Enhanced LMI Representations for L2-L~

performance ofpolytopic uncertain systems:continuous-time case

Juntao WangCollege of Civil Aviation and Safety Engineering

Shenyang Institute ofAeronautical Engineering, Shenyang 110136, P. R. ChinaWei-Jun Wu

National Key Laboratory ofAntenna and Microwave TechnologyXidian University, Xi'an 710171, P. R. China

II. LMI REPRESENTATIONS FOR L2 - Loo PERFORMANCE

Consider the following continuous-time systems

Definition 1: [6] The L2 - Loo norm ofthe transfer function

Tzw (s) is defined as

where x E Rn is the state, WE Rq is the disturbance signal of

finite energy in the space L2 [0, + 00) , Z E RP is the objective

signal and A, B, C are known constant matrices. Denote the

transfer function of the system (1) for w to z by

parameter-dependent Lyapunov function that assures robuststability and a guaranteed cost. To eliminate this limitation, byintroducing two auxiliary slack matrices, many new sufficientLMI representations of H oo and/or H 2 performance are

established in [11] and [12] for discrete-time systems and in [12]and [10] for continuous systems. Moreover, the LMIrepresentations of H 00 performance established in [10] is

proved to be necessary and sufficient by employing a descriptorsystem approach. These conditions have been applied to theanalysis and synthesis for polytopic uncertain systems andprovide less conservativeness.

Motivated by the underlying idea in [10] and [12], in this notewe present several new L2 - Loo performance criteria for

continuous-time systems. The proposed LMI representations areproved to be necessary and sufficient. Due to the introduction oftwo slack matrices, less conservativeness is intended to beachieved when used in the analysis and synthesis of polytopicuncertain systems. Two numerical examples are employed toillustrate the feasibility and advantage of the present criteria.

(1)

(2)

{X(t) = Ax(t) +Bw(t),

z(t) = Cx(t)

Tzw(s) = C(sI - A)-l B.

I. INTRODUCTION

The domain of robust analysis and robust control synthesisfor uncertain linear systems has been thoroughly investigated inthe last two decades. In the earlier work, the robust analysis andsynthesis for polytopic and/or norm-bounded uncertain systemsis based on the quadratic concept. It is well known that the maindrawback of the LMI representations in quadratic framework isthat the Lyapunov function used for testing systemperformances is involved in system matrices. Thus therequirement of a single Lyapunov function over the wholeuncertainty domain is imposed on the plant, which can lead tomuch conservativeness. To reduce the degree of conservatisminherent in quadratic framework, many scholars have attemptedto seek multiple or parameter-dependent Lyapunov functions. Asignificant breakthrough toward this direction is the work in [1].In this work, by introducing an auxiliary slack matrix theauthors presented a new LMI stability criterion of discrete-timesystems, which exhibits a kind of decoupling between theLyapunov matrix and the dynamic matrices. When applying themodified LMI criterion to robust analysis and synthesis forpolytopic uncertain system, the Lyapunov function is allowed tobe vertex-dependent, thus the design conservatism is reduced.Motivated by the idea in [1], many scholars have established aseries of LMI performance criteria with a kind of separationproperty between the Lyapunov matrices and the dynamicmatrices for continuous-time systems in [2]-[6] and fordiscrete-time systems in [7] - [9]. Moreover, these LMIrepresentations of performances have been applied to theanalysis and synthesis for polytopic uncertain systems [2] - [9].Although providing less conservative evaluations, theseconditions still impose some common matrices to obtain the

Abstract- Based on two recent results, several new criteria ofL2 - Loo performance for continuous-time linear systems are

established by introducing two slack matrices. When used inrobust synthesis of systems with polytopic uncertainties, it canreduce conservatism inherent in the earlier quadratic method andthe parameter-dependent Lyapunov function approach. Twonumerical examples are included to illustrate the feasibility andadvantage of the proposed representations.

1-4244-2386-6/08/$20.00 ©2008 IEEE

,sup)(22

2 ),0[0 L

L

LwLLzw w

zsT ∞

∞ ∞+∈∉− =

where

.d)(,)(sup21

2

2

0⎟⎠⎞

⎜⎝⎛== ∫

∞+

∞ttwwtzz L

tL

Lemma 1: [6] Given a scalar 0>γ , system (1) is asymptotically stable with γ<

∞−LLzwT2

if and only if there

exists a positive definite matrix Q satisfying the following LMIs:

,2ICQCT γ< (3)

.0<++ TT BBQAAQ (4) Theorem 1: Given a scalar 0>γ , system (1) is asymptotically stable with γ<

∞−LLzwT2

if and only if there

exist positive definite matrix Q and matrices F and G such that there holds:

,02

<⎥⎥⎦

⎤

⎢⎢⎣

⎡

−−

QQCCQI

Tγ (5)

.00

0<

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−−+−+−+

IBBGGAQFG

QAFGFF

T

TT

TTT

(6)

Proof: Obviously, (5) is equivalent to (3) by Schur Complement Lemma. So it suffices to show the equivalence between (6) and (4).

(6) ⇒ (4): Let

,00

0⎥⎦

⎤⎢⎣

⎡=

III

T

pre- and post-multiplying the both sides of (6) by T and ,TT respectively, gives (4).

(4) ⇒ (6) : Let

,0

,0

,000

⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡−

=⎥⎦

⎤⎢⎣

⎡=

BB

IAI

AI

E

[ ].:0

,:0 21

2

1 NNIA

IN

MM

III

M =⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡−

=

it is easily obtained that

.,0

0,

000

⎥⎦

⎤⎢⎣

⎡−

=⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡=

BB

BMI

ANAM

IMEN (7)

Let

,0~,~ >+= QQBBZ T then we have

.0~00 <

⎥⎥⎦

⎤

⎢⎢⎣

⎡

−−++=Ξ

QZBBQAAQ TT

(8)

It is easily obtained that

[ ] [ ] [ ]

TTTT

TTTTTT

TTTT

TTT

TT

T

TT

TTT

MBBAQQAM

MBBMMANZMM

MMZNAMMANQ

QQMEN

MENQ

QQNAM

BBB

BIZ

II

IIZI

IA

QQQII

QQQ

IA

BBBBBBBB

ZZZ

QQQAAQ

)(

)(

)(0~~0~~

0)()(0

00

0~~

000

000

0~~

00

20

0~~

22

22

++=

+−+

−+⎥⎥⎦

⎤

⎢⎢⎣

⎡

−−

+⎥⎥⎦

⎤

⎢⎢⎣

⎡

−−=

−⎥⎦

⎤⎢⎣

⎡−

+−⎥⎦

⎤⎢⎣

⎡−+−−⎥

⎦

⎤⎢⎣

⎡+

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎥⎥⎦

⎤

⎢⎢⎣

⎡

−−

⎥⎦

⎤⎢⎣

⎡+⎥

⎦

⎤⎢⎣

⎡

⎥⎥⎦

⎤

⎢⎢⎣

⎡

−−

⎥⎦

⎤⎢⎣

⎡=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

−−+⎥

⎦

⎤⎢⎣

⎡−

+⎥⎥⎦

⎤

⎢⎢⎣

⎡

−−+=Ξ

where

.~0

)(0

~~

22 ⎥⎦

⎤⎢⎣

⎡−

=−+⎥⎥⎦

⎤

⎢⎢⎣

⎡

−−=

ZQAQQ

MZNENQ

QQNQ TT

Therefore, inequality (8) becomes

.0)( <++ TTTT MBBAQQAM

A congruence transformation results in

.0<++ TTT BBAQQA (9)

Let

,0⎥⎦

⎤⎢⎣

⎡=

GFQ

Q

substituting the expression of Q into (9) gives

,0<⎥⎥⎦

⎤

⎢⎢⎣

⎡

−−−+−+−+

TTT

TTT

BBGGAQFGQAFGFF

which, by Schur Complement Lemma, is equivalent to (6).

Theorem 2: Given a scalar 0>γ , system (1) is asymptotically stable with γ<

∞−LLzwT2

if and only if there

exist a positive definite matrix P and matrices F and G satisfying

,02

<⎥⎥⎦

⎤

⎢⎢⎣

⎡

−−

PCCI

Tγ

.00

<⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−+−+−+

PBGGPAFG

PAFGQAAQ

T

TT

TTT

Proof: It is only shown that (11) is equivalent to (6). Pre- and

post-multiplying the both sides of (6) by diag ),,( 11 IQQ −− gives

,00

0

1

111

11111

11111

1111

<⎥⎥⎥

⎦

⎤

−−−

+−

⎢⎢⎢

⎣

⎡

+−+

−

−−−

−−−−−

−−−−−

−−−−

IQBBQGQGQ

QAQFQGQQ

AQFQQQGQQFQFQQ

T

T

TT

T

T

which implies (11). Theorem 3: Given a scalar 0>γ . System (1) is asymptotically stable with γ<

∞−LLzwT2

if and only if there

exist a positive definite matrix P and matrices F and G such that there holds:

,02

<⎥⎥⎦

⎤

⎢⎢⎣

⎡

−−

PCCI

Tγ (12)

.0<⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−−+−+−+

IGBFBBGGGAGFPBFGAFPFAAF

TT

TTT

TTTTT

(13)

Proof: It is obvious that (12) is equivalent to (3). It is only necessary to show the equivalence between (13) and (4).

(13) ⇒ (4): Let

,0

0⎥⎥⎦

⎤

⎢⎢⎣

⎡=

IBAIT T

T

then pre- and post-multiplying the both sides of (6) by T and

TT , respectively, gives (4). (4) ⇒ (13): Let ,1−= QP it follows from (9) that

.0<++ PBBPPAAP TTTT

Let

,0⎥⎦

⎤⎢⎣

⎡=

GFQ

P

substituting the expression of Q into (14) gives

[ ] ,0<⎥⎥⎦

⎤

⎢⎢⎣

⎡+

⎥⎥⎦

⎤

⎢⎢⎣

⎡

−−+−+−+ GBFB

BGBF

GGAGFQGAFQAFFA TT

T

T

TT

TTTT

which, by Schur Complement Lemma, is equivalent to (13).

In the case where the system (1) is subject to uncertainties in the form of a polytopic model

[ ]

[ ] [ ] ,,)()()(1

1

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

Γ∈==

Ω∈

∑=

ααααα iiii

r

i

CBACBA

CBA

(15)

where Γ is the unit simplex

,0,1:),,,(1

21⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

≥==Γ ∑=

ii

r

ir ααααα (16)

Theorem 1 can be readily used to derive the following robust performance criterion. Corollary 1: Consider the system (1) subject to the uncertainties (15). For a prescribed ,0>γ the system (1) is robustly stable with γ<

∞−LLzwT2

if there exist positive

definite matrix Q and matrices ,,,2,1,, riGF ii = satisfying

,02

<⎥⎥⎦

⎤

⎢⎢⎣

⎡

−−

QQCQCI

Ti

iγ

.00

0<

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−−+−+−+

IBBGGQAFG

QAFGFF

Ti

iTiiii

Ti

Ti

Tii

Tii

In quadratic framework, according to Lemma 1 the following result is given. Corollary 2: Consider the system (1) subject to the uncertainties (15). For a prescribed ,0>γ the system (1) is robustly stable with γ<

∞−LLzwT2

if there exists a positive

definite matrix Q satisfying

,02

<⎥⎥⎦

⎤

⎢⎢⎣

⎡

−−

QQCQCI

Ti

iγ

.0<⎥⎥⎦

⎤

⎢⎢⎣

⎡

−+

IBBQAQA

Ti

iTii

According to the result in [6], the following conclusion is obtained. Corollary 3: Consider the system (1) subject to the uncertainties (15). For a prescribed ,0>γ the system (1) is robustly stable with γ<

∞−LLzwT2

if there exist a positive

definite matrix P and matrices ,,,2,1, riGi = satisfying

,02

<⎥⎥⎦

⎤

⎢⎢⎣

⎡

−−

PCCI

Ti

iγ

.0

000000 <

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

−−

−++−−

IGBPG

PPGABGGPAGGG

iT

i

iT

Ti

Ti

Tii

Ti

III. STATE FEEDBACK CONTROL FOR POLTOPIC UNCERTAIN

SYSTEMS In this section, the result in Theorem 1 is applied to the ∞− LL2 control problem of polytopic uncertain continuous-time systems. Consider the system

,12

⎩⎨⎧

+=++=

DuCxzwBuBAxx

(17)

where u is the control input and other signals are defined as in section II . We assume that the system (17) is subject to uncertainties in the form of a polytopic model

,,00)()(

)()()(

0

12

1

12

12

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

Γ∈⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡=

Ω∈⎥⎦

⎤⎢⎣

⎡

∑=

αααα

ααα

ii

iiii

r

iDC

BBADC

BBA

DCBBA

where Γ is defined as (16). For (17), consider the state feedback control law .Kxu = The closed-loop system is

.)()( 12

⎩⎨⎧

+=++=

xDKCzwBxKBAx

(18)

The present problem is to find the state feedback gain matrix K such that the ∞− LL2 norm of the closed-loop transfer function

)(sTzw from disturbance w to objective output z is less

than a prescribed positive number .γ Theorem 4: Given ,0>γ there exists a state feedback law

Kxu = such that the closed-loop system (18) is asymptotically stable with γ<

∞−LLzw sT2

)( if and only if there exists a

positive definite matrix Q and matrices F , G and R satisfying the following LMIs:

,0)(

2<

⎥⎥⎦

⎤

⎢⎢⎣

⎡

−++−QDRCQ

DRCQIT

γ (19)

.00

0

1

12

2

<⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−−++−

++−+

IBBGGRBAQFG

BRQAFGFF

T

TT

TTTTT

(20)

In addition, a desired state feedback gain is given by

.1−= RQK (21) In view of the property of the simplex ,Γ the following conclusion is obtained. Theorem 5: Given ,0>γ there exists a state feedback gain K such that the system () is robustly asymptotically stable with

γ<∞−LLzw sT

2)( if there there a positive definite matrix Q ,

matrices riGF ii ,,2,1,, = and R satisfying

,0)(

2<

⎥⎥⎦

⎤

⎢⎢⎣

⎡

−++−QRDQC

RDQCIT

ii

iiγ (22)

.00

0

1

12

2

<⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−−++−

++−+

IBBGGRBQAFG

BRQAFGFF

Ti

iTiiiii

Ti

Ti

TTi

Tii

Tii

(23)

In addition, a desired control law is given by (21). Remark 1: In the above theorem, two invertible matrix

variables F and G are allowed to vary with vertices of the polytope, while the positive matrix variable Q must be identical for all vertices.

IV. ILLUSTRATE EXAMPLES

Example 1: Consider the example studied in [4] and [10]:

[ ],21,10

,1110

−=⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡−−+−

= CBgg

A

where ],[ hhg −∈ is an interval parameter. Table 1 lists the

minimum ∗γ of ∞− LL2 norm guaranteed by the way proposed in Corollary 1 which is obtained according to the

criteria developed in this paper. For comparison convenience, the minimum values of ∗γ based on Corollary 2 and Corollary 3 are also listed. Table 1 shows that our result is less conservative than the results in the quadratic framework and based on [Gao2003]. Example: Consider the state feedback ∞− LL2 control problem of the yaw angles of the following satellite system ([4], [10]):

Table 1: Minimum ∞− LL2 norm guaranteed by the proposed way

case 0=h 0=h .1 0=h .2

∗γ of Corollary 1 1.5812 1.7250 1.9118 ∗γ of Corollary 2 1.5848 1.7283 1.9135 ∗γ of Corollary 3 1.5820 2.7270 1.9120

,

0

00

1000

2.02.02.02.0

10000100

2

1

2

1

2

1

2

1

T

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

+

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

+⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−−−−

=

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

gw

ffff

θθθθ

θθθθ

where 1θ and 2θ are yaw angles of two rigid bodies joined by a flexible link, T is the control torque, w is the torque disturbance and the parameter f is the viscous damping of the link model. A finite element analysis gives the uncertainty ranges of the two parameters as:

].2.1,8.0[],04.0,0038.0[ ∈∈ gf The objective signal is ).01.0,col( 2 Tθ=z

Based on Corollary 5 , a minimum achievable ∗γ is 0.8425, and the corresponding control law is

[ ] ,9305.10964.01230.12921.0102 θ−−−−=T

with ).,,,col( 2121 θθθθθ = In quadratic framework, the

minimum is given by ,8427.0=∗γ and the corresponding control is

[ ] .9469.10971.01354.12953.0102 θ−−−−=T

This example shows again that our proposed criterion achieves less conservative results.

V. CONCLUSIONS

Several new enhanced L 2 − L criteria for continuous-time systems are established based on two recent results. The proposed conditions are necessary and sufficient. Furthermore, the presented criteria are applied to the analysis and synthesis of the systems with polytopic uncertainties. Due to the introduction of the more slack matrices, the proposed criteria can achieve less conservative results over results based on the traditional quadratic framework and the parameter-dependent Lyapunov function. The two numerical examples show the feasibility and advantage of the present criteria.

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