiX + XTAi ∗ ∗ ∗AF0ij + AT

i ATi Y + Y TAi + BF0jC2i + CT

2iBTF0j ∗ ∗

BT1i BT

1iY + DT21iB

TF0j −I ∗

C1iX − CF0j C1i 0 −γ2I

⎤⎥⎥⎥⎦ , 1 ≤ i ≤ j ≤ r

Hij =

⎡⎢⎢⎢⎣

He(Me4aij+Me4aji)2

ST(B1i+B1j)2

(ST−NT)(B1i+B1j)+BF jD21i+BF iD21j

2

CT1i−CT

F i+CT1j−CT

F j

2

CT1i+CT

1j

2

∗ −I 0∗ ∗ −γ2I

⎤⎥⎥⎥⎦ (30)



where

β7 = β1+2∥∥NT

a (N−1)T∥∥β5+2β6

(β3+

∥∥NTa (N−1)T

∥∥ β2

)+

(β3 +

∥∥NTa (N−1)T

∥∥β2

)2. (32)

Then, for ε ∈ (0, (1/λ∗)], the H∞-norm of the system (6) with

AFij = N−TAFij BFi = N−TBFi, 1 ≤ i ≤ j ≤ r(33)

is less than or equal to γ.Proof: See Appendix.

Remark 3: Theorem 1 presents sufficient conditions underwhich an upper bound (1/λ) of singularly perturbed parameter εand filter gains can be obtained. From (31) and ηi > 0, βi >0, and 1 ≤ i ≤ 6, we have found that λ can be minimized bysolving the following optimization problem:

minimize6∑

i=1

aiηi +6∑

i=1,i =4

biβi + b4

∥∥NTa N−T

∥∥subject to (18)−(29) (34)

where ai, bi, and 1 ≤ i ≤ 6 are positive weighting constants tobe chosen. However, the optimization problem cannot directlybe solved due to the term b4‖NT

a N−T‖. Consider

∥∥NTa N−T

∥∥ =∥∥∥∥[

0 0N21 0

] [N−1

11 −N−111 NT

21N−122

0 N−122

]∥∥∥∥=

∥∥∥∥[

0 0N21N

−111 −N21N

−111 NT

21N−122

]∥∥∥∥=

∥∥∥∥[

0 0N21N

−111 0

]+

[0 00 −N21N

−111 NT

21N−122

]∥∥∥∥≤

∥∥N21N−111

∥∥ +∥∥N21N

−111 NT

21N−122

∥∥≤‖N21‖

∥∥N−111

∥∥ + ‖N21‖2∥∥N−1

11

∥∥ ∥∥N−122

∥∥ .

From (22), we have

∥∥N−111

∥∥ ≤ η4

∥∥N−122

∥∥ ≤ η5 ‖N21‖2 ≤ η6.

Then, it follows that

∥∥NTa N−T

∥∥ ≤ √η6η4 + η4η5η6.

Combining it with ηi > 0, βi > 0, and 1 ≤ i ≤ 6, (34) can beindirectly minimized by solving the optimization problem

minimize6∑

i=1

(aiηi + biβi) subject to (18)−(29)

where ai, bi, and 1 ≤ i ≤ 6 are positive weighting constantsto be chosen. Since the constraints (18)–(29) are of LMIs,the optimization problem can be effectively solved via LMIControl Toolbox [25]. It should be pointed out that the upper

bound 1/λ obtained by solving the earlier optimization problemmay be conservative. After obtaining the filter gains, a less-conservative bound of ε can be obtained by Theorem 2 in thenext section.

B. Estimation of the Upper Bound of ε

In this section, assume that the filter has been designed.The following theorem gives a technique to estimate the up-per bound of singularly perturbed parameter ε subject to thefiltering-error system (6) with an H∞ performance bound.

Theorem 2: Consider the system (6), and let γ > 0 be a givenconstant. If there exist a scalar λ and matrices S, Sa, N , Na, J l

ii,1 ≤ i = l ≤ r, J l

ij , 1 ≤ i < j ≤ r, and 1 ≤ l ≤ r with

S =[

S11 0S21 S22

]N =

[N11 0N21 N22

]

Sa =[

0 ST21

0 0

]Na =

[0 NT

21

0 0

]

where S11, S22, N11, N22, J lii, and 1 ≤ i = l ≤ r are symmet-

ric matrices, satisfying the following LMIs:

Y1 < λS11 (35)[Y1 ST

21

S21 S22

]> 0 (36)

Y2 < −λN11 (37)[Y2 −NT

21

−N21 −N22

]> 0 (38)

Jjii + J i

ij +(J i

ij

)T> 0, 1 ≤ i < j ≤ r (39)

J ijj + Jj

ij +(Jj

ij

)T

> 0, 1 ≤ i < j ≤ r (40)

He(J l

ij + Jjil + J i

jl

)> 0, 1 ≤ i < j < l ≤ r (41)

[Eij ]r×r +[J l

ij

]r×r

< 0, 1 ≤ l ≤ r (42)

[Eaij ]r×r < −λ([Eij ]r×r +

[J l

ij

]r×r

), 1 ≤ l ≤ r

(43)

where

Eij =

⎡⎢⎢⎣

E11ij E12

ij E13ij E14

ij

∗ E22ij E23

ij E24ij

∗ ∗ E33ij E34

ij

∗ ∗ ∗ E44ij

⎤⎥⎥⎦ (44)

E11ij =

12He(STAi+STAj)

E12ij =

12He

(STAi+AT

FijN+CT2iB

TFjN+AT

i (S−N)

+ STAj +ATFjiN+CT

2jBTFiN+AT

j (S−N))



E13ij =

12ST(B1i+B1j)

E14ij =

12

(CT

1i−CTFi+CT

1j − CTFj

)E22

ij =12He

((S−N)TAi + NTBFjC2i + (S−N)TAj

+ NTBFiC2j

)E23

ij =12

[(ST−NT)(B1i+B1j)

+ NTBFjD21i+NTBFiD21j

]E24

ij =12

(CT

1i+CT1j

)E33

ij = −I

E34ij =0 E44

ij = −γ2I

Eaij =

⎡⎢⎢⎣

E11aij E12

aij E13aij E14

aij

∗ E22aij E23

aij E24aij

∗ ∗ E33aij E34

aij

∗ ∗ ∗ E44aij

⎤⎥⎥⎦ (45)

E11aij =

12

(ST

a Ai+STa Aj

)E12

aij =12

(ST

a Ai+ATFijNa+CT

2iBTFjNa+AT

i (Sa−Na)

× STa Aj +AT

FjiNa+CT2jB

TFiNa+AT

j (Sa−Na))

E13aij =

12ST

a (B1i+B1j)

E14aij = 0

E22aij =

12He

((Sa − Na)TAi+NT

a BFjC2i+(Sa−Na)TAj

+ NTa BFiC2j

)E23

aij =12

[(ST

a −NTa

)(B1i+B1j)+NT

a BFjD21i

+ NTa BFiD21j

]E24

aij =0 E33aij =0 E34

aij =0 E44aij = 0.

Then the H∞-norm of the system (6) is less than or equal to γfor ε ∈ (0, ε∗] with ε∗ = 1/λ.

Proof: From condition (35) and (36), we have

[λS11 ST

21

S21 S22

]> 0. (46)

For ε ∈ (0, 1/λ], from (46), it follows that[1ε S11 ST

21

S21 S22

]> 0, for ε ∈

(0,

1λ

].

Multiplying the earlier inequality by ε, then, yields

Sε =[

S11 εST21

εS21 εS22

]> 0, for ε ∈

(0,

1λ

]. (47)

Similarly, from condition (37) and (38), we can obtain

Nε =[

N11 εNT21

εN21 εN22

]< 0, for ε ∈

(0,

1λ

]. (48)

On the other hand, by (39)–(43) and applying Lemma 3, itfollows that:

r∑i=1

r∑j=1

αiαjEij < 0

r∑i=1

r∑j=1

αiαjEaij < −λ

r∑i=1

r∑j=1

αiαjEij

which implies that

r∑i=1

r∑j=1

αiαj(εEaij + Eij) < 0, ε ∈(

0,1λ

]. (49)

From (44) and (45), then (49) can be rewritten as (50), shownat the bottom of the page, where Me3a(α) is same as in (10).Applying Schur complement to (50), then it follows that:

Me3(α) < 0 (51)

where Me3(α) is same as in (9). By (47), (48), and (51) andapplying Lemma 2, the conclusion follows. Thus, the proof iscomplete.

Remark 4: By Theorem 2, an upper bound of ε can beobtained by solving the following optimization problem:

Minimize λ subject to (35)−(43).

It is a GEVP [15], which can be effectively solved using LMIControl Toolbox [25]. The problem of computing the boundof ε was considered in [18], where a method of finding aninterval [ε, ε], ε > 0 so that the system is stable for ε ∈ [ε, ε]was derived. However, the search for small ε is related toε-dependent computation, which cannot avoid the difficultieslinked with the stiffness of the equations.

Moreover, a method of finding an interval (0 ε∗] so thatthe system is stable and satisfies an H∞ performance index

⎡⎢⎢⎣

Me3a(α)+MTe3a(α) ST

ε E−1ε B1(α)(

STε −NT

ε

)E−1

ε B1(α)+NεE−1ε BF (α)D21(α)

CT1 (α)−CT

F (α)CT

1 (α)∗ −I 0∗ ∗ −γ2I

⎤⎥⎥⎦< 0, for ε ∈

(0,

1λ

](50)



for ε ∈ (0 ε∗] is given in [19]. However, the H∞ performanceindex in [19] is dependent on Lyapunov matrices and is dif-ferent from the general definition for H∞ performance index,which is independent of Lyapunov matrices and is exploited inTheorem 2. Some comparisons among the earlier methods areshown in Example 1.

C. Comparison With the Existing Results

In this section, first, based on Theorem 1, a filter-designcondition without the consideration of improving the upperbound of singular-perturbation parameter ε will be given asa corollary of Theorem 1. Then, a comparison between thecorollary and the existing result in [20] will be presented inTheorem 3.

Corollary 1:

1) For a given scalar γ > 0, if there exist matrices S, N ,AFij , BFi, CFi, 1 ≤ i, j ≤ r, J l

ii, 1 ≤ i = l ≤ r, J lij ,

1 ≤ i < j ≤ r, and 1 ≤ l ≤ r with

S =[

S11 0S21 S22

]N =

[N11 0N21 N22

]

where S11, S22, N11, N22, J lii, and 1 ≤ i = l ≤ r are

symmetric matrices, satisfying the following LMIs:

S11 > 0 S22 > 0, N11 < 0 N22 < 0 (52)

Jjii + J i

ij +(J i

ij

)T> 0, 1 ≤ i < j ≤ r (53)

J ijj + Jj

ij +(Jj

ij

)T

> 0, 1 ≤ i < j ≤ r

He(J l

ij + Jjil + J i

jl

)> 0, 1 ≤ i < j < l ≤ r

[Hij ]r×r +[J l

ij

]r×r

< 0, 1 ≤ l ≤ r (54)

where Hij is the same as in (30); then, there exists anε∗ > 0 such that the H∞-norm of the system (6) with (33)is less than or equal to γ for ε ∈ (0, ε∗].

2) For a given scalar γ > 0, if there exist matrices S, N ,AFij , BFi, CFi, 1 ≤ i, and j ≤ r with

S =[

S11 0S21 S22

]N =

[N11 0N21 N22

]

where S11, S22, N11, and N22 are symmetric matrices,satisfying the following LMIs:

S11 > 0, S22 > 0, N11 < 0, N22 < 0 (55)

Hii < 0, 1 ≤ i ≤ r (56)

Hij + Hji < 0, 1 ≤ i < j ≤ r (57)

where Hij is the same as in (30); then, there exists anε∗ > 0 such that the H∞-norm of the system (6) with (33)is less than or equal to γ for ε ∈ (0, ε∗]. Furthermore, ifthe conditions of 2) hold, then the ones of 1) hold.

Proof:

1) Assume that the conditions of 1) in Corollary 1 hold, thenfrom (52), we have that there exist ηi, 1 ≤ i ≤ 6 such that

1η1

I < S111η2

I < S22 ST21S21 < η3I

1η4

I < −N111η5

I < −N22 NT21N21 < η6I.

Applying Schur complement to the earlier inequalityyields (21) and (22).

Moreover, it is easily obtained that there exist βi, 1 ≤i = 4 ≤ 6 such that (24)–(27) and (29) hold.

On the other hand, if (54) holds, then there exists a verysmall scalar ρ > 0 such that

[Hij ]r×r +[J l

ij

]r×r

+ rρdiag[Ia, Ia, . . . , Ia]r×r < 0,

l = 1, . . . , r (58)

where

Ia =

⎡⎣ I2n×2n 0 0

0 Inw×nw0

0 0 Inz×nz

⎤⎦

and we can obtain the following inequality:

rρdiag[I0, I0, . . . , I0]r×r < rρdiag[Ia, Ia, . . . , Ia]r×r (59)

where

I0 =

⎡⎣ I2n×2n 0 0

0 0nw×nw0

0 0 0nz×nz

⎤⎦ .

Moreover, we can also obtain the following inequality:

[Rbij ]r×r < rρdiag[I0, I0, . . . , I0]r×r (60)

where Rbij = ρI0.From (58)–(60), it follows that

[Hij ]r×r +[J l

ij

]r×r

+ [Rbij ]r×r < 0, l = 1, . . . , r

i.e.,

[Hij + Rbij ]r×r +[J l

ij

]r×r

< 0, l = 1, . . . , r. (61)

Let Λ = ρI2n×2n. Then Rbij =

⎡⎣ Λ 0 0

0 0 00 0 0

⎤⎦.

Let Φij = Hij + Rbij , then (61) implies that (28)holds.

Because Λ = ρI2n×2n, then there exists β4 > 0 suchthat (23) holds.

Therefore, if the conditions of 1) in Corollary 1, thenthe conditions of Theorem 1 hold. Furthermore, fromTheorem 1, it follows that there exists an ε∗ > 0 such thatthe H∞-norm of the system (6) with (33) is less than orequal to γ for ε ∈ (0, ε∗].



2) Assume that the conditions of 2) in Corollary 1 hold, thenthere exist J l

ii = 0, 1 ≤ i = l ≤ r and J lij = −Hij1 ≤

i < j ≤ r, 1 ≤ l ≤ r such that

Jjii + J i

ij +(J i

ij

)T> 0, 1 ≤ i < j ≤ r

J ijj + Jj

ij +(Jj

ij

)T

> 0, 1 ≤ i < j ≤ r

He(J l

ij + Jjil + J i

jl

)> 0, 1 ≤ i < j < l ≤ r

[Hij ]r×r + [J lij ]r×r < 0, l = 1, . . . , r

which implies that (53) and (54) hold. Then, the con-ditions of 1) in Corollary 1 hold, then the conclusionfollows.

Corollary 1 provides two filter-design methods without theconsideration of improving the upper bound of ε. By the proofof Corollary 1, it can be seen that 1) in Corollary 1 is less con-servative than 2). Moreover, the equivalence between the exist-ing result given in [20] (i.e., Lemma 4) and 2) in Corollary 1will be given in the following theorem, which shows that thenew filter-design method given by 1) in Corollary 1 is lessconservative than the existing one [20].

Theorem 3: The conditions of Lemma 4 hold if, and only if,the conditions of 2) in Corollary 1 hold.

Proof: (=⇒): Assume that the conditions of Lemma 4hold.

Choose

S =X−1 N = X−1 − Y (62)

AFij =AF0ijX−1 − BF0jC2i − Y TAi

BFi =BF0i CFi = CF0iX−1. (63)

Because X and Y have the lower triangular structures, then

S11 = X−111

S22 = X−122

N11 = X−111 − Y11

N22 = X−122 − Y22. (64)

From (15), we have[Y11 00 Y22

]> 0

[X11 00 X22

]> 0.

Combining it and (64), then, yields

S11 = X−111 > 0 S22 = X−1

22 > 0.

Applying the Schur complement to (15) yields[Y11 00 Y22

]−

[X−1

11 00 X−1

22

]> 0.

Combining it and (64), then it follows that

N11 = X−111 − Y11 < 0

N22 = X−122 − Y22 < 0.

Pre- and postmultiplying (16) by diag[ST I I I] =diag[X−T I I I] and its transpose, then we can obtain

Ψii < 0, 1 ≤ i ≤ r

Ψij + Ψji < 0, 1 ≤ i < j ≤ r

where Ψij is solved as shown at the bottom of the page.Combining it and (62) and (63), then we have

Hii < 0, 1 ≤ i ≤ r

Hij + Hji < 0, 1 ≤ i < j ≤ r

i.e., (56) and (57) hold.Thus, the conditions of 2) in Corollary 1 hold.(⇐=): Assume that the conditions of 2) in Corollary 1 hold.

Then choose

X = S−1 Y = S − N (65)

AF0ij = AFijS−1 + BFjC2iS

−1 + (S − N)TAiS−1

BF0i = BFi CF0i = CFiS−1. (66)

From (55) and (65) and considering the lower triangular struc-ture of S and N , we have[

X−111 00 X−1

22

]> 0

[X−1

11 − Y11 00 X−1

22 − Y22

]=

[S11 − Y11 0

0 S22 − Y22

]

=[

N11 00 N22

]< 0.

Applying Schur complement to the earlier inequality, then itfollows that (15) holds.

Moreover, pre- and postmultiplying (56) and (57) bydiag[XT I I I] and its transpose, then we can obtain

Hii < 0, 1 ≤ i ≤ r

Hij + Hji < 0, 1 ≤ i < j ≤ r

Ψij =

⎡⎢⎢⎢⎣

STAi + ATi S ∗ ∗ ∗

AF0ijS + ATi S AT

i Y + Y TAi + BF0jC2i + CT2iB

TF0j ∗ ∗

BT1iS BT

1iY + DT21iB

TF0j −I ∗

C1i − CF0jS C1i 0 −γ2I

⎤⎥⎥⎥⎦



Fig. 1. Tunnel-diode-circuit diagram.

where Hij is solved as shown at the bottom of the page.Combining it and (65) and (66), then we have

Ψii < 0, 1 ≤ i ≤ r

Ψij + Ψji < 0, 1 ≤ i < j ≤ r

i.e., (16) holds. Thus, the conditions of Lemma 4 hold.

IV. EXAMPLES

In this section, two numerical examples are given. The firstone shows the effectiveness of the new design method with im-proving the upper bound of singular-perturbation parameter ε.The second one further shows the new given design method [1)in Corollary 1] without considering that the bound of singular-perturbation parameter ε can give less-conservative results thanthe existing method in [20] (Lemma 4).

Example 1: Consider a tunnel diode circuit shown in Fig. 1,where the tunnel diode is characterized by

iD(t) = 0.01vD(t) + 0.05v3D(t).

Assuming that the inductance εL is the parasitic parameter andletting x1(t) = vC(t) and x2(t) = iL(t) be the state variables,we have

Cx1(t) = −0.01x1(t) − 0.05x31(t) + x2(t) + 0.1w1(t)

εLx2(t) = − x1(t) − Rx2(t)

y(t) =Sx(t) + w2(t)

z(t) =x1(t)

where w(t) = [w1(t) w2(t)]T is the disturbance noise input,y(t) is the measurement output, z(t) is the controlled output,and S = [1 0] is the sensor matrix. The parameters of the circuitare C = 100 mF, R = 0.5 Ω, and εL = εH = 0.01 H. With

these parameters, the dynamic of the tunnel diode circuit canbe rewritten as

x1(t) = − 0.1x1(t) − 0.5x31(t) + 10x2(t) + w1(t)

εx2(t) = − x1(t) − 0.5x2(t)

y(t) =Sx(t) + w2(t)

z(t) =x1(t). (67)

For the sake of simplicity, we will use as few rules as possible.Assuming that |x1(t)| ≤ 3, the nonlinear network system (67)can be approximated by the following T-S fuzzy model:

Eεx(t) =2∑

i=1

αi(t)Aix(t) + B1w(t)

y(t) =2∑

i=1

αi(t)C2ix(t) + D21w(t)

z(t) = x1(t)

where αi(t) is the normalized time-varying fuzzy weightingfunction for each rule i = 1, 2, x(t) = [xT

1 (t), xT2 (t)]T

A1 =[−0.1 10−1 −0.5

]

A2 =[−4.6 10−1 −0.5

]

B1 =[

1 00 0

]

C1 = [ 1 0 ]

C21 = C22 = [ 1 0 ]

D21 = [ 0 1 ]

Eε =[

1 00 ε

].

Fig. 2 shows the plot of the membership functions for rules 1and 2.

First, we apply Theorem 1 to design filter with weightingscalars

a1 = 0.001 a2 = 0.0011 a3 = 0.001 a4 = 10 000

a5 = 10 000 a6 = 0.01 b1 = 0.1 b2 = 0.1

b3 = 0.8 b4 = 1 b5 = 0.0005 b6 = 0.07

Hij =

⎡⎢⎢⎢⎣

AiX + XTATi ∗ ∗ ∗

AFijX + BFjC2iX + (S − N)TAiX + ATi AT

i Y + Y TAi + BFjC2i + CT2iB

TFj ∗ ∗

BT1i BT

1iY + DT21iB

TFj −I ∗

C1iX − CFjX C1i 0 −γ2I

⎤⎥⎥⎥⎦



Fig. 2. Membership functions α(x1(t)) and (x1(t)).

and γ = 0.05. The corresponding gain matrices are given asfollows:

AF11 =[−3.7234 10.0593−0.7649 −0.5029

]

AF12 =[−4.1219 10.0578−0.7892 −0.5028

]

AF21 =[−7.6107 10.0578−0.6166 −0.5028

]

AF22 =[−5.2345 10.0590−0.7637 −0.5028

]

BF1 =[

3.1647−0.1557

]

BF2 =[−0.32560.0169

]

CF1 = [ 0.9372 −0.0001 ]

CF2 = [ 0.8578 −0.0001 ] . (68)

Next, applying Lemma 4, i.e., the approach in [20] [which isequivalent to 2) in Corollary 1] for designing a filter, the filtergain matrices are given as follows:

AF11 =[

60.0861 66.7726−1.1746 −0.5873

]

AF12 =[−6.0267 40.9452−0.7924 −0.3962

]

AF21 =[−5.9058 40.9459−0.7924 −0.3962

]

AF22 =[

39.9183 70.7685−1.1834 −0.5917

]

BF1 =[

0.1803−0.0000

]

BF2 =[

0.1803−0.0000

]

CF1 = [ 0.4318 −0.2871 ]

CF2 = [ 0.5857 −0.2058 ]

γ = 0.0497. (69)

TABLE IESTIMATED BOUNDS ε∗ WITH ε ∈ (0 ε∗] BY THEOREM 2

Since Example 1 is the simplest fuzzy model (i.e., fuzzy modelwith two fuzzy rules), the introduced variables J l

ij , 1 ≤ i, j,and l ≤ r of 1) in Corollary 1 do not play a role for reducingconservatism, and the computational results show that 1) inCorollary 1 and Lemma 4 [or 2) in Corollary 1] give the sameH∞ performance index.

In addition to the filter-design technique in Theorem 1,a new method for estimating the upper bound of singular-perturbation parameter ε is also given in Theorem 2. In orderto illustrate the effectiveness of both the new filter-designtechnique (Theorem 1) and the method for estimating the upperbound of ε (Theorem 2), the bounds of ε of the singularlyperturbed fuzzy system with the filter gains (68) and (69) willbe, respectively, estimated by Theorem 2 and the methods in[18] and [19].

Note that the type of bounds ε ∈ [ε ε] with ε > 0 is con-sidered in [18], where a weighting matrix Q is involved. The

weighting matrix is Q =[

CT1 (α)C1(α) −CT

1 (α)CF (α)−CT

F (α)C1(α) CTF (α)CF (α)

]for this example. Unfortunately, the condition of [18] is in-feasible for all ε or ε ∈ (0,+∞); the reason is that, in the

condition, block matrices

[AT

ijP Δ + ΔPAij + Q ΔP

P Δ − 12I

]

and

[AT

ijPΔ + ΔPAij + Q ΔP

PΔ − 12I

][18, Eq. (53)] have to

be less than zero, where Δ = diag[I I ε I εI], Δ =diag[I I ε I εI].

However, if a smaller Q is exploited, for example, 0.12 × Qis considered, then the condition in [18] is feasible, and thecorresponding results are given in Table II. Moreover, theobtained upper bounds by Theorem 2 and the methods in [19]are shown in Tables I and III, respectively.

For comparisons with the existing methods, we have thefollowing observations.

1) From Tables I–III, it can be seen that Theorem 1 is effec-tive to obtain larger upper bounds of singular-perturbationparameter than the existing filter-design method(Lemma 4).

2) For the methods for estimating the upper bounds ofsingular-perturbation parameter (Theorem 2 and the onesin [18] and [19]), Tables I–III show that Theorem 2can give larger estimated bounds of singular-perturbationparameter for this example than the existing methods in[18] and [19].

3) Note that the singular-perturbation parameter ε = 0.01 inthe example, for α1(t) = 1 and α2(t) = 0

Aeε(α) =[

E−1ε A1 0

E−1ε BF1C21 E−1

ε AF11

].

The eigenvalues of Aeε(α) with the designed filter gain(69) are −25.0500 + 19.4293i, −25.0500 − 19.4293i,



TABLE IIESTIMATED BOUNDS ε AND ε WITH ε ∈ [ε ε] BY THE METHOD OF [18]

TABLE IIIESTIMATED BOUNDS ε∗ WITH ε ∈ (0 ε∗] BY THE METHOD OF [19]

Fig. 3. Trajectory of ξ1(t) with the gain (69).

0.6771 + 65.6805i, and 0.6771 − 65.6805i. Therefore,the designed filter by Lemma 4 (the existing method) can-not be applied. The fact is also illustrated by the trajectoryof the state ξ1(t) of the filter in Fig. 3. Therefore, the newproposed method with improving the upper bounds of εis necessary.

Following these comparisons, some simulations will be givento illustrate the effectiveness of the new proposed method.

Assume that the initial state x(0) = [0 0]T and the distur-bance w(t) = sin(t)[1/(t + 1) 1/(t2 + 1)], the state ξ1(t) ofthe filter with the gain (69) is given in Fig. 3. It can be seenthat ξ1(t) is not convergent, which shows that the existingmethod [2) in Corollary 1, i.e., Lemma 4] does not work.The reason is that the existing method [2) in Corollary 1]does not address the issue of improving the bound of singular-perturbation parameter ε when the filter is designed, and thebound of singular-perturbation parameter ε of the singularlyperturbed system with the designed filter gain (69) is less thanthe practical singularly perturbed parameter ε = 0.01 for thisexample. Therefore, the resulting system is unstable.

Using the gain (68) given by Theorem 1 (the new proposedmethod), the trajectory of ze = z(t) − z(t), i.e., the filter error,is shown in Fig. 4; it is shown that the filter error ze = z(t) −z(t) tends to zero, which means that the filter estimation zensures that fast-tracking characteristic. For some instant T , the

ratio of√∫ T

0 zTe (t)ze(t)dt/

∫ T

0 wT(t)w(t)dt can show the in-

Fig. 4. Trajectory of ze = z(t) − z(t) with gain (68).

Fig. 5.

√∫ T

0zTe (t)ze(t)dt/

∫ T

0wT(t)w(t)dt with gain (68).

fluence of disturbance w(t) on the filter error ze = z(t) − z(t),and the plot of the ratio is shown in Fig. 5. It can be seen thatthe ratio tends to a constant value 0.016, which is less than theprescribed value γ = 0.05 (an obtained H∞ performance-indexrequirement). These simulations show that the new proposedmethod is effective.



Example 2: Consider the singularly perturbed fuzzy system(3) with r = 2

A1 =[

0 10−2 −8

]A2 =

[2 6.1−3 −9

]

B11 =[

0.6 01 0

]B12 =

[1 02 0

]C11 = [ 1 0 ] C12 = [ 1 −1 ]C21 = [0 3] C22 = [0 − 3]

D211 = [2 1] D212 = [0 1] Eε =[

1 00 ε

].

Applying 1) and 2) in Corollary 1 to the example, thenthe obtained optimal H∞ performance bounds are γ = 2.0760and γ = 2.2881, which implies that the slack variables J l

ij of1) in Corollary 1 are helpful for less-conservative results. SinceLemma 4 (the method in [20]) is equivalent to 2) in Corollary 1(see Theorem 3), the example also shows that the new technique[1) in Corollary 1] can give less-conservative results than theexisting method [20].

V. CONCLUSION

In this paper, we have investigated the problem of designingH∞ filters for fuzzy singularly perturbed systems with theconsideration of improving the bound of singular-perturbationparameter. An LMI-based approach is presented for simultane-ously designing the bound of the singularly perturbed parameterε and H∞ filters for a fuzzy singularly perturbed system. Whenthe bound of singularly perturbed parameter ε is not under con-sideration, the result reduces to an LMI-based design methodfor H∞ filtering of fuzzy singularly perturbed systems, which isless conservative than the existing one. Furthermore, a methodis given for evaluating the upper bound of singularly perturbedparameter subject to the constraint that the considered system isto be with a prescribed H∞ performance bound, and the upperbound can be obtained by solving a GEVP. Numerical examplesare given to illustrate the effectiveness of the proposed methods.

APPENDIX

Proof of Lemma 2:Proof: Let the symmetric matrix Pε in Lemma 1 have the

following decomposition:

Pε =[

Yε Nε

NTε ∗

]P−1

ε =[

Xε Mε

MTε ∗

](70)

where Mε and Nε are invertible.Denote

Γ1 =[

Xε IMT

ε 0

]Γ2 =

[I Yε

0 NTε

]. (71)

Then Γ1 and Γ2 are invertible, and the following equali-ties hold:

PεΓ1 = Γ2

MεNTε = I − XεYε (72)

ΓT1 PεΓ1 =

[Xε II Yε

]. (73)

Choose Sε = X−1ε and Mε = Xε. Then from (72), it fol-

lows that

Nε = X−1ε − Yε.

Since Γ1 is invertible, then from (73), it follows that Pε > 0is equivalent to [

Xε II Yε

]> 0. (74)

Applying Schur complement to (74), then we have proof that(8) holds. Therefore, (74) holds when (8) holds.

Denote

Me1(α) = PεAeε(α) + ATeε(α)Pε

+ PεBeε(α)BTeε(α)Pε +

1γ2

CTe (α)Ce(α).

Pre- and postmultiplying the earlier equality by ΓT1 and Γ1,

it follows that

Me2(α) = ΓT1 Me1(α)Γ1

=Me2a(α) + MTe2a(α)

+[

E−1ε B1(α)

YεE−1ε B1(α) + NεE

−1ε BF (α)D21(α)

]

×[

E−1ε B1(α)

YεE−1ε B1(α) + NεE

−1ε BF (α)D21(α)

]T

+1γ2

[ C1(α)Xε − CF (α)MTε C1(α) ]T

× [ C1(α)Xε − CF (α)MTε C1(α) ] (75)

where, from (8), it follows that

[Sε 00 I

]is invertible.

Pre- and postmultiplying (75) by

[Sε 00 I

]and its transpose,

then yields Me2a(α), shown at the bottom of the next page, and

Me3(α) =[

Sε 00 I

]Me2(α)

[Sε 00 I

]

=Me3a(α) + MTe3a(α)

+[

SεE−1ε B1


−1ε BF (α)D21(α)

]

×[

SεE−1ε B1(α)


−1ε BF (α)D21(α)

]T

+1γ2

[C1(α)−CF (α) C1(α) ]T

× [ C1(α)−CF (α) C1(α) ]

where Me3a(α) is given by (10). From (9) and the earlier equal-ity, it follows that Me3(α) < 0. Since Mε = Xε is invertible, Γ1

is invertible.



From

Me1(α) = Γ−T1

[Sε 00 I

]−1

Me3(α)[

Sε 00 I

]−1

Γ−11 < 0

then we have that Me1(α) < 0. Applying Lemma 1, the con-clusion follows. Thus, the proof is complete.

Proof of Theorem 1:Proof: In order to give a brief expression, we use the

following notations:

A =r∑

i=1

αiAi B1 =r∑

i=1

αiB1i C1 =r∑

i=1

αiC1i

C2 =r∑

i=1

αiC2i D21 =r∑

i=1

αiD21i

AF =r∑

i=1

r∑j=1

αiαjAFij BF =r∑

i=1

αiBFi

CF =r∑

i=1

αiCFi AF =r∑

i=1

r∑j=1

αiαjAFij

BF =r∑

i=1

αiBFi CF =r∑

i=1

αiCFi.

Let

AFij = NTAFij BFi = NTBFi.

The proof consists of four parts. In part 1, we prove that (76)and (77) hold. In parts 2 and 3, we prove that (91) and (93)holds, respectively. Finally, the proof is completed in part 4 byusing (76), (77), (91), and (93).

Part 1: First, from condition (21), we have

∥∥S−111

∥∥ < η1

∥∥S−122

∥∥ < η2

∥∥S21ST21

∥∥ < η3.

Combining it and (31), then, yields∥∥∥S− 1

211 ST

21S−122 S21S

− 12

11

∥∥∥ ≤ η1η2η3 ≤ λ∗

which implies

S− 1

211 ST

21S−122 S21S

− 12

11 ≤ η1η2η3I ≤ λ∗I.

Pre- and postmultiplying the earlier inequality by S1/211 and its

transpose, and applying the Schur complement, then we canobtain [

1ε S11 ST

21

ST21 S22

]> 0, for ε ∈

(0,

1λ∗

].

Multiplying the earlier inequality by ε, then it follows that

Sε =[

S11 εST21

εS21 εS22

]> 0, for ε ∈

(0,

1λ∗

]. (76)

Similarly, from condition (22), we have

Nε =[

N11 εNT21

εN21 εN22

]< 0, for ε ∈

(0,

1λ∗

]. (77)

Part 2: Denote

Me5ab =[

0NT

a (N−1)T

][ NTAF + NTBF C2 NTBF C2 ] .

(78)

Then

Me5ab ≤∥∥NT

a (N−1)T∥∥

×∥∥[

NTAF + NTBF C2 NTBF C2

]∥∥ I. (79)

Multiplying (27) by αiαj and summing them, we have⎡⎣ −β2

5I 0 ATF + CT

2 BF

0 −β25I CT

2 BTF

AF + BF C2 BF C2 −I

⎤⎦ < 0.

Applying the Schur complement to the earlier inequality,then it follows that[

ATF + CT

2 BTF

CT2 BT

F

][ AF + BF C2 BF C2 ] ≤ β2

5I

which implies ∥∥∥[AT

F + BF C2 BF C2

]∥∥∥ < β5

i.e., ∥∥[NTAT

F + NTBF C2 NTBF C2

]∥∥ < β5.

Combining it and (79), we have

Me5ab ≤∥∥NT

a (N−1)T∥∥ β5I. (80)

Multiplying (24) by αiαj and summing them, we obtain⎡⎣−β2

6I 0 STB1

∗ −β26I (ST − NT)B1 + BF D21

∗ ∗ −I

⎤⎦ < 0.

Applying Schur complement to the earlier inequality, then,yields

[STB1

(ST−NT)B1+BF D21

][STB1

(ST−NT)B1+BF D21

]T

≤β26I

Me2a(α) =[

E−1ε A(α)Xε

YεE−1ε A(α)Xε + NεE

−1ε AF (α)MT

ε + NεE−1ε BF (α)C2(α)Xε

E−1ε A(α)

YεE−1ε A(α) + NεE

−1ε BF (α)C2(α)

]



which implies that∥∥∥∥[

STB1

(ST − NT)B1 + BF D21

]∥∥∥∥ < β6. (81)

Denote

Me5b =[

STB1

(ST − NT)B1 + NTBF D21

]

×[

STa B1(

STa − NT

a

)B1 + NT

a BF D21

]T

.

Then

Me5b≤∥∥∥∥[

STB1


]∥∥∥∥×

∥∥∥∥∥[

STa B1(

STa − NT

a

)B1 + NT

a BF D21

]T∥∥∥∥∥ I

≤∥∥∥∥[

STB1


]∥∥∥∥×

(∥∥∥∥∥[

STa B1(

STa −NT

a

)B1

]T∥∥∥∥∥ +

∥∥∥∥∥[

0NT

a BF D21

]T∥∥∥∥∥)

I

≤∥∥∥∥[

STB1


]∥∥∥∥×

(∥∥∥∥∥[

STa B1(

STa −NT

a

)B1

]T∥∥∥∥∥+

∥∥NTa (N−1)T

∥∥‖BFD21‖)

I.

Combining it and (81), then, yields

Me5b≤β6

(∥∥∥∥∥[

STa B1(

STa −NT

a

)B1

]T∥∥∥∥∥+

∥∥NTa (N−1)T

∥∥‖BF D21‖)I.

(82)

Multiplying (25) by αiαj and summing them, then it fol-lows that [

−β22I ∗

BF D21 −I

]< 0

which implies that

‖BF D21‖ ≤ β2. (83)

Similarly, from (26), we can obtain∥∥∥∥[

STa B1(

STa − NT

a

)B1

]∥∥∥∥ ≤ β3. (84)

From (82)–(84), it follows that

Me5b ≤ β6

(β3 +

∥∥NTa (N−1)T

∥∥ β2

)I (85)∥∥∥∥

[ST

a B1(ST

a − NTa

)B1 + NT

a BF D21

]∥∥∥∥≤

∥∥∥∥[

SaB1(ST

a − NTa

)B1

]∥∥∥∥ +∥∥NT

a (N−1)T∥∥ ‖BF D21‖

= β3 +∥∥NT

a (N−1)T∥∥β2. (86)

Multiplying (29) by αi, 1 ≤ i ≤ r and summing them, thenwe have

Me5aa + MTe5aa =

r∑i=1

αi

(Me5aai + MT

e5aai

)

=[

STa A(α) ST

a A(α)(Sa − Na)TA(α) (Sa − Na)TA(α)

]

+[

STa A(α) ST

a A(α)(Sa−Na)TA(α) (Sa−Na)TA(α)

]T

< β1I. (87)

Now, consider Me5 as follows:

Me5 =Me5a + MTe5a + Me5b + MT

e5b

+ ε

[ST

a B1(ST

a − NTa

)B1 + NT

a BF D21

]

×[

STa B1(

STa − NT

a

)B1 + NT

a BF D21

]T

=Me5aa + MTe5aa + Me5ab + MT

e5ab + Me5b + MTe5b

+ ε

[ST

a B1(ST

a − NTa

)B1 + NT

a BF D21

]

×[

STa B1(

STa − NT

a

)B1 + NT

a BF D21

]T

(88)

and Me5a and Me5b, shown at the bottom of the page, whereMe5aa and Me5ab are, respectively, the same as in (78) and (87),

Me5a =[

STa A ST

a ANT

a AF + NTa BF C2 + (Sa − Na)TA (Sa − Na)TA + NT

a BF C2

]

Me5b =[

STB1


] [ST

a B1(ST

a − NTa

)B1 + NT

a BF D21

]T



and from (80) and (85)–(87), we have

Me5 ≤Me5aa + MTe5aa + Me5ab + MT

e5ab + Me5b + MTe5b

+∥∥∥∥[

STa B1(

STa + NT

a

)B1 − NT

a BF D21

]∥∥∥∥2)

I

=(β1+2

∥∥NTa (N−1)T

∥∥ β5+2β6

(β3+

∥∥NTa (N−1)T

∥∥ β2

)+

(β3 +

∥∥NTa (N−1)T

∥∥ β2

)2)I

=β7I

where β7 is the same as in (32).By the definition of λ∗ in (31), it follows that

β4β7 ≤ λ∗. (89)

On the other hand, from (23), we can obtain

β−14 I < Λ β4 > 0

which implies that

1‖Λ‖ < β4. (90)

Then, from (89) and (90), it yields

Λ− 12 Me5Λ− 1

2 ≤ ‖Me5‖‖Λ‖ I ≤ β7

‖Λ‖I ≤ β4β7I.

Combining it and (89), then we have

Λ− 12 Me5Λ− 1

2 ≤ λ∗I

which is equivalent to

Me5 ≤ λ∗Λ

which implies that

εMe5 ≤ Λ, for ε ∈(

0,1λ∗

](91)

where Me5 is the same as in (88).Part 3: From condition (28) and Lemma 3, we obtain

r∑i=1

r∑j=1

αiαjΦij < 0

i.e.,⎡⎢⎢⎣

Me4a+MTe4a+Λ STB1

(ST−NT)B1+BF D21

CT1 −CT

F

CT1

∗ −I 0∗ ∗ −γ2I

⎤⎥⎥⎦<0

(92)

where

Me4a =[

STA STAAF +BF C2+ (S−N)TA (S−N)TA+BF C2

].

Applying the Schur complement to (92), then, yields

Me4 + Λ < 0 (93)

where

Me4 =Me4a + MTe4a

+[

STB1

(ST − NT)B1 − NTBF D21

]

×[

STB1


]T

+1γ2

[C1 − CF C1 ]T[C1 − CF C1 ] < 0.

Part 4: From (91) and (93), it follows:

Me4 + εMe5 < 0, ε ∈(

0,1λ∗

]

where the definitions of Me5, Me4a, and Me4 are given in (88),(92), and (93), respectively.

Consider the solution for Me4 + εMe5, shown on thenext page.

Combining it and (33), then it follows that (94), shown on thepage following the next page, where Sε and Nε are the same asin (47) and (48).

Pre- and postmultiplying the earlier inequality by

Γ−T1

[Sε 00 I

]−T

and its transpose, then we have (95), shown on the page fol-lowing the next one. Consider Pε is same as in (70) and chooseXε = S−1

ε , Mε = Xε = S−1ε , then from (72), we have Nε =

Sε − Yε and (96), shown on the page following the next one.Therefore

Γ−T1

[Sε 00 I

]−T

(Me4 + εMe5)[

Sε 00 I

]−1

Γ−11

= Γ−T1 Me2(α)Γ−1

1 < 0, ε ∈(

0,1λ∗

]

where Me2(α) is same as in (75). Moreover, it follows thatΓ−T

1 Me2(α)Γ−11 = Me1(α) from (75). Then

Me1 < 0, ε ∈(

0,1λ∗

]. (97)

Finally, from (76) and (77), we have Sε > 0 and Nε < 0,respectively. Note that Xε = S−1

ε , Nε = Sε − Yε, then Xε > 0,X−1

ε − Yε < 0, applying Schur complement, it follows that[Xε II Yε

]> 0. Combining it and (73), then we can obtain

Pε > 0.Then, by Pε > 0, (97), and Lemma 1, it follows that the H∞-

norm of the system (6) with

AFi = N−TAFi BFi = N−TBFi

is less than or equal to γ for ε ∈ (0, (1/λ∗)].



Me4 + εMe5 =Me4a + MTe4a +

[STB1


] [STB1


]T

+1γ2

[C1 − CF C1]T[C1 − CF C1] + ε(Me5a + MT

e5a + Me5b + MTe5b

)

+ ε2[

STa B1(

STa − NT

a

)B1 + NT

a BF D21

] [ST

a B1(ST

a − NTa

)B1 + NT

a BF D21

]T

=[

STA STAAF +BF C2+ (S−N)TA (S−N)TA+BF C2

]+

[STA STA

AF +BF C2+ (S − N)TA (S−N)TA+BF C2

]T

+[

STB1


] [STB1


]T

+1γ2

[C1 − CF C1]T[C1 − CF C1]

+ ε

[ST

a A STa A

NTa AF + NT

a BF C2 + (Sa − Na)TA (Sa − Na)TA + NTa BF C2

]

+ ε

[ST

a A STa A

NTa AF + NT


]T

+ ε

[STB1


] [ST

a B1(ST

a − NTa

)B1 + NT

a BF D21

]T

+ ε

([STB1


] [ST

a B1(ST

a − NTa

)B1 + NT

a BF D21

]T)T

+ ε2[

STa B1(

STa − NT

a

)B1 + NT

a BF D21

] [ST

a B1(ST

a − NTa

)B1 + NT

a BF D21

]T

=

([STA STA

AF + BF C2 + (S − N)TA (S − N)TA + BF C2

]

+ ε

[ST

a A STa A

NTa AF + NT


] )

+

([STA STA

AF + BF C2 + (S − N)TA (S − N)TA + BF C2

]T

+ ε

[ST

a A STa A

NTa AF + NT


]T)

+1γ2

[C1 − CF C1]T[C1 − CF C1]

+

([STB1


] [STB1


]T

+ ε

[STB1


] [ST

a B1(ST

a − NTa

)B1 + NT

a BF D21

]T

+ ε

([STB1


] [ST

a B1(ST

a − NTa

)B1 + NT

a BF D21

]T)T

+ ε2[

STa B1(

STa − NT

a

)B1 + NT

a BF D21

] [ST

a B1(ST

a − NTa

)B1 + NT

a BF D21

]T)

< 0, ε ∈(

0,1λ∗

]



Me4 + εMe5 =[

STε E−1

ε A STε E−1

ε ANT

ε E−1ε AF + NT

ε E−1ε BF C2 + (Sε − Nε)TE−1

ε A (Sε − Nε)TE−1ε A + NT

ε E−1ε BF C2

]

+[

STε E−1

ε A STε E−1

ε ANT

ε E−1ε AF + NT



ε E−1ε BF C2

]T

+1γ2

[C1 − CF C1]T[C1 − CF C1]

+

([STB1


]+ ε

[ST

a B1(ST

a − NTa

)B1 + NT

a BF D21

] )

×([

STB1


]+ ε

[ST

a B1(ST

a − NTa

)B1 + NT

a BF D21

] )T

=[

STε E−1

ε A STε E−1

ε ANT

ε E−1ε AF + NT



ε E−1ε BF C2

]

+[

STε E−1

ε A STε E−1

ε ANT

ε E−1ε AF + NT



ε E−1ε BF C2

]T

+1γ2

[C1 − CF C1]T[C1 − CF C1]

+[

STε E−1

ε B1(ST

ε − NTε

)E−1

ε B1 + NTε E−1

ε BF D21

] [ST

ε E−1ε B1(

STε − NT

ε

)E−1

ε B1 + NTε E−1

ε BF D21

]T

< 0, ε ∈(

0,1λ∗

](94)

Γ−T1

[Sε 00 I

]−T

(Me4 + εMe5)[

Sε 00 I

]−1

Γ−11

= Γ−T1

([E−1

ε AS−1ε E−1

ε ANT

ε E−1ε AF S−1

ε + NTε E−1

ε BF C2S−1ε + (Sε − Nε)TE−1

ε AS−1ε (Sε − Nε)TE−1

ε A + NTε E−1

ε BF C2

]

+[

E−1ε AS−1

ε E−1ε A

NTε E−1

ε AF S−1ε + NT

ε E−1ε BF C2S

−1ε + (Sε − Nε)TE−1

ε AS−1ε (Sε − Nε)TE−1

ε A + NTε E−1

ε BF C2

]T

+1γ2

[C1S−1ε − CF S−1

ε C1]T[C1S−1ε − CF S−1

ε C1]

+[

E−1ε B1(

STε − NT

ε

)E−1

ε B1 + NTε E−1

ε BF D21

] [E−1

ε B1(ST

ε − NTε

)E−1

ε B1 + NTε E−1

ε BF D21

]T)

Γ−11

< 0, ε ∈(

0,1λ∗

](95)

Γ−T1

[Sε 00 I

]−T

(Me4 + εMe5)[

Sε 00 I

]−1

Γ−11

= Γ−T1

([E−1

ε AXε E−1ε A

NTε E−1

ε AF MTε + NT

ε E−1ε BF C2Xε + YεE

−1ε AXε YεE

−1ε A + NT

ε E−1ε BF C2

]

+[

E−1ε AXε E−1

ε ANT

ε E−1ε AF MT

ε + NTε E−1

ε BF C2Xε + YεE−1ε AXε YεE

−1ε A + NT

ε E−1ε BF C2

]T

+1γ2

[C1Xε − CF Mε C1]T[C1Xε − CF Mε C1]

+[

E−1ε B1

YεE−1ε B1 + NT

ε E−1ε BF D21

] [E−1

ε B1

YεE−1ε B1 + NT

ε E−1ε BF D21

]T)

Γ−11

< 0, ε ∈(

0,1λ∗

](96)



REFERENCES

[1] P. Kokotovic, H. K. Khalil, and J. O’Reilly, Singular PerturbationMethods in Control: Analysis and Design. New York: Academic, 1986.

[2] Z. Gajic and M. T. Lim, “A new filtering method for linear singularlyperturbed systems,” IEEE Trans. Autom. Control, vol. 39, no. 9, pp. 1952–1955, Sep. 1994.

[3] X. Shen and L. Deng, “Decomposition solution of H∞ filter gain insingularly perturbed systems,” Signal Process., vol. 55, no. 3, pp. 313–320, Dec. 1996.

[4] M. T. Lim and Z. Gajic, “Reduced-order H∞ optimal filtering for systemswith slow and fast modes,” IEEE Trans. Circuits Syst. I, Fundam. TheoryAppl., vol. 47, no. 2, pp. 250–254, Feb. 2000.

[5] T. Takagi and M. Sugeno, “Fuzzy identification of systems and its ap-plications to modeling and control,” IEEE Trans. Syst., Man, Cybern.,vol. SMC-15, no. 1, pp. 116–132, Jan. 1985.

[6] K. Tanaka and H. O. Wang, Fuzzy Control Systems Design and Analysis.Hoboken, NJ: Wiley, 2000.

[7] X. J. Ma, Z. Q. Sun, and Y. Y. He, “Analysis and design of fuzzy controllerand fuzzy observer,” IEEE Trans. Fuzzy Syst., vol. 6, no. 1, pp. 41–51,Feb. 1998.

[8] K. Tanaka, T. Hori, and H. O. Wang, “A multiple Lyapunov functionapproach to stabilization of fuzzy control systems,” IEEE Trans. FuzzySyst., vol. 11, no. 4, pp. 582–589, Aug. 2003.

[9] H.-N. Wu and K.-Y. Cai, “Robust fuzzy control for uncertain discrete-timenonlinear Markovian jump systems without mode observations,” Inf. Sci.,vol. 177, no. 6, pp. 1509–1522, Mar. 2007.

[10] W.-J. Wang, Y.-J. Chen, and C.-H. Sun, “Relaxed stabilization criteria fordiscrete-time T-S fuzzy control systems based on a switching fuzzy modeland piecewise Lyapunov function,” IEEE Trans. Syst., Man, Cybern. B,Cybern., vol. 37, no. 3, pp. 551–559, Jun. 2007.

[11] C. S. Tseng and B. S. Chen, “H∞ fuzzy estimation for a class of nonlineardiscrete-time dynamic systems,” IEEE Trans. Signal Process., vol. 49,no. 11, pp. 2605–2619, Nov. 2001.

[12] B. S. Chen, C. L. Tsai, and D. S. Chen, “Robust H∞ and mixedH2 filters for equalization designs of nonlinear communication systems:Fuzzy interpolation approach,” IEEE Trans. Fuzzy. Syst., vol. 11, no. 3,pp. 384–398, Jun. 2003.

[13] S. K. Nguang and W. Assawinchaichote, “H∞ filtering for fuzzy dynam-ical systems with D stability constraints,” IEEE Trans. Circuits Syst. I,Reg. Papers, vol. 50, no. 11, pp. 1503–1508, Nov. 2003.

[14] G. Feng, “Robust H∞ filtering of fuzzy dynamic systems,” IEEE Trans.Aerosp. Electron. Syst., vol. 41, no. 2, pp. 658–670, Apr. 2005.

[15] S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan, “Linear matrixinequalities in systems and control theory,” in SIAM Studies in AppliedMathematics. Philadelphia, PA: SIAM, 1994.

[16] H. Liu, F. Sun, and Z. Sun, “Stability analysis and synthesis of fuzzysingularly perturbed systems,” IEEE Trans. Fuzzy. Syst., vol. 13, no. 2,pp. 273–284, Apr. 2005.

[17] W. Assawinchaichote and S. K. Nguang, “H∞ fuzzy control design fornonlinear singularly perturbed systems with pole placement constraints:An LMI approach,” IEEE Trans. Syst., Man, Cybern. B, Cybern., vol. 34,no. 1, pp. 579–588, Feb. 2004.

[18] T. H. S. Li and K. J. Lin, “Stabilization of singularly perturbed fuzzysystems,” IEEE Trans. Fuzzy. Syst., vol. 12, no. 5, pp. 579–595, Oct. 2004.

[19] T.-H. S. Li and K.-J. Lin, “Composite fuzzy control of nonlinearsingularly perturbed systems,” IEEE Trans. Fuzzy. Syst., vol. 15, no. 2,pp. 176–187, Apr. 2007.

[20] W. Assawinchaichote and S. K. Nguang, “H∞ filtering for fuzzysingularly perturbed systems with pole placement constraints: An LMIapproach,” IEEE Trans. Signal Process., vol. 52, no. 6, pp. 1659–1667,Jun. 2004.

[21] H. Mukaidani, “A numerical algorithm for finding solution ofsign-indefinite algebraic Riccati equations for general multiparametersingularly perturbed systems,” Appl. Math. Comput., vol. 189, no. 1,pp. 255–270, Jun. 2007.

[22] N. R. Sandell, Jr., “Robust stability of systems with application to singularperturbations,” Automatica, vol. 15, no. 4, pp. 467–470, Jul. 1979.



[25] P. Gahinet, A. Nemirovski, A. J. Laub, and M. Chilali, LMI ControlToolbox. Natick, MA: MathWorks, Inc., 1995.

[26] A. Isidori, “H∞ control via measurement feedback for affine nonlinearsystems,” Int. J. Robust Nonlinear Control, vol. 4, no. 4, pp. 553–574,Apr. 1994.

Guang-Hong Yang (SM’04) received the B.S. andM.S. degrees in mathematics and the Ph.D. degreein control engineering from Northeastern Univer-sity, Shenyang, China, in 1983, 1986, and 1994,respectively.

From 1986 to 1995, he was a Lecturer/AssociateProfessor with Northeastern University. In 1996, hewas a Postdoctoral Fellow with the Nanyang Techno-logical University, Singapore, Singapore. From 2001to 2005, he was a Research Scientist/Senior ResearchScientist with the National University of Singapore,

Singapore. He is currently a Professor with the College of Information Scienceand Engineering, and Key Laboratory of Integrated Automation of ProcessIndustry (Ministry of Education), Northeastern University. His current researchinterests include fault-tolerant control, fault detection and isolation, nonfragilecontrol systems design, and robust control.

Dr. Yang is an Associate Editor for the International Journal of SystemsScience and the International Journal of Control, Automation, and Systems.He is also an Associate Editor of the Conference Editorial Board of the IEEEControl Systems Society.

Jiuxiang Dong was born on January 2, 1978, inShenyang, China. He received the B.S. degree inmathematics and applied mathematics and the M.S.degree in applied mathematics from Liaoning Nor-mal University, Dalian, China, in 2001 and 2004,respectively. He is currently working toward thePh.D. degree in the College of Information Sci-ence and Engineering and Key Laboratory of Inte-grated Automation of Process Industry (Ministry ofEducation), Northeastern University, Shenyang.

His research interests include fuzzy, robust, andfault-tolerant controls.



Automatica 43 (2007) 1597–1604www.elsevier.com/locate/automatica

Brief paper

Parameterization of all stabilizing H∞ static state-feedback gains:Application to output-feedback design

J. Gadewadikara,∗, Frank L. Lewisa, L. Xieb, V. Kucerac, M. Abu-Khalafa

aAutomation and Robotics Research Institute, University of Texas at Arlington, USAbSchool of Electrical and Electrical Engineering, Nanyang Technological University, Singapore

cCenter for Applied Cybernetics, and Control Engineering at Czech Technical University, Prague, Czech Republic

Received 8 December 2005; received in revised form 10 October 2006; accepted 5 February 2007Available online 6 July 2007

Abstract

This paper presents a simplified parameterization of all H∞ static state-feedback controllers in terms of a single algebraic Riccati equationand a free parameter matrix. As a special case, necessary and sufficient conditions for the existence of an static output-feedback gain are given.An efficient computational algorithm is given and its correctness proven. No initial stabilizing output-feedback gain is needed. The techniqueis used to design an H∞ lateral–directional command augmentation system for the F-16 aircraft. 2007 Elsevier Ltd. All rights reserved.

Keywords: Controller parameterization; Static output feedback; H∞ control

1. Introduction

The parameterization of all stabilizing controllers and thestatic output-feedback control problems are two of the mostresearched and written about issues in modern control. Staticoutput feedback is indeed a special case of the former problemwhere the controller gain is restricted to lie in some subspace.The computation of optimal H∞ controllers of prescribed orderand of static output-feedback controllers are non-convex prob-lems, and so are difficult to confront. Main objective of thiswork is to present a simpler parameterization to achieve dis-turbance attenuation with a static state-feedback compensator;the approach is bringing H∞ control, static compensation, andparameterization together. First, an introduction of the relevantwork is given below.

This paper was not presented at any IAFC meeting. This paper wasrecommended for publication in revised form by Associate Editor Sam Geunder the direction of Editor Miroslav Krstic.

∗ Corresponding author. Tel.: +1 817 272 5967; fax: +1 817 272 5989.E-mail addresses: [email protected] (J. Gadewadikar),

[email protected] (F.L. Lewis), [email protected] (L. Xie),[email protected] (V. Kucera), [email protected] (M. Abu-Khalaf).

0005-1098/$ - see front matter 2007 Elsevier Ltd. All rights reserved.doi:10.1016/j.automatica.2007.02.005

1.1. Parameterization of H∞ dynamic compensators

The parameterization of all stabilizing polynomial con-trollers was first given by Youla, Jabr, and Bongiorno (1976)and Kucera (1975, 1979) in terms of solutions to a Diophantineequation. The H∞ approach (Doyle, Glover, Khargonekar, &Francis, 1989) has shown its effectiveness in the design ofcontrollers for modern systems. A parameterization of H∞ dy-namic compensators for the state-feedback and full-informationcases was given by Zhou (1992) in terms of the Hamiltonianmatrix and the Youla–Kucera parameterization. Parameteriza-tion of H∞ dynamic compensators was given by Iwasaki andSkelton (1995). Henrion, Kucera, and Molina-Cristobal (2005)use the Youla–Kucera parameterization to provide a method fordesigning low order compensators by simultaneous optimiza-tion over the numerator and denominator of the compensator.Application is made to H∞ control.

1.2. Static output feedback

A survey of static output-feedback control is given bySyrmos, Abdallah, Dorato, and Grigoriadis (1997). Conditionsoften involve two Riccati equations coupled by a spectral


http://www.elsevier.com/locate/automatica

mailto:[email protected]





1598 J. Gadewadikar et al. / Automatica 43 (2007) 1597–1604

radius condition (Lewis & Syrmos, 1995). Moerder and Calise(1985) presented an algorithm for computing the optimal H2static output-feedback gain that is in standard use, along witha convergence proof. This algorithm is extended and used foraircraft control design in (Stevens & Lewis, 2003). An initialstabilizing output-feedback gain is needed, which is difficult todetermine for practical systems. Recent results address theseissues. Yu (2004) provides a solution for the optimal staticoutput-feedback problem in terms of two AREs plus a projectivecoupling condition, but the AREs are standard ones involvingstate feedback, not output feedback. An algorithm is presentedthat requires an initial stabilizing state-feedback gain, which iseasy to find using standard LQR methods.

A parameterization of all static output-feedback gains isgiven by Yasuda, Skelton, and Grigoriadis (1993) in termsof such equations. LMI approaches have been presented byIwasaki and Skelton (1995), where a parameterization wasalso presented. In Colaneri, Geromel, and Locatelli (1997)and Geromel, de Souza, and Skelton (1998), a parameteriza-tion of static output-feedback gains is given in terms of theblocks of a symmetric matrix derived from coupled matrix in-equalities. Also provided is a condition in terms of a matrixand its inverse. A numerical procedure is given to determinethese quantities. LMI conditions for optimal and H∞ staticoutput feedback are given by Shaked (2003). El Ghaoui andBalakrishnan (1994) propose an iterative algorithm for design-ing LTI controllers with prespecified structure based on LMIs.Sufficient LMI conditions for OPFB control problem are pre-sented in (Crusius and Trofino, 1999). Work by Scherer (1995)was one of the first that discussed the H∞ problem with mixedH2/H∞ objectives.

Trofino-Neto and Kucera (1993) solve the static output-feedback problem using inverse optimal control in terms of asingle ARE projected onto nullspace perpendicular of C and afree parameter matrix.

1.3. Motivation

Solutions to the parameterization of all stabilizing controllersand the output-feedback problems are generally complex, de-pending for instance on coupled Riccati equations, LMIs, or ona matrix and its inverse belonging to some sets.

Geromel and Peres (1985) give a sufficient condition for astabilizing static output feedback in terms of a single ARE witha free parameter matrix, plus a condition on the form of thegain matrix. A computational algorithm is proposed. Kuceraand de Souza (1995) provide necessary and sufficient condi-tions for stabilizing static output feedback control using theseconstructions. Gadewadikar, Lewis, and Abu-Khalaf (2006)extend these conditions to obtain necessary and sufficientconditions for the existence of an H∞ static output feedbackcontrol. Geromel, Yamakami, and Armentano (1989) providesimilar results for the discrete-time case for decentralized feed-back and output feedback. de Souza and Xie (1992) provide forthe discrete-time case a parameterization of all stabilizing H∞controllers in terms of similar constructions, and use that result

to solve the H∞ static output-feedback problem for discretesystems.

To summarize; a lot of work has been pursued, however sev-eral problems are still open, most of the approaches though the-oretically appealing, are hard to implement, are difficult to solvefor higher-order systems, and can impose numerical problems.In this paper we present a parameterization of all stabilizingH∞ controllers for continuous-time systems in terms of a sin-gle ARE and a free parameter matrix. The form of the result issimple and can be used to select the parameter matrix to satisfyadditional design criteria. The result is used to develop neces-sary and sufficient conditions for existence of a solution to theH∞ static output feedback problem for continuous-time sys-tems. An algorithm is provided that does not require an initialstabilizing output-feedback gain, but only an initial stabilizingstate-feedback gain, which is easy to find. The correctness ofthe algorithm is proven, namely that if it converges, it yieldsthe correct solution. The results are used to simply design anH∞ lateral–directional command augmentation system (CAS)for the F-16 aircraft given in Stevens and Lewis (2003).

2. Parameterization of all stabilizing static state-feedbackgains

2.1. System description and definitions

Consider the linear time-invariant (LTI) system in Fig. 1

x = Ax + Bu + Dd, y = Cx, (1)

x ∈ Rn, u ∈ Rm, y ∈ Rp

and a performance output z(t) that satisfies

‖z(t)‖2 = xTQx + uTRu, (2)

with QT =Q0, RT =R > 0. It is assumed that C has full rowrank, a standard assumption to avoid redundant measurements.

The system L2 gain is said to be bounded or attenuated by if∫ ∞

0 ‖z(t)‖2 dt∫ ∞0 ‖d(t)‖2 dt

=∫ ∞

0 (xTQx + uTRu) dt∫ ∞0 (dTd) dt

2 (3)

for any non-zero energy-bounded disturbance input d. Call ∗the minimum gain for which this occurs. Doyle et al. (1989)and Scherer (1992) show procedure to calculate ∗ using aquadratically convergent algorithm. For linear systems, there

z

y = Cx

x = Ax + Bu + Dd

u = −Kx

d

u

y

z

x2 = xT Qx + uT Ru

·

Fig. 1. System description.


J. Gadewadikar et al. / Automatica 43 (2007) 1597–1604 1599

are explicit formulae to compute ∗, see, e.g., Chen (2000).Throughout this paper we shall assume that is fixed and > ∗.

2.2. Bounded L2 gain design problem

Defining a constant state-feedback control as

u = −Kx, (4)

for the system described in Section 2.1, it is desired to find thestate-variable feedback (SVFB) gain K such that the closed-loop system Ac ≡ A−BK is asymptotically stable and the L2gain is bounded by a prescribed value > ∗.

The following main theorem shows necessary and sufficientconditions for the parameterization of all stabilizing H∞ staticstate-feedback gains. To the best of our knowledge, it providesa simpler parameterization than most other results in the liter-ature.

Theorem 1 (Parameterization of all H∞ SVFB gains). Assumethat (A, Q1/2) is detectable and (A, B) is stabilizable. Then Kis a state feedback that stabilizes system (1) and guarantees L2gain bounded by > ∗, if and only if there exists a parametermatrix L such that

K = R−1(BTP + L), (5)

where P = P T 0, is a solution to

PA + ATP + Q + 1

2 PDDTP − PBR−1BTP

+ LTR−1L = 0. (6)

Proof. Necessity: Suppose that there exists a state-feedbackgain K that stabilizes the closed loop system Ac ≡ A − BK

and satisfies L2 gain bounded by . Consider the equation

PAc + ATc P + 1

2 PDDTP + Q + KTRK = 0. (7)

Considering the definition (2), from Knobloch, Isidori, andFlockerzi (1993, Theorem 2.3.1), closed-loop stability and L2gain boundedness implies that Eq. (7) has a unique symmetricsolution such that P 0. Rearranging Eq. (7) and completingthe square will yield

PA + ATP + Q + 1

2 PDDTP − PBR−1BTP

+ (K − R−1BTP)TR(K − R−1BTP) = 0. (8)

Substituting the gain defined by (5) in (8) yields (6).Sufficiency: Define Q ≡ Q + KTRK + (1/2)PDDTP and

Q ≡ Q + KTRK . Suppose that (5) and (6) hold, then (7)follows, so that

PAc + ATc P + Q = 0. (9)

We claim that detectability of (A, Q1/2) implies detectabilityof (Ac, Q

1/2). Note that detectability of (A, Q1/2) implies thatA − LQ1/2 is stable for some L. One can write

A − LQ1/2 = Ac − LQ1/2 (10)

for (Q1/2)T ≡ [(Q1/2)T (R1/2K)T (1/)PD] and L ≡[L −BR−1/2 0]. It follows that if pair (A, Q1/2) is detectablethen pair (Ac, Q

1/2) is detectable as well. Moreover Q0,and P 0. Therefore, using the Lyapunov stability criteria(Wonham, 1985, Lemma 12.2), Eq. (9) implies closed-loopstability. Finally for z=[Q1/2x R1/2 u]T one has zTz=xTQx,and

PAc + ATc P + 1

2 PDDTP + Q = 0, (11)

and system L2 gain boundedness then follows from Van derSchaft (1992, Theorem 2).

A special case of this result shows when there exists a staticoutput-feedback (OPFB) gain F that stabilizes the system (1)with bounded L2 gain. OPFB requires a restricted form of thegain matrix K that has C as a right divisor. Define the controlinput for OPFB as

u = −Fy = −FCx = −Kx

which yields the closed-loop system A0 ≡ (A − BFC). Equa-tions for OPFB design are generally complicated, consisting ofthree coupled matrix equations, as given in Lewis and Syrmos(1995), Moerder and Calise (1985), and Yu (2004).

The next result provides simplified equations for OPFB de-sign. The corollary holds for a prescribed matrix Q and followsdirectly from Theorem 1.

Corollary 1 (Existence of H∞ OPFB for a given Q). Considera specified Q0 such that (A, Q1/2) is detectable, (A, B) isstabilizable, and a specified value of > ∗. Then there existsan OPFB gain F such that A0 ≡ (A − BFC) is asymptoti-cally stable with bounded L2 gain if and only if there exists aparameter matrix L such that

FC = R−1(BTP + L), (12)

where P = P T 0, is a solution to

PA + ATP + Q + 1

2 PDDTP

− PBR−1BTP + LTR−1L = 0. (13)

The next result gives necessary and sufficient conditions forthe existence of a stabilizing OPFB with bounded L2 gain. Itfollows from Corollary 1 by setting Q=CTC. It is a refinementof the result in Kucera and de Souza (1995), and extends thatresult to solve the H∞ OPFB problem.

Corollary 2 (Existence of H∞ OPFB). Assume that (A, B)

is stabilizable. Then, for a given > ∗, there exists an OPFBgain such that A0 ≡ (A − BFC) is asymptotically stable withL2 gain bounded by if and only if:

(i) (A, C) is detectable and there exist matrices L and P =P T 0 such that



(ii) FC = R−1(BTP + L).

(iii)

PA + ATP + CTC + 1

2 PDDTP

− PBR−1BTP + LTR−1L = 0. (14)

Proof. Necessity: To prove necessity, set Q = CTC, K = FC

and proceed with the necessity proof of Theorem 1. Note furtherthat stability of (A − BFC) implies detectability of (A, C).

Sufficiency: Suppose that (i)–(iii) hold, then the sufficiencyproof of Theorem 1 holds with Q = CTC0.

Remarks. Note that under the special form Q = CTC, theperformance output is given by

‖z‖2 = ‖y‖2 + ‖u‖2.

Note that a stabilizability condition on (A, B) is hidden in theassumption of existence of a solution P = P T 0 to (14).

3. Solution algorithm for H∞ output feedback

Most existing iterative algorithms for static OPFB designrequire the determination of an initial stabilizing OPFB gain,which can be very difficult for practical systems, and the solu-tion of three coupled design equations—two Riccati equationsand a spectral radius condition. Here is presented an algorithmto solve the two coupled design equations in Corollaries 1 and 2.The algorithm starts with a stabilizing SVFB gain, which caneasily be found using standard computational tools (e.g. LQR).It is shown that the algorithm, if it converges, solves the output-feedback problem.

Since C has full row rank, the right inverse is defined asC+ ≡ CT(CCT)−1, which is best computed using the SVD

C = USVT = U [S0 0][V T

1

V T2

]. (15)

Then CV =C[V1 V2]=[C1 0], so that CV 1=C1, and CV 2=0.Note that I = V1V

T1 + V2V

T2 . Therefore,

(i) C+ = V S+UT = V1(S0)−1UT.

(ii) V1VT1 = I −V2V

T2 =C+C is the projection onto nullspace

perpendicular of C.

(iii) V2VT2 = I − V1V

T1 = I − C+C is the projection onto the

nullspace of C.

The equation FC = R−1(BTP + L) has an exact solution F ifand only if

0 = (BTP + L)(I − C+C) = (BTP + L)V2VT2 .

Then the solution is given by

F = R−1(BTP + L)C+ = R−1(BTP + L)V1(S0)−1UT.

The following algorithm is based onYu (2004), where, however,three coupled equations must be solved. It solves for a statefeedback at each iteration, and then projects that SVFB ontonullspace perpendicular of C.

OPFB design algorithm.

1. Initialize: Fix ∗. Set n = 0, L0 = 0. Solve a standard(e.g. LQR) Riccati equation for given Q and R and obtain astabilizing SVFB gain as initial gain K0. Define closed-loopmatrix A0 = A − BK0.2. nth iteration: Solve ARE for P

Pn(An) + (An)TPn + Q + KT

n RKn

+ 1

2 PnDDTPn = 0, (16)

update K, project onto nullspace perpendicular of C

Kn+1 = R−1(BTPn + Ln)(I − V2VT2 ), (17)

update L

Ln+1 = RKn − BTPn, (18)

update closed-loop system matrix

An+1 = A − BKn+1. (19)

3. Check convergence. If converged, go to step 4 otherwise setn = n + 1 go to step 2.4. End. Set K = Kn+1 and compute OPFB gain F =KV 1(S0)

−1UT.The convergence can be checked using the norm of Pn+1 − Pn

e.g. ‖Pn+1 − Pn‖ < ε , here ε is a small number and operator‖ ‖ denotes the matrix norm.

The next result shows the correctness of the algorithm,namely, that if it converges, it provides the H∞ OPFB gain.

Lemma 1. If this algorithm converges, it provides the H∞OPFB gain.

Proof. Clearly, at convergence (7) holds for Pn+1 = Pn ≡ P .Substitution of Eq. (17) into Eq. (18) yields

Ln+1 = RR−1(BTPn + Ln)(I − V2vT2 ) − BTPn.

At convergence Ln+1 = Ln ≡ L, so that L = (BTP + L)(I −V2V

T2 ) − BTP , therefore

BTP + L = (BTP + L)C+C. (20)

This guarantees that there exists a solution F to (12) given byF = KC+ = R−1(BTP + L)C+ = KV 1(S0)

−1UT.

Remark. Note that the proof guarantees that, if the algorithmconverges, then there exists a solution F to (12).



4. H∞ output-feedback design for F-16 regulator

In this example we demonstrate the effectiveness of the pro-posed static H∞ OPFB design technique on a realistic designexample. It is desired to design a lateral–directional (e.g., rolldamper/yaw damper) CAS for the F-16 dynamics from Stevensand Lewis (2003) linearized at the nominal flight condition inTable 3.6-3 (Stevens & Lewis, 2003) (VT = 502 ft/s, 300 psfdynamic pressure, cg at 0.35c).

This CAS shown in Fig. 2 has two input channels, and in-cludes actuators and a washout filter. GW is the washout cir-cuit, the transfer function Ga represents an equivalent transferfunction for differential actuation of the left and right ailerons,and Gr is the rudder actuator. The design requires the selectionof eight control gains. The structured nature of the CAS meansthat OPFB, not SVFB, must be used.

The lateral states are sideslip , bank angle , roll p, and yawrate r . Additional states a and r are introduced by the aileronand rudder actuators a = (20.2/(s +20.2))ua, r = (20.2/(s +20.2))ur. A washout filter rw = (s/(s + 1))r is used, with r theyaw rate and rw the washed out yaw rate. The washout filterstate is denoted xw. The entire state vector is

x = [ p r a r xw]T.

The control inputs are the rudder and aileron servo in-puts so that u = [ua ur]T. The disturbance is given byd(t) = [da(t) dr(t) dn(t)]T, where da(t) affects the aileronactuation, dr(t) the rudder actuation, and dn(t) the washoutfilter state. Disturbances are applied to the control inputs,e.g. wind gusts, and to the measurements, i.e. measurementnoise.

The full-state variable model of the aircraft plus actuators,washout filter, disturbance, and control dynamics is of the form

ββφ

p

δδr

δaGa

Gr

ur

ua

GW

r

rw

r1

r2

+

+

2 x 4 OPFB Gain

matrix

Aircraft

Fig. 2. Lateral/directional augmentation system.

given by Eq. (1), with

A =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

−0.3220 0.0640 0.0364 −0.9917 0.003 0.0008 0

0 0 1 0.0037 0 0 0

−30.6492 0 −3.6784 0.6646 −0.7333 0.1315 0

8.5396 0 −0.0254 −0.4764 −0.0319 −0.0620 0

0 0 0 0 −20.2 0 0

0 0 0 0 0 −20.2 0

0 0 0 57.2958 0 0 −1

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

,

B =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

0 0

0 0

0 0

0 0

20.2 0

0 20.2

0 0

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

, D =⎡⎣0 0 0 0 1 0 0

0 0 0 0 0 1 0

0 0 0 0 0 0 1

⎤⎦

T

.

The output is

y =

⎡⎢⎢⎢⎣

rw

p

⎤⎥⎥⎥⎦ =

⎡⎢⎢⎢⎣

0 0 0 57.2958 0 0 −1

0 0 57.2958 0 0 0 0

57.2958 0 0 0 0 0 0

0 57.2958 0 0 0 0 0

⎤⎥⎥⎥⎦ x.

The factor of 57.2958 is added to convert angles from radiansto degrees. The feedback control is output feedback of the formu=Fy, so that the F is a 2 × 4 matrix. That is, eight feedbackgains must be selected.

In this example (A, C) are observable. For the computationof the H∞ output-feedback gain F it is necessary to select Q,R, and . The proposed algorithm makes it very easy and fast toperform the design for different values of Q, R, . We selectedQ = CTC and R = I . If the resulting gain F is not suitable interms of time responses and closed-loop poles, the elements ofQ and R can be changed and the design repeated. These valueswere found suitable in this example.

The gain parameter defines the desired L2 bound. For theinitial design, a fairly large is selected. If the algorithm con-verges, the parameter may be reduced. If is taken too smallthe algorithm will not converge since the ARE has no posi-tive semidefinite solution. After some design repetitions, whichwere performed very quickly using the algorithm; we found thesmallest value of the gain to be ∗=1.499. To compare this withpopular design methods, Chen (2000), smallest value of gainboth with State-feedback and Dynamic OPFB compensator is∗

s = 0.996. H∞ static output-feedback methods have bigger∗ than state-feedback methods because the static output feed-back is a subset of state feedback. The results for the gain ∗are

K =[

95.1780 −56.0021 −47.1518 −17.6280 0 0 0.3077

−19.7080 10.1009 9.3495 −50.5608 0 0 0.8825

],

F =[−0.3077 −0.8230 1.6612 −0.9774

−0.8825 0.1632 −0.3440 0.1763

].



Fig. 3. Trajectory of the gain K.

5 10 15 20 25

-1

0

1

L

1.5

0.5

-0.5

-1.5

-2

iteration

Fig. 4. Trajectory of the parameter matrix L.

5 10 15 20 25

0

2000

4000

6000

8000

10000

12000

14000

P

-2000

iteration

Fig. 5. Trajectory of algebraic Riccati equation solution P.

The resulting closed-loop poles are at

s = − 11.3762 ± 25.4311i, −15.1711,

− 4.1338, −1.3423 ± 1.0468i, −1.1350.

Convergence properties can be observed in Figs. 3–6. Note thatthe norm in Fig. 6 was used in the stopping criterion for thealgorithm

The resulting gains are applied to the system, and step distur-bances d(t) are introduced in simulations to verify disturbancerejection properties of the design. The resulting time responsesshown in Figs. 7 are very good. The design procedure basedon solving two coupled equations is significantly easier thanmethods based on solving three coupled equations, e.g. as de-scribed in Lewis and Syrmos (1995) and Moerder and Calise(1985). Moreover, the H∞ static OPFB controller generally

5 10 15 20 25

Fro

Norm

P

104

102

100

10-2

10-4

10-6

iteration

Fig. 6. Trajectory of Frobenius norm ‖Pn+1 − Pn‖ log scale.

0 1 2 3 4 5 6 7

0

1

2

0 1 2 3 4 5 6 7

0

1

Tim

e r

espon

se

Tim

e r

esponse

Time

Roll mode states φ and p

Time

1.5

0.5

-0.5

0.5

-0.5

Dutch-roll states β and r

Fig. 7. Closed loop response.

outperforms the H2 optimal OPFB controller when disturbancesare introduced.

Acknowledgments

The authors would like to acknowledge Carlos de Souza, forhis enthusiasm and ongoing insight into the problems of outputfeedback and controller parameterization.

This research was supported by NSF grant ECS-0501451 andARO grant W91NF-05-1-0314, and the Ministry of Educationof the Czech Republic project 1M0567.

References

Chen, B. M. (2000). Robust and H∞ Control. Berlin: Springer.Colaneri, P., Geromel, J. C., & Locatelli, A. (1997). Control Theory and

design, an RH2 and RH∞ viewpoint. San Diego: Academic PressInterscience.



Crusius, C. A. R., & Trofino, A. (1999). Sufficient LMI conditions for outputfeedback control problems. IEEE Transactions on Automatic Control,44(5), 1053–1057.

Doyle, J. H., Glover, K., Khargonekar, P., & Francis, B. (1989). State solutionsto standard H2 and H∞ control problems. IEEE Transactions on AutomaticControl, 34(8), 831–847.

de Souza, C. E., & Xie, L. (1992). On the discrete-time bounded reallemma with application in the characterization of static state feedbackH∞ controllers. Systems & Control Letters, 18(1), 61–71.

El Ghaoui, L., & Balakrishnan, V. (1994). Synthesis of fixed-structurecontrollers via numerical optimization. In Proceedings of IEEE Conferenceon Decision and Control, FL, pp. 2678–2683.

Gadewadikar, J., Lewis, F. L., & Abu-Khalaf, M. (2006). Necessary andsufficient conditions H∞ static output-feedback control. Journal ofGuidance Dynamics and Control, 29(4), 915–920.

Geromel, J. C., & Peres, P. L. D. (1985). Decentralised load frequency control.IEE Proceedings, 132(5), 225–230.

Geromel, J. C., de Souza, C. C., & Skelton, R. E. (1998). Static outputfeedback controllers: Stability and convexity. IEEE Transactions onAutomatic Control, 43(1), 120–125.

Geromel, J. C., Yamakami, A., & Armentano, V. A. (1989). Structuralconstrained controllers for discrete-time linear systems. Journal ofOptimization Theory and Applications, 61(1), 73–94.

Henrion, D., Kucera, V., & Molina-Cristobal, A. (2005). Optimizingsimultaneously over the numerator and denominator polynomials in theYoula–Kucera Parametrization. IEEE Transactions on Automatic Control,50(9), 1369–1374.

Iwasaki, T., & Skelton, R. E. (1995). All fixed order H∞ controllers: observerbased structure and covariance bounds. IEEE Transactions on AutomaticControl, 40(3), 512–516.

Knobloch, H. W., Isidori, A., & Flockerzi, D. (1993). Topics in control theory.Berlin, Germany: Birkhauser.

Kucera, V. (1975). Stability of discrete linear feedback systems. Preprints 6thIFAC Congress, Vol. 1, paper 44.1, Boston.

Kucera, V. (1979). Discrete linear control: The polynomial equation approach.Chichester: Wiley.

Kucera, V., & de Souza, C. E. (1995). A necessary and sufficient conditionfor output feedback stabilizability. Automatica, 31(9), 1357–1359.

Lewis, F. L., & Syrmos, V. L. (1995). Optimal control. 2nd ed., New York:Wiley-Interscience.

Moerder, D. D., & Calise, A. J. (1985). Convergence of a Numerical Algorithmfor calculating optimal output feedback gains. IEEE Transactions onAutomatic Control, 30(9), 900–903.

Scherer, C. W. (1992). H∞-optimization without assumptions on finite orinfinite zeros. SIAM Journal on Control Optimization, 30, 143–166.

Scherer, C. W. (1995). Multiobjective H2/H∞ Control. IEEE Transactionson Automatic Control, 40(6), 1054–1062.

Shaked, U. (2003). An LPD approach to robust H2 and H∞ staticoutput-feedback design. IEEE Transactions on Automatic Control, 48(5),866–872.

Stevens, B. L., & Lewis, F. L. (2003). Aircraft control and simulation. 2nded., New York: Wiley Interscience.

Syrmos, V. L., Abdallah, C. T., Dorato, P., & Grigoriadis, K. (1997). Staticoutput feedback—a survey. Automatica, 33(2), 125–137.

Trofino-Neto, A., & Kucera, V. (1993). Stabilization via static output feedback.IEEE Transactions on Automatic Control, 38(5), 764–765.

Van der schaft, A. J. (1992). L2 gain analysis of nonlinear systems andnonlinear state feedback H∞ control. IEEE Transactions on AutomaticControl, 37(6), 770–784.

Wonham, W. M. (1985). Linear multivariable control. Berlin, Germany:Springer.

Yasuda, K., Skelton, R., & Grigoriadis, K. (1993). Covariance controllers: anew parameterization of the class of all stabilizing controllers. Automatica,29, 785–788.

Youla, D. C., Jabr, H. A., & Bongiorno, J. J. (1976). Modern Wiener–Hopfdesign of optimal controllers, Part II: The multivariable case. IEEETransactions on Automatic Control, AC-21, 319–338.

Yu, J.-T. (2004). A convergent algorithm for computing stabilizing staticoutput-feedback gains. IEEE Transactions on Automatic Control, 49(12),2271–2275.

Zhou, K. (1992). On the parameterization of H∞ controllers. IEEETransactions on Automatic Control, 37(9), 1442–1445.

Jyotirmay Gadewadikar received his Bache-lor’s degree in Electronics and InstrumentationEngineering from SGS Institute of Technologyand Science, Indore, India in 1997. He thenworked for Tata Steel as a Senior Officer andwas involved with installation and commission-ing in Cold Rolling Mill Project. He receivedthe Master’s of Science in Electrical Engineer-ing in 2003. He then joined Automation andRobotics Research Institute (ARRI) for Systems& Controls Research, Prototyping & Implemen-tation of Control Algorithms on Electromechan-ical Systems, taught Control System Design

Capstone course, and was involved in the set up and development ofundergraduate controls lab, he was also at Qualcomm San Diego as anIntern. His research work includes H∞ control, output feedback control,helicopter UAV control, electromechanical systems, and Automated Testingand Measurement tools for optimization of Wireless TelecommunicationNetworks.

F.L. Lewis, Fellow IEEE, Fellow UK Instituteof Measurement & Control, PE Texas, UK Char-tered Engineer, is a Distinguished Scholar Pro-fessor and Moncrief-O’Donnell Chair at Uni-versity of Texas at Arlington’s Automation &Robotics Research Institute. He obtained theBachelor’s Degree in Physics/EE and the MSEEat Rice University, the M.S. in Aeronautical En-gineering from Univ. W. Florida, and the Ph.D.at Ga. Tech. He works in feedback control, in-telligent systems, and sensor networks. He is au-thor of five US patents, 174 journal papers, 286conference papers, and 11 books. He received

the Fulbright Research Award, NSF Research Initiation Grant, andASEE Terman Award. Received Outstanding Service Award from Dal-las IEEE section, selected as Engineer of the year by Ft. WorthIEEE Section. Listed in Ft. Worth Business Press Top 200 Leaders inManufacturing.

Lihua Xie received the B.E. and M.E. degreesin electrical engineering from Nanjing Univer-sity of Science and Technology in 1983 and1986, respectively, and the Ph.D. degree inelectrical engineering from the University ofNewcastle, Australia, in 1992.Dr. Xie is currently a Professor with the Schoolof Electrical and Electronic Engineering,Nanyang Technological University, Singapore.He held teaching appointments in the Depart-ment of Automatic Control, Nanjing Universityof Science and Technology from 1986 to 1989.His current research interests include robust

control and estimation, networked control systems, time delay systems, controlof disk drive systems and sensor networks. In these areas, he has publishedseveral papers and co-authored (with C. Du) the monograph H-infinity Controland Filtering of Two-dimensional Systems (Springer, 2002).Dr. Xie is currently an associate editor of IEEE Transactions on AutomaticControl, Automatica, IEEE Transactions on Circuits and Systems—II, andJournal of Control Theory and Applications. He is also a member of theEditorial Board of IEE Proceedings on Control Theory and Applications. Heserved as an associate editor of International Journal of Control, Automationand Systems from 2004 to 2006 and the Conference Editorial Board, IEEEControl Systems Society from 2000 to 2005. He is a Fellow of IEEE and aFellow of Institution of Engineers, Singapore.



Vladimír Kucera was born in Prague, CzechRepublic, in 1943. He graduated in ElectricalEngineering from the Czech Technical Univer-sity in Prague in 1966, and received the CSc.and DrSc. degrees in Control Engineering fromthe Czechoslovak Academy of Sciences in 1970and 1979, respectively.The research interests of V. Kucera include lin-ear systems, optimal and robust control. He hascontributed to the theory of the Riccati equa-tion, pioneered the use of polynomial equationsin control system design, and paved the way tothe parametrization of all stabilizing controllers.

From 1970 to 1990 he was a Research Scientist, and from 1990 to 1998 theDirector of the Institute of Information Theory and Automation, Academyof Sciences of the Czech Republic. Since 1995 he has been Professor ofControl Engineering, and from 2000 to 2006 the Dean of the Faculty ofElectrical Engineering, Czech Technical University in Prague.V. Kucera is the author of four books, including Discrete Linear Control:The Polynomial Equation Approach (Wiley, 1979) and Analysis and Designof Discrete Linear Control Systems (Prentice-Hall, 1991), and published 300research papers. He is Past President of IFAC, Fellow of IEEE, and Fellowof the Czech Academy of Engineering. He was awarded the National Prize ofthe Czech Republic in 1989, Automatica Best Paper Award in 1990, Medal ofthe Ministry of Education of the Czech Republic in 2000, Felber Gold Medal

of the Czech Technical University in 2006, and French Government Medal“Chevalier dans l’ordre des Palmes Académiques” in 2006. He is an HonoraryProfessor at the Northeastern University, Shenyang, China (1996) and receivedDoctor honoris causa degrees from Université Paul Sabatier, Toulouse (2003)and Université Henri Poincaré, Nancy (2005).

Murad Abu-Khalaf was born in Jerusalem,Palestine in 1977. He obtained his B.S. inElectronics and Electrical Engineering fromBogaziçi University in Istanbul, Turkey in1998, and the M.S. and Ph.D. in ElectricalEngineering from The University of Texas atArlington in 2000 and 2005, respectively. Hisresearch interest is in the areas of nonlinearcontrol, optimal control, neural network con-trol, and adaptive intelligent systems. He isthe author/co-author of one book, two bookchapters, eight journals papers and 15 refereedconference proceedings. He is a member of

IEEE member, and a member of Etta Kappa Nu honor society, and is listedin Who’s Who in America. His research interests are in the areas of nonlinearcontrol, optimal control, neural network control, adaptive intelligent systems.



Information Sciences 177 (2007) 3005–3015

www.elsevier.com/locate/ins

Static output feedback controller design for fuzzy systems:An ILMI approach

Dan Huang, Sing Kiong Nguang *

The Department of Electrical and Computer Engineering, The University of Auckland, Private Bag 92019 Auckland, New Zealand

Received 15 July 2004; received in revised form 4 February 2007; accepted 11 February 2007

Abstract

This paper examines the problem of static output feedback control of a Takagi–Sugeno (TS) fuzzy system. The exis-tence of a static output feedback control law is given in terms of the solvability of bilinear matrix inequalities. An iterativealgorithm based on the linear matrix inequality is proposed to compute the static output feedback gain. To reduce theconservatism of the design, the structural information of the membership function of the fuzzy rules is incorporated.Numerical examples are used to illustrate the validity of the methods. 2007 Elsevier Inc. All rights reserved.

Keywords: Static output feedback control; Takagi–Sugeno fuzzy model; Linear matrix inequality

1. Introduction

Over the past two decades, static output feedback problem has attracted considerable attentions of manyresearchers [2–4,11,12]. The problem can be stated as follows: given a system, find a static output feedback sothat the closed-loop system is stable. Normally, the existence of a full order output feedback control law isgiven in terms of the solvability of two convex problems. However, the synthesis of a static output feedbackgain or a fixed order controller is much more difficult. The main rationale is that the separation principle doesnot hold in such cases. A comprehensive survey on static output feedback can be found in [12]. The static out-put feedback problem is important in its own right, because static controllers are less expensive to be imple-mented and more reliable in practice. In [12] the authors show that any dynamic output feedback problem canbe transformed into a static output feedback problem. Hence, the static output feedback formulation is moregeneral than the full order dynamic output feedback formulation, that is, the static output formulation can beused to design a full order dynamic controller, but the converse is not true.

0020-0255/$ - see front matter 2007 Elsevier Inc. All rights reserved.

doi:10.1016/j.ins.2007.02.014

* Corresponding author.E-mail address: [email protected] (S.K. Nguang).


3006 D. Huang, S.K. Nguang / Information Sciences 177 (2007) 3005–3015


Recently, a great amount of effort has been devoted to describing a nonlinear system using a Takagi–Sugeno fuzzy model; see [1,5–10,13–17,20]. In this fuzzy model, local dynamics in different state space regionsare represented by local linear systems. The overall model of the nonlinear system is obtained by ‘‘blending’’ ofthese linear models through nonlinear fuzzy membership functions. Unlike conventional modelling techniqueswhich use a single model to describe the global behavior of a nonlinear system, fuzzy modelling is essentially amulti-model approach in which simple sub-models (typically linear models) are fuzzily combined to describedthe global behavior of a nonlinear system. The TS fuzzy model has been proved to be a very good represen-tation for a certain class of nonlinear dynamic systems. Based on this fuzzy model, a number of systematicmodel-based fuzzy control design methodologies have been developed.

The major drawback of the above mentioned papers [1,5–10,13–17,20] is that their design methodologies donot incorporate membership function characteristics, which may lead to conservative design methodologies.Motivated by this drawback and the simplicity of static output feedback controller, in this paper, input mem-bership function characteristics are incorporated into the static output feedback control design. To the best ofour knowledge, the problem of static output feedback for TS fuzzy models with membership functions’characteristics has not been studied in the literature. We show that the existence of a static output feedbackcontrol law can be expressed in terms of the solvability of bilinear matrix inequalities (BMIs). In order tocompute a solution to these BMIs, an iterative algorithm based on the linear matrix inequality has beendeveloped.

The rest of this paper is organized as follows. System description and notations are given in Section 2. Mainresults are presented in Section 3. The validity of our approach is demonstrated by examples in Section 4.Finally, conclusions are given in Section 5.

2. System description and notations

A fuzzy dynamic model has been proposed by Takagi and Sugeno [13] to represent local linear input/outputrelations of nonlinear systems. This fuzzy linear model is described by IF–THEN rules and has been shown tobe able to approximate a large class of nonlinear systems. As in [14,15,18], we examine a TS fuzzy model, inwhich the ith rule is formulated as follows:

Plant Rule i:IF z1ðtÞ is Mi1 and and zpðtÞ is Mip,THEN

_xðtÞ ¼ AixðtÞ þ BiuðtÞ;yðtÞ ¼ CixðtÞ;

ð2:1Þ

where i ¼ 1; . . . ; r, r is the number of fuzzy rules; zkðtÞ are premise variables, Mik are fuzzy sets, k ¼ 1; . . . ; p; pis the number of premise variables; xðtÞ 2 Rn; uðtÞ 2 Rm; yðtÞ 2 Rl denote state, control input, and output,respectively. Matrices Ai 2 Rnn; Bi 2 Rnm and Ci 2 Rln are of appropriate dimensions.

Remark 2.1. There are two major ways in TS fuzzy modelling. One is the TS fuzzy model identification[13,16,17] using input–output data, and the other is the TS fuzzy model construction, by the idea of sectornonlinearity [14,15,18].

By using center-average defuzzifer, product inference and singleton fuzzifer, the local models can be inte-grated into a global nonlinear model:

_xðtÞ ¼ AðhÞxðtÞ þ BðhÞuðtÞ;yðtÞ ¼ CðhÞxðtÞ;

ð2:2Þ

where

AðhÞ ¼Xr

i¼1

hiðzðtÞÞAi; BðhÞ ¼Xr

i¼1

hiðzðtÞÞBi; CðhÞ ¼Xr

i¼1

hiðzðtÞÞCi

D. Huang, S.K. Nguang / Information Sciences 177 (2007) 3005–3015 3007


and

zðtÞ ¼ ½z1ðtÞ; z2ðtÞ; . . . ; zpðtÞT;

xiðzðtÞÞ ¼Yp

k¼1

MikðzkðtÞÞ; xiðzðtÞÞP 0;Xr

i¼1

xiðzðtÞÞ > 0;

hiðzðtÞÞ ¼xiðzðtÞÞPri¼1xiðzðtÞÞ

; hiðzðtÞÞP 0;Xr

i¼1

hiðzðtÞÞ ¼ 1:

Here, MikðzkðtÞÞ denote the grade of membership of zkðtÞ in Mik.For the nonlinear plant represented by (2.2), the fuzzy output feedback controller is inferred as

uðtÞ ¼ KðhÞyðtÞ; ð2:3Þ
where KðhÞ ¼
Pri¼1hiðzðtÞÞKi. Ki in each plant rule is a local controller gain to be designed. Applying the fuzzy

controller (2.3) to the global fuzzy plant (2.2), we have the following closed-loop system:

_xðtÞ ¼ ðAðhÞ þ BðhÞKðhÞCðhÞÞxðtÞ: ð2:4Þ
For the sake of notational convenience, xðtÞ, uðtÞ, and hiðzðtÞÞ are denoted as x, u, and hi, respectively.
Note that in the symmetric block matrices, we use (*) as an ellipsis for terms that are induced by symmetry.Before ending this section, we recall the S-procedure which will be used in the proof.Denote the set G ¼ fgg and let FðgÞ;Y1ðgÞ; . . . ;YmðgÞ be some functional or functions. Further define

domain H:

H ¼ fg 2 G : Y1ðgÞ 6 0; . . . ;YmðgÞ 6 0g:
The S-procedure [19] states that FðgÞ < 0 8g 2H, if exist k1 P 0; . . . ; km P 0 such that
FðgÞ Xm

k¼1

kkYkðgÞ < 0:

3. Main results

In this section, we shall present our procedure for designing a static output feedback control gain for thesystem (2.4). A static output feedback design methodology without input membership function characteristicsis presented in Section 3.1. Section 3.2 provides a static output feedback design methodology with input mem-bership function characteristics.

3.1. Controller design without membership function characteristics

Before presenting the first result, the following Lemma is derived.

Lemma 3.1. The system (2.4) is asymptotically stable by means of a static output feedback if there exist

symmetric matrix P > 0 and matrices Kis satisfy the following conditions:

ATi P þ PAi PBiBT

i P þ Y ii ðÞT

BTi P þ KiCi I

" #< 0 ði ¼ 1; . . . ; rÞ; ð3:1Þ

12

ATi P þ PAi

PBiBTj P

þATj P þ PAj

PBjBTi P

0BBBBB@

1CCCCCAþ Y ij ðÞT

12ðBT

i P þ KiCj þ BTj P þ KjCiÞ I

2666666664

3777777775< 0 ði < jÞ; ð3:2Þ



Y 11 Y 12 Y 1r

Y 12 Y 22 Y 2r

..

. . .. ..

.

Y 1r Y 2r Y rr

0BBBB@1CCCCA > 0: ð3:3Þ

Proof. Let us choose Lyapunov function candidate as V ðxÞ ¼ xTPx. The time derivative of V ðxÞ along the sys-tem (2.4) is

_V ðxÞ ¼ _xTPxþ xTP _x

¼ xTfAðhÞ þ BðhÞKðhÞCðhÞgTPxþ xTPfAðhÞ þ BðhÞKðhÞCðhÞgx6 xTfðAðhÞ þ BðhÞKðhÞCðhÞÞTP þ P ðAðhÞ þ BðhÞKðhÞCðhÞÞ þ CðhÞTKðhÞTKðhÞCðhÞgx¼ xTfAðhÞTP þ PAðhÞ PBðhÞBðhÞTP þ ðBðhÞTP þ KðhÞCðhÞÞTðBðhÞTP þ KðhÞCðhÞÞgx: ð3:4Þ

From here, we can see that as long as

W ¼ AðhÞTP þ PAðhÞ PBðhÞBðhÞTP þ ðBðhÞTP þ KðhÞCðhÞÞTðBðhÞTP þ KðhÞCðhÞÞ < 0; ð3:5Þ
the fuzzy system (2.4) is stable.
Using the Schur complement, W is equivalent to the following quadratic matrix equality:

W ¼ AðhÞTP þ PAðhÞ PBðhÞBðhÞTP ðÞT

BðhÞTP þ KðhÞCðhÞ I

" #ð3:6Þ

¼Xr

i¼1

hi

Xr

i¼1

hj

ATi P þ PAi PBiBT

j P ðÞT

BTi P þ KiCj I

" #

¼Xr

i¼1

h2i

ATi P þ PAi PBiBT

i P ðÞT

BTi P þ KiCi I

" #

þ 2Xr

i<j

hihj

12ðAT

i P þ PAi PBiBTj P þ AT

j P þ PAj PBjBTi P Þ ðÞT

12ðBT

i P þ KiCj þ BTj P þ KjCiÞ I

" #

6 Xr

i¼1

h2i Y ii 2

Xr

i<j

hihjY ij

¼

h1

h2

..

.

hr

0BBBB@1CCCCA

T Y 11 Y 12 Y 1r

Y 12 Y 22 Y 2r

..

. . .. ..

.

Y 1r Y 2r Y rr

0BBBB@1CCCCA

h1

h2

..

.

hr

0BBBB@1CCCCA: ð3:7Þ

Thus, if (3.3) holds, (3.7) is less than zero, i.e., W < 0, and the fuzzy system (2.4) is stable. h

Note that the matrix inequalities (3.1) and (3.2) are bilinear matrix inequalities (BMI) and cannot be solvedwith ease by a convex optimization algorithm. An iterative algorithm (ILMI) based on the linear matrixinequality (LMI) has been employed to solve this BMI problem in [2,5].

To apply an ILMI method, we need to accommodate the negative quadratic term PBiBTi P by introducing

an additional design matrix variable X. Using the fact ðX P ÞTBðhÞBðhÞTðX P ÞP 0 for any X and P of thesame dimension, we obtain

X TBðhÞBðhÞTP þ PBðhÞBðhÞTX X TBðhÞBðhÞTX 6 PBðhÞBðhÞTP : ð3:8Þ

Plugging (3.8) into (3.6) and following the same procedure of the proof afterwards, we obtain the followingtheorem:



Theorem 3.1. The fuzzy system (2.4) is asymptotically stable by means of a static output feedback if there exist

symmetric matrix P > 0 and matrices Kis and a matrix X satisfy the following conditions:

Gii ðÞT

T ii I

" #< 0; ði ¼ 1; . . . ; rÞ; ð3:9Þ

Gij ðÞT

T ij I

" #< 0; ði < jÞ; ð3:10Þ

Y 11 Y 12 Y 1r

Y 12 Y 22 Y 2r

..

. . .. ..

.

Y 1r Y 2r Y rr

0BBBBBB@

1CCCCCCA > 0; ð3:11Þ

where

Gii ¼ ATi P þ PAi X TBiBT

i P PBiBTi X þ X TBiBT

i X þ Y ii;

Gij ¼1

2ðAT

i P þ PAi X TBiBTj P PBiBT

j X þ X TBiBTj X

þ ATj P þ PAj X TBjBT

i P PBjBTi X þ X TBjBT

i X Þ þ Y ij;

T ii ¼ BTi P þ KiCi;

T ij ¼1

2ðBT

i P þ KiCj þ BTj P þ KjCiÞ:

Remark 3.1. If the auxiliary variable X is fixed, the BMI (3.9) and (3.10) reduce to LMIs in P and Kis andcan be solved efficiently. However, in general, fixing the auxiliary variable X yields no solution to the LMIs.Thus, to relax those LMIs and make them feasible, a new term aP is introduced in (3.9) and (3.10) asfollows:

Gii aP ðÞT

T ii I

" #< 0 ði ¼ 1; . . . ; rÞ; ð3:12Þ

Gij aP ðÞT

T ij I

" #< 0 ði < jÞ: ð3:13Þ

The ILMI algorithm [2], as will be shown later, repeats the search for P and Kis and updates the auxiliaryvariable X by decreasing a until a becomes negative. If a < 0 is reached, then the BMIs (3.9) and (3.10) havea solution.

Iterative linear matrix inequality (ILMI) algorithm

Step 1: Solve the following LMIs that is jointly convex in ðQ; F Þ:
Q ¼ QT > 0; ð3:14Þ
QATi þ AiQ F TBiBT

i BiBTi F ðÞT

BTi F I

" #< 0 ði ¼ 1; . . . ; rÞ: ð3:15Þ

Set t ¼ 1 and X t ¼ FQ1 and choose a0 to be a big enough number.Step 2: Solve the following optimization problem in P and Kis using the auxiliary variable Xt determined inthe previous step to obtain at:



Minimize at

Subject to P > 0;

Gii atP ðÞT

T ii I

" #< 0 ði ¼ 1; . . . ; rÞ;

Gij atP ðÞT

T ij I

" #< 0 ði < jÞ;

eY > 0:

Step 3: If at < 0, P and Kis obtained in Step 2 are a feasible solution to the BMIs and stop.Step 4: Solve the following optimization problem in P and Kis using at and Xt determined in Step 2:

Minimize traceðP ÞSubject to P > 0;

Gii atP ðÞT

T ii I

" #< 0 ði ¼ 1; . . . ; rÞ;

Gij atP ðÞT

T ij I

" #< 0 ði < jÞ;

eY > 0:

Step 5: If kX t Pk < d, go to Step 6, where d is a predetermined small value. Else set t ¼ t þ 1 and X t ¼ P ,then go to Step 2.Step 6: The system may not have a feasible solution and stop.

Remark 3.2

1. Note that (3.9) and (3.11) imply

H ii ¼ ATi P þ PAi X TBiBT

i P PBiBTi X þ X TBiBT

i X < 0; ð3:16Þ
which provides an initial guess for X. By premultiplying and postmultiplying (3.16) with Q ¼ P1 and defin-ing F ¼ XQ, we can see that H ii < 0 is equivalent to LMI (3.15) by the Schur complement. What Step 1does is to seek a X corresponds to the necessary condition of (3.9) and uses it as an initial guess for theiterative algorithm.
2. aP is introduced in (3.9) and (3.10) to relax the LMIs. (3.12) and (3.13) correspond to the following Lyapu-nov inequality:

V ðxÞ ¼ xTPx; _V ðxÞ 6 aV ðxÞ:
If a is negative, the BMI (3.10) is feasible and the fuzzy system (2.4) is stable.
3. The optimization problem in Step 2 is a generalized eigenvalue minimization problem. This step guaranteesthe progressive reduction of at. Step 4 is to guarantee the convergence of the algorithm.

4. Sometimes, at+1 can be greater than at in Step 2 which may be due to the implementation problem of theLMI program. In this situation, we must set atþ1 ¼ at. Owing to the effect of some numerical errors in Step4, the optimization problem in Step 4 may be infeasible. In such case, let at ¼ at þ Dat, where D is a smallpositive real number, and go to Step 4 to solve the optimization problem again.

3.2. Controller design with membership function characteristics

In the previous result, the membership function characteristics have been neglected. This information iscrucial in many cases and may render less conservative stability conditions. Based on the so-called (outer)



ellipsoidal approximation algorithm, a new result is derived in this subsection by incorporating the member-ship function characteristics. Before proceeding with the development of the controller design technique, thefollowing definition is needed.

Definition 3.1. Define Rij as the region where the fuzzy rule i and fuzzy rule j are activated:

Rij fxjhiðxÞhjðxÞ > 0g:
This paper assumes that each region Rijði; j 62 In) can be outer approximated by a union of ellipsoids Eijk for
k ¼ 1; . . . ;m; where m is the number of ellipsoids. This assumption implies that there exist matrices Eijk; f ijk

such that

Rij [mk¼1

Eijk; where Eijk ¼ fxjkEijkxþ fijkk 6 1g: ð3:17Þ

Notice that each Eijk can also be represented as LMI forms:

Eijk ¼x

1

T ETijkEijk ðÞT

f TijkEijk ð1 f T

ijkfijkÞ

" #x

1

6 0: ð3:18Þ

Remark 3.3. Suppose that Rij ¼ fxjd1 6 cTx 6 d2g, then it is easy to see that we can take Eij1 ¼ 2cT=ðd2 d1Þand fij1 ¼ ðd2 þ d1Þ=ðd2 d1Þ.

Using this definition and Theorem 3.1, we have the following theorem.

Theorem 3.2. The fuzzy system (2.4) is asymptotically stable by means of a static output feedback if fori ¼ 1; . . . ; r, i < j 6 r, and k ¼ 1; . . . ;m, there exist symmetric matrix P > 0 and matrices Kis and scalars

kijk P 0, and matrix X satisfy the following conditions:

Gii ðÞT

T ii I

" #< 0;

Gij ðÞT

T ij I

" #< 0

9>>>>>=>>>>>;when 0 2 Eijk; ð3:19Þ

Gii kiikSiik ðÞT ðÞT

T ii I ðÞT

kiikY iik 0 kiikZiik

264375 < 0

Gij kijkSijk ðÞT ðÞT

T ij I ðÞT

kijkY ijk 0 kijkZijk

264375 < 0

9>>>>>>>>>=>>>>>>>>>;when 0 62 Eijk; ð3:20Þ

Y 11 Y 12 Y 1r

Y 12 Y 22 Y 2r

..

. . .. ..

.

Y 1r Y 2r Y rr

0BBBB@1CCCCA > 0; ð3:21Þ

where

Sijk ¼ ETijkEijk;

Y ijk ¼ f TijkEijk;

Zijk ¼ ð1 f TijkfijkÞ:



Proof. Following from the proof of Lemma 3.1, it can be shown that

_V ðxðtÞÞ 6Xr

i¼1

h2i xTðtÞðGii þ T T

ii T ii Y iiÞxðtÞ þ 2Xr

i<j

hihjxTðtÞðGij þ T TijT ij Y ijÞxðtÞ:

Notice that in Theorem 3.1, conditions (3.9) and (3.10), respectively, imply that

xTðtÞ½Gii þ T Tii T iixðtÞ < 0; xðtÞ 6¼ 0; xðtÞ 2 Rn

and

xTðtÞ½Gij þ T TijT ijxðtÞ < 0; xðtÞ 6¼ 0; xðtÞ 2 Rn:

These requirements may lead to conservatism. In actually fact, we only need

xTðtÞ½Gii þ T Tii T iixðtÞ < 0; xðtÞ 6¼ 0; xðtÞ 2 Rii

and

xTðtÞ½Gij þ T TijT ijxðtÞ < 0; xðtÞ 6¼ 0; xðtÞ 2 Rij:

One way to introduce this condition into the stability conditions is through the use of S-procedure. When0 2 Eijk, that is, Eijk contains the origin, the term ð1 f T

ijkfiikÞ becomes positive which implies that for the LMIsto hold, kijk must be zero; see (3.20). However, the LMIs are no longer strictly feasible. Hence, to avoid non-strictly feasible, when 0 2 Eijk, the membership function characteristics are not incorporated. Therefore, weobtained (3.19) which is the same as in Theorem 3.1. When 0 62 Eijk, that is, Eijk does not contain the origin,by the use of the S-procedure, we have

xTðtÞ½Gii þ T Tii T iixðtÞ kiikEiik < 0; k ¼ 1; . . . ;m ð3:22Þ

and

xTðtÞ½Gij þ T TijT ijxðtÞ kijkEijk < 0; k ¼ 1; . . . ;m: ð3:23Þ

By Schur complements, we have conditions given in (3.20). Notice that conditions given in (3.19) imply thatthe origin is globally stable. h

Remark 3.4. Clearly from (3.22) and (3.23), when kijk 7!0, the conditions given in Theorem 3.2 become thesame as in Theorem 3.1. Therefore, if there exists a solution to Theorem 3.1, there always exists a solutionto Theorem 3.2, but the conserve is not true. Hence, Theorem 3.2 is less conservative than Theorem 3.1.

The ILMI algorithm can be applied to Theorem 3.2.

4. Numerical examples

In this section, two design examples and their computer simulation are provided to illustrate the validity ofthe designs obtained in the previous section.

Example 1 (Lorenz chaotic system). To design a fuzzy static output feedback controller, the Lorenz chaoticsystem needs to be represented by a TS fuzzy model. An exact TS fuzzy modelling [15] is employed toconstruct a TS fuzzy model for the Lorenz chaotic system. The method utilises the concept of sectornonlinearity. For more details, see [18,14]. The following Lorenz chaotic system with the input term will beconsider in the sequel:

_x1ðtÞ ¼ ax1ðtÞ þ ax2ðtÞ þ uðtÞ;_x2ðtÞ ¼ cx1ðtÞ x2ðtÞ x1ðtÞx3ðtÞ;_x3ðtÞ ¼ x1ðtÞx2ðtÞ bx3ðtÞ;

ð4:1Þ

where a ¼ 10; b ¼ 8=3; c ¼ 28, x1ðtÞ; x2ðtÞ and x3 are the state variables, and uðtÞ is the control input.Assume that x1ðtÞ 2 ½NN , the nonlinear terms x1ðtÞx3ðtÞ and x1ðtÞx2ðtÞ can be expressed as



x1x3ðtÞ ¼ h1ðx1ðtÞÞ½Nx3ðtÞ h2ðx1ðtÞÞ½Nx3ðtÞ
and
x1x2ðtÞ ¼ h1ðx1ðtÞÞ½Nx2ðtÞ þ h2ðx1ðtÞÞ½Nx2ðtÞ;
where
h1ðx1ðtÞÞ ¼x1ðtÞ þ N

2Nand h2ðx1ðtÞÞ ¼

x1ðtÞ þ N2N

:

Then, using the above membership functions, we can have the following TS fuzzy model which exactly rep-resents (4.1) under the assumption on bounds of the state variable x1ðtÞ 2 ½NN .

_xðtÞ ¼X2

i¼1

hiðzðtÞÞAixðtÞ þX2

i¼1

hiðzðtÞÞBiuðtÞ;

yðtÞ ¼X2

i¼1

hiðzðtÞÞCixðtÞ;ð4:2Þ

where xðtÞ ¼ ½x1ðtÞx2ðtÞx3ðtÞT,

A1 ¼a a 0

c 1 N

0 N b

264375; A2 ¼

a a 0

c 1 N

0 N 8=3

264375; B1 ¼ B2 ¼

1

0

0

264375; C1 ¼ C2 ¼ ½ 1 0 0 :

The TS fuzzy model (4.2) exactly represents (4.1) under the assumption on bounds of the state variablex1ðtÞ 2 ½NN where N > 0. However, this assumption is not strict because of two reasons. It is well know thatstate variables of the chaotic system are bounded. In addition, N can be set to any value. Even if the nonlinearequations of the Lorenz chaotic system are unknown, recently developed fuzzy modelling techniques [13,16,17]using observed data can be utilised to obtain fuzzy models.

Through the simulations of (4.1), we learn that x1ðtÞ 2 ½30; 30, hence N is set be 30. Using Theorem 3.1,we obtain the static output feedback gains as K1 ¼ 1220, K2 ¼ 1278:3, and K3 ¼ 1751:7. Simulationresult is shown in Fig. 1 for the initial condition ½x1; x2; x3 ¼ ½10;10;10.

Example 2 (Nonlinear mass-spring-damper system). Consider a nonlinear mass-spring-damper mechanical sys-tem with a nonlinear spring:

0 2 4 6 8 10—15

—10

—5

0

5

10

Time (sec)

Stat

es

x1(t)x2(t)x3(t)

Fig. 1. Lorenz’s chaotic system.

—1 —0.5 0 0.5 1 1.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

X2(t)

Deg

ree

of m

embe

rshi

p

h1(x2(t))

h2(x2(t))

Membership function

0 10 20 30 40 50 60 70 80 90 100—0.5

—0.4

—0.3

—0.2

—0.1

0

0.1

0.2

0.3

0.4

Time (sec)

The

stat

es

x1(t)x2(t)

Simulation result

a b

Fig. 2. Nonlinear mass-spring-damper system: (a) membership function; (b) simulation result.



_x1ðtÞ ¼ 0:1125x1ðtÞ 0:02x2ðtÞ 0:67x32ðtÞ þ uðtÞ;

_x2ðtÞ ¼ x1ðtÞ;yðtÞ ¼ x2ðtÞ;

ð4:3Þ

where x2ðtÞ is the spring’s displacement and x1ðtÞ ¼ _x2ðtÞ. The term 0:67x32 is due to the nonlinearity of the

spring. The spring is attached to a wall, therefore the spring’s displacement x2ðtÞ is physically constrainedby the length of the spring and the wall. The length of the spring could be any value, in this paper, we assumex2ðtÞ 2 ½1; 1:5. The lower limit is due to the minimum length that the spring can be compressed. Same asExample 1, the concept of sector nonlinearity [15] is employed to construct an exact TS fuzzy model forthe mass-spring-damper system. Using the fact that x2ðtÞ 2 ½1; 1:5, this nonlinear term can be expressed as

0:67x32ðtÞ ¼ h1ðx2ðtÞÞ½0x2ðtÞ h2ðx2ðtÞÞ½1:5075x2ðtÞ;

where h1ðx2ðtÞÞ ¼ 1 x2ðtÞ2:25

and h2ðx2ðtÞÞ ¼ x2ðtÞ2:25

; see Fig. 2a.

Using h1ðx2ðtÞÞ and h2ðx2ðtÞÞ, we obtain the following TS fuzzy model which exactly represents (4.3) underthe assumption on bounds of the state variable x2ðtÞ 2 ½11:5:

_xðtÞ ¼X2

i¼1

hiðzðtÞÞAixðtÞ þX2

i¼1

hiðzðtÞÞBiuðtÞ;

yðtÞ ¼X2

i¼1

hiðzðtÞÞCixðtÞ;ð4:4Þ

where xðtÞ ¼ x1ðtÞx2ðtÞ½ T,

A1 ¼0:1125 0:02

1 0

; A2 ¼

0:1125 1:5275

1 0

; B1 ¼ B2 ¼

1

0

; C1 ¼ C2 ¼ ½ 0 1 :

From the membership functions given in Fig. 2a, we have R11 ¼S1

k¼1E11k, R12 ¼S1

k¼1E13k, and R22 ¼S1k¼1E33k, where E111 ¼ E121 ¼ E221 ¼ ðx2ðtÞ 1:5Þðx2ðtÞ þ 1Þ 6 0. In the form given in (3.18), we obtain

T 111 ¼ T 121 ¼ T 221 ¼ ½00:8; f 111 ¼ f121 ¼ f221 ¼ 0:2:

Applying Theorem 3.2, a solution to the BMIs given in Theorem 3.2 is found with a ¼ 0:0015 to be

P ¼3:3633 0:36961

0:36961 2:6421
and the static output feedback gains
K1 ¼ 0:75351; K2 ¼ 0:75348:



Fig. 2b depicts the simulation result with the initial condition x0 ¼ ½0:1;0:5. One can see that the closed sys-tem is asymptotically stable.

5. Conclusion

Static output feedback control designs for a TS fuzzy system have been provided in this paper. The exis-tence of a static output feedback control law has been expressed in terms of the solvability of bilinear matrixinequalities. To compute a solution to the BMIs, an iterative algorithm based on the linear matrix inequalityhas been proposed. To reduce the conservatism of the design, the structural information of the rule base hasbeen incorporated. Numerical examples have been provided to illustrate the validity of our design.

References

[1] W. Assawinchaichote, S.K. Nguang, P. Shi, Robust H1 fuzzy filter design for uncertain nonlinear singularly perturbed systems withMarkovian jumps: an LMI approach, Information Sciences 177 (7) (2007) 1699–1714.

[2] Y.Y. Cao, J. Lam, Y.X. Sun, Static output feedback stabiliztion: an ILMI approach, Automatica 34 (12) (1998) 1641–1645.[3] W. Chang, J.B. Park, Y.H. Joo, G.R. Chen, Static output-feedback fuzzy controller for Chen’s chaotic system with uncertainties,

Information Sciences 151 (2003) 227–244.[4] I.N. Kar, Design of static output feedback controller for uncertain systems, Automatica 35 (1) (1999) 169–175.[5] E. Kim, H. Lee, New approaches to relaxed quadratic stability condition of fuzzy control systems, IEEE Transactions on Fuzzy

Systems 8 (5) (2000) 523–534.[6] J. Li, H.O. Wang, D. Niemann, K. Tanaka, Dynamic parallel distributed compensation for Takagi–Sugeno fuzzy systems: An LMI

approach, Information Sciences 123 (3–4) (2000) 201–221.[7] S.K. Nguang, W. Assawinchaichote, H1 filtering for fuzzy dynamical systems with pole placement constraints, IEEE Transactions

on Circuits and Systems – I 50 (11) (2003) 1503–1508.[8] S.K. Nguang, P. Shi, Robust H1 output feedback control design for fuzzy dynamic systems with quadratic D stability constraints: an

LMI approach, Information Sciences 176 (15) (2006) 2161–2191.[9] S.K. Nguang, P. Shi, H1 fuzzy output feedback control design for nonlinear systems: an LMI approach, in: Proc. 40th IEEE Conf.

on Decision and Control, Orlando, USA, 2001, pp. 2501–2506.[10] S.K. Nguang, P. Shi, Stabilisation of a class of nonlinear time-delay systems using fuzzy models, in: Proc. 39th IEEE Conf. on

Decision and Control, Sydney, Australia, 2000, pp. 4415–4419.[11] E. Prempain, I. Postlethwaite, Static output feedback stabilisation with H1 performance for a class of plants, Systems and Control

Letter 43 (6) (2001) 159–166.[12] V.L. Syrmos, C.T. Abdallah, P. Dorato, K. Grigoriadis, Static output feedback: a survey, Automatica 33 (2) (1997) 125–137.[13] T. Takagi, M. Sugeno, Fuzzy identification of systems and its applications to medeling and control, IEEE Transactions of System,

Man, and Cybernetics 15 (1) (1985) 116–132.[14] K. Tanaka, T. Ikeda, H.O. Wang, Fuzzy regulators and fuzzy observers: Relaxed stability conditions and LMI-based design, IEEE

Transactions on Fuzzy Systems 6 (2) (1998) 250–265.[15] T. Taniguchi, K. Tanaka, H. Ohtake, H.O. Wang, Model construction, rule reduction, and robust compensation for generalized form

of Takagi–Sugeno fuzzy ystems, IEEE Transactions on Fuzzy Systems 9 (4) (2001) 525–538.[16] L.X. Wang, A Course in Fuzzy Systems and Control, Prentice-Hall, Englewood Cliffs, NJ, 1997.[17] L. Wang, R. Langari, Building sugeno-type models using fuzzy discretization and orthogonal parameter estimation techniques, IEEE

Transactions on Fuzzy Systems 3 (4) (1995) 454–458.[18] H.O. Wang, K. Tanaka, M.F. Griffin, An approach to fuzzy control of nonlinear systems: stability and design issues, IEEE

Transactions on Fuzzy Systems 4 (1) (1996) 14–23.[19] V.A. Yakubovich, S-procedure in nonlinear control theory, Vestnik Leningradskogo Universiteta, Ser. Matematika (1971) 62–77.[20] S.S. Zhou, G. Feng, J. Lam, S.Y. Xu, Robust H1 control for discrete-time fuzzy systems via basis-dependent Lyapunov functions,

Information Sciences 174 (3–4) (2005) 197–217.


AN LMI-BASED ALGORITHM FOR DESIGNING SUBOPTIMALSTATIC H2/H∞ OUTPUT FEEDBACK CONTROLLERS∗

F. LEIBFRITZ†

SIAM J. CONTROL OPTIM. c© 2001 Society for Industrial and Applied MathematicsVol. 39, No. 6, pp. 1711–1735

Abstract. We consider the problem of designing a suboptimal H2/H∞ feedback control law fora linear time-invariant control system when a complete set of state variables is not available. Thisproblem can be necessarily restated as a nonconvex optimization problem with a bilinear, multiob-jective functional under suitably chosen linear matrix inequality (LMI) constraints. To solve such aproblem, we propose an LMI-based procedure which is a sequential linearization programming ap-proach. The properties and the convergence of the algorithm are discussed in detail. Finally, severalnumerical examples for static H2/H∞ output feedback problems demonstrate the applicability ofthe considered algorithm and also verify the theoretical results numerically.

Key words. static controllers, output feedback, suboptimal control, robust control, linearsystems, linear matrix inequalities, nonconvex programming

AMS subject classifications. 90C22, 90C26, 93A99, 93B36, 93B51, 93B52, 93C05, 93D09,49N99, 65K05

PII. S0363012999349553

1. Introduction. The static or reduced fixed order dynamic output feedbackcontrol problem that meets desired performance and/or robustness specifications isan active research area of the control community. In this paper we consider the com-putational design of static H2/H∞ output feedback controllers. This is an importantexample of a nonconvex control problem. It consists of determining a static outputfeedback gain which achieves a certain nominal (suboptimal) performance measuresubject to a robustness constraint. The static output feedback problems are impor-tant, since it is not always possible to have full access to the state vector and acontroller must be used which is based only on the available observations. Moreover,they are important because other problems are reducible to some variations of thestatic output feedback problem and relevant when a simple controller must be useddue to cost and reliability.

During the past decade, control problems with combined H2 and H∞ designcriteria have gained a great deal of attention. Concerning continuous-time systems,[7] provides the solution of standardH2 andH∞ control problems in terms of algebraicRiccati equations, where both state feedback and full order compensator-based outputfeedback are considered. The design of feedback controllers that satisfy both H∞and H2 specifications is interesting because it offers robust stability and nominalperformance. In 1989, Bernstein and Haddad [2] introduced a mixedH2/H∞ problem.Their approach is to minimize an auxiliary cost subject to an H∞ norm constraint,and this cost yields an upper bound on the H2 norm. The work of [2] is extendedin [46] and [6], where another mixed H2/H∞ problem is addressed. The systemconsidered therein is dual to the Bernstein–Haddad setup (see [45]). Other relatedworks on the design of H2/H∞ controllers by state or full order output feedback canbe found, for example, in [23], [33], [35], [38], and [40]. Only [44] considers a mixedH2/H∞ problem for the static output feedback case. The solvability conditions and

∗Received by the editors January 7, 1999; accepted for publication (in revised form) September5, 2000; published electronically February 28, 2001.

http://www.siam.org/journals/sicon/39-6/34955.html†FB-IV Mathematics, University of Trier, D-54286 Trier, Germany ([email protected]).

1711


1712 F. LEIBFRITZ

the algorithms discussed in the above literature are based on coupled Riccati and/orLyapunov equations. Recently, linear matrix inequalities (LMIs) have attained muchattention in control engineering [3], [5], [36], since many control problems can beformulated in terms of LMIs and thus solved via convex programming approaches.For example, this includes H∞ [4], [11], [12], [25], H2 [18], [37], and mixed H2/H∞[19], [20], [27], [29], [39]. However, the resulting controllers are state feedback or oforder nx equal to the plant. The difficulties arise if we want to design a static (orreduced fixed order) output feedback controller. Then the problem of determining astatic output feedback controller including H2 and H∞ can be restated as a linearalgebra problem, which involves two coupled LMIs. In this case, the solution of oneshould be the inverse of the other. The problem is then no longer convex [12], [25],[31], [30], [42], [43], and finding a solution numerically to these nonconvex problemsis a difficult task.

In this paper we will develop an LMI-based computational procedure for solving amixed H2/H∞ problem by a static output feedback controller, which is an extensionof the algorithm proposed by Leibfritz [31] for the design of stabilizing static H2 andH∞ output feedback gains. The suboptimal static H2/H∞ output feedback problemconsidered in this paper can be necessarily rewritten to a nonconvex, multiobjectiveprogramming problem. In particular, this problem consists of minimizing a (noncon-vex) functional of the form J (P,Q, Y ) = Tr(PQ) + Tr(Y ) subject to suitably chosenLMI constraints. Then, using the solution of this problem (if any exists), the existenceof a static H2/H∞ output feedback gain F can be tested by solving a suitably chosenLMI feasibility problem in F . Therefore, the problem of finding a suboptimal staticH2/H∞ output feedback gain reduces to an optimization problem with a nonconvexobjective over a convex set and an LMI feasibility problem.

Similar to Leibfritz [31], we will derive the so-called sequential linear program-ming matrix method (SLPMM) for solving the resulting nonconvex programmingproblem. This approach is motivated by successive minimization of a linearizationof J (P,Q, Y ) subject to LMI constraints as proposed by [1] for general nonconvexbilinear programming problems. Note that the SLPMM algorithm is closely related tothe cone complementarity linearization method developed by [9] for solving the staticoutput feedback stabilization problem. As shown in [31], the theoretical advantagesof the SLPMM algorithm over the cone complementarity algorithm are the following.First, the SLPMM algorithm always generates a strictly decreasing sequence of theobjective function values which is bounded below by an integer nx, and thus it isconvergent. Second, the sequence of iterates generated by the SLPMM is containedin a compact level set, and therefore it is always bounded. Finally, if a correspondingbilinear matrix problem is nonempty, then every accumulation point of the generatedsequence solves this bilinear matrix problem. In this case, it is always possible toreconstruct a static output feedback gain from this solution. On the other hand, ifthere exists no matrix pair satisfying the bilinear matrix problem, then the SLPMMalgorithm always terminates with an objective function value of a bilinear matrixinequality minimization problem which is greater than nx. This indicates for theconsidered plant that there exists no static output feedback controller. But in thiscase, one can construct a reduced fixed order dynamic controller from the computedsolution of the bilinear matrix inequality minimization problem. For more details, werefer to Leibfritz [31]. In contrast to these strong theoretical convergence results, theauthors in [9] can guarantee only that the sequence of a linear approximation of theobjective function values is a monotonically (nonstrict) decreasing sequence, which


SUBOPTIMAL STATIC H2/H∞ OUTPUT FEEDBACK DESIGN 1713

is bounded below by 2nx. Moreover, in the last few years a number of numericalprocedures have been proposed for solving static H2 output feedback problems. Forexample, the LMI-based methods also include the Min-Max algorithm of Geromel,de Souza, and Skelton [16], [17], the XY-centering algorithm of Iwasaki and Skelton[26], and the alternating projection method of Grigoriadis and Skelton [21], but theconvergence of these algorithms is not always guaranteed. Particularly, the Min-Maxalgorithm guarantees only sequences of upper and lower bounds to the maximal andminimal eigenvalues of PQ, which are strictly decreasing, and increasing sequencesunder strong technical assumptions. In addition, for ensuring the global convergenceof the Min-Max algorithm, they must assume that the generated sequences are con-tained in a compact set. On the other hand, it may occur that the Min-Max algorithmgenerates an unbounded sequence even if the bilinear matrix problem is nonempty.In this case the Min-Max method breaks down [9], [17]. Since the XY-centering al-gorithm is closely related to the Min-Max procedure, the global convergence of thisapproach can be shown only under similar technical assumptions that are as strong asthe ones imposed for the Min-Max algorithm [26, Theorem 2]. Finally, the alternatingprojection method is guaranteed to converge only locally. These observations motivateus to derive the SLPMM algorithm for more complicated problems such as the staticH2/H∞ problem. For this problem class, we will show that the SLPMM procedurebehaves theoretically as well as numerically in a similar way to that described above.

The paper is organized as follows. Section 2 defines the considered system real-ization and describes the static H2/H∞ output feedback problem considered in thispaper. Section 3 contains the necessary and sufficient conditions for the existenceof stabilizing static H∞ output feedback controllers. Moreover, the formulation ofthe LMI-based nonconvex optimization problem, which must be necessarily solvableif the static H2/H∞ output feedback problem has a solution, can be found therein.Section 3.1 presents the main part of this paper. Therein we motivate the nonconvexmultiobjective programming problem, and, similarly as in [31], we derive the SLPMMalgorithm for finding a numerical solution of this problem class. Thereafter, we dis-cuss the properties and global convergence of this procedure. Finally, in section 4,we present several examples for the design of suboptimal static H2/H∞ output feed-back control laws, which will demonstrate the applicability of the SLPMM algorithmapplied to this problem class. We also verify numerically the theoretical results anddemonstrate the design of reduced fixed order dynamic controllers if the algorithmterminates with an optimal value greater than nx + Tr(Y ∗).

We will use the following notation. Ir denotes the (r × r) identity matrix. Theset of real symmetric (n × n) matrices is denoted by Sn, and S+

n describes the coneof symmetric positive definite (n × n) matrices. For A ∈ Sn, A 0 (A 0) meansthat A is positive definite (semidefinite). Similarly, A ≺ 0 (A 0) denotes thatA is negative definite (semidefinite). For A,B ∈ Sn, A B (A B) denotes theusual Loewner ordering [24]. The symbol Tr(A) =

∑ni=1 aii is the trace operator of a

matrix A ∈ Rn×n. ||A||F is the Frobenius norm of a matrix. Finally, ||Tzw||∞ denotes

the H∞ norm of a proper and real rational stable transfer matrix, i.e., Tzw ∈ RH∞[10], and ||Tzw||H2

is the usual H2 norm of a strictly proper and real rational stabletransfer matrix, i.e., Tzw ∈ RH2 [10].

2. The static H2/H∞ output feedback problem. In this section, we focuson the staticH2/H∞ output feedback problem as formulated by Doyle et al. [6] for thefull order dynamic output feedback case. However, we take a “suboptimal” approachfor designing a static output feedback controller which is similar to [29]. The problem


1714 F. LEIBFRITZ

solved by Doyle et al. has been shown to be a dual problem of Bernstein and Haddad [2]in some sense; see, for example, Yeh, Banda, and Chang [45]. Khargonekar and Rotea[29] have obtained a nice solution to the dual problem for a class of suboptimal fullorder output feedback compensators. For example, a convex optimization approachis proposed to solve the full order output feedback dual problem. However, the staticoutput feedback case considered in this paper is much more difficult, since the problemis then no longer convex.

Consider a finite dimensional linear time–invariant plant ΣP with the state spacerealization

ΣP

x(t) = Ax(t) +B0w0(t) +B1w1(t) +B2u(t), x(0) = 0,z(t) = C1x(t) +D10w0(t) +D11w1(t) +D12u(t),y(t) = C2x(t) +D20w0(t) +D21w1(t),

where x(t) ∈ Rnx is the state, wi(t) ∈ R

nw , i = 0, 1, are the disturbance inputs,u(t) ∈ R

nu is the control input, z(t) ∈ Rnz is the regulated output, and y(t) ∈ R

ny isthe measured output of the control system. The static output feedback controller ΣC

with the state space realization

ΣC

u(t) = Fy(t)

is to be designed, where F ∈ Rnu×ny denotes the unknown static output feedback

gain. Substituting ΣC into the plant, ΣP yields the corresponding closed loop systemgiven by

Σcl

x(t) = AFx(t) +B0Fw0(t) +BFw1(t), x(0) = 0,z(t) = CFx(t) +D0Fw0(t) +DFw1(t),

where the closed loop matrices AF , BF , CF , DF , B0F , and D0F are defined as follows:

AF = A+B2FC2, BF = B1 +B2FD21, CF = C1 +D12FC2, DF = D11 +D12FD21,B0F = B0 +B2FD20, D0F = D10 +D12FD20.

Throughout the whole paper, the following assumptions are imposed on the system.Assumption 2.1.(1) The pair (A,B2) is stabilizable, and the pair (A,C2) is detectable.(2) The data matrices in ΣP , especially the following, are real constant matrices:

A ∈ Rnx×nx , B1 ∈ R

nx×nw , B2 ∈ Rnx×nu , C1 ∈ R

nz×nx , C2 ∈ Rny×nx ,

D11 ∈ Rnz×nw , D12 ∈ R

nz×nu , D21 ∈ Rny×nw , B0 ∈ R

nx×nw ,D10 ∈ R

nz×nw , D20 ∈ Rny×nw .

(3) F ∈ Rnu×ny , nu < nx, ny < nx, and rank(B2) = nu, rank(C2) = ny.

We start by considering the analysis problem. Assume the static output feedbacklaw ΣC is fixed such that the closed loop system Σcl is internally stable. For example,there exists a static output feedback gain in the set

Fs = F ∈ Rnu×ny | AF is Hurwitz,(2.1)

the so-called stability set. Let Tzw1(s) = CF (sI−AF )−1BF +DF (Tzw0(s) = CF (sI−

AF )−1B0F +D0F ), s ∈ C, denote the closed loop transfer matrix from w1 to z (fromw0 to z). The concepts of H∞ norm and H2 norm/cost are well known (cf. [7]).Therefore, we will omit detailed discussion and content ourselves with starting the



following definitions for reference. Since all coefficient matrices are assumed to be realand F ∈ Fs is fixed, the transfer matrix Tzw1 ∈ RH∞ and the H∞ norm of Tzw1 isdefined by

||Tzw1 ||∞ = supω∈R

σmax(Tzw1(iω)),(2.2)

where σmax(Tzw1(·)) denotes the largest singular value of Tzw1and i denotes the

imaginary unit. Recall that Tzw0 ∈ RH2 and the H2 norm of Tzw0 is finite, forexample, ||Tzw0 ||H2 < ∞, if and only if D0F ≡ 0. In this case, if Lo denotes theobservability Gramian of the pair (AF , CF ), the H2 norm of Tzw0

can be computedby

||Tzw0||2H2

= Tr(BT0FLoB0F );(2.3)

for example, Lo satisfies Lyap(Lo, F ) = ATFLo + LoAF + CT

FCF = 0.Now let the scalar γ > 0 be given and assume that ||Tzw1 ||∞ < γ. Define RF =

Inw−γ−2DTFDF , and then it is a standard fact (see, for example, [30], [47]), that there

exist a unique real symmetric matrix X and a gain F ∈ Rnu×ny such that RF 0,

Ric(X,F ) = ATFX +XAF + γ−1CT

FCF(2.4)

+ γ−1(XBF + γ−1CTFDF )R−1

F (BTFX + γ−1DT

FCF ) = 0,

and AF + γ−1BFR−1F (BT

FX + γ−1DTFCF ) is Hurwitz. Moreover, X satisfies (cf. [2])

0 Lo X P,

where P fulfills Ric(P, F ) 0. Thus, if the H2 norm of Tzw0is finite, then we have

||Tzw0 ||2H2= Tr(BT

0FLoB0F ) ≤ Tr(BT0FXB0F ) ≤ Tr(BT

0FPB0F ).(2.5)

Similarly as in [2], [46], [29], or [35], these inequalities motivate us to define thefollowing auxiliary H2/H∞ cost function for the linear time–invariant closed loopsystem Σcl:

Cγ(P, F ) = Tr(BT0FPB0F ),(2.6)

which is an upper bound on ||Tzw0||2H2

if and only if D0F ≡ 0. Moreover, Cγ(P, F ) ≤Tr(Y ) whenever the symmetric matrices P 0 and Y 0, Y ∈ R

nw×nw , satisfyRic(P, F ) ≺ 0 and

ψ(F, P, Y ) :=

[Y BT

0FPPB0F P

] 0.(2.7)

Note that (2.7) is equivalent to Y BT0FPB0F 0 and can be stated also as follows:[

Y BT0 P

PB0 P

]+

[0PB2

]F

[D20 0

]+

[DT

20

0

]FT

[0 BT

2 P] 0.(2.8)

Thus, for given P ∈ S+nx

, the matrix inequality (2.8) is an LMI in Y and F .By this discussion, it is immediate that ||Tzw1 ||∞ < γ and ||Tzw0 ||2H2

< Cγ(P, F ) ifand only ifD0F ≡ 0. Hence, the Riccati inequality Ric(P, F ) ≺ 0 leads to anH∞ normbound γ and an H2 cost upper bound Cγ(P, F ). If D0F = 0, then ||Tzw0 ||2H2

= ∞. In


1716 F. LEIBFRITZ

this case, we do not interpret the auxiliary cost function Cγ(P, F ) as an upper boundon the H2 cost, but we can interpret it as a robust performance measure similarto the results of [46, Theorem 4] and [6, Theorem 1]. The results concerning theoptimization of the auxiliary static H2/H∞ performance measure over the set of allstabilizing static controller gains F satisfying ||Tzw1 ||∞ < γ can be obtained from thefollowing characterization. The proof of this result is very similar to [29, Lemma 2.1]and is thus omitted (see also [6, Theorems 1, 2]).

Lemma 2.2. Consider the stable closed loop system Σcl and let Tzw1(Tzw0

) denotethe closed loop transfer matrix from w1 to z (from w0 to z). Let γ > 0 be given andsuppose that Tzw0 is strictly proper, i.e., D0F ≡ 0. Let Ric(·) be defined by (2.4).Then there exists a gain F ∈ R

nu×ny satisfying ||Tzw1 ||∞ < γ if and only if thereexists a pair (P, F ), F ∈ R

nu×ny , P ∈ S+nx, such that RF 0 and Ric(P, F ) ≺ 0. In

this case,

Cγ(P, F ) = infTr(BT0FPB0F ) | (P, F ) satisfy RF 0,Ric(P, F ) ≺ 0, P 0.(2.9)

With the result of Lemma 2.2, our goal is to minimize the auxiliary static H2/H∞performance measure over all stabilizing static output feedback gains F that enforcethe H∞ constraint. From the previous discussion, this is equivalent to minimizingTr(Y ) over all matrices F , P , and Y satisfying Ric(P, F ) ≺ 0, RF 0, P 0, Y 0,and ψ(F, P, Y ) 0. Thus, we state the suboptimal static H2/H∞ output feedbackproblem considered in this paper as the following nonconvex optimization problem:

min Tr(Y ), subject to (s.t.) P 0, RF 0, Ric(P, F ) ≺ 0, Y 0,ψ(F, P, Y ) 0.

(2.10)

In the following sections, we will discuss a procedure for finding solutions to thisproblem. Note, a solution (P ∗, F ∗, Y ∗) of (2.10), if any exists, is suboptimal in thesense that

Coptγ (X,F ) = minTr(BT

0FXB0F ) | X 0, RF 0,Ric(X,F ) = 0≤ Tr(BT

0F∗P ∗B0F∗) ≤ Tr(Y ∗),(2.11)

where Coptγ (X,F ) denotes the minimal value of the “optimal” static H2/H∞ output

feedback problem.Finally, note that the results in the following sections can be easily extended to

the following static H2/H∞ output feedback problem formulation:

min Tr(Y ), s.t. P 0, RF 0,Ric(P, F ) ≺ 0, Y 0, ψ(F, P, Y ) 0,Lyap(P, F ) ≺ 0, D0F = 0.

(2.12)

Here, the goal is to minimize an upper estimate of the optimal static H2 performancesubject to the H∞ constraint.

3. Suboptimal static H2/H∞ output feedback design; LMI approach.This paragraph is devoted to the computational design of a suboptimal static H2/H∞output feedback controller. For deriving the LMI-based formulation of the staticH2/H∞ output feedback problem, we need the following result which can be found,for example, in [25].

Lemma 3.1. Let B ∈ Rn×m, rank(B) = m < n, C ∈ R

r×n, rank(C) = r < n, andΩ ∈ R

n×n be given. Then there exists F ∈ Rm×r satisfying BFC + (BFC)T + Ω ≺ 0

if and only if N(BT )TΩN(BT ) ≺ 0 and N(C)TΩN(C) ≺ 0 hold, where N(BT ) ∈



Rn×(n−m), N(C) ∈ R

n×(n−r), denoting any matrices whose columns form orthonormalbases of the null spaces of BT , C, respectively.

By involving Lemma 3.1 and the strict bounded real lemma [47], the suboptimalstatic H∞ output feedback problem, i.e., find a static output feedback gain F , if anyexists, such that AF is Hurwitz and ||Tzw1 ||∞ < γ, can be transformed to a problemof solving two LMIs coupled through a bilinear matrix equation. Similar results canbe found, for example, in [12], [25], and [30].

Theorem 3.2 (existence of static H∞ controllers). Let Fs = ∅, γ > 0 be givenand consider the closed loop system Σcl with w0 = 0. Then the following are equiva-lent.

(i) There exists a static output feedback gain F ∈ Rnu×ny such that AF is a

Hurwitz matrix and ||Tzw1||∞ < γ.

(ii) There exists a pair (F, P ), F ∈ Rnu×ny , P ∈ S+

nx, satisfying

BFC + (BFC)T + Ω ≺ 0,(3.1)

Ω =

ATP + PA PB1 CT

1

BT1 P −γInw

DT11

C1 D11 −γInz

,B =

PB2

0D12

, C = [C2 D21 0] .

(iii) There exist matrices P ∈ S+nxand Q ∈ S+

nxsatisfying PQ = I and

NTQ

[AQ+QAT + γ−1B1B

T1 (C1Q+ γ−1D11B

T1 )T

(C1Q+ γ−1D11BT1 ) γ−1D11D

T11 − γInz

]NQ ≺ 0,(3.2)

NTP

[ATP + PA+ γ−1CT

1 C1 PB1 + γ−1CT1 D11

(PB1 + γ−1CT1 D11)

T γ−1DT11D11 − γInw

]NP ≺ 0,(3.3)

where NQ := N([BT2 DT

12]) and NP := N([C2 D21]), denoting any matrices whosecolumns form orthonormal bases of the null spaces of [BT

2 DT12] and [C2 D21],

respectively.Proof. Combining the strict bounded real lemma [47], Theorem 3.1, and using a

Schur complement argument yields the desired result with Q = P−1.The inequalities (3.2) and (3.3) are LMIs in Q and P , respectively, and are there-

fore convex. But finding P 0 and Q 0 satisfying PQ = I, (3.2), and (3.3) togetheris a difficult task since

ΦH∞(P,Q, γ) := (P,Q) ∈ S+nx| (P,Q) satisfying PQ = I, (3.2), (3.3)(3.4)

is a nonconvex set, i.e., elements in ΦH∞(P,Q, γ) must be inverse to each other. Anumerical algorithm for determining a pair (P,Q) ∈ ΦH∞(P,Q, γ) can be found, forexample, in Leibfritz [31].

Similarly as in Theorem 3.2, we can eliminate the matrix variable F from (2.8)by using [25, Theorem 1], which is a generalization of Lemma 3.1. In particular, wehave the following lemma.

Lemma 3.3. Let Y ∈ Rnw×nw , Y 0, P ∈ S+

nx, and F ∈ R

nu×ny . Moreover,let B0 ∈ R

nx×nw , B2 ∈ Rnx×nu , and D20 ∈ R

ny×nw be given. Suppose [0 PB2]T ∈

R(nw+nx)×nu , rank([0 PB2]

T ) ≤ nu, and [D20 0] ∈ Rny×(nw+nx), rank([D20 0]) ≤

ny. Then the following statements are equivalent.(i) There exists a triple (F, P, Y ) satisfying (2.8).


1718 F. LEIBFRITZ

(ii) There exist matrices Y ∈ Rnw×nw , Y 0, P ∈ S+

nx, and Q ∈ S+

nxsatisfying

PQ = I and

Q(Q,Y ) = N([0 BT2 ])T

[Y BT

0

B0 Q

]N([0 BT

2 ]) 0,(3.5)

P(P, Y ) = N([D20 0])T[Y BT

0 PPB0 P

]N([D20 0]) 0,(3.6)

where N([0 BT2 ]) and N([D20 0]) denote any matrices whose columns form orthonor-

mal bases of the null spaces of [0 BT2 ] and [D20 0], respectively.

Proof. Observing that

N

([0PB2

]T)

=

[Inw

00 P−1

]N

([0B2

]T)

and using [25, Theorem 1], we obtain the desired result with Q = P−1.The set

Φ(P,Q, Y ) := (P,Q) ∈ S+nx, Y ∈ Snw | (P,Q, Y )

satisfying Y 0, PQ = I, (3.5), (3.6)(3.7)

is not convex due to the coupling condition PQ = I.Obviously, a suboptimal static H2/H∞ controller in the sense of the previous

section exists if and only if condition (ii) of Theorem 3.2 and (2.8) hold for the samegain matrix F. Assuming that there exist matrices F, P ∈ S+

nx, Y ∈ Snw satisfying

(3.1) and (2.8), there exist matrices (P,Q) ∈ S+nx

and Y ∈ Snwsatisfying condition

(iii) of Theorem 3.2 and condition (ii) of Lemma 3.3. Thus, the static H2/H∞ outputfeedback problem can be necessarily transformed to the following bilinear optimizationproblem:

min Tr(Y ), s.t. (P,Q, Y ) ∈ ΦH∞(P,Q, γ) ∩ Φ(P,Q, Y ).(3.8)

Now suppose that there exists a solution triple (P ∗, Q∗, Y ∗) of (3.8) (not necessarilyunique); then there exists a suboptimal static H2/H∞ output feedback controller ifand only if there exists an F satisfying (3.1) and (2.8) for P = P ∗ and Y = Y ∗.Note, for given (P ∗, Q∗, Y ∗), this is an LMI feasibility problem in F . Therefore, thesuboptimal static H2/H∞ problem is solvable if and only if the bilinear optimizationproblem (3.8) has a solution and the corresponding LMI feasibility problem in F isnonempty. Hence, these observations lead to the following necessary and sufficientconditions for the existence of static H2/H∞ output feedback controllers.

Theorem 3.4. Let Fs = ∅, γ > 0 be given, and consider the closed loop systemΣcl. Then the following statements are equivalent.

(i) There exists a triple (F, P, Y ), F ∈ Rnu×ny , P ∈ S+

nx, Y ∈ Snw

, and Y 0satisfying (3.1) and (2.8).

(ii) There exist matrices P ∈ S+nx, Q ∈ S+

nx, Y ∈ Snw , and Y 0 satisfying

PQ = I, (3.2), (3.3), (3.5), (3.6), and, for all such fixed (P, Y ), there is amatrix F ∈ R

nu×ny satisfying (3.1) and (2.8).Due to this result, the suboptimal static H2/H∞ problem can be solved by first

finding a solution of (3.8) which is also a necessary condition for the solvability of(2.10). Secondly, if (3.8) has a solution, the static H2/H∞ output feedback gain, ifany exists, can be obtained by solving the corresponding LMI feasibility problem in F .Finally, all gains F which can be reconstructed in this way will be called suboptimaldue to the relation Tr(Y ∗) ≥ Cγ(P ∗, F ) ≥ Copt

γ (X,F ).



3.1. Bilinear LMI-based algorithm. In this subsection we describe a bilinear,multiobjective LMI-based algorithm for finding a triple (P,Q, Y ) satisfying approxi-matively (3.8). As defined by (3.4) and (3.7), the sets ΦH∞(P,Q, γ) and Φ(P,Q, Y )are not convex and not closed. To make these sets convex, the coupling constraintPQ = I can be weakened to the following well-known semidefinite programming(SDP)-relaxation:

P Q−1 0 ⇐⇒ M(P,Q) :=

[P II Q

] 0.(3.9)

Replacing PQ = I in the nonconvex sets ΦH∞(P,Q, γ) and Φ(P,Q, Y ) by M(P,Q) 0 yields a convex approximation of these sets. Moreover, for computational purposes,we prefer to have closed sets. Introducing a positive scalar β > 0 and replacingthe closed loop matrix AF by A + βI + B2FC2 in Theorem 3.2, we can rewrite theexistence conditions of Theorem 3.2 to the following bilinear matrix feasibility problem(cf. Leibfritz [31]).

Find (P,Q) 0, such that PQ = I, Qβ(Q, γ) 0, Pβ(P, γ) 0,(3.10)

where, for given β > 0 and γ > 0, we define

Qβ(Q, γ) = NTQ

[AQ+QAT + 2βQ+ γ−1B1B

T1 (C1Q+ γ−1D11B

T1 )T

C1Q+ γ−1D11BT1 γ−1D11D

T11 − γInz

]NQ,

(3.11)

Pβ(P, γ) = NTP

[ATP + PA+ 2βP + γ−1CT

1 C1 PB1 + γ−1CT1 D11

(PB1 + γ−1CT1 D11)

T γ−1DT11D11 − γInw

]NP ,(3.12)

and NQ, NP are defined as in Theorem 3.2. With these definitions, we replace thenonconvex set ΦH∞(P,Q, γ) by the bilinear approximation

ΦH∞(P,Q, γ, β) = (P,Q) ∈ S+nx| (P,Q) satisfying PQ = I,Qβ(Q, γ) 0,

Pβ(P, γ) 0(3.13)

and redefine the bilinear optimization problem (3.8) as

min Tr(Y ), s.t. (P,Q, Y ) ∈ ΦH∞(P,Q, γ, β) ∩ Φ(P,Q, Y ).(3.14)

Then, replacing PQ = I by M(P,Q) 0 in (3.7) and (3.13) yields the followingclosed and linear approximations to the nonconvex and open sets ΦH∞(P,Q, γ) andΦ(P,Q, Y ), respectively:

Xβ(P,Q, γ) = (P,Q) ∈ Snx |M(P,Q) 0,Qβ(Q, γ) 0,Pβ(P, γ) 0(3.15)

and

X (P,Q, Y ) = (P,Q) ∈ Snx, Y ∈ Snw

|M(P,Q) 0, Y 0, Q(Q,Y ) 0,

P(P, Y ) 0.(3.16)

Observe that the condition PQ = I, which is equivalent to Tr(PQ) = nx, is satisfiedif and only if rank(M(P,Q)) ≡ nx, i.e., the nx smallest eigenvalues of the positive


1720 F. LEIBFRITZ

semidefinite (2nx × 2nx) matrix M(P,Q) are equal to zero. In fact, the constraintPQ = I characterizes the boundary of the set

(P,Q) ∈ Snx | P Q−1 0 = (P,Q) ∈ Snx|M(P,Q) 0.

Thus, a feasible matrix triple to the following nonconvex bilinear matrix feasibilityproblem

Find (P,Q) 0, Y 0 such that

PQ = I, Qβ(Q, γ) 0, Pβ(P, γ) 0, Q(Q,Y ) 0, P(P, Y ) 0(3.17)

can be obtained by searching for boundary points of the linear and closed set Xβ(P,Q, γ)

∩ X (P,Q, Y ), defined by (3.15) and (3.16), respectively. This suggests the followingnonconvex bilinear matrix inequality minimization problem:

min Tr(PQ), s.t. (P,Q, Y ) ∈ Xβ(P,Q, γ) ∩ X (P,Q, Y ).(3.18)

Note that there exists a feasible triple (P,Q, Y ) satisfying (3.17) if and only if theoptimal value of (3.18) is equal to nx. Since we are interested in a solution triple(P,Q, Y ) of the nonconvex problem (3.14), this observation motivates us to define thefollowing bilinear, multiobjective programming problem:

min Tr(PQ) + Tr(Y ), s.t. (P,Q, Y ) ∈ Xβ(P,Q, γ) ∩ X (P,Q, Y ).(3.19)

This problem combines the objective functionals of (3.14) and (3.18). Obviously,minimizing Tr(PQ) enforces a solution of (3.19) to be close to or on the boundaryof the feasible set of (3.19), while minimizing Tr(Y ) drives a solution of (3.19) to besuboptimal for the corresponding static H2/H∞ output feedback problem.

If the triple (P ∗, Q∗, Y ∗) is a boundary solution of (3.19) satisfying P ∗Q∗ = I,then we know that (P ∗, Q∗, Y ∗) is contained in the feasible set of (3.14) and alsosatisfies

nx + Tr(Y ∗) = minTr(PQ) + Tr(Y ) | (P,Q, Y ) ∈ ΦH∞(P,Q, γ, β) ∩ Φ(P,Q, Y ).

Thus, the optimal value of (3.14) fulfills

minTr(Y ) | (P,Q, Y ) ∈ ΦH∞(P,Q, γ, β) ∩ Φ(P,Q, Y ) − nx ≤ Tr(Y ∗)− nx= minTr(PQ) + Tr(Y ) | (P,Q, Y ) ∈ Xβ(P,Q, γ) ∩ X (P,Q, Y ) − nx.

Hence, an optimal solution of (3.19) yields an upper bound to the optimal value of(3.14) and at least a suboptimal solution of (3.14) if and only if the minimal triple(P ∗, Q∗, Y ∗) of (3.19) satisfies P ∗Q∗ = I. Finally, if and only if P ∗Q∗ = I, then thecorresponding suboptimal static H2/H∞ output feedback gain can be reconstructedfrom (P ∗, Q∗, Y ∗) if and only if the following LMI feasibility problem in F is nonempty.

Find F ∈ Rnu×ny , such that BFC + (BFC)T + Ω 0 and[

Y ∗ BT0 P

∗

P ∗B0 P ∗

]+

[0

P ∗B2

]F

[D20 0

]

+

[DT

20

0

]FT

[0 BT

2 P∗ ]

0,

(3.20)



where

Ω =

ATP ∗ + P ∗A+ 2βP ∗ P ∗B1 CT

1

BT1 P

∗ −γInwDT

11

C1 D11 −γInz

,B =

P ∗B2

0D12

, C = [C2 D21 0] .

In this case, the discussion in section 3 implies the existence of a suboptimal staticH2/H∞ controller. It also guarantees that AF is Hurwitz, ||Tzw1 ||∞ < γ, andTr(Y ∗) ≥ Tr(BT

0FP∗B0F ) ≥ Copt

γ (X,F ). Thus, such an F is indeed a suboptimalstatic H2/H∞ output feedback gain.

In what follows we explain a numerical procedure for determining an optimalsolution of the bilinear, multiobjective programming problem (3.19). This problem isnot convex since the functional Tr(PQ) is, in general, not convex, but it is bilinear.Therefore, in problem (3.19) we minimize a combination of a bilinear and linear matrixfunctional over a closed convex set. To solve such a problem, a sequential linearizationprogramming approach as proposed by [1] for general nonconvex bilinear programmingproblems can be used. In particular, the idea of this approach is very simple. Insteadof solving the nonconvex problem (3.19) directly, we linearize the bilinear part of theobjective functional. Then we minimize successively the resulting (linearized) LMIconstrained semidefinite programming problems.

Algorithm 1 (SLPMM).For given β > 0, let Xβ(P,Q, γ) ∩ X (P,Q, Y ) = ∅.

(0) Determine (P 0, Q0, Y 0) ∈ Xβ(P,Q, γ) ∩ X (P,Q, Y ).

For k = 0, 1, 2, . . . do(1) Determine (Uk, V k, Zk) as the unique solution of

min Tr(PQk + P kQ) + Tr(Y ),

s.t. (P,Q, Y ) ∈ Xβ(P,Q, γ) ∩ X (P,Q, Y ).(3.21)

(2) If Tr(UkQk + P kV k) + Tr(Zk) = 2Tr(P kQk) + Tr(Y k), −→ Stop.(3) Compute α ∈ [0, 1] by solving

minα∈[0,1] Tr((P k + α(Uk − P k))(Qk + α(V k −Qk)))

+ Tr(Y k + α(Zk − Y k)).(3.22)

(4) Set P k+1 = (1 − α)P k + αUk, Qk+1 = (1 − α)Qk + αV k, and Y k+1 =(1− α)Y k + αZk.

This algorithm is similar to the SLPMM procedure proposed by Leibfritz [31] forthe design of stabilizing static H2 and suboptimal static H∞ output feedback con-trollers. The initialization step of Algorithm 1 is an LMI feasibility problem, and(3.21) is an SDP problem with a linear objective functional under LMI constraints.There are many algorithms available for solving such kinds of problems. For example,interior point methods developed for SDPs can be used (cf. [13]). Algorithm 1 termi-nates if the first order necessary minimum principle is satisfied at a (local) minimumof (3.19) [34, section 6.1]. For example, define

J (P,Q, Y ) := Tr(PQ) + Tr(Y ).(3.23)

Note that J (P,Q, Y ) is continuous and differentiable on the cone of symmetric ma-trices. Hence,

J ′(P,Q, Y )(U, V, Z) = Tr(UQ+ PV ) + Tr(Z),J ′′(P,Q, Y )(U, V, Z)(H,G,W ) = Tr(HV + UG)


1722 F. LEIBFRITZ

for any U, V,H,G ∈ Snxand Z,W ∈ Snw

. If (P ∗, Q∗, Y ∗) denotes a local minimum ofJ over the closed convex set Xβ(P,Q, γ)∩ X (P,Q, Y ), then for any feasible direction(P − P ∗, Q−Q∗, Y − Y ∗) = (δP, δQ, δY ) at (P ∗, Q∗, Y ∗) we obtain

J ′(P ∗, Q∗, Y ∗)(δP, δQ, δY ) = Tr(PQ∗ + P ∗Q) + Tr(Y )− 2Tr(P ∗Q∗)− Tr(Y ∗) ≥ 0

(3.24)

for all (P,Q, Y ) ∈ Xβ(P,Q, γ) ∩ X (P,Q, Y ) [34, section 6.1, Proposition 1]. Numeri-cally, for a sufficiently small scalar ε > 0, we terminate the algorithm if

τk := Tr(UkQk + P kV k) + Tr(Zk)− 2Tr(P kQk)− Tr(Y k) ≥ −ε, k ≥ 0.(3.25)

Moreover, if (P k, Qk, Y k) ∈ Xβ(P,Q, γ)∩X (P,Q, Y ) and (Uk, V k, Zk) ∈ Xβ(P,Q, γ)∩X (P,Q, Y ) do not satisfy (3.24), we determine a step size parameter α ∈ [0, 1] byminimizing (3.22) with respect to α. Finally, the new iterates (P k+1, Qk+1, Y k+1) areconvex combinations of (P k, Qk, Y k) and (Uk, V k, Zk), respectively, and therefore,they are also contained in Xβ(P,Q, γ) ∩ X (P,Q, Y ). Note that the points (Uk −P k, V k − Qk, Zk − Y k), k ≥ 0, are descent directions for J at (P k, Qk, Y k) unlessτk ≥ 0 for some k ≥ 0.

For proofing the convergence of Algorithm 1, we need the following result. Sup-pose for fixed β > 0 that there exist matrices (P 0, Q0, Y 0) ∈ Xβ(P,Q, γ)∩X (P,Q, Y );then we can define the following level set:

Γ(P 0, Q0, Y 0) := (P,Q, Y ) ∈ Xβ(P,Q, γ) ∩ X (P,Q, Y ) | J (P,Q, Y )≤ J (P 0, Q0, Y 0).(3.26)

For this level set, we conclude the following lemma.Lemma 3.5. Let β > 0 be given, Xβ(P,Q, γ) ∩ X (P,Q, Y ) be nonempty, and let

(P 0, Q0, Y 0) ∈ Xβ(P,Q, γ)∩ X (P,Q, Y ) be given. Then the level set Γ(P 0, Q0, Y 0) iscompact.

Proof. Using [24, Theorem 7.4.10], the closeness of Xβ(P,Q, γ), X (P,Q, Y ), andthe definition of Γ(P 0, Q0, Y 0) shows the desired result.

With the compactness of the level set Γ(P 0, Q0, Y 0), it is straightforward to showthe existence of an optimal solution of (3.19) in this level set. In particular, we havethe following lemma.

Lemma 3.6. Let Xβ(P,Q, γ) ∩ X (P,Q, Y ) = ∅ and (P 0, Q0, Y 0) ∈ Xβ(P,Q, γ) ∩X (P,Q, Y ) be given. Then there exists an optimal solution (P ∗, Q∗, Y ∗) of (3.19) inthe level set Γ(P 0, Q0, Y 0).

Proof. By the compactness of Γ(P 0, Q0, Y 0), the continuity of J , and the theoremof Bolzano–Weierstrass, the result follows immediately.

Thus, Lemma 3.6 ensures the existence of a solution of the multiobjective pro-gramming problem (3.19) at least in the compact level set Γ(P 0, Q0, Y 0).

The following lemma is needed in the proof of the Theorem 3.8 given below.Lemma 3.7. Let a > 0, b < 0, and c ≥ 1 be given. Then there exists ρ∗ ≥ 0 such

that the optimal solution of minα∈[0,1] (c+ α b+ (α2/2) a) satisfies

0 < α∗ = −b+ ρ∗a

≤ 1.

Moreover, ρ∗ = 0 if and only if a ≥ −b.Proof. The result can be proven by using the convexity of the objective function

and the Karush–Kuhn–Tucker theorem.



The next result states the basic convergence properties of Algorithm 1. It also jus-tifies the fact that we need only the existence of a solution of (3.19) in Γ(P 0, Q0, Y 0).Before stating these properties, we define the boundary of the set Xβ(P,Q, γ) ∩X (P,Q, Y ), given by

∂X (P,Q, Y ) := (P,Q, Y ) ∈ Xβ(P,Q, γ) ∩ X (P,Q, Y ) | PQ = I.(3.27)

Obviously,

∂X (P,Q, Y ) = ΦH∞(P,Q, γ, β) ∩ Φ(P,Q, Y ) ⊆ Xβ(P,Q, γ) ∩ X (P,Q, Y ).

Hence, the level set Γ(P 0, Q0, Y 0) also contains all (P,Q, Y ) on the boundary ofXβ(P,Q, γ)∩X (P,Q, Y ). Therefore, similarly as in Lemma 3.6, we can conclude thatthere also exists an optimal solution of the nonconvex optimization problem (3.14) inthe level set Γ(P 0, Q0, Y 0).

Theorem 3.8. Let β > 0 and (P 0, Q0, Y 0) ∈ Xβ(P,Q, γ) ∩ X (P,Q, Y ) be given.

Furthermore, assume that (P k, Qk, Y k) ⊂ Xβ(P,Q, γ)∩ X (P,Q, Y ) is generated byAlgorithm 1. Then the sequence (P k, Qk, Y k) is well defined and for all k ≥ 0, wehave

nx + Tr(Y ∗) ≤ nx + Tr(Y k+1) ≤ J (P k+1, Qk+1, Y k+1) < J (P k, Qk, Y k)(3.28)

unless J ′(·)(·) = 0, where (P ∗, Q∗, Y ∗) solves (3.19). Thus, J (P k, Qk, Y k) con-verges to J ≥ nx + Tr(Y ∗) and for all k ≥ 0, (P k, Qk, Y k) ∈ Γ(P 0, Q0, Y 0), i.e.,(P k, Qk, Y k) is bounded. Finally, J (P k, Qk, Y k) = nx + Tr(Y k) if and only if(P k, Qk, Y k) ∈ ∂X (P,Q, Y ) and J ′(·)(·) = 0 for some k.

Proof. For all k ≥ 0, we define the following abbreviations:

Sk = (P k, Qk, Y k), δSk = (Uk−P k, V k−Qk, Zk−Y k), and T k = (Uk, V k, Zk).

If Algorithm 1 terminates at Sk ∈ Xβ(P,Q, γ) ∩ X (P,Q, Y ), k ≥ 0, then

J ′(Sk)(Sk) = J ′(Sk)(T k) = minS∈Xβ(P,Q,γ)∩X (P,Q,Y )

J ′(Sk)(S)

and J ′(P k, Qk, Y k)(P − P k, Q − Qk, Y − Y k) ≥ 0 for all (P,Q, Y ) ∈ Xβ(P,Q, γ) ∩X (P,Q, Y ). Thus, if the algorithm does not terminate in step (2), then

J ′(Sk)(δSk) = Tr(UkQk + P kV k) + Tr(Zk)− 2Tr(P kQk)− Tr(Y k) < 0,(3.29)

i.e., δSk is a descent direction. Using Sk ∈ Xβ(P,Q, γ)∩X (P,Q, Y ), T k ∈ Xβ(P,Q, γ)∩X (P,Q, Y ), and the convexity of Xβ(P,Q, γ)∩ X (P,Q, Y ), steps (3) and (4) of Algo-rithm 1 imply that

(P k+1, Qk+1, Y k+1) ∈ Xβ(P,Q, γ) ∩ X (P,Q, Y )

for all α ∈ [0, 1]. Hence, the sequence Sk = (P k, Qk, Y k) is well defined byAlgorithm 1.

To show the strictly decreasing property of J (P k, Qk, Y k), note that for α ∈[0, 1], the construction of the algorithm implies

J (Sk+1) = J (Sk) + α∗ J ′(Sk)(δSk) +α2∗

2J ′′(Sk)(δSk)(δSk)

≤ J (Sk) + α J ′(Sk)(δSk) +α2

2J ′′(Sk)(δSk)(δSk)

≤ J (Sk) + J ′(Sk)(δSk) +1

2J ′′(Sk)(δSk)(δSk),(3.30)

where α∗ ∈ [0, 1] denotes the optimal value of (3.22).


1724 F. LEIBFRITZ

Assuming J ′′(Sk)(δSk)(δSk) = 0 and using (3.29), from (3.30) we obtain J (Sk+1)< J (Sk) for all k ≥ 0 unless J ′(Sk)(δSk) ≡ 0. Moreover, if J ′′(Sk)(δSk)(δSk)< 0, then we can conclude that J (Sk+1) < J (Sk) for all k ≥ 0. Finally, ifJ ′′(Sk)(δSk)(δSk) > 0, then the problem is convex. Defining a = J ′′(·)(·)(·),b = J ′(·)(·), and c = J (·), Lemma 3.7 implies the existence of ρ∗ ≥ 0 such thatthe solution α∗ of (3.22) satisfies

0 < α∗ = − J′(Sk)(δSk) + ρ∗

J ′′(Sk)(δSk)(δSk)≤ 1.

But this implies

1

2α2∗ J ′′(Sk)(δSk)(δSk) = −1

2α∗ J ′(Sk)(δSk)− 1

2α∗ρ∗.(3.31)

Using α∗ ∈ (0, 1], ρ∗ ≥ 0, (3.29), (3.30), and (3.31), we obtain J (Sk+1) < J (Sk)unless J ′(Sk)(δSk) ≡ 0 for all k ≥ 0. Hence, J (P k, Qk, Y k) is a strictly decreasingsequence unless J ′(Sk)(δSk) ≡ 0. Since Tr(Y k) ≥ 0 and Tr(P kQk) ≥ nx for all k ≥ 0,the sequence J (P k, Qk, Y k) is also bounded below by nx + Tr(Y ∗). Therefore, itconverges to a limit J ≥ nx+Tr(Y ∗). Moreover, by the definition of the compact levelset Γ(P 0, Q0, Y 0), we know that (P k, Qk, Y k) ⊂ Γ(P 0, Q0, Y 0), and thus that thegenerated sequence (P k, Qk, Y k) is bounded. Finally, assume that (P k, Qk, Y k) ∈∂X (P,Q, Y ) and J ′(Sk)(δSk) = 0 for some k. Note that this is fulfilled if and onlyif P kQk = I and the minimal point T k of (3.21) satisfy Uk = P k, V k = Qk, andZk = Y k for some k. Thus, J (P k, Qk, Y k) = nx + Tr(Y k) ≥ nx + Tr(Y ∗) if and onlyif (P k, Qk, Y k) ∈ ∂X (P,Q, Y ) and J ′(Sk)(δSk) = 0 for some k.

Theorem 3.8 ensures that J (P k, Qk, Y k) is a strictly decreasing sequence whichis bounded below by nx + Tr(Y ∗) if and only if (P k, Qk, Y k) ∈ ∂X (P,Q, Y ) andJ ′(Sk)(δSk) < 0. On the other hand, J (P k, Qk, Y k) = nx +Tr(Y k) ≥ nx +Tr(Y ∗) ifand only if (P k, Qk, Y k) ∈ ∂X (P,Q, Y ) and J ′(Sk)(δSk) = 0 for some k ≥ 0. Thenwe know that P kQk = I and Uk = P k, V k = Qk, Zk = Y k for some k. Hence, in thiscase, the SLPMM algorithm terminates in step (2) at a point satisfying the couplingcondition PQ = I.

The SLPMM algorithm may be interpreted as a modified version of the cone com-plementarity algorithm, expect that in each iteration we must also compute a stepsize parameter. But this further computational work is essential for the strictly de-creasing property of the SLPMM. Indeed, the novel aspect of the SLPMM algorithmis that it always generates a strictly decreasing sequence J (P k, Qk, Y k). More-over, this approach guarantees the boundedness of the iterates (P k, Qk, Y k) for allk ≥ 0. In contrast to this, the cone complementarity algorithm, the XY-centeringalgorithm, and other related computational methods in the literature do not sharethese properties.

The strictness in the inequality (3.28) and the boundedness of (P k, Qk, Y k) isessential to prove the global convergence of the SLPMM algorithm. The followingtheorem states the global convergence of our method under rather weak assumptionscompared to the existing algorithms in the literature.

Theorem 3.9. Let β > 0 and (P 0, Q0, Y 0) ∈ Xβ(P,Q, γ) ∩ X (P,Q, Y ) be given.

Furthermore, assume that (P k, Qk, Y k) ⊂ Xβ(P,Q, γ)∩ X (P,Q, Y ) is generated byAlgorithm 1. Then the following hold.

(i) The algorithm terminates at some (P k, Qk, Y k) satisfying (3.24), or everyaccumulation point (P , Q, Y ) of (P k, Qk, Y k) is stationary, i.e., (P , Q, Y ) satisfy(3.24).



(ii) Every accumulation point (P , Q, Y ) of (P k, Qk, Y k) satisfies the nonconvexbilinear matrix feasibility problem (3.17) and solves the nonconvex optimization prob-lem (3.14) if and only if the boundary set ∂X (P,Q, Y ) defined in (3.27) is nonempty.

Proof. (i) This result follows immediately by making straightforward modifica-tions to the proof of Leibfritz [31, Theorem 3.7].

(ii) Note that ∂X (P,Q, Y ) = ΦH∞(P,Q, γ, β) ∩ Φ(P,Q, Y ) ⊂ Γ(P 0, Q0, Y 0) andthat Γ(P 0, Q0, Y 0) is compact. Theorem 3.8 implies limk→∞ J (P k, Qk, Y k) = J ≥nx + Tr(Y ∗) =: J ∗. Since (P k, Qk, Y k) ∈ Γ(P 0, Q0, Y 0), we also know that thefunctions J (P,Q, Y ) = Tr(PQ) + Tr(Y ) and J (P,Q, Y ) = Tr(Y ) attain the (global)minimum values on the compact level set Γ(P 0, Q0, Y 0); i.e., J ∗ = J (P ∗, Q∗, Y ∗) ≡nx + Tr(Y ∗) and J ∗ = J (P ∗, Q∗, Y ∗) ≡ Tr(Y ∗), respectively.

Suppose J > J ∗ = nx + Tr(Y ∗). From the compactness of Γ(P 0, Q0, Y 0) = ∅and the continuity of J , it follows that for every ε > 0, there exist a δ > 0 and avicinity Uδ(P

∗, Q∗, Y ∗) such that Uδ(P∗, Q∗, Y ∗) ∩ Γ(P 0, Q0, Y 0) = ∅ and

0 ≤ J (P,Q, Y )− J ∗ < ε

for all (P,Q, Y ) ∈ Uδ(P∗, Q∗, Y ∗)∩Γ(P 0, Q0, Y 0). Choosing ε := J −nx−Tr(Y ∗) > 0

implies

J (P,Q, Y ) < ε+ nx + Tr(Y ∗) = J .

But this implies the existence of an integer k such that for all k ≥ k and (P k, Qk, Y k) ∈Uδ(P

∗, Q∗, Y ∗) ∩ Γ(P 0, Q0, Y 0) we have J (P k, Qk, Y k) < J , which contradicts thatthe sequence J (P k, Qk, Y k) converges monotonically to J . Hence J = J ∗ =nx + Tr(Y ∗); i.e.,

limk→∞

J (P k, Qk, Y k) = J ∗ ⇐⇒ limk→∞

(J (P k, Qk, Y k)− nx) = J ∗.(3.32)

Since (P k, Qk, Y k) ⊂ Γ(P 0, Q0, Y 0) and Γ(P 0, Q0, Y 0) is compact, we conclude theexistence of a convergent subsequence of (P k, Qk, Y k); i.e.,

limj→∞

(P kj , Qkj , Y kj ) = (P , Q, Y ) ∈ Γ(P 0, Q0, Y 0).

Suppose that this accumulation point is not globally optimal. Then J (P , Q, Y ) >J ∗ = nx + Tr(Y ∗) > J ∗ and the sequence J (P k, Qk, Y k) − nx − Tr(Y ∗) do nottend to zero. This contradicts (3.32). Hence, ∂X (P,Q, Y ) = ∅ if and only if everyaccumulation point satisfies (3.17) and solves

minTr(PQ) + Tr(Y ) | (P,Q, Y ) ∈ ΦH∞(P,Q, γ, β) ∩ Φ(P,Q, Y ) = nx + Tr(Y ∗)= nx + minTr(Y ) | (P,Q, Y ) ∈ ΦH∞(P,Q, γ, β) ∩ Φ(P,Q, Y ),

which in turn is equivalent to the solvability of (3.14).From Theorem 3.8, we know that the sequence J (P k, Qk, Y k) is strictly de-

creasing unless J ′(·)(·) = 0 and is bounded below by nx + Tr(Y ∗), and hence con-verges. If in the limit Tr(PQ) = nx, then ∂X (P,Q, Y ) = ∅. In this case, Theorem3.9 guarantees the existence of an accumulation point which satisfies the nonconvexbilinear matrix feasibility problem (3.17). Moreover, this point is also a solution ofthe nonconvex optimization problem (3.14), and therefore, in the limit the neces-sary condition for the existence of a static H2/H∞ output feedback gain is fulfilled.


1726 F. LEIBFRITZ

Note that for reconstructing such a gain, it is not necessary that the computed so-lution of the multiobjective problem (3.19) is the global solution of this problem. Itsuffices that this solution is contained in the boundary set ∂X (P,Q, Y ) defined by(3.27), i.e., it satisfies PQ = I. Then, this solution triple is at least an upper boundto the global optimal value of (3.14), and the corresponding gain, if any exists, isat least suboptimal. Thus, if Algorithm 1 terminates with a boundary solution of(3.19), i.e., J (P k, Qk, Y k) = nx + Tr(Y k) for some k ≥ 0, then Theorem 3.8 im-plies (P k, Qk, Y k) ∈ ∂X (P,Q, Y ) for some k. Moreover, (P k, Qk, Y k) ∈ ∂X (P,Q, Y )satisfies

minTr(Y ) | (P,Q, Y ) ∈ ΦH∞(P,Q, γ, β) ∩ Φ(P,Q, Y ) ≤ Tr(Y k).

Then, using the results of the previous section, a corresponding suboptimal staticH2/H∞ output feedback controller exists if and only if the LMI feasibility problem(3.20) in F is nonempty. On the other hand, it may occur that Algorithm 1 terminatesat a triple (P k, Qk, Y k) satisfying the first order necessary minimum principle, but(P k, Qk, Y k) ∈ ∂X (P,Q, Y ). In this case, we know that Tr(P kQk) > nx and a staticgain F can not be reconstructed from this triple. But we can reconstruct a reduceddynamic output feedback control law of order nc ≤ nx−maxnu, ny from this tripleby using a well-known system augmentation technique. For more details, we refer thereader, for example, to Leibfritz [31].

4. Numerical examples. In this section, several examples are given for testpurposes in order to test the SLPMM approach. We present examples for the designof suboptimal static H2/H∞ output feedback controllers.

The SLPMM algorithm as well as the reconstruction of the controller gain Ffrom the corresponding LMIs have been implemented making use of MATLAB 5.0facilities. Particularly, for determining a feasible solution (P 0, Q0, Y 0) ∈ Xβ(P,Q, γ)∩X (P,Q, Y ) in Algorithm 1, step (0), we have used the LMI control toolbox [14] routineFEASP, which finds a solution to a given system of LMIs, if any exists. Moreover, forsolving the linearized minimization problem (3.21) of Algorithm 1, step (1), which is asemidefinite programming problem, we have taken the LMI control toolbox procedureMINCX. This solver is an implementation of Nesterov and Nemirovski’s projectivemethod for minimizing a linear objective function under LMI constraints as describedin [13]. For MINCX we have adjusted the desired relative accuracy on the optimalvalue to 10−12.

We terminated Algorithm 1 if, for a sufficiently small scalar ε > 0, the condition(3.25) was fulfilled. Moreover for the computation of a step size α ∈ [0, 1] we haveused the MATLAB function FMIN.

Finally, we have taken the LMI control toolbox function FEASP applied to theLMI feasibility problem (3.20) for reconstructing the static output feedback controllergain F .

The following data are given in the tables: the iteration counter k of the SLPMMalgorithm; if k = 0, j denotes the total number of iterations for finding a feasible ini-tial point (P 0, Q0, Y 0) by FEASP; else j is the total number of iterations for solvingthe semidefinite programming problem (3.21) by MINCX with relative accuracy of10−12; the objective function value J (P k, Qk, Y k) = Tr(P kQk)+Tr(Y k) of the corre-sponding multiobjective optimization problem (3.19); the function values Tr(P kQk)and Tr(Y k); and τk, which indicates if the first order necessary minimum principle isapproximatively satisfied.



Table 4.1Convergence behavior of the SLPMM algorithm for the VTOL helicopter model: static H2/H∞

case.

k j J (Pk, Qk, Y k) Tr(PkQk) Tr(Y k) τk

0 6 1209.96168 1156.3660285 53.5956538 —1 53 4.21991536 4.0001534809 0.21976188 −2.058e+ 032 45 4.21973458 4.0000000000 0.21973458 −1.808e− 043 18 4.21973458 4.0000000000 0.21973458 0.000e+ 00

For all test examples we have chosen the data matrices B0, D10, D11, and D20 asfollows:

B0 = B1, D10 = D11 = 0, D20 = 0.

This choice guarantees D0F = 0. Therefore, if a static H2/H∞ controller gain Fexists for the corresponding test problem, it is an upper estimate of ||Tzw0 ||2H2

thatenforces ||Tzw1

||∞ < γ.Example 1. A state space model of the longitudinal motion of a VTOL helicopter

is considered as in [28]. The dynamic equations of the helicopter model are linearizedaround a nominal solution where the given dynamic equation is computed for typicalloading and flight conditions of the VTOL helicopter at a certain airspeed [41]. Thelinearization results in a fourth order linear time-invariant state equation with twocontrol and two unknown input components. The data matrices of the linearizedmodel are given by

A =

−0.0366 0.0271 0.0188 −0.45550.0482 −1.0100 0.0024 −4.02080.1002 0.3681 −0.7070 1.4200

0 0 1.0000 0

, B1 =

0.0468 00.0457 0.00990.0437 0.0011

−0.0218 0

,

B2 =

0.4422 0.17613.5446 −7.5922

−5.5200 4.49000 0

,

C1 = 1√2

[2 0 0 00 1 0 0

], C2 =

[0 1 0 0

], D12 = 1√

2

[1 00 1

],

D21 =[0.00039 0.00174

].

The goal is to design a suboptimal static H2/H∞ output feedback controller ΣC bythe SLPMM Algorithm 1 according to the discussion of the previous sections. In therun of Algorithm 1 we have chosen β = 0.01, γ = 0.423722, and ε = 10−10. Table4.1 demonstrates the convergence behavior of the SLPMM. It illustrates numericallythat J (P k, Qk, Y k) is a strictly decreasing sequence if (P k, Qk, Y k) ∈ ∂X (P,Q, Y )and τk < 0, which converges to nx + Tr(Y ∗) = 4 + 0.21973458. Thus, we have∂X (P,Q, Y ) = ∅, and equality holds if and only if (P k, Qk, Y k) ∈ ∂X (P,Q, Y ) andτk = 0 for some k according to Theorem 3.8. Moreover, Theorem 3.9 guaranteesthe existence of an accumulation point of P k, Qk, Y k) which also solves the non-convex optimization problem (3.14). Therefore, a suboptimal static H2/H∞ outputfeedback controller exists if and only if the LMI feasibility problem (3.20) in F isnonempty. The resulting static gain matrix which satisfies (3.20) and the closed loop


1728 F. LEIBFRITZ

eigenvalues are

F =[−0.978987 19.43483

]T, λ(AF ) =

[−152.04 −0.0989 −0.2990± 0.949i

].

Moreover, ||Tzw1 ||∞ = 0.2937866 < γ and Tr(BT0FPB0F ) = 0.02869950 ≤ Tr(Y ) =

0.21973458, which shows that F is a suboptimal static H2/H∞ output feedback gain.To compare this solution with the optimal one, we have calculated Copt

γ (X,F ) =0.0231044 with an algorithm proposed by Leibfritz [32], [30]. This shows, that (3.19)provides a good suboptimal solution to the problem under consideration. As it can bealso verified, the sequence (P k, Qk, Y k) ⊂ Γ(P 0, Q0, Y 0) and is thus bounded. Forexample, we have

||P k|| = 82.26, 49.82, 49.82, 49.82,||Qk|| = 41.16, 1.159, 1.159, 1.159,||Y k|| = 26.86, 0.1099, 0.1099, 0.1099,

which demonstrates numerically the boundedness of the generated sequence accordingto Theorem 3.8. The whole computation needs 2.47 CPU seconds on a SUN Ultra 60.

Finally, for this example, we demonstrate numerically that the extended staticH2/H∞ output feedback problem (2.12) can be also solved by the LMI approachas discussed in section 3. The SLPMM terminates after four outer iterations withan approximate solution (P,Q, Y ) of the corresponding bilinear programming prob-lem, i.e., Tr(PQ) = 4.0000001. Then, using this solution, the corresponding LMIfeasibility problem in F was found to be nonempty. Thus, a static H2/H∞ outputfeedback gain F satisfying (2.12) exists, and the resulting gain matrix is given byF = [−0.457876 17.38107]T .

Example 2 (transport airplane). This example studies the longitudinal motion ofa modern transport airplane under VMIN flight conditions [15]. The linearized statespace model yields the following data matrices:

A =

−0.06254 0.01888 0 −0.56141 −0.02751 0 0.06254 −0.00123 00.01089 −0.99280 0.99795 0.00097 −0.07057 0 −0.01089 0.06449 00.07743 1.67540 −1.31111 −0.00030 −4.25030 0 −0.07743 −0.10883 0

0 0 1 0 0 0 0 0 00 0 0 0 −20.0000 20 0 0 00 0 0 0 0 −30 0 0 00 0 0 0 0 0 −0.88206 0 00 0 0 0 0 0 0 −0.88206 0.008820 0 0 0 0 0 0 −0.00882 −0.88206

,

B1 =

0 0 0 10 0 0 00 0 0 00 0 0 00 0 0 00 0 0 0

1.3282 0 0 00 1.62671 0 00 −68.75283 0 0

, B2 =

00000

30000

, C1 = 1√

2

−0.005190.476040.00098

−0.000310.03378

00.00519

−0.030860

T

,

CT2 =

−0.00519 00.47604 00.00098 0

−0.00031 00.03378 0

0 00.00519 0

−0.03086 00 0

, DT

21 =

0 00 00 00 1

,

and D12 = [1/√

2]. The goal is the design of a robust static control law. Definingβ = 10−2, γ = 0.152032 and ε = 10−12, Table 4.2 illustrates the convergence behaviorof the SLPMM for this example. According to Theorem 3.8, it shows numerically



Table 4.2Convergence behavior of the SLPMM algorithm for the transport airplane: static H2/H∞ case.

k j J (Pk, Qk, Y k) Tr(PkQk) Tr(Y k) τk

0 18 1.766e+ 14 1.766e+ 14 7.622e+ 07 —1 58 131.706754 83.966174369 47.74057934352 −3.533e+ 142 49 9.80419099 9.4602075306 0.343983455585 −1.689e+ 023 56 9.39445809 9.0504993630 0.343958730681 −2.843e− 014 86 9.34395876 9.0000000338 0.343958722207 −4.318e− 025 24 9.34395876 9.0000000338 0.343958722207 0.000e+ 00

the strictly decreasing property of the objective function values of (3.19) and thenonemptiness of the boundary of the set Xβ(P,Q, γ) ∩ X (P,Q, Y ), i.e., in the limitwe achieve Tr(PQ) = nx. Thus, ∂X (P,Q, Y ) = ∅ and Theorem 3.9 ensures theexistence of an accumulation point of P k, Qk, Y k) which also solves the nonconvexoptimization problem (3.14). Therefore, a suboptimal static H2/H∞ output feedbackcontroller exists if and only if the LMI feasibility problem (3.20) in F is nonempty.Solving the LMI feasibility problem (3.20) results in the feasible gain

F =[

2.286293 0.001023].

The real parts of the closed loop poles range between −0.02686 and −33.4213. More-over, ||Tzw1 ||∞ = 0.08088935 and Tr(BT

0FPB0F ) = 0.05135721 ≤ Tr(Y ) = 0.34395872.Once again, comparing this solution with the optimal one, we have calculatedCopt

γ (X,F ) = 0.0502692 with the algorithm of [32] for solving the optimal staticH2/H∞ output feedback problem (2.11). Finally, the whole computation needs 53.53CPU seconds. Again, this example demonstrates numerically the theoretical proper-ties of our algorithm and shows that this approach can be used for the design of asuboptimal static H2/H∞ output feedback gain.

Example 3 (Euler–Bernoulli beam). This example consists of a simple supportedEuler–Bernoulli beam as discussed in [22] (see also [18]). Following [22], the statespace model is given by ΣP , with matrices as follows:

A = diag

[0 1

−r4 −0.02 r2

], r = 1, . . . , 5

,

BT2 = C2 =

[0 0.9877 0 −0.309 0 −0.891 0 0.5878 0 0.7071

], B1 =

[B2 010×1

],

C1 =

[0.809 0 −0.9511 0 0.309 0 0.5878 0 −1 0

0 0 0 0 0 0 0 0 0 0

],

D12 =

[0

0.5

], D21 =

[0 1.9

].

Choosing β = 0.01, γ = 3.59251, and ε = 10−10, Algorithm 1 terminates after 28.59CPU seconds, and (3.20) provides the static H2/H∞ output feedback controller gainF = −0.6221049 with Tr(BT

0FPB0F ) = 0.5742053 ≤ Tr(Y ) = 4.4202 and ||Tzw1 ||∞ =2.179593. Moreover, the closed loop poles are

λ(AF ) =[−0.395± 24.9i −0.257± 15.9i −0.326± 8.98i −0.305± 0.95 −0.059± 3.99i

].

Using the algorithm proposed by [30], [32] yields the optimal cost value Coptγ (X,F ) =

0.185282 of (2.11) with Fopt = −1.036985 and ||Tzw1 ||∞ = 2.14176.


1730 F. LEIBFRITZ

101

100

101

102

105

104

103

102

101

100

101

Frequency (rad/sec)

Sin

gula

r Val

ues

(db)

Fig. 4.1. Singular value plot.

0 10 20 30 40 50 60 70 80 90 100

5

0

5

10

15

Time t

Pen

alty

out

put z

(t)

/ Obs

erve

d ou

tput

y(t

)

Fig. 4.2. Phase portraits of the regulated output z (dashed line) and the observed output y(solid line) for the Euler–Bernoulli beam under worst case disturbances: Controlled case.

The SLPMM algorithm has generated the sequences

J (P k, Qk, Y k)4k=0 = 229.5443, 14.4202522, 14.4202000, 14.4202000, 14.4202000,Tr(P kQk)4k=0 = 180.32, 10.000029, 10.0000000001, 10.0000000000, 10.0000000000,

τk4k=1 = −3.281 · 102,−5.224 · 10−5,−1.278 · 10−10, 0.000,

which underline the theoretical results stated in Theorems 3.8 and 3.9.Figure 4.1 shows the singular value response of the corresponding closed loop

system for the problem under consideration. In Figure 4.2, the phase portraits forthe regulated output variable z (dashed line) and the observed output variable y



0 10 20 30 40 50 60 70 80 90 100

4

3

2

1

0

1

2

3

4

5

6

Time t

Wor

st c

ase

inpu

t w(t

)

Fig. 4.3. Worst case disturbances.

0 10 20 30 40 50 60 70 80 90 100

250

200

150

100

50

0

50

100

150

200

250

Time t

Unc

ontr

olle

d P

enal

ty o

utpu

t z(t

) / O

bser

ved

outp

ut y

(t)

Fig. 4.4. Phase portraits of the regulated output z (dashed line) and the observed output y(solid line) for the Euler–Bernoulli beam under worst case disturbances: Uncontrolled case.

(solid line) are displayed for the computed suboptimal static H2/H∞ output feedbackcontroller under the corresponding worst case inputs w, as illustrated in Figure 4.3.These plots demonstrate that the computed controller gain yields an asymptoticallystable closed loop system and satisfies the robustness constraint, even if the worst-case input affects the system. In contrast to this, the uncontrolled system outputs,as shown in Figure 4.4, oscillate away from the stable equilibrium of the system.Especially, the worst case inputs support this behavior and drive the system to beunstable.

Example 4 (reduced order control). In this example we consider a three–mass-spring system and illustrate our approach for the design of a reduced order compen-


1732 F. LEIBFRITZ

sator if the SLPMM algorithm terminates at a point (P,Q, Y ) satisfying Tr(PQ) > nx.The system under consideration consists of three unit masses connected by two linearsprings with stiffness constant κ. There we have assumed that only the position ofthe third mass is measured and a control force acts on the first mass. The linear statespace realization of this system is given by ΣP with the following data matrices:

A =

0 1 0 0 0 0−κ 0 κ 0 0 00 0 0 1 0 0κ 0 −2κ 0 κ 00 0 0 0 0 10 0 κ 0 −κ 0

, B1 =

000001

, B2 =

010000

, C2 =

000010

T

, D12 =

0000001

,

C1 = [I6 0]T , D21 = 0, and the state components xi, xi+1, i = 1, 2, 3, denote theposition and the velocity of the mass i, respectively. It is well known that this linearsystem is not stabilizable by a static output feedback gain, i.e., Fs = ∅, but it isstabilizable by a reduced order controller of order nc = 3, where

ΣCD xc(t) = Acxc(t) +Bcy(t), u(t) = Ccxc(t) +Dcy(t)

denotes the compensator of order nc ≤ nx with Ac ∈ Rnc×nc , Bc ∈ R

nc×ny , Cc ∈R

nu×nc ,Dc ∈ Rnu×ny , and xc(t) ∈ R

nc . Using Algorithm 1 with β = 0.1, γ = 50.0869,and ε = 10−8, we can compute a reduced order compensator of the form ΣCD forthis example as follows. Since Fs = ∅, the SLPMM algorithm provides a solutiontriple (P,Q, Y ) of (3.19) satisfying Tr(PQ) > nx, i.e., (P,Q, Y ) ∈ ∂X (P,Q, Y ). Inparticular, the algorithm has generated the following sequences:

J (P k, Qk, Y k)5k=0 = 10192.8935, 48.3013702, 38.2048006, 37.5019180, 36.7597188,

36.7596674,Tr(P kQk)5k=0 = 9786.7919, 39.2714948, 30.9417654, 30.8152693, 30.7274670,

30.7274665,Tr(Y k)5k=0 = 406.1016, 9.0298754, 7.2630352, 6.6866487, 6.0322518,

6.0322009,τk5k=1 = −1.84 · 104,−9.50,−1.24 · 10−2,−9.12 · 10−3,−3.19 · 10−9.

Obviously, J (P k, Qk, Y k) is a strictly decreasing sequence which tends to Tr(P 5Q5)+ Tr(Y 5) = 30.7274665 + 6.0322009 > nx + Tr(Y ∗), and Algorithm 1 terminates af-ter six iterations satisfying approximatively the first order necessary condition. Since∂X (P,Q, Y ) = ∅, we know that there exists no static gain for the problem underconsideration. But it is possible to reconstruct a reduced order compensator fromthe computed solution triple (P,Q, Y ) of (3.19). For example, first we compute theeigenvalues of P −Q−1 0. If there are m eigenvalues of P −Q−1 which are less thanor equal to ε > 0, ε $ 1, then there exists an ncth order dynamic output feedbackcontrol law of the form ΣCD such that the eigenvalues of the augmented closed loopsystem matrix AF = A + B2F C2 have negative real parts [8, Theorem 3.1], wherenc = nx −m and the augmented system matrices are defined by [12]:

(4.1)

A =

[A 00 0nc

], B1 =

[B1

0

], B2 =

[0 B2

Inc 0

], C1 = [C1 0],

C2 =

[0 Inc

C2 0

], D12 = [0 D12], D21 =

[0D21

], F =

[Ac Bc

Cc Dc

],



B0 = B1, D20 = [0 0ny×nw]T , and D10 = D11 = D11. Second, we decompose the

positive semidefinite matrix P −Q−1 as UUT +E by a singular value decomposition,where U ∈ R

nx×nc , E ∈ Rnx×nx , ||E|| ≤ ε, and we define

P =

[P UUT Inc

], Y = Y.(4.2)

By [36, Lemma 7.5] we have P 0. Then, replacing in (3.20) the system matricesby their augmented counterparts, the corresponding LMI feasibility problem in F hasa solution. Choosing ε = 10−10 yields m = 3, nc = 3, and the resulting third ordercompensator gain

F =

[Ac Bc

Cc Dc

]=

−0.23619 −0.59416 1.85859 −2.31612

0.52718 −0.14806 1.38309 −1.58075−0.51506 −0.91840 −0.64521 2.59614−0.05706 −0.01872 0.55757 −1.17127

.

The real parts of closed loop eigenvalues of AF ranges between−0.00963 and−0.02874.

Moreover, Tr(BT0FP B0F ) = 6.032201 and ||Tzw1

||∞ = 46.46845 < γ, which shows nu-

merically that F is a suboptimal third order H2/H∞ output feedback compensatorgain. Summing up, this example demonstrates that we always can construct from thecomputed solution of Algorithm 1 at least a reduced order compensator gain whichsatisfies the design criteria. Finally, the whole computation time needs 9.62 CPUseconds on a SUN Ultra 60.

REFERENCES

[1] K. P. Bennett and O. L. Mangasarian, Bilinear separation of two sets in n–space, Comput.Optim. Appl., 2 (1993), pp. 207–227.

[2] D. S. Bernstein and W. M. Haddad, LQG control with an H∞ performance bound: A Riccatiequation approach, IEEE Trans. Automat. Control, 34 (1989), pp. 293–305.

[3] S. P. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan, Linear Matrix Inequalities inSystem and Control Theory, SIAM Stud. Appl. Math. 15, SIAM, Philadelphia, 1994.

[4] M. Chilali and P. Gahinet, H∞ design with pole placement constraints: An LMI approach,IEEE Trans. Automat. Control, 41 (1996), pp. 358–367.

[5] J. Doyle, A. Packard, and K. Zhou, Review of LFTs, LMIs and µ, in Proceedings of the30th Conference on Decision and Control, Brighton, England, 1991, pp. 1227–1232.

[6] J. Doyle, K. Zhou, K. Glover, and B. Bodenheimer, Mixed H2 and H∞ performanceobjectives II: Optimal control, IEEE Trans. Automat. Control, 39 (1994), pp. 1575–1587.

[7] J. C. Doyle, K. Glover, P. P. Khargonekar, and B. A. Francis, State-space solutionsto standard H2 and H∞ control problems, IEEE Trans. Automat. Control, 34 (1989), pp.831–847.

[8] L. El Ghaoui and P. Gahinet, Rank minimization under LMI constraints: A framework foroutput feedback problems, in Proceedings of the European Control Conference, Groningen,The Netherlands, 1993, pp. 1176–1179.

[9] L. El Ghaoui, F. Oustry, and M. AitRami, A cone complementarity linearization algorithmfor static output feedback and related problems, IEEE Trans. Automat. Control, 42 (1997),pp. 1171–1176.

[10] B. Francis, A Course in H∞ Control Theory, Lecture Notes in Control and Inform. Sci. 88,Springer-Verlag, New York, London, Paris, 1987.

[11] P. Gahinet, Explicit controller formulas for LMI-based H∞ synthesis, Automatica J. IFAC,32 (1996), pp. 1007–1014.

[12] P. Gahinet and P. Apkarian, A linear matix inequality approach to H∞ control, Internat.J. Robust Nonlinear Control, 4 (1994), pp. 421–448.

[13] P. Gahinet and A. Nemirovski, The projective method for solving linear matrix inequalities,Math. Programming, 77 (1997), pp. 163–190.


1734 F. LEIBFRITZ

[14] P. Gahinet, A. Nemirovski, A. J. Laub, and M. Chilali, LMI Control Toolbox; For Usewith MATLAB, The Math Works Inc., Natick, MA, 1995.

[15] D. Gangsaas, K. R. Bruce, J. D. Blight, and U.-L. Ly, Application of modern synthesisto aircraft control: Three case studies, IEEE Trans. Automat. Control, 31 (1986), pp.995–1014.

[16] J. C. Geromel, C. C. de Souza, and R. E. Skelton, LMI numerical solution for outputfeedback stabilization, in Proceedings of the American Control Conference, Baltimore, MD,1994, pp. 40–44.

[17] J. C. Geromel, C. C. de Souza, and R. E. Skelton, Static output feedback controllers:Stability and convexity, IEEE Trans. Automat. Control, 43 (1998), pp. 120–125.

[18] J. C. Geromel and P. B. Gapski, Synthesis of positive real H2 controllers, IEEE Trans.Automat. Control, 42 (1997), pp. 988–992.

[19] J. C. Geromel, P. L. D. Peres, and S. R. Souza, A convex approach to the mixed H2/H∞control problem for discrete-time uncertain systems, SIAM J. Control Optim., 33 (1995),pp. 1816–1833.

[20] A. Giusto, A. Trofino-Neto, and E. B. Castelan, H∞ and H2 design techniques for aclass of prefilters, IEEE Trans. Automat. Control, 41 (1996), pp. 864–871.

[21] K. M. Grigoriadis and R. E. Skelton, Low order control design for LMI problems usingalternating projection methods, Automatica J. IFAC, 32 (1996), pp. 1117–1125.

[22] W. M. Haddad, D. S. Bernstein, and Y. W. Wang, Dissipative H2/H∞ controller synthesis,IEEE Trans. Automat. Control, 49 (1994), pp. 827–831.

[23] W. M. Haddad, V. Kapila, and D. S. Bernstein, Robust H∞ stabilization via parameterizedlyapunov bounds, IEEE Trans. Automat. Control, 42 (1997), pp. 241–248.

[24] R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge University Press, Cambridge,UK, 1985.

[25] T. Iwasaki and R. E. Skelton, All controllers for the general H∞ control problem: LMIexistence conditions and state space formulas, Automatica J. IFAC, 30 (1994), pp. 1307–1317.

[26] T. Iwasaki and R. E. Skelton, The XY-centering algorithm for the dual LMI problem: Anew approach to fixed-order control design, Internat. J. Control, 62 (1995), pp. 1257–1272.

[27] I. Kaminer, P. P. Khargonekar, and M. A. Rotea, Mixed H2/H∞ control for discrete timesystems via convex optimization, Automatica J. IFAC, 29 (1993), pp. 57–70.

[28] L. H. Keel, S. P. Bhattacharyya, and J. W. Howze, Robust control with structured pertur-bations, IEEE Trans. Automat. Control, 33 (1988), pp. 68–77.

[29] P. P. Khargonekar and M. A. Rotea, Mixed H2/H∞ control: A convex optimization ap-proach, IEEE Trans. Automat. Control, 36 (1991), pp. 824–837.

[30] F. Leibfritz, Static Output Feedback Design Problems, Shaker Verlag, Aachen, Germany, 1998.[31] F. Leibfritz, Computational Design of Stabilizing Static Output Feedback Controllers, Tech-

nical Report Trierer Forschungsberichte Mathematik/Informatik 99–01, Universitat Trier,Trier, Germany, 1999.

[32] F. Leibfritz, Static Output Feedback Design by Using a Newton-SQP Interior Point Method,Technical Report Trierer Forschungsberichte Mathematik/Informatik 99–03, UniversitatTrier, Trier, Germany, 1999.

[33] D. J. N. Limebeer, B. D. O. Anderson, and B. Hendel, A Nash game approach to mixedH2/H∞ control, IEEE Trans. Automat. Control, 39 (1994), pp. 69–82.

[34] D. G. Luenberger, Introduction to Linear and Nonlinear Programming, Addison-Wesley,Menlo Park, London, Sydney, 1973.

[35] D. Mustafa, Combined H∞/LQG control via the optimal projection equations: On minimizingthe LQG cost bound, Internat. J. Robust Nonlinear Control, 1 (1991), pp. 99–109.

[36] A. Packard, K. Zhou, P. Pandey, and G. Becker, A collection of robust control problemsleading to LMIs, in Proceedings of the 30th Conference on Decision and Control, Brighton,England, 1991, pp. 1245–1250.

[37] M. A. Rotea, The generalized H2 control problem, Automatica J. IFAC, 29 (1993), pp. 373–385.

[38] M. A. Rotea and P. P. Khargonekar, H2-optimal control with an H∞-constraint: The statefeedback case, Automatica J. IFAC, 27 (1991), pp. 307–316.

[39] M. A. Rotea and P. P. Khargonekar, Generalized H2/H∞ control, in Robust ControlTheory, IMA Vol. Math. Appl. 66, B. A. Francis and P. P. Khargonekar, eds., Springer-Verlag, New York, 1995, pp. 81–103.

[40] A. Saberi, B. M. Chen, S. Peddapullaiah, and U.-L. Ly, Simultaneous H2/H∞ optimalcontrol: The state feedback case, Automatica J. IFAC, 29 (1993), pp. 1611–1614.



[41] S. N. Singh and A. R. Coelho, Nonlinear control of mismatched uncertain linear systems andapplication to control of aircraft, Journal of Dynamic Systems, Measurement and Control,106 (1984), pp. 203–210.

[42] R. E. Skelton, J. Stoustrup, and T. Iwasaki, The H∞ control problem using static outputfeedback, Internat. J. Robust Nonlinear Control, 4 (1994), pp. 449–455.

[43] V. L. Syrmos, C. T. Abdallah, P. Dorato, and K. Grigoriadis, Static output feedback—asurvey, Automatica J. IFAC, 33 (1997), pp. 125–137.

[44] I. Yaesh and U. Shaked, Minimum entropy static output feedback control with an H∞ normperformance bound, IEEE Trans. Automat. Control, 42 (1997), pp. 853–858.

[45] H.-H. Yeh, S. S. Banda, and B.-C. Chang, Necessary and sufficient conditions for mixed H2

and H∞ optimal control, IEEE Trans. Automat. Control, 37 (1992), pp. 355–358.[46] K. Zhou, K. Glover, B. Bodenheimer, and J. Doyle, Mixed H2 and H∞ performance

objectives I: Robust performance analysis, IEEE Trans. Automat. Control, 39 (1994), pp.1564–1574.

[47] K. Zhou and P. P. Khargonekar, An algebraic Riccati equation approach to H∞ optimiza-tion, Systems Control Lett., 11 (1988), pp. 85–91.



Information Sciences 179 (2009) 3041–3058


Contents lists available at ScienceDirect

Information Sciences

journal homepage: www.elsevier .com/locate / ins

H1 control design for fuzzy discrete-time singularly perturbed systemsvia slow state variables feedback: An LMI-based approach

Jiuxiang Dong, Guang-Hong Yang *

College of Information Science and Engineering, Northeastern University, Shenyang 110004, PR ChinaKey Laboratory of Integrated Automation of Process Industry (Ministry of Education), Northeastern University, Shenyang 110004, PR China

a r t i c l e i n f o

Article history:Received 15 May 2008Received in revised form 6 February 2009Accepted 16 March 2009

Keywords:Takagi–Sugeno (T–S) fuzzy systemsDiscrete-time systemsNonlinear singularly perturbed systemsH1 controlLinear matrix inequalitiesReduced-order control

0020-0255/$ - see front matter 2009 Elsevier Incdoi:10.1016/j.ins.2009.03.012

* Corresponding author. Address: College of Info13504182968; fax: +86 24 83681939.

E-mail addresses: [email protected](G.-H. Yang).

a b s t r a c t

This paper addresses the H1 control problem via slow state variables feedback for discrete-time fuzzy singularly perturbed systems. At first, a method of evaluating the upper boundof singular perturbation parameter with meeting a prescribed H1 performance boundrequirement is given. Subsequently, two methods for designing H1 controllers via slowstate variables feedback are presented in terms of solutions to a set of linear matrixinequalities (LMIs). In particular, one of them can be used to improve the upper boundof the singular perturbation parameter . Finally, two numerical examples are given toillustrate the effectiveness of the proposed methods.

2009 Elsevier Inc. All rights reserved.

1. Introduction

Slow and fast dynamic phenomena in control systems often occur due to the presence of small ‘‘parasitic” parameters,such as motor control systems, electronic circuits, magnetic-ball suspension systems, and so on. In a state space framework,such systems are commonly modeled by a mathematical description of singular perturbations, where a small parameter isexploited to determine the degree of separation between slow and fast parts of the dynamical system. Due to the very smallsingular perturbation parameter , the analysis and synthesis approaches for normal systems often lead to ill-conditionedresults. Therefore, a so-called reduction technique with a two-step design methodology [17] is widely adopted for overcom-ing the difficulty. Firstly, through the separate stabilization of two lower dimensional subsystems in two different timescales, a composite stabilizing controller is synthesized from separate stabilizing controllers of the two subsystems, wherethe controller could be determined without the knowledge of the small singular perturbation parameter. In the past severaldecades, many control problems of singularly perturbed systems have attracted considerable attentions, see the survey pa-per [23] and the references therein.

In control theory, a well-known convex optimization technique, i.e., linear matrix inequality (LMI) technique, has beenextensively exploited to solve control problems [3]. In contrast to Riccati approaches, linear matrix inequalities (LMIs)can be formulated as convex optimization problems that are amenable to compute solution and can be solved effectively[3]. Another good feature of LMIs is their ability of adding constraints to the parametrical optimization problem provided

. All rights reserved.

rmation Science and Engineering, Northeastern University, Shenyang 110004, PR China. Tel.: +86

, [email protected] (J. Dong), [email protected], [email protected]





http://www.sciencedirect.com/science/journal/00200255

http://www.elsevier.com/locate/ins

3042 J. Dong, G.-H. Yang / Information Sciences 179 (2009) 3041–3058


they are themselves linear with respect to unknowns [12]. In particular, for H1 synthesis, it has the merit of eliminating theregularity restrictions attached to the Riccati-based solutions [11]. Motivated by the merits of the LMI formulations, someLMI-based controller design approaches for singularly perturbed systems have been developed in [7–10,28] recently.

On the other hand, there has been a great deal of interest in using Takagi–Sugeno (T–S) fuzzy models to approximate non-linear systems, and many control problems of nonlinear systems have been widely studied based on T–S fuzzy systems, see[14,15,26,29,30] and the references therein. In particular, the controller design conditions for the state feedback [19], staticoutput feedback [21], and dynamic output feedback [2] cases are exploited in terms of solutions of LMIs for fuzzy singularlyperturbed systems. Moreover, the methods of designing H1 state feedback controllers with pole placement constraints aregiven in [1]. In most cases, the ‘‘fast” dynamics of singularly perturbed systems are not adequately modeled and are, therefore,neglected in order to simplify the design [24,6]. In practice the fast variables are not directly measurable sometimes, for exam-ple, the flexible variables (modeled as fast variables) of the flexible link manipulators [22]. Therefore, the study on the prob-lem of designing state feedback controllers by only using slow state variables of singularly perturbed systems is necessary.

Because most of the existing synthesis techniques for singularly perturbed systems are independent of for avoiding toobtain ill-conditioned results, it is of great importance to find the bound of for ensuring the stability of the closed-loopsystems. As a result, it has attracted increasing interest in the past several decades. In [5], an approach to characterizeand compute the stability bound is presented for continuous-time singularly perturbed systems. By considering criticalstability criteria with a bialternate product, systematic approaches to determine the exact stability bound of discrete-timesingularly perturbed systems are given in [13,18]. Moreover, an algorithm for finding the upper bound of the singular per-turbation parameter for D-stability is presented in [16]. However, by the authors’ knowledge, the topic of evaluating theupper bounds of singular perturbation parameters for nonlinear discrete-time singularly perturbed systems with meetingH1 performance requirements has not been studied. In this paper, the topic will be addressed by using the two lemmas thatare proposed for linear singularly perturbed systems in [8], which are given in Appendices A and B.

In this paper, a method of evaluating the upper bound of the singular perturbation parameter for discrete-time fuzzysingularly perturbed systems with meeting a prescribed H1 performance bound requirement is given. Furthermore, twoH1 controller design methods via slow state variables feedback are presented in terms of solutions to a set of LMIs. In contrastto the conventional design methods [23], the new design methods are with twofold advantages. One is that the two designmethods are based on LMIs, which can eliminate the regularity restrictions attached to the Riccati-based solution. The otheris that one of the two methods can be used to improve the upper bound of singularly perturbed parameter at the stage ofcontrol design, which implies that the tradeoff between the H1 performance index and the upper bound of the singularperturbation parameter is considered in the design. Thus, the new controller design method can overcome the disadvantagethat the allowable upper bound of the singular perturbation parameter of the closed-loop system with the controllersdesigned by the existing ones is too small to be used. This paper is organized as follows. Section 2 presents system descriptionand some preliminaries. In Section 3, a sufficient condition is derived for evaluating the upper bound of subject to a pre-scribed H1 performance constraint. Moreover, two new LMI-based H1 controller design methods are presented. In particular,one of them can improve the upper bound of the singular perturbation parameter by designing controllers. The validity ofthese approaches is illustrated by two numerical examples in Section 4. Finally, Section 5 concludes the paper.

Notation: Rn denotes the set which consists of real n-vectors (n 1 matrices). For a symmetric block matrix, (*) is used forthe blocks induced by symmetry, for example,

M11 M21 M22 M31 M32 M33

264375 ¼ M11 MT

21 MT31

M21 M22 MT32

M31 M32 M33

264375

The superscript T stands for matrix transposition and the notation MT denotes the transpose of the inverse matrix of M.

2. System description and some preliminaries

A class of nonlinear singularly perturbed systems under consideration are described by the following fuzzy system model:

Plant Rule i :

IF v1ðkÞ is Mi1 and v2ðkÞ is Mi2; . . . ; vpðkÞ is Mip; THEN

x1ðkþ 1Þ ¼ Ai11x1ðkÞ þ Ai

12x2ðkÞ þ Biw1wðkÞ þ Bi

u1uðkÞx2ðkþ 1Þ ¼ Ai

21x1ðkÞ þ Ai22x2ðkÞ þ Bi

w2wðkÞ þ Biu2uðkÞ

zðkÞ ¼ Ciz1x1ðkÞ þ Ci

z2x2ðkÞ þ DizwwðkÞ þ Di

zuuðkÞ

ð1Þ

where i ¼ 1;2; . . . ; r, r is the number of IF–THEN rules. Mil, 1 6 i 6 r, 1 6 l 6 p are fuzzy sets. v iðkÞ are the premise variables,x1ðkÞ 2 Rn1 and x2ðkÞ 2 Rn2 are respectively the slow and fast state vectors, uðkÞ 2 Rnu is the control input, wðkÞ 2 Rnw is thedisturbance, zðkÞ 2 Rnz is the controlled output, the matrices Ai

11, Ai12, Ai

21, Ai22, Bi

w1, Biw2, Bi

u1, Biu2, Ci

z1, Ciz2, Di

zw and Dizu are of

appropriate dimensions. > 0 is a singular perturbation parameter, which determines the degree of separation betweenthe ‘‘slow” and ‘‘fast” modes of the system [1].

J. Dong, G.-H. Yang / Information Sciences 179 (2009) 3041–3058 3043


Denote

-iðvðkÞÞ ¼Yp

j¼1

lijðv jðkÞÞ

lijðv jðkÞÞ is the grade of membership of v jðkÞ in Mij, where it is assumed that
Xr
i¼1

-iðvðkÞÞ > 0; -iðvðkÞÞP 0 i ¼ 1;2; . . . ; r

Let aiðvðkÞÞ ¼-iðvðkÞÞPri¼1-iðvðkÞÞ

, then

0 6 aiðvðkÞÞ 6 1; andXr

i¼1

aiðvðkÞÞ ¼ 1 ð2Þ

aiðvðkÞÞ, i ¼ 1; . . . ; r are said to be normalized membership functions. Then, the T–S fuzzy model of (1) is inferred as follows:

x1ðkþ 1Þ ¼Xr

i¼1

aiðvðkÞÞ Ai11x1ðkÞ þ Ai


u1uðkÞ

x2ðkþ 1Þ ¼Xr

i¼1

aiðvðkÞÞ Ai21x1ðkÞ þ Ai


u2uðkÞ

zðkÞ ¼Xr

i¼1

aiðvðkÞÞ Ciz1x1ðkÞ þ Ci

z2x2ðkÞ þ DizwwðkÞ þ Di

zuuðkÞ

ð3Þ

which can be rewritten as follows:

xðkþ 1Þ ¼ AðaðkÞÞExðkÞ þ BwðaðkÞÞwðkÞ þ BuðaðkÞÞuðkÞzðkÞ ¼ CzðaðkÞÞExðkÞ þ DzwðaðkÞÞwðkÞ þ DzuðaðkÞÞuðkÞ

ð4Þ

where

AðaðkÞÞ ¼Xr

i¼1

aiðvðkÞÞAi; BwðaðkÞÞ ¼

Xr

i¼1

aiðvðkÞÞBiw;

BuðaðkÞÞ ¼Xr

i¼1

aiðvðkÞÞBiu; CzðaðkÞÞ ¼

Xr

i¼1

aiðvðkÞÞCiz;

DzwðaðkÞÞ ¼Xr

i¼1

aiðvðkÞÞDizw; DzuðaðkÞÞ ¼

Xr

i¼1

aiðvðkÞÞDizu

xðkÞ ¼x1ðkÞx2ðkÞ

E ¼

In1n1 00 In2n2

Ai ¼

Ai11 Ai

12

Ai21 Ai

22

" #Bi

w ¼Bi

w1

Biw2

" #

Biu ¼

Biu1

Biu2

" #Ci

z ¼ ½Ciz1Ci

z2

ð5Þ

In this paper, the concept of parallel distributed compensation (PDC) is used to design fuzzy controllers, i.e., the designedfuzzy controller shares the same fuzzy sets with the fuzzy model in the premise parts. The more details can be found in[25]. For the fuzzy model (1), the following slow state feedback controller is adopted, where Ki

a, 1 6 i 6 r are the parametersto be designed.

Control Rule i :

IF v1ðkÞ is Mi1 and v2ðkÞ is Mi2; . . . ;vpðkÞ is Mip

THEN uðkÞ ¼ Kiax1ðkÞ ð6Þ

Because the control rules are the same as the plant rules, the fuzzy controller can be obtained as follows:

uðkÞ ¼Xr

i¼1

aiðvðkÞÞKiax1ðkÞ ¼

Xr

i¼1

aiðvðkÞÞKi x1ðkÞx2ðkÞ

ð7Þ

where

Ki ¼ Kia 0

ð8Þ



Combining (7) and (4), then the resulting closed-loop system is given as follows:

xðkþ 1Þ ¼ AðaðkÞÞE þ BuðaðkÞÞKðaðkÞÞð ÞxðkÞ þ BwðaðkÞÞwðkÞzðkÞ ¼ CzðaðkÞÞE þ DzuðaðkÞÞKðaðkÞÞð ÞxðkÞ þ DzwðaðkÞÞwðkÞ

ð9Þ

where

KðaðkÞÞ ¼Xr

i¼1

aiðvðkÞÞ Kia 0

ð10Þ

The replacement of x2 by n2 in system (9) will result in the equivalent system

nðkþ 1Þ ¼ E AðaðkÞÞ þ BuðaðkÞÞKðaðkÞÞð ÞnðkÞ þ EBwðaðkÞÞwðkÞzðkÞ ¼ CzðaðkÞÞ þ DzuðaðkÞÞKðaðkÞÞð ÞnðkÞ þ DzwðaðkÞÞwðkÞ

ð11Þ

where

nðkÞ ¼x1ðkÞn2ðkÞ
The H1 norm in [20] for nonlinear discrete-time systems is applicable for nonlinear discrete-time singularly perturbed sys-tems. The definition is given as follows:
Definition 1 [20]. Given a real number c > 0, it is said that the H1 norm of the closed-loop system (11) is less than or equalto c (i.e., the exogenous signals are locally attenuated by c) if there exists a neighborhood U of x ¼ 0 such that for everypositive integer N and for every w 2 l2ð½0;NÞ;Rnw Þ for which the state trajectory of the closed-loop system (11) startingxð0Þ ¼ 0 remains in U for all k 2 ½0;N, the response z 2 l2ð½0;N;Rnz Þ of (11) satisfies
XN
i¼0

kzkk26 c2

XN

i¼0

kwkk2; for all N

In this paper, the following problems will be addressed.

2.1. Evaluation of the upper bound of with meeting stability and H1 performance bound requirement

Let c > 0 be a given constant and the gains Ki be given. Find an > 0 as big as possible such that the system (1) with (7) isasymptotically stable and its H1-norm is less than or equal to c for any 2 ð0; .

2.2. H1 controller designs without the consideration of improving the upper bound of

Let c > 0 be a given constant. Find gains Ki ði ¼ 1; . . . ; rÞ, and there exists a positive scalar , such that the system (1) with(7) is asymptotically stable and its H1-norm is less than or equal to c for any 2 ð0; .

2.3. H1 controller design with the consideration of improving the upper bound of

Let c > 0 be a given constant and > 0 be a prescribed upper bound of the singular perturbation parameter . Find gainsKi

a ði ¼ 1; . . . ; rÞ and an > 0 such that the system (1) with (7) is asymptotically stable and its H1-norm is less than or equalto c for any 2 ð0; .

The following lemmas will be used in this sequel.

Lemma 2. If there exists a symmetric positive-definite matrix PðaðkÞÞ such that the following LMIs hold,

U11ðk; kþ 1Þ U21ðk; kþ 1Þ U22ðk; kþ 1Þ

< 0; f or 2 ð0; ð12Þ

where AðaðkÞÞ, BwðaðkÞÞ, BuðaðkÞÞ, CzðaðkÞÞ, DzwðaðkÞÞ and DzuðaðkÞÞ are the same as in (5), KðaðkÞÞ is the same as in (10), and

U11ðk; kþ 1Þ ¼ PðaðkÞÞ þ AðaðkÞÞ þ BuðaðkÞÞKðaðkÞÞð ÞT

EPðaðkþ 1ÞÞE AðaðkÞÞ þ BuðaðkÞÞKðaðkÞÞð Þ

þ 1c

CzðaðkÞÞ þ DzuðaðkÞÞKðaðkÞÞð ÞT

CzðaðkÞÞ þ DzuðaðkÞÞKðaðkÞÞð ÞU21ðk; kþ 1Þ ¼ BT

wðaðkÞÞEPðaðkþ 1ÞÞE AðaðkÞÞ þ BuðaðkÞÞKðaðkÞÞð Þ

þ 1c

DTzwðaðkÞÞ CzðaðkÞÞ þ DzuðaðkÞÞKðaðkÞÞð Þ

U22ðk; kþ 1Þ ¼ BTwðaðkÞÞEPðaðkþ 1ÞÞEBwðaðkÞÞ þ

1c

DTzwðaðkÞÞDzwðaðkÞÞ cI

ð13Þ



Then for each singular perturbation parameter 2 ð0; , the closed-loop system (11) is asymptotically stable and its H1 norm isless than or equal to c.

Proof. See Appendix A. h

Lemma 3 [8]. For a given positive scalar , if the following conditions are satisfied,

a P 0 ð14Þa2 þ b þ c < 0 ð15Þc < 0 ð16Þ

where a, b and c are constants, then

a2 þ bþ c < 0; f or 2 ½0; ð17Þ

Proof. See Appendix B. h

Lemma 4 [8]. For a given positive scalar and matrices T1, T2, T3, if following conditions are satisfied,

T1 P 0 ð18Þ2T1 þ T2 þ T3 < 0 ð19ÞT3 < 0 ð20Þ

then

2T1 þ T2 þ T3 < 0; f or 2 ½0; ð21Þ

Proof. See Appendix C. h

3. Main results

In this section, a method of evaluating the upper bound of singularly perturbed parameter subject to the stability of theclosed-loop system with meeting H1 performance bound requirements is presented. Moreover, two sufficient conditions fordesigning H1 controllers are given. In particular, one of them can improve the upper bound of the singular perturbationparameter by designing controllers.

3.1. Computation of stability bound of subject to an H1 performance bound constraint

Firstly, the following preliminary lemma is needed.

Lemma 5. If there exists a symmetric matrix

QðaðkÞÞ ¼ Q 11ðaðkÞÞ QT21ðaðkÞÞ

Q 21ðaðkÞÞ Q22ðaðkÞÞ

" #;

such that the following inequalities hold, " #
2
0 00 Q 22ðaðkÞÞ

0 0 0 0 0 0 0 0 0

2666664

3777775þ 0 Q T

21ðaðkÞÞQ21ðaðkÞÞ 0

0 0 0 0 0 0 0 0 0

26666664

37777775

þ

W11ðk; kþ 1Þ 0 cI

W31ðk; kþ 1Þ BwðaðkÞÞ Qðaðkþ 1ÞÞÞ W41ðk; kþ 1Þ DzwðaðkÞÞ 0 cI

2666437775 < 0; f or 2 ð0; ð22Þ

where
W11ðkÞ ¼
Q11ðaðkÞÞ 00 0

SðaðkÞ;aðkþ 1ÞÞ STðaðkÞ;aðkþ 1ÞÞ

W31ðkÞ ¼ AðaðkÞÞ þ BuðaðkÞÞKðaðkÞÞð ÞSðaðkÞ;aðkþ 1ÞÞW41ðkÞ ¼ CzðaðkÞÞ þ DzuðaðkÞÞKðaðkÞÞð ÞSðaðkÞ;aðkþ 1ÞÞ

ð23Þ

then for each singular perturbation parameter 2 ð0; , the system (9) is asymptotically stable and its H1 norm is less than orequal to c.



Proof. See Appendix D. h

Based on Lemma 5, a method of evaluating the upper bound of the singular perturbation parameter with a prescribedH1 performance bound constraint is given in the following theorem.

Theorem 6. For a given positive scalar , if there exist matrices Q i ¼ ðQiÞT , Sij, 1 6 i; j 6 r

Q i ¼Q i

11 ðQ i21Þ

T

Q i21 Q i

22

" #; Sij ¼

S11 0Sij

21 Sij22
satisfying the following LMIs
Miil < 0; 1 6 i; l 6 r ð24aÞ1

r 1Miil þ

12ðMijl þMjilÞ < 0; 1 6 i – j 6 r; 1 6 l 6 r ð24bÞ

Kiil < 0; 1 6 i; l 6 r ð25aÞ1

r 1Kiil þ

12ðKijl þKjilÞ < 0; 1 6 i – j 6 r; 1 6 l 6 r ð25bÞ

where

Mijl ¼

Q i11 0

0 0

" # Sil ðSilÞT

0 cI

Ai þ BiuKj

Sjl Bi

w Ql

Ciz þ Di

zuKj

Sjl Dizw 0 cI

266666666664

377777777775

Kijl ¼ 2

0 0

0 Q i22

" #0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

266666664

377777775þ

0 ðQ i21Þ

T

Q i21 0

" #0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

2666666664

3777777775

þ

Q i11 0

0 0

" # Sil ðSilÞT

0 cI

Ai þ BiuKj

Sjl Bi

w Ql

Ciz þ Di

zuKj

Sjl Dizw 0 cI

266666666664

377777777775
then for each singular perturbation parameter 2 ð0; , the system (9) is asymptotically stable and its H1 norm is less than orequal to c.
Proof. See Appendix E. h

Remark 7. Theorem 6 presents a method of estimating the upper bound of singularly perturbed parameter subject to thestability of the closed-loop system (9) while satisfying an H1 performance bound requirement. An upper bound of can beobtained by solving the following optimization problem:

Minimize subject to ð24Þ and ð25Þ

which can be effectively solved by using the LMI Control Toolbox [11].

3.2. H1 controller design

In this subsection, two LMI-based methods of designing H1 controllers are given. The two methods are with the merit ofeliminating the regularity restrictions attached to the Riccati-based solutions. In particular, one of them can improve theupper bound of by designing controllers.



3.2.1. Design without the consideration of improving the upper bound of In the following, an LMI-based design method without considering the improvement of the upper bound of the singular

perturbation parameter is given.

Theorem 8. If there exist matrices Qi ¼ ðQiÞT , Sil, Li, 1 6 i; l 6 r, with

Q i ¼Q i

11 ðQ i21Þ

T

Q i21 Q i

22

" #; Sil ¼

S11 0Sil

21 Sil22

; Li ¼ Li

a 0

ð26Þ

satisfying the following LMIs,

Caiil < 0; 1 6 i; l 6 r ð27aÞ1

r 1Caiil þ

12ðCaijl þ CajilÞ < 0; 1 6 i – j 6 r; 1 6 l 6 r ð27bÞ

where

Caijl ¼

Q i11 0

0 0

" # Sil ðSilÞT

0 cI AiSjl þ Bi

uLj Biw Q l

CizSjl þ Di

zuLj Dizw 0 cI

266666664

377777775 < 0; f or 1 6 i; j; l 6 r

then there exists a sufficient small > 0 such that for 2 ð0; , the closed-loop system (9) with

Kia ¼ Li

aS111 ; 1 6 i 6 r ð28Þ

is asymptotically stable and its H1 norm is less than or equal to c.

Proof. The proof is easily obtained from Theorem 6 and omitted. h

Remark 9. Theorem 8 presents a sufficient condition for designing H1 controllers for discrete-time fuzzy singularly per-turbed systems. The method is based on LMIs, and with the merit of eliminating the regularity restrictions attached tothe Riccati-based solutions. Example 12 in Section 4 will illustrate the effectiveness of the method. However, in the method,the issue of improving the upper bound of the singular perturbation parameter is not addressed. As a result, the obtainedcontroller might give a very small stability bound so that the resulting closed-loop system is unstable for a practical singularperturbation parameter , see Example 14 in Section 4. In order to overcome the difficulty, another new method with theconsideration of improving the upper bound of the singular perturbation parameter while satisfying H1 performance con-straints will be proposed in the next subsection.

3.2.2. Design with the consideration of improving the upper bound of In this subsection, a new LMI-based H1 controller design method with the consideration of improving the upper bound of

the singular perturbation parameter is given as follows:

Theorem 10. For a given positive scalar , if there exist matrices Q i ¼ ðQiÞT , Sil, Li, 1 6 i; l 6 r, with

Q i ¼Q i

11 ðQ i21Þ

T

Q i21 Q i

22

" #; Sil ¼

S11 0Sil

21 Sil22

; Li ¼ Li

a 0

satisfying (27b) and the following LMIs,

Cbiil < 0; 1 6 i; l 6 r ð29aÞ1

r 1Cbiil þ

12ðCbijl þ CbjilÞ < 0; 1 6 i – j 6 r; 1 6 l 6 r ð29bÞ

where

Cbijl ¼ 2

0 00 Q i

22

0 0 0

0 0 0 00 0 0 00 0 0 0

2666664

3777775þ 0 ðQi

21ÞT

Q i21 0

" #0 0 0

0 0 0 00 0 0 00 0 0 0

26666664

37777775þQ i

11 00 0

" # Sil ðSilÞT

0 cI AiSjl þ Bi

uLj Biw Q l

CizSjl þ Di

zuLj Dizw 0 cI

266666664

377777775
Then for 2 ð0; , the closed-loop system (9) with (28) is asymptotically stable and its H1 norm is less than or equal to c.



Proof. The proof is easily obtained from Theorem 6 and omitted. h

Remark 11. In this section, Theorem 6 provides an LMI-based condition for estimating the upper bound of singularly per-turbed parameter of discrete-time singularly perturbed systems subject to H1 performance bound constraints. In contrastto the existing techniques for estimating stability bounds, the new method can be used to estimate the bound of singularlyperturbed parameter subject to H1 performance bound constraints. Moreover, two controller design methods are respec-tively given by Theorems 8 and 10. The two methods can eliminate the regularity restrictions attached the existing Riccati-based solution [23], the fact is shown by Example 12. In particular, the method given by Theorem 10 can be used to improvethe upper bound of singularly perturbed parameter by designing controllers, which is illustrated by Example 14.

4. Example

In Example 12, a discrete-time fuzzy singularly perturbed system is obtained by discretizing a tunnel diode circuit, whichis borrowed from [2]. Because the example does not satisfy the regularity property, the conventional Riccati-based method isnot applicable. The new LMI-based method given by Theorem 8 will be applied to the example for illustrating itseffectiveness.

Moreover, Example 14 is given for better illustrating the effectiveness of the method given by Theorem 10 (i.e., the H1controller design method with the consideration of improving the upper bound of singularly perturbed parameter ).

Example 12. Consider a tunnel diode circuit (Fig. 1), where the diode current is iDðtÞ and the diode voltage vDðtÞ, and theysatisfy that

iDðtÞ ¼ 0:2vDðtÞ 0:05v3DðtÞ

Moreover, C, R and L denote the capacitor, the resistance and the inductance, respectively. vCðtÞ, vRðtÞ and vLðtÞ are thecapacitor voltage, the resistance voltage and the inductor voltage, respectively. iCðtÞ, iRðtÞ and iLðtÞ are the capacitor current,the resistance current and the inductor current, respectively. uðtÞ is the input voltage, wðtÞ is the disturbance. ConsideringFig. 1 and applying the Kirchoff voltage and current law, we have that

CdðvCðtÞÞ

dt¼ icðtÞ; RiLðtÞ ¼ vRðtÞ

LdðiLðtÞÞ

dt¼ vLðtÞ; iCðtÞ ¼ iLðtÞ iDðtÞ

vLðtÞ ¼ uðtÞ vRðtÞ vCðtÞ þwðtÞ

ð30Þ

Let x1ðtÞ ¼ vCðtÞ, x2ðtÞ ¼ iLðtÞ. Combining them and (30), then it follows that

C _x1ðtÞ ¼ 0:2x1ðtÞ þ 0:05x31ðtÞ þ x2ðtÞ

L _x2ðtÞ ¼ x1ðtÞ Rx2ðtÞ þ uðtÞ þwðtÞzðtÞ ¼ x1ðtÞ þ 0:1wðtÞ

ð31Þ

where zðtÞ is the controlled output. Assume that the parameters of the circuit are C ¼ 100 mF, L ¼ 1 mH and R = 20 X. Withthese parameters, (31) can be rewritten as follows:

Fig. 1. The tunnel diode circuit in Example 12.



_x1ðtÞ ¼ 2x1ðtÞ þ 0:5x31ðtÞ þ 10x2ðtÞ

_x2ðtÞ ¼ 0:1x1ðtÞ 2x2ðtÞ þ 0:1uðtÞ þ 0:1wðtÞzðtÞ ¼ x1ðtÞ þ 0:1wðtÞ

ð32Þ

where xðtÞ ¼ ½xT1ðtÞ; xT

2ðtÞT and ¼ 104. Assume that jxðtÞj 6 3 and model the nonlinear system (32) by the sector nonlinear-

ity approach in [14], then the following T–S fuzzy model can be obtained:

E _xðtÞ ¼X2

i¼1

aiðtÞ AicxðtÞ þ Bi

cwwðtÞ þ BicuuðtÞ

zðtÞ ¼

X2

i¼1

aiðtÞ CiczxðtÞ þ Di

czwwðtÞ

where

A1c ¼

2 100:1 2

; A2

c ¼6:9 100:1 2

; B1

cw ¼ B2cw ¼

00:1

B1

cu ¼ B2cu ¼

00:1

; C1

cz ¼ C2cz ¼ ½1 0; D1

czw ¼ D2czw ¼ 0:1; E ¼

1 00
and aiðtÞ is the normalized time-varying fuzzy weighting function for each rule i ¼ 1;2, satisfies a1ðtÞ ¼ 1 x2
1ðtÞ9 ,

a2ðtÞ ¼ 1 a1ðtÞ.

From (32), it can be seen that x2ðtÞ, i.e., the inductor current iLðtÞ, changes very fast and exhibits a large range of variation.Therefore, it is very difficult to measure x2ðtÞ. On the other hand, the slow variable x1ðtÞ changes slowly and exhibits a smallrange of variation, it can be easily measured. Therefore, the slow state feedback controller is needed for this example.

Inhere, we discretize the model with a sampling period T ¼ 0:3 s and a zero-order holder, then the following discrete-time singularly perturbed model is obtained:

xðkþ 1Þ ¼X2

i¼1

aiðkÞ AidExðkÞ þ Bi

dwwðkÞ þ BiduuðkÞ

zðkÞ ¼

X2

i¼1

aiðkÞ CidzxðkÞ þ Di

dzwwðkÞ

where

A1d ¼

1:5683 7:84150:0784 0:3921

; A2

d ¼6:8210 34:10390:3410 1:7051

; B1

dw ¼0:18940:0405

B2

dw ¼0:45470:0273

; B1

du ¼0:18940:0405

; B2

du ¼0:45470:0273

; C1

dz ¼ C2dz ¼ ½1 0;

D1dzw ¼ D2

dzw ¼ 0:1

Due to DdzwðaðkÞÞ– 0, the regularity property is not satisfied, the conventional Riccati-based methods [23] are not applicable.Applying Theorem 8 (the H1 controller design method without the consideration of improving the upper bound of the sin-gular perturbation parameter ), then the following results are obtained

Q 1 ¼0:4220 0:03080:0308 995:8465

; Q 2 ¼

0:4220 0:02930:0293 994:0820

;

L1a ¼ 4:4462; L2

a ¼ 6:5378;

K1a ¼ 10:5348; K2

a ¼ 15:4908; copt ¼ 0:51639

With the obtained controller gains, the upper bound of the singular perturbation parameter of the tunnel diode circuit sys-tem are estimated as ¼ 0:1772 by using Theorem 6 under the H1 performance constraint c ¼ 0:5614. It is bigger than thepractical parameter ¼ 104. Therefore, the designed controller can be used and some simulation results are shown in Figs.

2–4 with an initial state xð0Þ ¼ ½0:51T , and wðkÞ ¼ 1; 5 6 k 6 100; others:

From Figs. 2, and 3, it can be seen that the resulting closed-loop system is asymptotically stable and with a good H1 per-

formance, which further shows the effectiveness of the proposed condition in Theorem 8.

Remark 13. Note that Theorem 8 is applied for designing an H1 controller in Example 12 and the upper bound of singularlyperturbed parameter for the resulting closed-loop system is more than the practical , hence the controller is applicable.Since the controller design condition in Theorem 8 is without considering the tradeoff of the H1 performance bound c andthe stability bound , the obtained controller by Theorem 8 might be not applicable for some singularly perturbed systems.For solving the problem, Theorem 10 can be applied as an alternative, which is shown in Example 14.

0 5 10 15 20 25 30−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Time (sec)

x(k)

x1x2

Fig. 2. Trajectories of xðkÞ in Example 12.

0 5 10 15 20 25 30−0.5

0

0.5

1

1.5

2

2.5

3

3.5

Time (sec)

z(k)

, w(t)

zw

Fig. 3. Trajectories of zðkÞ and wðkÞ in Example 12.

Fig. 4. Membership functions aiðx1ðkÞÞ for Example 14.





Example 14. Consider a numerical example, which is described by (4) with

Table 1Upper b

c ¼ 1:5c ¼ 2c ¼ 2:5

A1 ¼

0:6 1 2 10 2 0 3

0:1 0 0:3 12:5 0:1 0:2 0:4

2666437775; A2 ¼

0:7 1 3 0:80 1 0 2

0:1 0 0:3 18 0:5 0:1 0:6

2666437775

B1w ¼

00:20

0:2

2666437775; B2

w ¼

00:30

0:1

2666437775; B1

u ¼

0:510

0:8

2666437775; B2

u ¼

0:310

0:4

2666437775

C1z ¼ C2

z ¼1 0 0 00 1 0 0

; D1

zw ¼01

; D2

zw ¼0

1:2

; D1

zu ¼ D2zu ¼

00:1

E ¼

1 0 0 00 1 0 00 0 00 0 0

2666437775

where ¼ 0:05 and the membership functions aiðx1ðkÞÞ, i ¼ 1;2 are given in Fig. 4.

Applying Theorem 8 to the example (the H1 controller design method without the consideration of improving the upperbound of the singular perturbation parameter ), we can obtain the following controller gains and the optimal H1 perfor-mance bound.

K1a ¼ ½0:1544 2:1685; K2

a ¼ ½0:0764 1:9145copt ¼ 1:3334

ð33Þ

ounds of .

Theorem 8 Theorem 10

0.0014 0.05270.0049 0.05560.0073 0.0576

0 5 10 15 20 25 30−3

−2

−1

0

1

2

3x 1015

Time (sec)

x(k)

x1x2

Fig. 5. Trajectories of xðkÞ via the gain (33) in Example 14.



Moreover, by using Theorem 10 with ¼ 0:05 (the H1 controller design method with the consideration of improving theupper bound of the singular perturbation parameter ), we can obtain

K1a ¼ ½0:2233 2:1834; K2

a ¼ ½0:7358 1:8368copt ¼ 1:4284

ð34Þ

Then the allowable upper bound of the singular perturbation parameter of the closed-loop system with the controller gain(33) or (34) can be estimated by using Theorem 6. The obtained results are shown in Table 1.

From Table 1, it can be seen that,for the larger H1 performance indices,the larger upper bounds of the singular perturba-tion parameter are achieved by using Theorem 10, which shows that the method of Theorem 10 is effective for improvingthe upper bound of the singular perturbation parameter .

0 5 10 15 20 25 30−6

−4

−2

0

2

4

6

8

Time (sec)

x(k)

x1x2

Fig. 6. Trajectories of xðkÞ via the gain (34) in Example 14.

0 5 10 15 20 25 300

5

10

15

20

25

30

Time (sec)

The

ratio

1.3572

Fig. 7.ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPk

i¼0zT ðiÞzðiÞ=Pk

i¼0wT ðiÞwðiÞq

with the gain (34) in Example 14.



Note that ¼ 0:05, then for the case of a1ðkÞ ¼ 0, a2ðkÞ ¼ 1, the eigenvalues of

AðaðkÞÞE þ BuðaðkÞÞKðaðkÞÞ ¼ A2E þ B2uK2 ¼ A2E þ B2

u½K2a0

with the controller gain (33) designed by using Theorem 8 are 1.1190, 0.6411, 0.4496, 0.1641, which implies that theresulting closed-loop system is unstable. Therefore,the controller cannot be applied. On the other hand, Table 1 showsthe achieved upper bounds of of the closed-loop system with the controller gains (34) under different H1 performancerequirements. The upper bounds are more than ¼ 0:05. The fact shows that Theorem 10 can be used to effectively improvethe upper bound of the singular perturbation parameter at the stage of controller design.

What it follows, some simulation results will be given in order to further validate the effectiveness of Theorem 10. As-sume that the initial state xð0Þ ¼ ½2 8 0 0T and the disturbance

wðkÞ ¼2; 3 6 k 6 100; others
Figs. 5 and 6 show the trajectories of the state xðkÞ of the closed-loop system with the controller gains (33) and (34), respec-
tively. Fig. 7 shows the ratio offfiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPk

i¼0zTðiÞzðiÞ=Pk

i¼0wTðiÞwðiÞq

in the simulation.From Fig. 5, it can be seen that the closed-loop system with the controller gain (33), which is obtained by Theorem 8, is

unstable. It can be seen that from Figs. 6 and 7 that the controller gain (34), which is obtained by Theorem 10, can guaranteethe stability and meet the H1 performance requirement. These simulation results further show the advantage of the methodgiven by Theorem 10, i.e., Theorem 10 can be used to effectively improve the upper bound of the singular perturbationparameter at the stage of controller design.

5. Conclusion

In this paper, the H1 control problem via slow state variables feedback for discrete-time fuzzy singularly perturbed systemshas been investigated. Two LMI-based methods for designing H1 controllers via slow state variables feedback are presented,and one of them can be used to improve the upper bound of the singular perturbation parameter , which can overcome thedisadvantage in the conventional design methods where the designed controller might not be used because the resulting allow-able upper bound of the singular perturbation parameter is too small. Moreover, a method of evaluating the upper bound of asingular perturbation parameter with meeting a prescribed H1 performance bound requirement is given in terms of solutionsto a set of LMIs. The effectiveness of the proposed methods has been illustrated by the numerical examples.

Acknowledgements

This study was supported in part by the Funds for Creative Research Groups of China (No. 60821063), the State Key Pro-gram of National Natural Science of China (Grant No. 60534010), National 973 Program of China (Grant No. 2009CB320604),the Funds of National Science of China (Grant No. 60674021), the 111 Project (B08015) and the Funds of PhD program ofMOE, China (Grant No. 20060145019).

Appendix A. Proof of Lemma 2

Proof. Consider the system (11) (which is equivalent to the system (9)), and let PðaðkÞÞ > 0. Choose Lyapunov function

VðkÞ ¼ cnTðkÞPðaðkÞÞnðkÞ

then

Vðkþ 1Þ VðkÞ þ zTðkÞzðkÞ c2wTðkÞwðkÞ¼ c AðaðkÞÞ þ BuðaðkÞÞKðaðkÞÞð ÞnðkÞ þ BwðaðkÞÞwðkÞð ÞT EPðaðkþ 1ÞÞE AðaðkÞÞ þ BuðaðkÞÞKðaðkÞÞð ÞnðkÞðþ BwðaðkÞÞwðkÞÞ cnTðkÞPðaðkÞÞnðkÞ þ CzðaðkÞÞ þ DzuðaðkÞÞKðaðkÞÞð ÞnðkÞð þ DzwðaðkÞÞwðkÞÞT CzðaðkÞÞððþ DzuðaðkÞÞKðaðkÞÞÞnðkÞ þ DzwðaðkÞÞwðkÞÞ c2wTðkÞwðkÞ

¼ cnðkÞwðkÞ

TU11ðk; kþ 1Þ UT

21ðk; kþ 1ÞU21ðk; kþ 1Þ U22ðk; kþ 1Þ

" #nðkÞwðkÞ

where /11ðk; kþ 1Þ, /21ðk; kþ 1Þ, /22ðk; kþ 1Þ are the same as in (13).From (12) and the above equality, then it follows that

Vðkþ 1Þ VðkÞ þ zTðkÞzðkÞ c2wTðkÞwðkÞ 6 0; for 2 ð0;



From the above inequality, we have that the system (11) with 2 ð0; is asymptotically stable in the disturbance-free case.For the initial condition xð0Þ ¼ 0 and every positive integer N, sum the above inequality from k ¼ 0 to N, then we can obtain

VðNÞ Vð0Þ þX1k¼0

zTðkÞzðkÞ c2X1k¼0

wTðkÞwðkÞ ¼ VðNÞ þX1k¼0

zTðkÞzðkÞ c2X1k¼0

wTðkÞwðkÞ 6 0; for 2 ð0;

which implies that
XN
k¼0

zTðkÞzðkÞ 6 c2XN

k¼0

wTðkÞwðkÞ; for 2 ð0;

Combining it and Definition 1, it follows that the H1 norm of the closed-loop system (11) is less than or equal to c. h

Appendix B. Proof of Lemma 3

Proof. Consider the following two cases:

(i): If a ¼ 0, then from (15) and (16), (17) obviously holds.(ii): If a > 0, we consider the following quadratic function of ,

yðÞ ¼ a2 þ bþ c ð35Þ

Since a > 0, yðÞ is convex function of [4]. From (15) and (16), it follows that yðÞ < 0 and yð0Þ < 0, which further impliesthat yðÞ < 0 for 2 ½0; , i.e., when a > 0, (17) holds. Thus, the proof is complete. h

Appendix C. Proof of Lemma 4

Proof. For all nonzero vector xðkÞ, pre- and post-multiplying (18)–(20) by xTðkÞ and its transpose, then we have

xTðkÞT1xðkÞP 0 ð36Þ

2xTðkÞT1xðkÞ þ xTðkÞT2xðkÞ þ xTðkÞT3xðkÞ < 0 ð37Þ

xTðkÞT3xðkÞ < 0 ð38Þ

Denote axk¼ xTðkÞT1xðkÞ, bxk

¼ xTðkÞT2xðkÞ, cxk¼ xTðkÞT3xðkÞ. Substituting axk

, bxkand cxk

into (36)–(38), then it follows that

axkP 0 ð39Þ

axk2 þ bxk

þ cxk< 0 ð40Þ

cxk< 0 ð41Þ

From (39)–(41) and applying Lemma 3, we can obtain axk2 þ bxk

þ cxk< 0 for 2 ½0; , i.e., for all nonzero vector xðkÞ,

2xTðkÞT1xðkÞ þ xTðkÞT2xðkÞ þ xTðkÞT3xðkÞ < 0; for 2 ½0;

which implies that matrix inequality (21) holds. Thus, the proof is complete. h

Appendix D. Proof of Lemma 5

Proof. (22) can be rewritten as follows,

W11ðkÞ 0 cI

W31ðkÞ BwðaðkÞÞ Qðaðkþ 1ÞÞ W41ðkÞ DzwðaðkÞÞ 0 cI

2666437775 < 0; for 2 ð0; : ð42Þ

where W31ðkÞ, W41ðkÞ are the same as in (23) and

W11ðkÞ ¼ EQðaðkÞÞE SðaðkÞ;aðkþ 1ÞÞ STðaðkÞ;aðkþ 1ÞÞ

where E is same as in (5).



From (42), we can obtain that W11ðkÞ < 0. Considering the block (3,3) of (42), then yields QðaðkÞÞ > 0. Combiningit and W11ðkÞ < 0, then we have SðaðkÞ;aðkþ 1ÞÞ þ STðaðkÞ;aðkþ 1ÞÞ > 0. Pre- and post-multiply (42) by

STðaðkÞ;aðkþ 1ÞÞ 0 0 00 I 0 00 0 I 00 0 0 I

26643775 and

S1ðaðkÞ;aðkþ 1ÞÞ 0 0 00 I 0 00 0 I 00 0 0 I

26643775, respectively. Then it follows that

eW11ðkÞ 0 cI

AðaðkÞÞ þ BuðaðkÞÞKðaðkÞÞ BwðaðkÞÞ Qðaðkþ 1ÞÞ CzðaðkÞÞ þ DzuðaðkÞÞKðaðkÞÞ DzwðaðkÞÞ 0 cI

266664377775 < 0; for 2 ð0; ð43Þ

where
eW11ðkÞ ¼ STðaðkÞ;aðkþ 1ÞÞEQðaðkÞÞES1ðaðkÞ;aðkþ 1ÞÞ S1ðaðkÞ;aðkþ 1ÞÞ STðaðkÞ;aðkþ 1ÞÞ
Let P1ðaðkÞÞ ¼ EQðaðkÞÞE, therefore,

PðaðkÞÞ > 0 ð44Þ
then
STðaðkÞ;aðkþ 1ÞÞ PðaðkÞÞ

P1ðaðkÞÞ S1ðaðkÞ;aðkþ 1ÞÞ PðaðkÞÞ

P 0

which implies that

STðaðkÞ;aðkþ 1ÞÞP1ðaðkÞÞS1ðaðkÞ;aðkþ 1ÞÞ S1ðaðkÞ;aðkþ 1ÞÞ STðaðkÞ;aðkþ 1ÞÞP PðaðkÞÞ

Combining it with (43), we can obtain

PðaðkÞÞ 0 cI

AðaðkÞÞ þ BuðaðkÞÞKðaðkÞÞ BwðaðkÞÞ E1 P1ðaðkþ 1ÞÞE1

CzðaðkÞÞ þ DzuðaðkÞÞKðaðkÞÞ DzwðaðkÞÞ 0 cI

2666437775 < 0; for 2 ð0; ð45Þ

Applying Schur complement lemma to (45), then we have

PðaðkÞÞ 0 cI


264375

þ 1c

CTz ðaðkÞÞ þ KTðaðkÞÞDT

zuðaðkÞÞDT

zwðaðkÞÞ0

264375 CzðaðkÞÞ þ DzuðaðkÞÞKðaðkÞÞ DzwðaðkÞÞ 0½

¼ 11ðkÞ 21ðkÞ 22ðkÞ


2643750; for 2 ð0;

where

11ðkÞ ¼ PðaðkÞÞ þ 1c

CzðaðkÞÞ þ DzuðaðkÞÞKðaðkÞÞð ÞT

CzðaðkÞÞ þ DzuðaðkÞÞKðaðkÞÞð Þ

21ðkÞ ¼1c

DTzwðaðkÞÞ CzðaðkÞÞ þ DzuðaðkÞÞKðaðkÞÞð Þ

22ðkÞ ¼ cI þ 1c

DTzwðaðkÞÞDzwðaðkÞÞ

Applying Schur complement lemma to the above inequality, again, then it follows that

11ðkÞ T21ðkÞ

21ðkÞ 22ðkÞ

þ ATðaðkÞÞ þ KTðaðkÞÞBT

uðaðkÞÞBT

wðaðkÞÞ

EPðaðkþ 1ÞÞE AðaðkÞÞ þ BuðaðkÞÞKðaðkÞÞ BwðaðkÞÞ½

¼ U11ðk; kþ 1Þ UT21ðk; kþ 1Þ

U21ðk; kþ 1Þ U22ðk; kþ 1Þ

< 0; for 2 ð0; ð46Þ

where U11ðk; kþ 1Þ, U21ðk; kþ 1Þ and U22ðk; kþ 1Þ are the same as in (13).



Then applying Lemma 2 to (46), we have that, for each singular perturbation parameter 2 ð0; , the system (11) isasymptotically stable and its H1 norm is less than c. Then the conclusion follows. h

Appendix E. Proof of Theorem 6

Proof. Considering the block (3,3) of (24a) and from (24a), it follows that

Qi > 0 1 6 i 6 r

which also implies that

Qi22 > 0 1 6 i 6 r ð47Þ

Let

T1i ¼

0 00 Q i

22

0 0 0

0 0 0 00 0 0 00 0 0 0

2666664

3777775

T2i ¼

0 ðQ i21Þ

T

Q i21 0

" #0 0 0

0 0 0 00 0 0 00 0 0 0

26666664

37777775T3il ¼ Miil

From (47), it follows that
T1i P 0 ð48aÞ
From (25a), we have

2T1i þ T2i þ T3il < 0 ð48bÞ
From (24a), we have
T3il < 0 ð48cÞ

Applying Lemma 4 to (48a), then yields

2T1i þ T2i þ T3il < 0; for 2 ½0; ; 1 6 i; l 6 r

i.e.,

Kiil < 0; for 2 ½0; ; 1 6 i; l 6 r ð49aÞ

where

Kiil ¼ 2

0 00 Q i

22

0 0 0

0 0 0 00 0 0 00 0 0 0

2666664

3777775þ 0 ðQ i

21ÞT

Q i21 0

" #0 0 0

0 0 0 00 0 0 00 0 0 0

26666664

37777775þQ i

11 00 0

" # Sil ðSilÞT

0 cI Ai þ Bi

uKi

Sil Biw Q l

Ciz þ Di

zuKi

Sil Dizw 0 cI

26666666664

37777777775
Similarly, from (47), (24b) and (25b), we can also obtain
1r 1

Kiil þ12ðKijl þKjilÞ < 0; for 2 ½0; ; 1 6 i – j 6 r; 1 6 l 6 r ð49bÞ

where

Kijl ¼ 2

0 00 Q i

22

0 0 0

0 0 0 00 0 0 00 0 0 0

2666664

3777775þ 0 ðQ i

21ÞT

Q i21 0

" #0 0 0

0 0 0 00 0 0 00 0 0 0

26666664

37777775þQ i

11 00 0

" # Sil ðSilÞT

0 cI AiSjl þ Bi

uKjSjl Biw Q l

CizSjl þ Di

zuKjSjl Dizw 0 cI

266666664

377777775



Applying the parameterized linear matrix inequality (PLMI) technique in [27] to (49a), then yields
Xr
i¼1

Xr

j¼1

aiðkÞajðkÞKijl < 0; 1 6 l 6 r

Multiplying the above inequality by alðkþ 1Þ and summing them from l ¼ 1 to r, then we can obtain
Xr
i¼1

Xr

j¼1

Xr

l¼1

aiðkÞajðkÞalðkþ 1ÞKijl < 0 ð50Þ

Note that

Xr

j¼1

Xr

l¼1

ajðkÞalðkþ 1ÞKjSjl ¼Xr

j¼1

Xr

l¼1

ajðkÞalðkþ 1Þ KjaS11 0Sjl

21 Sjl22

" #¼

Prj¼1

ajðkÞKjaS11 0

Prj¼1

Prl¼1

ajðkÞalðkþ 1ÞSjl21

Prj¼1

Prl¼1


2666437775

¼Prj¼1

ajðkÞKja 0

S11 0Prj¼1

Prl¼1


Prj¼1

Prl¼1


24 35¼ KðaðkÞÞ S11 0

S21ðaðkÞ;aðkþ 1ÞÞ S22ðk; kþ 1Þ

¼ KðaðkÞÞSðaðkÞ;aðkþ 1ÞÞ

where KðaðkÞÞ is the same as in (10) and

SðaðkÞ;aðkþ 1ÞÞ ¼S11 0Pr

j¼1

Prl¼1


Prj¼1

Prl¼1


24 35
Then Xr
i¼1

Xr

j¼1

Xr

l¼1

aiðkÞajðkÞalðkþ 1ÞBiuKjSjl ¼ BuðaðkÞÞKðaðkÞÞSðaðkÞ;aðkþ 1ÞÞ

Xr

i¼1

Xr

j¼1

Xr

l¼1

aiðkÞajðkÞalðkþ 1ÞDizuKjSjl ¼ DzuðaðkÞÞKðaðkÞÞSðaðkÞ;aðkþ 1ÞÞ

Therefore, (50) implies that (22) holds. Combining it and Lemma 5, it follows that for the singular perturbation parameter 2 ð0; , the system (9) is asymptotically stable and its H1 norm is less than c. Then the conclusion follows. h

References

[1] W. Assawinchaichote, S.K. Nguang, H1 fuzzy control design for nonlinear singularly perturbed systems with pole placement constraints: an LMIapproach, IEEE Transactions on Systems, Man and Cybernetics, Part B 34 (1) (2004) 579–588.

[2] W. Assawinchaichote, S.K. Nguang, P. Shi, H1 output feedback control design for uncertain fuzzy singularly perturbed systems: an LMI approach,Automatica 40 (12) (2004) 2147–2152.

[3] S. Boyd, L.E. Ghaoui, E. Feron, V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory, Society for Industrial and Applied Mathematics(SIAM), Philadelphia (PA), 1994.

[4] S. Boyd, L. Vandenberghe, Convex Optimization, Cambridge University Press, 2004.[5] L. Cao, H.M. Schwartz, Complementary results on the stability bounds of singularly perturbed systems, IEEE Transactions on Automatic Control 49 (11)

(2004) 2017–2021.[6] Y. Chen, Y. Liu, Summary of singular perturbation modeling of multi-time scale power systems, in: Transmission and Distribution Conference and

Exhibition, IEEE/PES, Asia and Pacific, 2005.[7] H.L. Choi, J.W. Son, J.T. Lim, Stability analysis and control of non-standard nonlinear singularly perturbed system, IEE Proceedings-Control Theory and

Applications 153 (6) (2006) 703–708.[8] J. Dong, G.H. Yang, Robust H1 control for standard discrete-time singularly perturbed systems, IET Control Theory and Applications 1 (4) (2007) 1141–

1148.[9] J. Dong, G.H. Yang, H1 control for nonstandard discrete-time singularly perturbed systems, Automatica 44 (2008) 1385–1393.

[10] E. Fridman, A descriptor system approach to nonlinear singularly perturbed optimal control problem, Automatica 37 (4) (2001) 543–549.[11] P. Gahinet, A. Nemirovski, A.J. Laub, M. Chilali, LMI Control Toolbox, The MathWorks, Inc., Natick, MA, 1995.[12] G. Garcia, J. Daafouz, J. Bernussou, The infinite time near optimal decentralized regulator problem for singularly perturbed systems: a convex

optimization approach, Automatica 38 (8) (2002) 1397–1406.[13] R. Ghosh, S. Sen, K.B. Datta, Method for evaluating stability bounds for discrete-time singularly perturbed systems, IEE Proceedings-Control Theory and

Applications 146 (2) (1999) 227–233.[14] K. Tanaka, H.O. Wang, Fuzzy Control Systems Design and Analysis: A Linear Matrix Inequality Approach, John Wiley & Sons Inc., New York, NY, USA,

2001.[15] C. Gong, B. Su, Robust L2—L1 filtering of convex polyhedral uncertain time-delay fuzzy systems, International Journal of Innovative Computing

Information and Control 4 (4) (2008) 793–802.[16] F.-H. Hsiao, J.-D. Hwang, S.-T. Pan, D-stability problem of discrete singularly perturbed systems, International Journal of Systems Science 34 (3) (2003)

227–236.



[17] P.V. Kokotovic, J. O’Reilly, H.K. Khalil, Singular Perturbation Methods in Control: Analysis and Design, Academic Press Inc., Orlando, FL, USA, 1986.[18] T.-H.S. Li, J.-S. Chiou, F.-C. Kung, Stability bounds of singularly perturbed discrete systems, IEEE Transactions on Automatic Control 44 (10) (1999)

1934–1938.[19] T.H.S. Li, K.-J. Lin, Stabilization of singularly perturbed fuzzy systems, IEEE Transactions on Fuzzy Systems 12 (5) (2004) 579–595.[20] W. Lin, C.I. Byrnes, H1-control of discrete-time nonlinear systems, IEEE Transactions on Automatic Control 41 (4) (1996) 494–510.[21] H. Liu, F. Sun, Y. Hu, H1 control for fuzzy singularly perturbed systems, Fuzzy Sets and Systems 155 (2) (2005) 272–291.[22] I. Lizarraga, V. Etxebarria, Combined PD-H1 approach to control of flexible link manipulators using only directly measurable variables, Cybernetics and

Systems 34 (1) (2003) 19–31.[23] D.S. Naidu, Singular perturbations and time scales in control theory and applications: an overview, Dynamics of Continuous Discrete and Impulsive

Systems-Series B-Applications and Algorithms 9 (2) (2002) 233–278.[24] V.R. Saksena, J. Cruz, Stabilization of singularly perturbed linear time-invariant systems using low-order observers, IEEE Transactions on Automatic

Control 26 (2) (1981) 510–513.[25] K. Tanaka, T. Ikeda, H.O. Wang, Fuzzy regulators and fuzzy observers: relaxed stability conditions and LMI-based designs, IEEE Transactions on Fuzzy

Systems 6 (2) (1998) 250–265.[26] C.-S. Ting, An observer-based approach to controlling time-delay chaotic systems via Takagi–Sugeno fuzzy model, Information Sciences 177 (20)

(2007) 4314–4328.[27] H.D. Tuan, P. Apkarian, T. Narikiyo, Y. Yamamoto, Parameterized linear matrix inequality techniques in fuzzy control system design, IEEE Transactions

on Fuzzy Systems 9 (2) (2001) 324–332.[28] J. Wang, H. Li, Nonlinear PI control of a class of nonlinear singularly perturbed systems, IEE Proceedings-Control Theory and Applications 152 (5)

(2005) 560–566.[29] Y. Wang, Z. Sun, H1 control of networked control system via LMI approach, International Journal of Innovative Computing Information and Control 3

(2) (2007) 343–352.[30] H.-N. Wu, K.-Y. Cai, Robust fuzzy control for uncertain discrete-time nonlinear Markovian jump systems without mode observations, Information

Sciences 177 (6) (2007) 1509–1522.

www.rsc.org/faradayRegistered Charity Number 207890

Faraday Discussions

This is an Accepted Manuscript, which has been through the RSC Publishing peer review process and has been accepted for publication.

Accepted Manuscripts are published online shortly after acceptance, which is prior to technical editing, formatting and proof reading. This new free service from RSC Publishing allows authors to make their results available to the community, in citable form, before publication of the edited article. This Accepted Manuscript will be replaced by the edited and formatted Advance Article as soon as this is available.

More information about Accepted Manuscripts can be found on the Faraday Discussions website at http://blogs.rsc.org/fd/2011/04/05/am/

To cite this manuscript please use its permanent Digital Object Identifier (DOI®), which is identical for all formats of publication.

For more information about Faraday Discussions, visit the website at www.rsc.org/faraday or contact the Editor, Philip Earis, at [email protected]

Accepted Manuscript

Please note that technical editing may introduce minor changes to the text and/or graphics contained in the manuscript submitted by the author(s) which may alter content, and that the standard Terms & Conditions and the ethical guidelines that apply to the journal are still applicable. In no event shall the RSC be held responsible for any errors or omissions in these Accepted Manuscripts or any consequences rising from the use of any information contained in them.

Poster abstract deadline 17 June 2011Early bird registration 17 June 2011Registration deadline 15 July 2011

Register now to attend this exciting meeting!www.rsc.org/fd154

This manuscript will be presented at the forthcoming Faraday Discussion meeting

Dow

nloa

ded

by W

ashi

ngto

n U

nive

rsity

in S

t. L

ouis

on

05 O

ctob

er 2

011

Publ

ishe

d on

14

July

201

1 on

http

://pu

bs.r

sc.o

rg |

doi:1

0.10

39/C

1FD

0007

1CView OnlineC:\work2011\shamov\yulia7\c1fd00071c.pdf 133

www.rsc.org/fd154

www.rsc.org/fd154

http://www.rsc.org/help/termsconditions.asp

http://www.rsc.org/publishing/journals/guidelines/

http://blogs.rsc.org/fd/2011/04/05/am/

www.rsc.org/faraday

www.rsc.org/faraday

http://dx.doi.org/10.1039/c1fd00071c

Protic pharmaceutical ionic liquids and solids: Aspects of Protonics

Jelena Stoimenovski,a Pamela M. Dean,a Ekaterina I. Izgorodina,a Douglas R. MacFarlane* a

a School of Chemistry, Monash University, Clayton, Victoria 3800, Australia. *E-mail: [email protected]; Tel: +61 3 9905 4540

A series of new protic compounds based on active pharmaceutical ingredients have been synthesised and characterised. Some of the salts synthesised produced ionic liquids, while others that were associated with rigid molecular structures tended to produce high melting points. The “protonic” behaviour of these compounds was found to be a major determinant of their properties. Indicator studies, FTIR-ATR and transport properties (Walden plot) were used to probe the extent of proton transfer and ion association in these ionic liquids. While proton transfer was shown to have taken place in all cases, the Walden plot indicated strong ion association in the primary amine based examples due to hydrogen bonding. This was further explored via crystal structures of related compounds, which showed that extended hydrogen bonded clusters tend to form in these salts. These clusters may dictate membrane transport properties of these compounds in vivo.

1. Introduction Many drug substances on the market are organic acids or bases that have an ability to form salts.1 Converting a drug from its acid or base form to a salt form can often allow a useful manipulation or optimisation of the drug’s properties, such as its absorption, pharmacodynamics or pharmacokinetics.2, 3 In addition, by keeping an active ion constant and changing the counterion of the drug salt, either by combining with another active, or other biocompatible ion, one can further alter its chemical and biological properties without changing the structure of the active ingredient. Different salt forms of an active are now accepted to be different drugs, with individual chemical and biological profiles, resulting in varying clinical efficacies and safety.1 Turning the pharmaceutical salt into an ionic liquid (IL) is a further conceptual step that has been discussed recently in this regard.4 ILs are salts with melting points below 100 oC and are typically comprised of cations and anions that have either relatively low symmetry or delocalised charge; these structural properties tend to provide a low, or sometimes nonexistent melting point.5, 6 A wide range of different physical and chemical properties can be designed into the liquid by simply varying the cations and anions of which it is comprised.6 From solvents for chemical reactions,7 to electrolytes for electrochemical cells,8 ILs are now of intense interest in a range of applications in different areas. Recently, recognising that more than half of pharmaceuticals currently on the market are organic salts, ILs have also been proposed as a novel approach to drug design and optimisation in the pharmaceutical world.4, 9-11 In 2007, Rogers and co-workers, in conjunction with our group, filed a

Far

aday

Dis

cuss

ion

s A

ccep

ted

Man

usc

rip

t

Dow

nloa

ded

by W

ashi

ngto

n U

nive

rsity

in S

t. L

ouis

on

05 O

ctob

er 2

011

Publ

ishe

d on

14

July

201

1 on

http

://pu

bs.r

sc.o

rg |

doi:1

0.10

39/C

1FD

0007

1C

View Online

C:\work2011\shamov\yulia7\c1fd00071c.pdf 134


series of patents in which the preparation of ILs from known active pharmaceutical ingredients (APIs) was described.12 The concept involved the replacement of what was usually a “ballast” counterion, for example sodium or bromide, with an alternative counterion that lowered the melting point of the salt such that it was liquid at ambient temperature, thus offering a range of different delivery modalities. If the “counterion” itself has active pharmaceutical properties then the compound can be considered a dual active, potentially presenting a synergistic combination of effects in a single compound. Other groups have been investigating ILs for related applications including anti-microbial13 and anti-bacterial activity.14-16 However, unlike the broader ILs field where the design or choice of cation and anion is, to an extent, flexible such that the desired low melting point can be achieved, this is not a case with pharmaceutical ILs. In this field the identity of at least one of the ions is specified, hence lowering the melting point of its salts becomes a challenging goal. This challenge has been explored over the last few years using an “anti-crystal engineering” approach,17 aided by a greater understanding of the lattice energy of crystalline organic salts as revealed by the Madelung constant.18 It was found, not surprisingly, that the extent of specific intermolecular interactions, such as hydrogen bonding, directly influences the phase behaviour of the product. Hence the advantageous combinations of active ion and counterion that can produce the low melting point required to form an IL will typically be those that avoid or disrupt all of these identifiable interactions.

Protic Ionic Liquids – the proton transfer conundrum

Protic Ionic Liquids, PILs, are a subclass of ILs formed by reacting a Brønsted acid with a Brønsted base. Unlike aprotic ILs, PILs exhibit Brønsted acidity due to the exchangeable proton, and can hence be used as solvents for acid catalysed reactions;19-21 for the same reason, they are also found to be suitable proton conducting electrolytes for fuel cells.22 Given their ease of preparation, being simply the product of an acid (HA) - base (B) neutralisation reaction (Scheme 1),

HA + B BH+ + A-

Scheme 1. Neutralisation reaction between an acid and a base

this IL family is of significant potential in the broad, multifaceted pharmaceutical application. Many pharmaceuticals are in fact delivered as simple salts of organic bases, or metal ion salts of organic acids. However, the properties manifested by PILs depend predominantly on the degree of proton transfer from the acid to the base as well as the hydrogen bonding in which the proton is involved; the possible outcomes are summarised in Figure 1 and all of these outcomes have been observed in recent examples. For the PIL to be considered a “pure” salt and not a complex mixture of acid, base and salt, the extent of proton transfer should be at least 99 %.23, 24 Despite the fact that in aqueous solution the proton transfer between the acids and bases usually involved would be strongly complete, the situation is more complex in the “neat” salt for complex reasons. Yoshizawa et. al.25 originally revealed the problem by correlating observed boiling points with differences in pKa

aq values (∆pKaaq)

between the acid and the amine base. On the basis of a comparison of Walden

Far

aday

Dis

cuss

ion

s A

ccep

ted

Man

usc

rip

t

Dow

nloa

ded

by W

ashi

ngto

n U

nive

rsity

in S

t. L

ouis

on

05 O

ctob

er 2

011

Publ

ishe

d on

14

July

201

1 on

http

://pu

bs.r

sc.o

rg |

doi:1

0.10

39/C

1FD

0007

1C

View Online



products (explained further below) it appeared that ∆pKaaq › 10 was needed for the full

proton transfer to be observed (i.e. to achieve the situation in Figure 1c rather than the acid-base mixture of Figure 1a).25

Figure 1. The four possible outcomes in PILs (a) mostly un-ionised acid and base (b) hydrogen bonded acid-base pairs (no proton transfer) (c) mostly ionised PIL (proton transfer occurred, independent ions), (d) associated PIL (proton transfer occurred but ions associated through hydrogen bonding).

To further investigate this phenomenon we explored a series of carefully chosen primary and tertiary amines of very similar pKa

aq values.26 We found that restricted proton transfer was clearly exhibited by the tertiary amine base systems, and ∆pKa

aq › 10 was indeed required to produce substantial proton transfer. On the other hand, the primary amine systems tended to be fully ionised even at ∆pKa

aq > 5. In other words, the primary amine based ILs tend to behave much more like aqueous systems. The strong difference in behaviour appears to be related to the hydrogen bond donor ability of the protonated base cation. In the tertiary amine case, only the single, transferred proton is available to hydrogen bond with the anion, whereas the primary base cation can offer multiple hydrogen bond sites. Free energy of solvation calculations from COSMO-RS confirmed this difference in the solvation environment in the two cases.26 However, clearly, additional intermolecular interactions stemming from long alkyl chains, π---π interactions and other hydrogen bonding sites in either ion, for example –OH functionality, can easily disturb these trends. Hence, introduction of such characteristics into tertiary amine based ILs could induce a stronger degree of proton transfer. These phenomena are crucial to attempts to form PILs based on APIs, since the lack of complete proton transfer may render the formulation nothing more than an acid/base/salt mixture. On the other hand, as we will show, the H-bonding introduced, involving both the transferred proton and also the residual conjugate base anion, can also produce strong ion-association in the PIL, potentially producing a further dimension of property variation in the product. We attempt to describe this complex set of interconnected phenomena surrounding the labile proton, its energetics and dynamics, in terms of a single unified concept – protonics. Exploring these phenomena in the context of APIs is the main theme of this paper. Pharmaceutically Active PROTIC Ionic Liquids A number of groups including us have investigated the use of PILs of various compositions as an approach to pharmaceutically active ILs.27, 28 For example, Bica et. al.27 described a series of salts of aspirin, e.g. lidocainium acetylsalicylate, Tg=-14°C, where lidocaine is a topical anaesthetic and aspirin is a broad analgesic. In the

Far

aday

Dis

cuss

ion

s A

ccep

ted

Man

usc

rip

t

Dow

nloa

ded

by W

ashi

ngto

n U

nive

rsity

in S

t. L

ouis

on

05 O

ctob

er 2

011

Publ

ishe

d on

14

July

201

1 on

http

://pu

bs.r

sc.o

rg |

doi:1

0.10

39/C

1FD

0007

1C

View Online



present work we have focussed on salt formation and the issue of proton transfer in a number of families of pharmaceutically active acids and bases. For the purposes of a systematic study, a series of acids and bases (Figure 2) were selected with slight structural variations and physico-chemical properties, with the objective of understanding the factors controlling melting point, H-bonding and proton transfer. A library of nine compounds and four oligomeric PILs was synthesised and characterised. Benzoic, salicylic and gentisic acids were chosen for this study, as these are frequently encountered in pharmaceutical compounds/formulations. Benzoic acid, BzH, a common preservative and pharmaceutical aid (antifungal), is used in a range of products on the market including ointments, mouthwashes and cosmetics. A derivative of benzoic acid, salicylic acid (SalH), is a widely recognized and used keratolytic, also possessing anti-inflammatory, analgesic and antipyretic properties as a sodium salt. Gentisic acid, GenH, another benzoic acid derivative, is used in the pharmaceutical industry as an analgesic and anti-inflammatory.29 The three pharmaceutical bases were chosen on the basis of their pKa

aq values and

their structure (Figure 2). Tuaminoheptane, NTH2, a primary amine base with a pKaaq

value of 10.50, is used as a nasal decongestant. Amantadine, NAH2, also a primary amine base with a pKa

aq value of 10.10, is an antiviral/anti-Parkinsonian drug. 2-

Pyrrolidinoethanol, (EtOH)pyr, a “generally recognised as safe compound”, is a tertiary amine with a pKa

aq value of 9.44.29

Benzoic acid (BzH) Salicylic acid (SalH) Gentisic acid (GenH) (1) (2) (3)

Tuaminoheptane Amantadine 2-Pyrrolidinoethanol (NTH2) (NAH2) (EtOH)pyr (4) (5) (6)

Figure 2. Acids and bases used in this study

2. Experimental

2.1 General

Tuaminoheptane (NTH2), amantadine (NAH2), 2-pyrrolidinoethanol ((EtOH)pyr) and gentisic acid (GenH) were purchased from Aldrich Chemical Company. Benzoic acid

Far

aday

Dis

cuss

ion

s A

ccep

ted

Man

usc

rip

t

Dow

nloa

ded

by W

ashi

ngto

n U

nive

rsity

in S

t. L

ouis

on

05 O

ctob

er 2

011

Publ

ishe

d on

14

July

201

1 on

http

://pu

bs.r

sc.o

rg |

doi:1

0.10

39/C

1FD

0007

1C

View Online



(BzH) was purchased from Sigma-Aldrich, while salicylic acid (SalH) was supplied by Chem Supply. The synthesis procedure and the characterisation of the resulting compounds can be found in the Electronic Supplementary Information.

2.2 Instrumental

Nuclear Magnetic Resonance (NMR) data were recorded on either a Bruker Avance 400 (9.4 Tesla magnet) with a 5mm broadband autotunable probe with Z-gradients and BACS 60 tube autosampler or on a Bruker DPX-200 spectrometer. All samples were measured as solutions in deuterated dimethylsulfoxide supplied by Cambridge Isotope Laboratories. Multiplicities are reported according to their relative splittings, as a singlet (s), doublet (d), doublet of doublets (d(d)), triplet (t), pentet (p), sextet, multiplet (m) or a broad signal (b). Electrospray Mass Spectrometry (ESI) was carried out on the Micromass Platform II API QMS Electrospray Mass Spectrometer with a cone voltage of 25 V or 35 V, using methanol as the mobile phase. Analyses were conducted in both positive (ESI+) and negative (ESI-) modes. Density data was collected using an Anton Paar DMA 5000 density meter. The density of the compounds was determined using the “oscillating U-tube principle”. The viscosity measurements were performed using a falling ball technique on an Anton Paar AMVn viscosity meter. Conductivity measurements were carried out on a locally designed dip cell probe containing two platinum wires covered in glass. 0.01M KCl solution at 25 oC was used to determine the cell constant. The conductivities were obtained by measuring the complex impedance spectra between 1 MHz and 0.01 Hz on a Solatron SI 1296 dielectric interface. Where crystals were obtained, the reflection intensity data were measured on a Bruker X8 APEX KAPPA CCD single crystal X-ray diffractometer ((EtOH)pyr+ Gen-; NAH3

+ Gen-; NAH3+ Sal-; NTH3

+ Gen-; (EtOH)pyr+ Bz-; (EtOH)pyr+ Sal-) and a Nonius KAPPA CCD single crystal X-ray diffractometer (NAH3

+ Bz-), using graphite-monochromated Mo KR radiation (λ) 0.71073 Å. Crystals were coated with Paratone N oil (Exxon Chemical Co.,TX, U.S.A.) immediately after isolation and cooled in a stream of nitrogen vapor on the diffractometer. Structures were solved by direct methods using the program SHELXS-97 and refined by full matrix least-squares refinement on F2 using SHELXL-97.25 All non-hydrogen atoms were revealed in the E-map and subsequent difference electron density maps and thus placed and refined anisotropically. All hydrogen atoms were observed in difference syntheses and were either refined isotropically or placed in geometrically idealized positions and constrained to ride on their parent atoms with C-H distances in the range 0.95-1.00 Å and Uiso(H)) xUeq(C), where x) 1.5 for methyl and 1.2 for all other C atoms.30 Specific details of refinement are contained in the relevant CIF files. The crystal structures of most of the synthesised salts are described here for the first time (CCDC Refcodes: 802629 for NTH3

+ Gen-, 802627 for NAH3+ Gen-, 802625 for NAH3

+ Sal-, 802628 for (EtOH)pyrH+ Bz-, 802630 for (EtOH)pyrH+ Sal- and 802624 for (EtOH)pyrH+ Gen-. The crystal structures for NAH3

+ Bz- has been reported before and can be found in reference 31. However, we have solved the structure of this compound and submitted the report as part of the supplementary information.

Far

aday

Dis

cuss

ion

s A

ccep

ted

Man

usc

rip

t

Dow

nloa

ded

by W

ashi

ngto

n U

nive

rsity

in S

t. L

ouis

on

05 O

ctob

er 2

011

Publ

ishe

d on

14

July

201

1 on

http

://pu

bs.r

sc.o

rg |

doi:1

0.10

39/C

1FD

0007

1C

View Online



Differential scanning calorimetry (DSC) scans were carried out at a heating/cooling rate of 10 oC/min in the range of -150 oC to 200 oC using a T.A Instruments Q100 differential scanning calorimeter. Thermal scans below room temperature were calibrated with the cyclohexane solid-solid transition and melting point at -87.0 °C and 6.5 °C respectively. Thermal scans above room temperature were calibrated with the indium, tin, lead and zinc. Transition temperatures are reported using the peak maximum of the thermal transition. The FTIR-ATR spectra for all the samples were obtained using Bruker IFS Equinox FTIR system, fitted with a Golden GateTM single bounce diamond. The FTIR spectrometer was connected to a computer equipped with OPUS 6.0 software, which was used for the analysis of spectra.

Far

aday

Dis

cuss

ion

s A

ccep

ted

Man

usc

rip

t

Dow

nloa

ded

by W

ashi

ngto

n U

nive

rsity

in S

t. L

ouis

on

05 O

ctob

er 2

011

Publ

ishe

d on

14

July

201

1 on

http

://pu

bs.r

sc.o

rg |

doi:1

0.10

39/C

1FD

0007

1C

View Online



3. Results and Discussion 3.1 Protic Pharmaceutical Salts Table 1 summarises the protic salts made in this work. Of the thirteen compounds synthesised, nine satisfy the definition of an IL by having melting points below 100 oC. Five of these are room temperature ILs. Salts containing the NAH2 base all possess high melting points. ILs containing the (EtOH)pyrH+ ion all have melting points below 100 oC, while those based on the NTH3

+ cation have no obvious trend amongst their melting points. When reacted with the same acid, the two primary amines, NTH2 and NAH2, give rise to compounds with significantly different melting points. The compounds based on the NTH3

+ cation, a seven carbon molecule, have melting points of 121±2oC and below, while those based on the NAH3

+ cation, a nine carbon molecule, possess melting points of 216±2oC and above. These differences can be understood from a thermodynamic point of view. Given that the change in Gibbs free energy of fusion (∆Gf) at the melting point is zero by definition, the enthalpy (∆Hf) and the entropy (∆Sf) of fusion of the compound determine Tm (equation 1).

∆Gf =0 = ∆Hf - Tm∆Sf => Tm = ∆Hf /∆Sf Eqn 1 Being a long-chain molecule, the NTH3

+ cation has more degrees of rotational freedom that can become active on melting and hence exhibits a larger ∆Sf,; this in turn leads to compounds with a lower melting point. On the other hand, NAH3

+, a rigid multi-ringed structure, has fewer rotational degrees of freedom that can become active on melting, resulting in a smaller ∆Sf and hence higher melting point of the resulting compounds. Thus, in order to synthesise low melting PILs, the more flexible anion/cation structures are desirable.

Far

aday

Dis

cuss

ion

s A

ccep

ted

Man

usc

rip

t

Dow

nloa

ded

by W

ashi

ngto

n U

nive

rsity

in S

t. L

ouis

on

05 O

ctob

er 2

011

Publ

ishe

d on

14

July

201

1 on

http

://pu

bs.r

sc.o

rg |

doi:1

0.10

39/C

1FD

0007

1C

View Online



* Represents compounds with two moles of acid, in some cases two different acids, to one mole of base

Table 1. The protic pharmaceutical compounds prepared in this study including a number that are ILs Far

aday

Dis

cuss

ion

s A

ccep

ted

Man

usc

rip

t

Dow

nloa

ded

by W

ashi

ngto

n U

nive

rsity

in S

t. L

ouis

on

05 O

ctob

er 2

011

Publ

ishe

d on

14

July

201

1 on

http

://pu

bs.r

sc.o

rg |

doi:1

0.10

39/C

1FD

0007

1C

View Online



3.2 Oligomeric Compositions

Another domain of PILs that has been explored recently are the protic oligomers, whereby several equivalents of acid were reacted with one equivalent of base to form H-bonded acid-oligomer based ILs (Scheme 2).32 For example, in the case of 2HA + B, the anionic species present appears to be [A-H-A]-. Proton transfer at this stoichiometry can be stronger than at the 1:1 stoichiometry.

(A-H)x + B Hx-1(A-)x + B+H

Scheme 2. Anion oligomer based PIL formation32

Bica et. al. have reported pharmaceutical ILs based on such oligomeric ions; these compositions allow one to alter the dose of one active ingredient over the other active ingredient in the case of dual protic pharmaceutical ILs.33 Recognising that such 2:1 mixtures may also represent “buffer” type properties,34 they may also have a useful property in controlling pH. A similar approach was investigated in Table 1 in which a second mole of acid, in some cases a different acid, was incorporated into the formulation, in principle forming the “dimer” acid/anion species. In the case of the formulations involving two different acids HA1 and HA2, there is no intrinsic reason why the dimeric species should be uniquely A1-H-A2; the real situation will be a dynamic equilibrium between all of the possible species, with this overall composition.

3.3 Physical Properties, Proton Transfer and Ion Association

As discussed above, certain important physicochemical properties of pharmaceutical acids/bases can be altered by converting them into salts of an appropriate counter ion. For example, in this study the salts of the originally water insoluble/slightly soluble acids and bases have been rendered water soluble in most cases. In the case of the dimer-acid based ILs, an additional acid molecule decreases the water solubility of the compounds. This in turn demonstrates the ease of water solubility manipulation, an important factor in drug delivery. The effects of proton transfer and ion association on physical properties are illustrated by the two salicylate compounds, one based on a primary amine, NTH3

+ Sal-, and the other on a tertiary amine, (EtOH)pyrH+ Sal-. Both compounds have low melting points. Indicator studies, the basis of which was described in detail in our previous paper,26 were used to show that both the primary amine, NTH2, as well as the tertiary amine, (EtOH)pyr, have the capability to completely deprotonate the polyprotic indicator-acid, phenol red. This is shown in Figure 3 where the amines have both turned a fuchsia colour in the presence of the indicator acid, which originally exists as an orange coloured solution. One would observe a yellow colour if only a partial deprotonation occurred. The proton transfer behaviour of the (EtOH)pyr, a tertiary amine, is therefore different to the behaviour of the simple tertiary amine cases studied previously,26 presumably as a result of the presence of the hydroxyl group.

Far

aday

Dis

cuss

ion

s A

ccep

ted

Man

usc

rip

t

Dow

nloa

ded

by W

ashi

ngto

n U

nive

rsity

in S

t. L

ouis

on

05 O

ctob

er 2

011

Publ

ishe

d on

14

July

201

1 on

http

://pu

bs.r

sc.o

rg |

doi:1

0.10

39/C

1FD

0007

1C

View Online



Figure 3. The primary amine, NTH2, and tertiary amine, (EtOH)pyr, with indicator phenol red

Further evidence of proton transfer in this case is seen in the FTIR-ATR spectra of (EtOH)pyrH+ Sal-, NTH3

+ Sal- and the starting materials (EtOH)pyr, NTH2 and SalH displayed in Figure 4. The spectra of the PILs (shown in red) are notably different to those of the starting materials. A peak at 1655 cm-1 in the SalH spectrum (shown in blue), corresponding to the C=O stretch of the acid, is absent in both PIL spectra, which is indicative of complete proton transfer in the ILs. Also, the disappearance of, or shift in, other peaks when compared to the original starting materials are all consistent with strong proton transfer in these PILs.

Far

aday

Dis

cuss

ion

s A

ccep

ted

Man

usc

rip

t

Dow

nloa

ded

by W

ashi

ngto

n U

nive

rsity

in S

t. L

ouis

on

05 O

ctob

er 2

011

Publ

ishe

d on

14

July

201

1 on

http

://pu

bs.r

sc.o

rg |

doi:1

0.10

39/C

1FD

0007

1C

View Online



Figure 4. FTIR-ATR spectra of (a) (EtOH)pyrH+ Sal- (red), (EtOH)pyr (green) and SalH (blue) and (b) NTH3

+ Sal- (red), NTH2 (green) and SalH (blue)

The effect of ion association is subtly revealed by the bulk properties in these systems. The density data (Figure 5) reveals two distinctly different groups of liquids, where the tertiary amine based PILs (including the 2:1 mixtures) appear to be denser than the primary amine based examples. This in turn implies that ions in the tertiary amine based PILs pack more tightly than in the primary amine based PILs. As will be shown below, the latter have a tendency to form open hydrogen bonded structures that produce lower densities.

Far

aday

Dis

cuss

ion

s A

ccep

ted

Man

usc

rip

t

Dow

nloa

ded

by W

ashi

ngto

n U

nive

rsity

in S

t. L

ouis

on

05 O

ctob

er 2

011

Publ

ishe

d on

14

July

201

1 on

http

://pu

bs.r

sc.o

rg |

doi:1

0.10

39/C

1FD

0007

1C

View Online



Figure 5. Density data of the synthesised PILs

The ionic conductivity (Figure 6) also shows two distinctly different groups of compounds; the tertiary amine based PILs have a conductivity approximately one order of magnitude greater than the primary amine based ILs, despite the fact that both families appear to be substantially proton transferred. On the other hand, their viscosities (Figure 6) show no distinct differences. Of all the compounds analysed in this study, the primary amine based PIL, NTH3

+ Sal-, displayed the highest viscosity followed by the tertiary amine based PIL, (EtOH)pyrH+ Sal-.

Figure 6. Conductivity (full symbols) and viscosity data (hollow symbols) of the synthesised PILs

Far

aday

Dis

cuss

ion

s A

ccep

ted

Man

usc

rip

t

Dow

nloa

ded

by W

ashi

ngto

n U

nive

rsity

in S

t. L

ouis

on

05 O

ctob

er 2

011

Publ

ishe

d on

14

July

201

1 on

http

://pu

bs.r

sc.o

rg |

doi:1

0.10

39/C

1FD

0007

1C

View Online



The transport property and density data can be combined into the Walden plot, to provide a revealing perspective on proton transfer and ion association. The Walden plot25, 35, 36 is a method of comparing transport property data using the well-known Walden rule that states:

Λη = k Eqn 2

where Λ is the molar conductivity and η is the viscosity. An estimate of the constant (k) can be obtained from data for a 0.1M aqueous solution of KCl, which is thought to be a good example of a fully dissociated salt. The Walden rule then appears as a straight line on a plot of log(molar conductivity) versus log(1/viscosity). Previous discussion of this in the literature35 has shown that many ILs fall quite close to the Walden rule line, as do many molten salts. ILs that fall well below the line are thought of as exhibiting signs of ion association in a way that inhibits conduction, but not viscosity. Additionally, in the protic systems incomplete proton transfer can also produce transport properties that fall well below the line. As a reminder of the impact of the logarithmic scale of Figure 7, the dashed line indicates the situation where the molar conductivity is only 10% of that expected from the Walden rule. Such systems are clearly showing a low degree of proton transfer and/or strong signs of ion-association.

Figure 7. Walden plot of transport property data for the synthesised PILs. Each data set represents a temperature series for the compound above its melting point. Also included are some data points for a number of simple inorganic salts at temperatures just above their respective melting points.37

Far

aday

Dis

cuss

ion

s A

ccep

ted

Man

usc

rip

t

Dow

nloa

ded

by W

ashi

ngto

n U

nive

rsity

in S

t. L

ouis

on

05 O

ctob

er 2

011

Publ

ishe

d on

14

July

201

1 on

http

://pu

bs.r

sc.o

rg |

doi:1

0.10

39/C

1FD

0007

1C

View Online



In the Walden plot in Figure 7 the PILs based on the NTH3+ cation display a

significant deviation from the ideal line, whereas those containing the (EtOH)pyrH+ cation show better concordance with the Walden rule. The latter is consistent with the infra-red evidence, which indicates strong proton transfer and might be considered a normal example of a PIL. However, the NTH3

+ salts appear surprisingly low on the Walden plot, given that the infra-red spectra indicated strong proton transfer. This perhaps indicates that these salts, although proton transferred, are highly associated in the liquid state. Ion pairs and larger clusters that have zero net charge are unable to conduct and therefore cannot contribute to the molar conductivity of the material. The NTH3 Sal case in Figure 7 is exhibiting only 1% of the conductivity that would be expected on the basis of the Walden rule; this indicates that the overwhelming majority of the ions in this case are “tied up” in ion pairs and clusters. We will return to the nature of association in these salts later, however, we will now present some data showing how association is of some importance in the membrane transport properties of these pharmaceutically active PILs.

3.4 Membrane Transport Properties

One of the benefits of synthesising IL forms of known actives is the potential to modulate the membrane permeation properties of an active. Tuning and controlling membrane transport is the key to a variety of novel delivery methods including slow, continuous-release transdermal approaches. Recent work by us38 and others39 provides some insight into the membrane transport properties of these pharmaceutically active IL compounds. It is a well established fact that salts do not readily cross the lipid bilayer unless there are active transport mechanisms available. Hence, it is not immediately obvious that IL forms of an active will be readily membrane permeable. On the other hand, if the IL has a propensity to form neutral species, ion-pairs or clusters then membrane transport may be enhanced as compared to an inorganic salt of the active. To further investigate this hypothesis in respect of these protic examples, we have carried out model membrane transport studies following the method of Tantishaiyakul et. al..39 A full publication describing these results will be presented elsewhere38; we present here an overview of the trends observed. Typical data, for example for NTH3

+ Sal- and its parent acid and base, as well as its simple salt forms that are used commercially, show that while the component acid is rapidly transported through the membrane, the sodium salt is not, nor is the sulfate salt (the form normally used pharmaceutically). On the other hand, the neat IL form, despite being a salt, is found to rapidly transport through the membrane over a period of one or two hours. Therefore the IL form is clearly more membrane permeable than the active ingredients that were the starting materials. Interestingly, the IL is not readily transported when dissolved in propylene glycol; this indicates that the situation is not simply one of providing an alternate, but independent, counter-ion. Indeed, Tantishaiyakul et. al. have speculated that ion-pairing can play a significant role in the transport of these types of compounds.39 It is interesting to note that NTH3

+ Sal- shows signs of strong ion pairing/aggregation by sitting low in the Walden plot and hence this aspect of the behaviour of these PILs may be of importance in their membrane transport properties. The nature of this ion pairing/aggregation is further explored in the next sections. It is also possible that reverse proton transfer to form the neutral acid and base at the point of dissolution into the membrane is the basis of transport – partially the result of a relatively smaller pKa gap than the inorganic salts. At this stage it is not possible to discriminate between these hypotheses, however doing so will allow us to understand how

Far

aday

Dis

cuss

ion

s A

ccep

ted

Man

usc

rip

t

Dow

nloa

ded

by W

ashi

ngto

n U

nive

rsity

in S

t. L

ouis

on

05 O

ctob

er 2

011

Publ

ishe

d on

14

July

201

1 on

http

://pu

bs.r

sc.o

rg |

doi:1

0.10

39/C

1FD

0007

1C

View Online



modulating the counter acid/base in these systems can be used to control their interaction with membranes.

3.5 Ion association - what can we learn from crystal structures?

Crystal structures can also provide some insight into the existence of proton transfer and ion association, and thereby serve as a guide to the existence of these phenomena in related liquid phases. Crystal structures for several of the compounds synthesised here have been solved and can be found in the supplementary information. The two ion-pairs shown in Figure 8 are the asymmetric units of the crystal structures of the tertiary amine based PIL, (EtOH)pyrH+ Sal-, and of the primary amine based PIL, NTH3

+ Gen-. As can be seen from the displayed structures, the proton transfer has occurred in both cases. However, we should note that proton transfer in the solid state is likely to be stronger since it incurs a smaller entropic penalty than in the liquid and therefore produces a larger free energy change. This occurs because proton transfer in the neat liquid case causes a sharp loss of the configurational degrees of freedom that are characteristic of the liquid state and have a substantially negative ∆S.

Figure 8a also shows that the hydroxyl group of the (EtOH)pyrH+ cation stabilises a resonating carbonyl group in the Sal- anion, through hydrogen bonding, thus supporting the proton transfer. The O. . .H distances observed in this pair are 1.805 and 1.937 Å, comparable to the hydrogen bond distances in water.

(a) (b)

Figure 8. Crystal structures (a) of the tertiary amine based IL, (EtOH)pyrH+ Sal- (m.p. 49±2 oC); N-H. . .O and O-H. . .O distances are shown in red; (b) of the primary amine based salt, NTH3

+ Gen- (m.p. 121±2 oC) (N-H. . .O distance is shown in red)

The asymmetric unit of NTH3

+ Gen-, shown in Figure 8b, displays relatively short interionic N-H. . .O contacts of 1.811 Å, indicating strong hydrogen bonding between the ions. Additional O-H. . .O and C-H. . .O hydrogen bonding is also observed between the cations.

Although the proton transfer is clearly complete in both cases, in accord with the FTIR-ATR data, a large difference between their positions in the Walden plot (> 1 orders of magnitude) demands explanation - an explanation that cannot be based on lack of proton transfer. Figure 9 provides a possible answer; NTH3

+ Gen- organises itself in the crystal in a cyclic cluster-like complex consisting of two cations and two anions.

Far

aday

Dis

cuss

ion

s A

ccep

ted

Man

usc

rip

t

Dow

nloa

ded

by W

ashi

ngto

n U

nive

rsity

in S

t. L

ouis

on

05 O

ctob

er 2

011

Publ

ishe

d on

14

July

201

1 on

http

://pu

bs.r

sc.o

rg |

doi:1

0.10

39/C

1FD

0007

1C

View Online



Figure 9. Cluster-like complex conformation of the primary amine based IL, NTH3

+ Gen-

If a cyclic arrangement such as this, with its overall zero charge, is reasonably long lived in the liquid state then the molar conductivity will be lower than expected, despite the strong degree of proton transfer. Part of the origin of this cluster-like complex may lie in the inter-ring interactions as seen in Figure S2. π. . .π stacking between phenyl rings in the anions imposes a certain arrangement of ions in the crystal structure that results in a strong hydrogen-bonded lattice.

Figure 10. Cluster-like complex confirmation of the primary amine based IL, NAH3

+ Gen-

Another example of this phenomenon was found in the analysis of the NAH3

+ Gen- crystal structure (m.p. = 250±2 oC) containing the NAH3

+ ion (Figure 10). In this case, a cyclic cluster-like complex is formed with both oxygens of the carboxylic group partaking in hydrogen bonding. Due to spatial constraints, one of the H. . .O bonds is longer than the other by 0.173 Å. A closer inspection of the packing diagram reveals that the cyclic arrangement extends beyond the cluster-like complex, with oxygens of the COO- group and the hydrogens of the NH3

+ group forming distinct channels of hydrogen-bonded interactions (Figure S3). Extended associated structures would also have the effect of placing these compounds low on the Walden plot,

Far

aday

Dis

cuss

ion

s A

ccep

ted

Man

usc

rip

t

Dow

nloa

ded

by W

ashi

ngto

n U

nive

rsity

in S

t. L

ouis

on

05 O

ctob

er 2

011

Publ

ishe

d on

14

July

201

1 on

http

://pu

bs.r

sc.o

rg |

doi:1

0.10

39/C

1FD

0007

1C

View Online



despite there being a high degree of proton transfer. Associations such as these would appear to also be crucial to the rapid membrane transport of these salts. Recently we have developed a generalised approach for the calculation of Madelung constants based on crystal structures.18 The Madelung constant, M, represents a measure of the net electrostatic energy in ionic solids and thus determines the stability of the ionic lattice with respect to an isolated ion pair. For stable lattices M is expected to be more than 1, with larger values indicating greater stability of the lattice. The Madelung constant can also be viewed as a geometric parameter; it provides us with a static perspective on cation-anion arrangements that might persist to an extent in the liquid state. Values of the Madelung constant closer to 1 represent ionic solids dominated by ion-pair interactions, whereas larger values indicate an extended lattice of inter-ionic interactions. In this study Madelung constants and electrostatic lattice energies (Table 2) were calculated for the PILs whose crystal structures were resolved in this work. Inspection of Table 2 reveals values that are relatively low; by comparison, a simple symmetrical organic salt such as [NMe4][BF4] has a Madelung constant close to that of NaCl (1.75). Ion association in the solid state is therefore lowering M considerably in all cases. Values for the (EtOH)pyrH+ based salts are closer to 1, indicating that the ion pairs are somewhat more isolated in this case. The electrostatic lattice energies also seem to generally follow the trend in measured melting points, with NAH3

+ based salts having the largest lattice energies and highest melting points. As discussed elsewhere,40 this indicates that the electrostatic part of the lattice energy plays a dominant role in the overall energetics of melting of these compounds, despite the obvious influence of H-bonding and ion association in other properties. Table 2. Madelung constants, electrostatic lattice energies and melting points of protic pharmaceutical salts prepared in this work.

M Electrostatic Lattice energy,

kJ mol-1

Tm, °C

NTH3+ Gen- 1.2307 -551 121

NAH3+ Bz- 1.2320 -624 256

NAH3+ Sal- 1.2566 -625 216

NAH3+ Gen- 1.2358 -616 250

(EtOH)pyrH Bz- 1.1731 -546 28 (EtOH)pyrH Sal- 1.0748 -493 49 (EtOH)pyrH Gen- 1.0831 -487 97

3.6 Perspectives on Protonics

In a number of fields of application, including ionic liquids and solids for fuel cell electrolytes,41 protein stabilising ionic liquids34 and water electrolysis42 the focus is on understanding the energetics and dynamics of protons and how they impact on the properties of the substance. Just as an understanding of the electronic structure of a molecule is vital to understanding its bonding and reactivity, we have found it useful

Far

aday

Dis

cuss

ion

s A

ccep

ted

Man

usc

rip

t

Dow

nloa

ded

by W

ashi

ngto

n U

nive

rsity

in S

t. L

ouis

on

05 O

ctob

er 2

011

Publ

ishe

d on

14

July

201

1 on

http

://pu

bs.r

sc.o

rg |

doi:1

0.10

39/C

1FD

0007

1C

View Online



to adopt a holistic view of protonic behaviour in these contexts in order to fully understand its subtleties. Much of the discussion above has concerned the location of the proton in these protic pharmaceutical systems, the consequences of its transfer from acid to base and the impact of its H-bonding, or lack of, in determining properties. The neat IL provides a rather unique environment in which to observe proton transfer, since there is an almost complete lack of a dielectric solvent to shield the charges from one another and the liquid is so concentrated that the charges are never more than one ion-pair diameter apart. Unlike water and other solvents, in which the protonic properties are heavily dominated by the properties of the solvent, in PILs the IL species themselves are the only acid and base present. The proton energy levels involved are therefore more directly revealed and manipulated, but they are also strongly impacted by the molecular structure of the species involved. The distinction between the H-bonded acid-base complex and the H-bonded, proton-transferred ion-pair becomes very slight in a PIL. In fact, it amounts to simply the matter of where the proton sits on a potential energy surface between the two heavy atoms involved (usually N and O). This is illustrated in Figure 11 for the case of acetic acid and methylpyrrolidine in the gas phase, where the proton transfer amounts to only a ~0.4A° shift in an otherwise H-bonded complex. Charge is usually a significant distinction between these two states in a solvent based system, but becomes less so in the PIL, because H-bonding in the neutral complex involves some generation of a partial charge on each atom, while, on the other hand, H-bonding in the ion-pair acts to diminish the unit charge that notionally exists on the heavy atoms. In the case of acetic acid and methylpyrrolidine these minor changes in position and charge do not produce a sufficiently significant free energy change to create a substantial population of the proton transferred species, in either the gas phase or the ionic liquid.26 As we will discuss in more detail below, further solvation interactions are required to push this proton transfer along.

Figure 11. Ab initio calculations for acetic acid and methylpyrrolidine ΔpKa

aq = 5. The potential energy surface was generated as a relaxed scan of the O-H bond stretch at the MP2/6-31G+(d,p) level of theory. The optimised points on the PES were then

Far

aday

Dis

cuss

ion

s A

ccep

ted

Man

usc

rip

t

Dow

nloa

ded

by W

ashi

ngto

n U

nive

rsity

in S

t. L

ouis

on

05 O

ctob

er 2

011

Publ

ishe

d on

14

July

201

1 on

http

://pu

bs.r

sc.o

rg |

doi:1

0.10

39/C

1FD

0007

1C

View Online



improved at the MP2/aug-cc-pVTZ level of theory. All calculations were performed using GAUSSIAN 0343

Angell and coworkers44, 45 have discussed the use of a proton free-energy scale, first introduced by Gurney46, to assist in understanding the energetics in these materials. In this perspective, the proton site in an acid moiety is considered as one of high proton chemical potential, and the available site on the base moiety as being one of low proton chemical potential. The energy scale involved bears many features in common with electrochemical potential scales; in particular the chemical potential of the proton is the difference in free energies between that of the acid HA and its conjugate base A-. The highest occupied proton level (HOPL) in the system thus corresponds to the species having the highest free energy change available from a proton donation process HA A- + H+. The lowest unoccupied proton level (LUPL) similarly corresponds to the site offering the largest free energy change from a proton addition process. A number of scenarios are illustrated in Figure 12. Note that the LUPL of the base, B, is also the HOPL of the corresponding protonated species BH+.

Figure 12. Proton Energy levels in (a) a simple, solvent-free acid-base mixture (b) a more complex case in which either the acid or base is multi-protic and (c) a buffer case where the HOPL is ~50% occupied and hence can accept or donate protons to an added strong acid or base to create a buffer action.34

The impact of hydrogen bonding and other solvation interactions on these proton energy levels is the key aspect we focus upon here, being dominated by the absence of, or limited presence of, H-bond donating species in the acid or base. Recognizing that the proton energy levels in Figure 12 actually represent a free energy change in a process that involves both the protonated and the unprotonated species, it is necessary

Far

aday

Dis

cuss

ion

s A

ccep

ted

Man

usc

rip

t

Dow

nloa

ded

by W

ashi

ngto

n U

nive

rsity

in S

t. L

ouis

on

05 O

ctob

er 2

011

Publ

ishe

d on

14

July

201

1 on

http

://pu

bs.r

sc.o

rg |

doi:1

0.10

39/C

1FD

0007

1C

View Online



to further deconstruct this diagram to explicitly identify the free energy of each species and then separately consider the effect of solvation on each. Figure 13 shows such an energy level diagram. The diagram begins by considering the isolated gas phase species and, as discussed above, in some cases there may be no proton transfer at all, as illustrated. Then, as shown in Figure 13, providing a source of solvation or additional hydrogen bonding, for example by a primary ammonium cation, impacts on the energy levels and hence ultimately on the proton transfer.26 In the case of the protic systems described here, other H-bonding sites in the molecules have additional solvating effects and hence further alter the respective positions of the proton energy levels available.

Figure 13. Free energy diagram showing the component molecular free energies that make up the proton energy levels in the gas phase and the effect of solvation on these.

Where proton transfer does not take place in the liquid state, despite the expectation from aqueous pKa data that it should, another phenomenon related to H-bonding can occur: un-ionised acid-base complex formation, as shown in Figure 1b. A number of examples from the lidocaine:fatty acid family of liquids have been described recently in the context of pharmaceutically active compounds where the ∆pKa

aq values were ~3 and it was shown that proton transfer did not take place.47 In this situation the lack of H-bonding or solvation interactions strongly alters the positions of the individual ionised species in Figure 13 and thereby the HOPL and LUPL levels, to the point that proton transfer is not energetically favoured. However, a complete suppression of the melting point indicated formation, instead, of a hydrogen bonded acid-base complex. Such un-ionised complexes can be thought of as a liquid analogue of the “co-crystals” that are much studied in the pharmaceutical field. In the liquid co-crystal case the association of the two components produces a stabilising interaction in the liquid state that results in a sharp lowering of the melting point.

Far

aday

Dis

cuss

ion

s A

ccep

ted

Man

usc

rip

t

Dow

nloa

ded

by W

ashi

ngto

n U

nive

rsity

in S

t. L

ouis

on

05 O

ctob

er 2

011

Publ

ishe

d on

14

July

201

1 on

http

://pu

bs.r

sc.o

rg |

doi:1

0.10

39/C

1FD

0007

1C

View Online



4. Conclusions

A number of protic ionic liquids and salts based on pharmaceutical acids and bases have been synthesised and characterised. FTIR-ATR and Walden plot studies were used to assess the proton transfer and determine the degree of ionicity in the synthesised compounds. It was found that additional hydrogen bonding functional groups in the tertiary amine case can promote proton transfer. In the case of primary amines, formation of cluster-like species containing protonated base/deprotonated acid pairs can result in salts possessing higher melting points and decreased ionicity. These findings are supported by crystal structures and Madelung constant calculations. Nonetheless, for application purposes, PILs with lower ionicity are more desirable for drug delivery as they may cross the membrane barrier more easily than the more ionised species. Membrane studies are currently being carried out in order to further investigate this phenomenon and are showing promising results. 5. References 1. P. Heinrich Stahl, C. G. Wermuth and Editors, Handbook of Pharmaceutical

Salts; Properties, Selection, and Use, VHCA and Wiley-VCH, 2008. 2. S. M. Berge, L. D. Bighley and D. C. Monkhouse, J. Pharm. Sci., 1977, 66, 1-

19. 3. G. Davies, The Pharmaceutical Journal, 2001, 266, 322-323. 4. J. Stoimenovski, D. R. MacFarlane, K. Bica and R. D. Rogers, Pharm. Res.,

2010, 27, 521-526. 5. P. M. Dean, J. M. Pringle and D. R. MacFarlane, Phys. Chem. Chem. Phys.,

2010, 12, 9144-9153. 6. R. D. Rogers and K. R. Seddon, Ionic Liquids: Industrial Applications to

Green Chemistry, American Chemical Society, 2003. 7. T. Welton, Chem. Rev., 1999, 99, 2071-2083. 8. F. Endres, D. MacFarlane, A. Abbott and Editors, Electrodeposition from

Ionic Liquids, Wiley-VCH, 2008. 9. W. L. Hough, M. Smiglak, H. Rodriguez, R. P. Swatloski, S. K. Spear, D. T.

Daly, J. Pernak, J. E. Grisel, R. D. Carliss, M. D. Soutullo, J. J. H. Davis and R. D. Rogers, New J. Chem., 2007, 31, 1429-1436.

10. W. L. Hough and R. D. Rogers, Bull. Chem. Soc. Jpn., 2007, 80, 2262-2269. 11. W. L. Hough-Troutman, M. Smiglak, S. Griffin, W. M. Reichert, I. Mirska, J.

Jodynis-Liebert, T. Adamska, J. Nawrot, M. Stasiewicz, R. D. Rogers and J. Pernak, New J. Chem., 2009, 33, 26-33.

12. Application: WO Pat., 2006-US39454 2007044693, 2007. 13. A. Cieniecka-Roslonkiewicz, J. Pernak, J. Kubis-Feder, A. Ramani, A. J.

Robertson and K. R. Seddon, Green Chem., 2005, 7, 855-862. 14. J. Pernak and J. Feder-Kubis, Chem.-Eur. J., 2005, 11, 4441-4449. 15. J. Pernak, K. Sobaszkiewicz and J. Foksowicz-Flaczyk, Chem.-Eur. J., 2004,

10, 3479-3485. 16. J. Pernak, I. Goc and I. Mirska, Green Chem., 2004, 6, 323-329. 17. P. M. Dean, J. Turanjanin, M. Yoshizawa-Fujita, D. R. MacFarlane and J. L.

Scott, Cryst. Growth Des., 2009, 9, 1137-1145. 18. E. I. Izgorodina, U. L. Bernard, P. M. Dean, J. M. Pringle and D. R.

MacFarlane, Cryst. Growth Des., 2009, 9, 4834-4839. 19. E. Janus, I. Goc-Maciejewska, M. Lozynski and J. Pernak, Tetrahedron Lett.,

2006, 47, 4079-4083.

Far

aday

Dis

cuss

ion

s A

ccep

ted

Man

usc

rip

t

Dow

nloa

ded

by W

ashi

ngto

n U

nive

rsity

in S

t. L

ouis

on

05 O

ctob

er 2

011

Publ

ishe

d on

14

July

201

1 on

http

://pu

bs.r

sc.o

rg |

doi:1

0.10

39/C

1FD

0007

1C

View Online



20. Y. Du and F. Tian, J. Chem. Res., 2006, 486-489. 21. Z. Duan, Y. Gu, J. Zhang, L. Zhu and Y. Deng, J. Mol. Catal. A Chem., 2006,

250, 163-168. 22. M. A. B. H. Susan, A. Noda, S. Mitsushima and M. Watanabe, Chem.

Commun. (Cambridge, U. K.), 2003, 938-939. 23. B. Nuthakki, T. L. Greaves, I. Krodkiewska, A. Weerawardena, M. I. Burgar,

R. J. Mulder and C. J. Drummond, Aust. J. Chem., 2007, 60, 21-28. 24. D. R. MacFarlane and K. R. Seddon, Aust. J. Chem., 2007, 60, 3-5. 25. M. Yoshizawa, W. Xu and C. A. Angell, J. Am. Chem. Soc., 2003, 125,

15411-15419. 26. J. Stoimenovski, E. I. Izgorodina and D. R. MacFarlane, Phys. Chem. Chem.

Phys., 2010, 12, 10341-10347. 27. K. Bica, C. Rijksen, M. Nieuwenhuyzen and R. D. Rogers, Phys. Chem.

Chem. Phys., 2010, 12, 2011-2017. 28. K. Bica and R. D. Rogers, Chem. Commun. (Cambridge, U. K.), 2010, 46,

1215-1217. 29. R. E. Buntrock, The Merck Index: An Encyclopedia of Chemicals, Drugs, and

Biologicals, Fourteenth Edition, edited by Maryadele J. O'Neil, Patricia E. Heckelman, Cherie B. Koch, and Kristin J. Roman, 2007.

30. G. M. Sheldrick, Acta Crystallogr., Sect. A Found. Crystallogr., 2008, A64, 112-122.

31. W. N. Zheng and B. Wang, Acta Crystallogr., Sect. E Struct. Rep. Online, 2009, E65, o2769.

32. K. M. Johansson, E. I. Izgorodina, M. Forsyth, D. R. MacFarlane and K. R. Seddon, PCCP, 2008, 10, 2972-2978.

33. K. Bica and R. D. Rogers, Chem. Commun. (Cambridge, U. K.), 46, 1215-1217.

34. D. R. MacFarlane, R. Vijayaraghavan, H. N. Ha, A. Izgorodin, K. D. Weaver and G. D. Elliott, Chem. Commun. (Cambridge, U. K.), 2010, 46, 7703-7705.

35. D. R. MacFarlane, M. Forsyth, E. I. Izgorodina, A. P. Abbott, G. Annat and K. Fraser, PCCP, 2009, 11, 4962-4967.

36. K. J. Fraser, E. I. Izgorodina, M. Forsyth, J. L. Scott and D. R. MacFarlane, Chem. Commun., 2007, 3817-3819.

37. S. I. Smedley, The Interpretation of Ionic Conductivity in Liquids, Plenum Press, New York, 1980.

38. J. Stoimenovski and D. R. MacFarlane, In preparation, 2011. 39. V. Tantishaiyakul, N. Phadoongsombut, W. Wongpuwarak, J.

Thungtiwachgul, D. Faroongsarng, K. Wiwattanawongsa and Y. Rojanasakul, Int. J. Pharm., 2004, 283, 111-116.

40. U. L. Bernard, E. I. Izgorodina and D. R. MacFarlane, J. Phys. Chem. C, 2010, 114, 20472-20478.

41. U. Rana and et.al., Chem. Commun., 2011, In press. 42. B. Winther-Jensen, K. Fraser, C. Ong, M. Forsyth and D. R. MacFarlane, Adv.

Mater. (Weinheim, Ger.), 2010, 22, 1727-1730. 43. M. J. T. Frisch, G. W.; Schlegel, H. B.; Scuseria, G. E.; Robb, M. A.;

Cheeseman, J. R.; Montgomery, J. A.; Jr., T. V.; Kudin, K. N.; Burant, J. C.; Millam, J. M.; Iyengar, S. S.; Tomasi, J.; Barone, V.; Mennucci, B.; Cossi, M.; Scalmani, G.; Rega, N.; Petersson, G. A.; Nakatsuji, H.; Hada, M.; Ehara, M.; Toyota, K.; Fukuda, R.; Hasegawa, J.; Ishida, M.; Nakajima, T.; Honda, Y.; Kitao, O.; Nakai, H.; Klene, M.; Li, X.; Knox, J. E.; Hratchian, H. P.;

Far

aday

Dis

cuss

ion

s A

ccep

ted

Man

usc

rip

t

Dow

nloa

ded

by W

ashi

ngto

n U

nive

rsity

in S

t. L

ouis

on

05 O

ctob

er 2

011

Publ

ishe

d on

14

July

201

1 on

http

://pu

bs.r

sc.o

rg |

doi:1

0.10

39/C

1FD

0007

1C

View Online



Cross, J. B.; Bakken, V.; Adamo, C.; Jaramillo, J.; Gomperts, R.; Stratmann, R. E.; Yazyev, O.; Austin, A. J.; Cammi, R.; Pomelli, C.; Ochterski, J. W.; Ayala, P. Y.; Morokuma, K.; Voth, G. A.; Salvador, P.; Dannenberg, J. J.; Zakrzewski, V. G.; Dapprich, S.; Daniels, A. D.; Strain, M. C.; Farkas, O.; Malick, D. K.; Rabuck, A. D.; Raghavachari, K.; Foresman, J. B.; Ortiz, J. V.; Cui, Q.; Baboul, A. G.; Clifford, S.; Cioslowski, J.; Stefanov, B. B.; G. Liu, A. L.; Piskorz, P.; Komaromi, I.; Martin, R. L.; Fox, D. J.; Keith, T.; Al-Laham, M. A.; Peng, C. Y.; Nanayakkara, A.; Challacombe, M.; Gill, P. M. W.; Johnson, B.; Chen, W.; Wong, M. W.; Gonzalez, C.; Pople, J. A., GAUSSIAN 03, Revision C.02;, 2004, Gaussian, Inc., Wallingford CT.

44. J. A. Bautista-Martinez, L. Tang, J. P. Belieres, R. Zeller, C. A. Angell and C. Friesen, J. Phys. Chem. C, 2009, 113, 12586-12593.

45. J.-P. Belieres and C. A. Angell, J. Phys. Chem. B, 2007, 111, 4926-4937. 46. R. W. Gurney, Ionic Processes in Solution, 1953. 47. K. Bica, J. Shamshina, W. L. Hough, D. R. MacFarlane and R. D. Rogers,

Chem. Commun. (Cambridge, U. K.), 2011, 47, 2267-2269.

Far

aday

Dis

cuss

ion

s A

ccep

ted

Man

usc

rip

t

Dow

nloa

ded

by W

ashi

ngto

n U

nive

rsity

in S

t. L

ouis

on

05 O

ctob

er 2

011

Publ

ishe

d on

14

July

201

1 on

http

://pu

bs.r

sc.o

rg |

doi:1

0.10

39/C

1FD

0007

1C

View Online



Documents

~$Binder1