Decentralized compensation of controller uncertainty in the steady-state bumpless transfer under the state/output feedback

INTERNATIONAL JOURNAL OF ROBUST AND NONLINEAR CONTROLInt. J. Robust Nonlinear Control 2009; 19:1083–1104Published online 15 August 2008 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/rnc.1360

Decentralized compensation of controller uncertainty in thesteady-state bumpless transfer under the

state/output feedback‡

Kai Zheng§ and Joseph Bentsman∗,†

Department of Mechanical Science and Engineering, University of Illinois at Urbana-Champaign,1206 W Green St., Urbana, IL 61801, U.S.A.

SUMMARY

The state/output feedback topology required for bumpless transfer under controller uncertainty is consid-ered. A steady-state diagonal dominance condition for a subsystem of this topology is derived, which isshown to be sufficient for the existence of the stabilizing decentralized mismatch compensator parameters.Recent results on the input/output steady-state gain structure of the one-degree-of-freedom and the two-degrees-of-freedom infinite horizon LQ bumpless transfer topologies are used to show that the diagonaldominance condition applies to topologies generated by a broad class of controllers, thus completingthe state/output feedback bumpless transfer technique. The topologies are also shown to be robust withrespect to uncertainty in the mismatch compensator. Copyright q 2008 John Wiley & Sons, Ltd.

Received 28 July 2006; Revised 19 November 2007; Accepted 2 June 2008

KEY WORDS: steady-state bumpless transfer; controller uncertainty; state/output feedback; decentralizedcontrollers; integral controllers

∗Correspondence to: Joseph Bentsman, Department of Mechanical Science and Engineering, University of Illinois atUrbana-Champaign, 1206 W Green St., Urbana, IL 61801, U.S.A.

†E-mail: [email protected]‡The work in this paper was done when Kai Zheng was with the University of Illinois at Urbana-Champaign.Currently Dr Zheng is with ArcelorMittal Steel Company U.S.A. R&D, 3001 E Columbus Drive, East Chicago,46322.§E-mail: [email protected]

Contract/grant sponsor: Grainger Center for Electric Machinery and Electromechanics at UIUCContract/grant sponsor: National Science Foundation; contract/grant numbers: CMS-0324630, ECS-0501407,DMI-0500453Contract/grant sponsor: Electric Power Research Institute

Copyright q 2008 John Wiley & Sons, Ltd.

1084 K. ZHENG AND J. BENTSMAN

1. INTRODUCTION

Recently, a considerable effort has been dedicated to bumpless transfer or seamless switchingbetween controllers in closed-loop control applications [1–6]. This effort has been characterizedby the synthesis of the transfer operators for controllers whose actual dynamics matches exactlythe nominal controller representation used in the transfer operator synthesis. The full informationset for bumpless transfer consists, in this case, of the full state of the offline controller along withthe online controller input and output. Among the approaches proposed, the infinite horizon LQbumpless transfer technique of Turner and Walker [6] stands out as a very convenient tool for thesteady-state bumpless transfer synthesis in industrial multi-input multi-output (MIMO) applicationsdue to its widely accessible and numerically reliable linear quadratic (LQ) computational settingas well as excellent convergence properties inherent in the LQ full state feedback.

In [7], however, it is pointed out that the variable scan times of the programmable logic controllerslargely used in industrial processes frequently cause random parameter variations, resulting incontroller uncertainty. In [8, 9], moreover, a significant controller uncertainty is also found inmegawatt/throttle pressure control of a boiler/turbine unit. This controller uncertainty is shown togive rise to the divergence of the output of the offline controller from that of the online one underthe topology of [6], thereby precluding controller transfer. Analysis of this problem carried out in[9] demonstrates that this convergence loss arises due to the use of the nominal state of the offlinecontroller, in lieu of its true state, by the transfer operator. To address this problem, the conceptof full information set for bumpless transfer under controller uncertainty is introduced in [9] andthis set is shown to consist of the nominal state and the actual output of the offline controlleralong with the actual input and output of the online controller. This observation resulted in anovel state/output feedback bumpless transfer topology and the corresponding design technique,developed in [8, 9] and demonstrated therein to solve the problem of high-quality steady-statebumpless transfer from the industry standard low-order nonlinear multi-loop proportional integralderivative (PID)-based controllers to modern MIMO robust controllers in the megawatt/throttlepressure control of a boiler/turbine unit.

The topology of [9] consists of two feedback loops: the inner loop that employees the nominalstate of the offline controller and is designed using the technique of [6], and the outer loop thatcompensates for the controller/model mismatch by employing the offline controller output. Ablock diagram showing the outer loop is given in Figure 1, where e′ and u′ are online controllerinput and output, respectively, K and � denote the mismatch compensator and its output, u is theoffline controller output, and G1 and G2 represent, respectively, the transfer functions from theinput e′ of the online controller and the output � of the mismatch compensator to the output uof the offline controller. When this loop is broken and K = I , � becomes u′. Hence, G1 and G2

Figure 1. Simplified offline controller subsystem in the state/output feedback bumpless transfer topologywith the outer feedback loop broken.

Copyright q 2008 John Wiley & Sons, Ltd. Int. J. Robust Nonlinear Control 2009; 19:1083–1104DOI: 10.1002/rnc

DECENTRALIZED COMPENSATION OF CONTROLLER UNCERTAINTY 1085

Figure 2. Reduced offline controller subsystem in the LQ bumpless transfer topology.

can be determined using Figure 2. The latter is exactly the LQ bumpless transfer topologyproposed in [6].

Thus, the structure of G2 is independent of controller uncertainty and is determined only bythe nominal offline controller and the transfer operator computed through the LQ technique of[6]. The transparency of the transfer operator implementation and tuning in industrial MIMOapplications dictates that the mismatch compensator structure be the simplest possible, preferablydiagonal and consisting of decoupled scalar PID-type controllers. In [8, 9] G2 is numerically foundto have a diagonal steady-state structure for a given specific boiler/turbine controller. This, inturn, is determined to be sufficient for the existence of the stabilizing decentralized mismatchcompensator, and the procedure based on loop shaping and convex programming is developed in[8, 9] to compute the required compensator parameter values.

The questions, however, remain whether the diagonal steady-state structure of G2 correspondingto the specific boiler/turbine controller considered in [8, 9] holds for a broad class of controllers,and if the latter is true, whether this property could be established without computing G2. Then,the decentralized mismatch compensator design of [8, 9] would become applicable to the entireclass of such controllers, and its applicability could be established directly just by examining theoffline controller properties. Moreover, it is demonstrated in [9] that the bumpless transfer operatorsynthesized by the technique proposed therein robustly stabilizes the offline controller subsystemwith respect to uncertainty in G2. Note that the mismatch compensator, usually implemented thesame way as G2, might suffer from uncertainty as well. Reference [9], however, does not addressthe robustness of the bumpless transfer topology with respect to uncertainty in the mismatchcompensator.

These questions are answered in the present work, thereby completing the methodology of[9] and making the technique proposed broadly applicable. Specifically, a steady-state diagonaldominance condition, encompassing a much wider class of offline controllers than the steady-statedecoupledness one previously introduced, is first proposed and shown to be sufficient for theexistence of a family of decentralized mismatch compensators, all of which are shown to stabilizethe state/output feedback topology with respect to uncertainty in both G2 and the compensatorsthemselves. Furthermore, the robustly stabilizing decentralized compensator parameters are readilyobtained using the convex optimization algorithm proposed in [9].

Then, the recent results in [10] on the input/output structure of the one-degree-of-freedom(1DOF) and the two-degrees-of-freedom (2DOF) infinite horizon LQ bumpless transfer topologiesare applied to demonstrate that G2 is indeed steady-state diagonally dominant for essentially theentire class of strictly proper controllers with the input dimension being no less than the output one.It is also shown that when the offline controller input dimension is less than its output one, stabi-lizing mismatch compensators of integral type do not exist due to G2 being singular. To compen-sate controller uncertainty in the latter case, while still preserving the structural transparency,the simple decentralized non-integral mismatch compensator structure is proposed and shown toadmit stabilizing parameters for essentially all controllers in this class. This completely solves



the problem of establishing the existence of the stabilizing decentralized mismatch compensatorsin the state/output feedback topology required for bumpless transfer under controller uncertaintysolely on the basis of the offline controller nominal model.

The paper is organized as follows: the 1DOF infinite horizon LQ bumpless transfer topologyand its supporting theory and input/output structure results are summarized in Section 2. Section 3summarizes the topology and theory of the 1DOF state/output feedback topology. In Section 4the diagonal steady-state structure of G2 in LQ bumpless transfer topology is formally linked tothe design of bumpless transfer under controller uncertainty for 1DOF topology and existenceresults for the decentralized controller uncertainty compensation are obtained. Section 5 extends theexistence results to 2DOF state/output feedback topology. The conclusion is given in Section 6.

Notation: Throughout this paper, I and 0 denote identity and zero matrices, respectively, withtheir dimension being clear from the context. In is used to denote the identity matrix of n×n,when it is necessary to indicate the dimension. A�B denotes that A is defined to be equal toB. The superscript ᵀ denotes matrix transposition. [ai j ]i, j=1,...,n denotes an n×n matrix, the i throw and j th column entry of which is ai j . For consistency with the terminology in [9], the term‘steady-state gain’ is used to denote the ‘static gain’ of a transfer function. The qualifier ‘infinitehorizon’ for LQ bumpless transfer is dropped, but assumed, throughout the paper.

2. LQ BUMPLESS TRANSFER: TOPOLOGY, THEORY, AND INPUT/OUTPUTSTRUCTURE

2.1. Topology and theory [6]The 1DOF LQ bumpless transfer topology proposed in [6] is shown in Figure 3. As seen from thisfigure, the static feedback matrix F maps the offline controller state x(t) and the online controlleroutput u′(t) and input e′(t), which is the plant output error signal, into the offline controller input�(t), forcing the offline controller output u(t) to reach the desired value needed for the bumplesstransfer. The derivation of matrix F (cf. [6]) is carried out through the quadratic minimization ofthe functional that includes the difference between two sets of signals, the output signals of bothcontrollers and the input signals driving the controllers, given by

J0(u,�)= 1

2

∫ ∞

0[zu(t)ᵀWuzu(t)+ze(t)

ᵀWeze(t)]dt (1)

where zu(t)=u(t)−u′(t), ze(t)=�(t)−e′(t), and Wu and We are constant positive-definiteweighting matrices of appropriate dimensions used to tune the design.

The computation of F is carried out as follows. First, a controllable and observable state-spacerealization (A, B,C,D) of the offline controller to be implemented is selected. The controllerdynamics along with the controller input �(t) and output u(t) is then given by

x = Ax+B�

u = Cx+D�(2)

where A, B,C, and D are matrices of dimension n×n,n×m,k×n, and k×m, respectively. Thevector � is computed as �=F[xᵀ (u′)ᵀ (e′)ᵀ]ᵀ. Assuming that the online controller output u′ and



input e′ are both constant in the steady state, F can be computed as

F=�

⎡⎢⎣(BᵀX+DᵀWuC)ᵀ

(−DᵀWu+BᵀM(CᵀWu+CᵀWuD�Dᵀ+XB�DᵀWu))ᵀ

(−We+BᵀM(CᵀWuD�We+XB�We))ᵀ

⎤⎥⎦ᵀ

(3)

where X is the positive semi-definite solution of the ARE X A+ AᵀX+X BX+C=0 with �=−(DᵀWuD+We)

−1, M=( Aᵀ+X B)−1, A= A+B�DᵀWuC , B= B�Bᵀ, and C=CᵀWu(I +D�DᵀWu)C . If D=0, then the expression for F noticeably simplifies.

2.2. Steady-state input/output structure [10]

The offline controller subsystem consisting of the offline controller and the feedback matrix F inFigure 3, denoted by G, can be represented by the loop configuration shown in Figure 4.

Partitioning G into (G1,G2) according to u=G1e′+G2u′, the topology of Figure 4 can besimplified to that shown in Figure 2.

In [10], the explicit formulas for the steady-state gains G1(0) and G2(0) of G1(s) and G2(s),respectively, for strictly proper controllers are expressed as

G1(0)=C(A−BW−1e BᵀX)−1BW−1

e [−We−Bᵀ(A−BW−1e BᵀX)−ᵀXB] (4)

and

G2(0)=C(A−BW−1e BᵀX)−1BW−1

e Bᵀ(A−BW−1e BᵀX)−ᵀCᵀWu (5)

respectively.

Figure 3. The one-degree-of-freedom LQ bumpless transfer topology.

Figure 4. Offline controller subsystem in the one-degree-of-freedom LQ bumpless transfer topology.



The use of the result from [11]—the symmetry of G2(0)—immediately leads to the followingtheorem [10].Theorem 1Suppose that the offline controller in Figure 4 is strictly proper and its input dimension is no lessthan its output one. Suppose further that the offline controller has no zeros at the origin. Then,G2(0) is positive definite for Wu = I and any choice of positive definite We.

Furthermore, G1(0) and G2(0) are shown in [10] to be zero/asymptotically zero andidentity/asymptotically identity, respectively, for almost all the strictly proper controllers with theinput dimension being no less than the output one. These results are summarized in Theorems2 and 3. Specifically, Theorem 2 guarantees that for controllers with an integrator in each oftheir output channels, the steady-state gains G1(0) of G1 and G2(0) of G2 are zero and identity,respectively, regardless of the choice of positive-definite weighting matrices Wu and We.

Theorem 2Suppose the offline controller in Figure 4 contains an integrator in each of its output channels, i.e.the offline controller is realizable in the following controllable and observable form:

x1 = Asx1+Bs�

x2 = Csx1+Ds�

u = x2

(6)

where � is the input to the controller, u is the output of the controller, (xᵀ1 xᵀ2 )ᵀ is the state vector,and As, Bs,Cs, and Ds are matrices of dimension n×n,n×m,k×n, and k×m, respectively. Letthe feedback matrix F in Figure 4 be calculated via minimization of the performance index (1).Suppose further that k�m. Then, for any choice of positive-definite weighting matrices Wu andWe, G2(0)= I and G1(0)=0.

In the case when the controller does not contain an integrator in each of its output channels,G2(0) and G1(0) are, in general, not identity and zero, respectively, for an arbitrary choice ofweighting matrices We and Wu . Nevertheless, under mild conditions, G2(0) and G1(0) are asymp-totically identity and zero, respectively, as We approaches zero. These conditions are summarizedin Theorem 3.

Theorem 3Let the offline controller in Figure 4 be realized in the following controllable and observable form:

x = Ax+B�

u = Cx(7)

where A, B, and C are matrices of dimension n×n,n×m, and k×n, respectively, with k�m�n.Let the feedback matrix F in Figure 4 be calculated via minimization of the performance index(1) for positive-definite weighting matrices Wu = Im and We=�2 Im . Suppose that (1) the offline

controller does not have zeros on the imaginary axis, i.e.(A− j�I

CB0

)is the full row rank for any

real � and (2) B is full column rank and CB is the full row rank. Then lim�→0+G2(0)= I andlim�→0+G1(0)=0.



For controllers with the input dimension being less than the output one, i.e. m<k in (7), itis shown in [10] that G2(0) is rank deficient, or singular, for any controller structure and anyweighting matrices We and Wu . Thus, in this case G2(0) can never be positive definite, let aloneidentity or asymptotically identity.

3. BUMPLESS TRANSFER UNDER CONTROLLER UNCERTAINTY: THE STATE/OUTPUTFEEDBACK TOPOLOGY

This section summarizes the 1DOF state/output feedback topology proposed in [9] for bumplesstransfer under controller uncertainty.

In the case of 1DOF, it is shown in [9], both analytically and computationally, that the presence ofthe offline controller uncertainty creates in the topology of Figure 3 a non-vanishing online/offlinecontroller output error, precluding bumpless transfer. This problem is addressed by the state/outputfeedback topology proposed in [9] shown in Figure 5.

The transfer operator in this topology is seen to combine two distinct feedback controllers in anested configuration, the inner one representing the state feedback and the outer one representingthe output feedback, respectively. This configuration permits retaining the infinite horizon LQdesign of [6] for the inner loop, while employing, under certain conditions, a simple integralcontrol law for the outer loop. To see this, it is first observed in [9] that the offline controllersubsystem in Figure 5 with no offline controller uncertainty can be rearranged to that shown inFigure 6. The latter can then be simplified to that shown in Figure 7. Then, it is clear that the goalof the mismatch compensator design is to select compensator K such that the offline controlleroutput u converges to constant reference input, u′, sufficiently fast for the application of interest,

Figure 5. The state/output feedback bumpless transfer topology.

Figure 6. Offline controller subsystem in the state/output feedback topology.



Figure 7. Simplified offline controller subsystem in the state/output feedback bumpless transfer topology.

Figure 8. Controller/model mismatch compensator for steady-state bumplesstransfer under controller uncertainty.

under a constant output disturbance, G1e′. Observing that the stability of G2 is guaranteed by theARE, a decentralized integrator controller structure KI given by

K =KI =⎡⎢⎣k11

· · ·knn

⎤⎥⎦=

⎡⎢⎢⎢⎢⎢⎣k1s

· · ·kns

⎤⎥⎥⎥⎥⎥⎦ (8)

depicted also in Figure 8, i.e. consisting of decoupled SISO integrator controllers, is adopted in[9] for K .

The design of KI is approached via an MIMO loop shaping method as follows. Let matrix F inthe subsystem G of Figure 6 be calculated using (3). Let the target loop transfer matrix L given by

L=

⎡⎢⎢⎢⎢⎢⎣�1s

· · ·�ns

⎤⎥⎥⎥⎥⎥⎦ (9)

where �1, . . . ,�n>0 are the integrator weights, be the desired loop gain of the closed loop inFigure 7. Also define S=(I +L)−1, and �=G2K −L , and introduce the target-loop-gain-basedfunction:

S�=(I +L)−1(G2K −L) (10)

Then, if ‖S�‖H∞ =�<1, the topology in Figure 9 is proved (cf. Theorem 3 in [9]) to be robustlystable for all � such that ‖�‖H∞�(1−�)/(1+�). The design of stabilizing parameters of KI isthen carried out using convex optimization.



Figure 9. Offline controller subsystem with uncertainty in G2.

The retained infinite horizon LQ design in combination with the MIMO loop shaping methodis shown in [9] to guarantee the convergence of the output of the offline controller to that of theonline controller even under significant controller uncertainty, provided stabilizing KI exists.

A preliminary sufficient condition for the existence of stabilizing parameters for KI , establishedthrough the MIMO loop shaping, is given by Theorem 2 in [9]. This condition consists in requiringG2 to be steady-state decoupled. The definition of the latter property is given as follows.

Definition 1A real rational square n×n transfer matrix G with entries gir , i,r =1,2, . . . ,n, is referred to assteady-state decoupled, if each diagonal entry of G is non-zero at zero frequency, i.e. gii ( j0) �=0, i=1,2, . . . ,n, and each off-diagonal entry of G is zero at zero frequency, i.e. gir ( j0)=0 if i �=r .

The above condition of steady-state decoupledness on G2 is apparently quite restrictive. More-over, this condition cannot be examined a priori without knowing the state feedback matrix F (cf.Figure 6), rendering the applicability of the proposed state/output feedback topology unclear.

4. EXISTENCE OF THE STABILIZING DECENTRALIZED MISMATCHCOMPENSATORS

In this section, first, the above steady-state decoupledness condition required for the existenceof stabilizing mismatch compensators is relaxed to a much weaker one, namely, a steady-statediagonal dominance condition. Then, the results on the structure specificity of G2(0) summarized inSection 3 are applied to establish the existence of stabilizing decentralized mismatch compensatorsin the state/output feedback topology for essentially all strictly proper controllers.

4.1. The steady-state diagonal dominance condition

This condition is introduced in the following definition.

Definition 2A real rational square n×n transfer matrix G is referred to as the steady-state diagonally dominant,if there exists a constant diagonal matrix E such that

max(G(0)E− I )<1 (11)

where max(·) denotes the maximum singular value function.



It is easy to see from a comparison between the definition of the steady-state decouplednessand diagonal dominance that the former condition requires all the off-diagonal entries to be zeroin the steady state, whereas the latter only requires them to be small relative to diagonal entries inthe steady state. Thus, the steady-state diagonal dominance condition is much weaker, and henceencompasses a much wider class of offline controllers, than the steady-state decoupledness one.Another way to see this is to note that the quantity in the left-hand side of (11) is zero for anysteady-state decoupled transfer matrix if matrix E is chosen to be

E=

⎡⎢⎢⎢⎢⎢⎣1

g11(0)

· · ·1

gnn(0)

⎤⎥⎥⎥⎥⎥⎦ (12)

where gii ( j0) �=0, i=1,2, . . . ,n are the steady-state values of the diagonal entries of G.

Furthermore, if G(0) is positive definite, then E can be selected as E=(c0/vmax(G(0)))I ,where vmax(G(0)) is the largest eigenvalue of G(0) and 0<c0<2. With this E, it follows that0<vi (G(0)E)<2, where vi (G(0)E) is the i th eigenvalue of G(0)E , and hence −1<vi (G(0)E−I )<1, i=1,2, . . . ,n. Note that G(0)E− I is symmetric, implying that max(G(0)E− I )=max(‖vi (G(0)E− I )‖, i=1,2, . . . ,n)<1. This result is stated in the following lemma.

Lemma 4A real rational square n×n transfer matrix G is steady-state diagonally dominant if G(0) is positivedefinite.

The following theorem shows that G2 being steady-state diagonally dominant is sufficientto guarantee the existence of the decentralized mismatch compensator KI in (8) such that thestate/output topology in Figure 5 is robustly stable.

Theorem 5Let matrix F in the subsystem G of Figure 6 be calculated using (3). Suppose that G2 in Figure 7is steady-state diagonally dominant. Then, there exist a set of constants kui , i=1, . . . ,n, and afamily of MIMO integral controllers KI in (8) with parameters 0<ki sgn(ki )�|kui |, i=1, . . . ,n,where sgn(·) is the sign function, such that for each KI in the family the topology in Figure 7 isrobustly stable with respect to uncertainty in both G2 and the mismatch compensator.

ProofTo prove the statements of this theorem, it suffices to show that

(1) as mentioned above, there exists a target loop transfer matrix L in (9) and a decentralizedintegral controller KI in (8) with parameters kui , i=1, . . . ,n, such that ‖S�‖H∞<1, whereS� is given in (10), i.e. KI with parameters kui , i=1, . . . ,n, guarantees nominal stability;

(2) the same KI guarantees robust stability with respect to uncertainty in both G2 and themismatch compensator;

(3) the results in (1) and (2) hold for all KI with parameters 0<ki sgn(ki )�|kui |, i=1, . . . ,n.

The proof is carried out in four parts. First, it is claimed that both ‖G2‖H∞ and ‖S�‖H∞ arefinite. Indeed, since F is calculated from (3), G in Figure 6 is stable, implying the stability of G2



in Figure 7. Moreover, it is trivial to see that (1+L)−1L is stable and

(1+L)−1G2K =

⎡⎢⎢⎢⎢⎢⎢⎣

s

s+�1

· · ·s

s+�n

⎤⎥⎥⎥⎥⎥⎥⎦G2

⎡⎢⎢⎢⎢⎢⎢⎣

k1s

· · ·kns

⎤⎥⎥⎥⎥⎥⎥⎦

=

⎡⎢⎢⎢⎢⎢⎢⎣

1

s+�1

· · ·1

s+�n

⎤⎥⎥⎥⎥⎥⎥⎦G2

⎡⎢⎢⎣k1

· · ·kn

⎤⎥⎥⎦

is also stable due to the stability of G2. Thus, S� is stable.Then, a target loop gain L in (9) and an MIMO integral controller KI in (8) with parameters

kui , i=1, . . . ,n, are constructed such that ‖S�‖H∞<1, where S� is given in (10). For the sake ofsimplicity, assume n=3. Let the diagonal entries of matrix S be sii , i=1,2,3. In addition, let theentries of matrix G2 be gip, i, p=1,2,3. Then,

S� = S(G2K −L)

=

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

s11

(g11k11− �1

s

)s11g12k22 s11g13k33

s22g21k11 s22

(g22k22− �2

s

)s22g23k33

s33g31k11 s33g32k22 s33

(g33k33− �3

s

)

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

=

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

�1s+�1

(g11

ku1�1

−1

)ku2

s+�1g12

ku3s+�1

g13

ku1s+�2

g21�2

s+�2

(g22

ku2

�2−1

)ku3

s+�2g23

ku1s+�3

g31ku2

s+�3g32

�3s+�3

(g33

ku3�3

−1

)

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦(13)

Choose kui , i=1,2,3, according to

kui =ci�i , i=1,2,3 (14)



Then, (13) takes the form

S�=

⎡⎢⎢⎢⎢⎢⎢⎢⎣

�1s+�1

(c1g11(s)−1)�2

s+�1c2g12(s)

�3s+�1

c3g13(s)

�1s+�2

c1g21(s)�2

s+�2(c2g22(s)−1)

�3s+�2

c3g23(s)

�1s+�3

c1g31(s)�2

s+�3c2g32(s)

�3s+�3

(c3g33(s)−1)

⎤⎥⎥⎥⎥⎥⎥⎥⎦(15)

Define �max=max(�i , i=1,2,3) and �min=min(�i , i=1,2,3). It is trivial to see that∣∣∣∣ �ij�+�p

∣∣∣∣� ∣∣∣∣ �max

j�+�min

∣∣∣∣for i, p=1,2,3. Then, it follows from (15) that for every ��0,

max(S�( j�))�∣∣∣∣ �max

s+�min

∣∣∣∣max(DG( j�))

where

DG(s)�

⎡⎢⎣c1g11(s) c2g12(s) c3g13(s)

c1g21(s) c2g22(s) c3g23(s)

c1g31(s) c2g32(s) c3g33(s)

⎤⎥⎦− I =G2(s)E− I

and

E=⎡⎢⎣c1

c2

c3

⎤⎥⎦Let d0=max(DG(0)). Then, the steady-state diagonal dominance of G2 implies that d0<1. Since‖G(s)‖H∞, and hence ‖G2(s)‖H∞ are finite, there exists a constant d1>0 such that

max(DG( j�))� |d0+ j�d1|for every ��0. Thus,

max(S�( j�)) �∣∣∣∣ �max

j�+�min

∣∣∣∣max(DG( j�))

�∣∣∣∣�max(d0+ j�d1)

j�+�min

∣∣∣∣�min(d0�max/�min,�maxd1)

< 1 (16)



for every ��0, and hence ‖S�‖H∞<1, if �i , i=1,2,3, is chosen such that

0 < �max<1/d1 and

1 � �max/�min<1/d0(17)

Note that d0<1 implies that 1/d0>1, guaranteeing that the above set in (17) for �i>0, i=1,2,3,is non-empty, and hence that the constructed KI with parameters kui , i=1,2,3, given by (14)together with (17) provides nominal stability of the offline controller subsystem in (6).

Next, it is claimed that KI constructed above stabilizes the offline controller subsystem inFigure 6 with respect to uncertainty in both G2 and the mismatch compensator. To demonstratethis, assume, without loss of generality, that uncertainty is multiplicative. Note that KI is diagonal,implying that the uncertainty in KI is diagonal as well, and hence commutes with KI . This furtherimplies that having uncertainty in KI is equivalent to having output multiplicative uncertainty inG2. Theorem 3 in [9], however, shows that any mismatch compensators satisfying ‖S�‖H∞ =�<1robustly stabilize the offline controller subsystem in Figure 6 with respect to output multiplicativeuncertainty � in G2 satisfying ‖�‖H∞<(1−�)/(1+�).

Finally, for the case of n=3, it only remains to show that for each KI in (8) with parameters0<ki sgn(ki )�|kui |, i=1, . . . ,n, there exist �i>0, i=1,2,3, such that ‖S�‖H∞<1.

Indeed, let ci =ki/�i , i=1,2,3, and choose �′i , i=1,2,3, such that k′

i =ci�′i , i=1,2,3. Then

0<�′i<�i , i=1,2,3 (18)

Using k′i and �′

i , i=1,2,3, in (15), the latter becomes

S�=

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

�′1

s+�′1(c1g11(s)−1)

�′2

s+�′1c2g12(s)

�′3

s+�′1c3g13(s)

�′1

s+�′2c1g21(s)

�′2

s+�′2(c2g22(s)−1)

�′3

s+�′2c3g23(s)

�′1

s+�′3c1g31(s)

�′2

s+�′3c2g32(s)

�′3

s+�′3(c3g33(s)−1)

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦Note that

∣∣∣∣ �′i

j�+�′i

∣∣∣∣ =∣∣∣∣ 1

j�/�′i +1

∣∣∣∣�

∣∣∣∣ 1

j�/�i +1

∣∣∣∣ in light of (18)

=∣∣∣∣ �ij�+�i

∣∣∣∣Copyright q 2008 John Wiley & Sons, Ltd. Int. J. Robust Nonlinear Control 2009; 19:1083–1104

DOI: 10.1002/rnc


for all ��0 and for i=1,2,3. This immediately implies that

max(S�) = max

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

�′1

s+�′1

(g11( j�)

g11(0)−1

)�′2

s+�′1

g12( j�)

g22(0)

�′3

s+�′1

g13( j�)

g33(0)

�′1

s+�′2

g21( j�)

g11(0)

�′2

s+�′2

(g22( j�)

g22(0)−1

)�′3

s+�′2

g23( j�)

g33(0)

�′1

s+�′3

g31( j�)

g11(0)

�′2

s+�′3

g32( j�)

g22(0)

�′3

s+�′3

(g33( j�)

g33(0)−1

)

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠

� max

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

�1s+�1

(g11( j�)

g11(0)−1

)�2

s+�1

g12( j�)

g22(0)

�3s+�1

g13( j�)

g33(0)

�1s+�2

g21( j�)

g11(0)

�2s+�2

(g22( j�)

g22(0)−1

)�3

s+�2

g23( j�)

g33(0)

�1s+�3

g31( j�)

g11(0)

�2s+�3

g32( j�)

g22(0)

�3s+�3

(g33( j�)

g33(0)−1

)

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠< 1

for all ��0, concluding the proof for the case of n=3.Note in the above argument that the structure of the diagonal and the off-diagonal elements of

S� is independent of its dimension. In the general case of n�2, the S� in (13) is simply replaced by[�p

s+�i

(gip

kup�p

−�i p

)]i,p=1,...,n

where �i p ={1 if i= p

0 otherwise

}

is the Kronecker delta function, the S� in (15) is replaced by [�p/(s+�i )(cpgip(s)−�i p)]i,p=1,...,n ,and the �max and �min are calculated by �max=max(�i , i=1, . . . ,n) and �min=min(�i , i=1, . . . ,n),and then the same argument and conclusion as above apply. This completes the proof. �

To stabilize the offline controller subsystem under large controller uncertainty, small valuesof ‖S�‖H∞ are desired (cf. Theorem 3 in [9]). To quantify the possible values of ‖S�‖H∞ , thefollowing result is obtained from the proof of the above theorem.

Corollary 6Suppose that all the assumptions of Theorem 5 are satisfied. Then ‖S�‖H∞ , where S� is given in(10), can be made arbitrarily close to max(G(0)E− I ).

ProofChoose �>0 and let �max=�min=(d0+�)/d1 in (16). The latter then becomes

max(S�( j�))�min(d0,d0+�)�d0+�

for every ��0. The claim validity then follows from noting that � is arbitrary and max(G(0)E−I )=d0. �



It is seen that for general steady-state diagonally dominant G, it might not be possible to make‖S�‖H∞ arbitrarily close to 0. Nevertheless, it is possible to do so for steady-state decoupled G,as max(G(0)E− I )=0 for matrix E in (12).

With the condition on G2 relaxed, the structural results for G2(0) summarized in Section 2can then be applied to establish the desired existence result, as presented in the following twosubsections.

4.2. Controller input dimension no less than the output one

In this case, existence of the stabilizing KI is easily established by sequentially applying the resultsfrom Theorem 1, Lemma 4, and Theorem 5.

Corollary 7Suppose that the offline controller input dimension in Figure 6 is no less than its output one, i.e.the dimension of � is no less than that of u. Suppose further that the offline controller has nozeros at the origin. Then, there exist a set of constants kui , i=1, . . . ,n, and a family of MIMOintegral controllers KI in (8) with parameters 0<ki sgn(ki )�|kui |, i=1, . . . ,n, such that for eachKI in the family the topology in Figure 7 is robustly stable with respect to uncertainty in both G2and the mismatch compensator. Moreover, the values of ku1 , . . . ,k

un can be found by the convex

optimization procedure proposed in [9].If guaranteed robustness with respect to uncertainty in G2(s) and KI is desired, Corollary 6

shows that G2(0) is required to be identity, or close to it. The latter condition, nevertheless, isalmost guaranteed if the offline controller contains an integrator in each of its output channels, orthe weighting matrix We is chosen to be close to zero, as summarized in Theorems 2 and 3. Thisresult is formalized in the following corollary.

Corollary 8Suppose that the assumptions of either Theorem 2 or Theorem 3 are satisfied. Then, there exist a setof constants kui , i=1, . . . ,n, and a family of MIMO integral controllers KI in (8) with parameters0<ki sgn(ki )�|kui |, i=1, . . . ,n, such that for each KI in the family the topology in Figure 9 isrobustly stable with respect to any ‖�‖H∞<<1. Moreover, the values of ku1 , . . . ,k

un can be found

by the convex optimization procedure proposed in [9].

4.3. Controller input dimension less than the output one

As summarized in Section 2, G2(0) is singular for any offline controller structure and any choiceof weighting matrices Wu and We when the offline controller input dimension is less than its outputone. In this case, Theorem 5 is inconclusive. Nevertheless, using Theorem 1 in [12], the followingtheorem asserts the non-existence of the stabilizing mismatch compensators of the integral type.

Theorem 9In the state/output feedback topology in Figure 5, stabilizing mismatch compensators of the formK (s)=K ′(s)/s, where K ′(s) is a real rational transfer matrix such that K (s) is proper, exist onlyif the offline controller input dimension is no less than its output one, i.e. the dimension of � isno less than that of u in Figure 6.



ProofSubstituting K ′(s)/s for K (s) in Figure 7 and merging K ′(s) with G2(s) give the diagram inFigure 10. Suppose K ′(s) is chosen such that the offline controller subsystem in Figure 10 isstable. Then, an application of Theorem 1 in [12] yields

det(K ′(0)G2(0)) �=0

immediately implying

det(G2(0)) �=0

since unstable pole-zero cancellation is not allowed for internal stability. It then follows that theoffline controller input dimension must be no less than its output one.

Theorem 9 essentially asserts that non-integral mismatch compensators must be utilized tocompensate for controller uncertainty when the offline controller input dimension is less than itsoutput one. An obvious method is to use various advanced MIMO control synthesis techniquessuch as H∞ synthesis. This, however, will produce mismatch compensators of non-transparentstructure and higher order.

To obtain a simple and transparent decentralized mismatch compensator similar to (8), it isproposed in the current work that a decentralized structure with each diagonal entry involvingonly first-order transfer function is used for the mismatch compensator. Using this structure, thesimplest mismatch compensator is given by

KN =⎡⎢⎣k11

· · ·knn

⎤⎥⎦=

⎡⎢⎢⎢⎢⎢⎣k1

s+k1

· · ·kn

s+kn

⎤⎥⎥⎥⎥⎥⎦ (19)

where it is seen that only one extra parameter is added. It is also seen from (19) that when is small, the mismatch compensator KN is of high steady-state gain. Moreover, lim→0 KN =KI ,

implying that KN approaches KI as goes to zero.After the structure of the mismatch compensator is specified, the question remains whether

and k1, . . . ,kn exist such that KN stabilizes the offline controller subsystem and such that theoffline controller output u tracks that of the online one u′ with small tracking error. The latter, as is

Figure 10. Simplified offline controller subsystem for K (s) in the form of K ′(s)/s in the state/outputfeedback bumpless transfer topology.



well-known, is in general achieved by high-gain controllers. Observe that the steady-state gain ofKN is I/. Hence, the above question amounts to the question of whether k1, . . . ,kn and arbitrarilysmall values of exist such that KN stabilizes the offline controller subsystem in Figure 7. Thelatter question is answered in the affirmative in the present work as follows.

First, it is observed that

KN =

⎡⎢⎢⎢⎢⎢⎣k1/s

1+k1/s

· · ·kn/s

1+kn/s

⎤⎥⎥⎥⎥⎥⎦= (I +KI )

−1KI

implying that the offline controller subsystem in Figure 7, with K substituted by KN , can bedescribed by the diagram in Figure 11. The latter is then easily seen to be equivalent to that inFigure 12.

The latter figure clearly shows that the above question reduces to the question of whether adecentralized mismatch compensator KI exists for G ′

2(s)�G2(s)+I. Since G2(0) is positivesemidefinite for Wu = I and any choice of positive definite We (cf. (5)), G ′

2(0)=G2(0)+I ispositive definite for any >0. This, in view of Lemma 4, implies that G ′

2(s) is steady-state

Figure 11. Simplifed offline controller subsystem for non-integral mismatch compensators.

Figure 12. Equivalent offline controller subsystem for non-integral mismatch compensator.



diagonally dominant for the above choice of weighting matrices Wu and We. The latter fact, withthe aid of Theorem 5, immediately guarantees the existence of stabilizing KI in Figure 12, givenany >0, or the existence of KN in (19) with arbitrarily small values of . This is formalized bythe following theorem.

Theorem 10Suppose the offline controller input dimension is less than its output one. Then, there exists amismatch compensator of the form (19) such that the offline controller subsystem in Figure 7 or 11is stable.

Remark 1When the offline controller input dimension is less than its output one, the range Roff of theoffline controller steady-state gain matrix is a subspace with dimension less than n. As a result,the tracking error |u−u′| cannot be made arbitrarily small by using small values of for anyonline controller output u′ /∈ Roff. The latter fact is true for the topology in Figure 11 as well asany other bumpless transfer topologies.

5. TWO DEGREES OF FREEDOM

5.1. 2DOF LQ bumpless transfer topology and theory [6]When the controllers are 2DOF, the offline controller, also assumed to be strictly proper, can berealized as

x = Ax+B1�+B2y

u = Cx(20)

The performance index in this case is chosen [6] as

J0(u,�)= 1

2

∫ ∞

0[zu(t)ᵀWuzu(t)+ze(t)

ᵀWeze(t)]dt

where zu(t)=u(t)−u′(t) and ze(t)=�(t)−r(t). In accordance with this performance index, a2DOF LQ bumpless transfer scheme, shown in Figure 13, is proposed in [6]. Note that the offlinecontroller input � is computed differently than its 1DOF counterpart. Specifically, � in this case isgiven by

�=F(xᵀ yᵀ (u′)ᵀ rᵀ)ᵀ (21)

where the transfer operator F is synthesized as

F=−W−1e

⎛⎜⎜⎜⎜⎜⎝(Bᵀ1 X)ᵀ

(−Bᵀ1 (Aᵀ−XB1W−1e Bᵀ1 )−1XB2)

ᵀ

(Bᵀ1 (Aᵀ−XB1W−1e Bᵀ1 )−1CᵀWu)

ᵀ

(−We−Bᵀ1 (Aᵀ−XB1W−1e Bᵀ1 )−1XB1)

ᵀ

⎞⎟⎟⎟⎟⎟⎠

ᵀ



Figure 13. 2DOF LQ bumpless transfer topology.

and X is the positive semi-definite stabilizing solution of the ARE

X A+AᵀX−XB1W−1e Bᵀ1 X+CᵀWuC=0

5.2. Steady-state input/output structure

To analyze the steady-state input/output structure of the 2DOF topology of Figure 16 describedabove, the subsystem in the topology in Figure 13 corresponding to the offline controller and thesimplified diagram of this subsystem are shown in Figures 14 and 15, respectively, where the transferfunction G in Figure 14 is decomposed according to u=G(rᵀ yᵀ (u′)ᵀ)ᵀ=G1r r+G1y y+G2u′shown in Figure 15.

Similarly, the explicit form of G1r ,G1y, and G2 is given as follows:

G1r (s) =C(s I −A+B1W−1e BT

1 X)−1B1W−1e (We+Bᵀ1 (Aᵀ−XB1W

−1e Bᵀ1 )−1XB1)

G1y(s) =C(s I −A+B1W−1e BT

1 X)−1[B1W−1e Bᵀ1 (Aᵀ−XB1W

−1e Bᵀ1 )−1X+ I ]B2

G2(s) =C(s I −A+B1W−1e BT

1 X)−1B1W−1e Bᵀ1 (−Aᵀ+XB1W

−1e Bᵀ1 )−1CᵀWu

Their corresponding steady-state gains are then

G1r (0) =C(A−B1W−1e BT

1 X)−1B1W−1e (−We−Bᵀ1 (Aᵀ−XB1W

−1e Bᵀ1 )−1XB1)=N B1 (22)

G1y(0) =C(A−B1W−1e BT

1 X)−1[−B1W−1e Bᵀ1 (Aᵀ−XB1W

−1e Bᵀ1 )−1X− I ]B2=N B2 (23)

G2(0) =C(A−B1W−1e BT

1 X)−1B1W−1e Bᵀ1 (Aᵀ−XB1W

−1e Bᵀ1 )−1CᵀWu (24)

respectively, where N�C(A−B1W−1e BT

1 X)−1[−B1W−1e Bᵀ1 (Aᵀ−XB1W−1

e Bᵀ1 )−1X− I ].It is then observed that G2(0) in (24) and G1r (0) in (22) are identical to their 1DOF counterparts,

i.e. G2(0) in (5) and G1(0) in (4), respectively, if the B matrix in (5) and (4) is replaced by B1. Thisobservation immediately leads to a corollary, asserting the conditions in the 2DOF LQ bumplesstransfer, under which G2(0) is identity/asymptotically identity and G1r (0) is zero/asymptoticallyzero, given as follows.



Figure 14. 2DOF offline controller subsystem in LQ bumpless transfer.

Figure 15. 2DOF reduced offline controller subsystem in LQ bumpless transfer.

Corollary 11Suppose the assumptions in either Theorem 2 or 3 with matrices A, B, and C therein replaced byA, B1, and C, respectively, of the 2DOF controller in (20), are satisfied. Then,

(1) the steady-state gain from u′ to u in Figure 15, i.e. G2(0) in (24), is identity/asymptoticallyidentity;

(2) the steady-state gain from r to u in Figure 15, i.e. G1r (0) in (22), is zero/asymptoticallyzero.

The result for G1y(0), on the other hand, is less straightforward, since it has no counterpart in1DOF topology. Nevertheless, exploiting the similarity between G1y(0) and G1r (0), the followingcorollaries give the conditions for which G1y(0)=0.

Corollary 12Suppose that the assumptions in Theorem 2, with matrices A, B, and C therein replaced by A, B1,

and C, respectively, of the 2DOF controller in (20), are satisfied. Then, the steady-state gain fromy to u in Figure 15, i.e. G1y(0) in (23), is zero.

Corollary 13Suppose that the assumptions in Theorem 3 with matrices A, B, and C therein replaced by A, B1,

and C, respectively, of the 2DOF controller in (20), are satisfied. Suppose further that img(B2)⊂img(B1), where img(•) denotes the image space of a matrix. Then, the steady-state gain from yto u in Figure 15 is asymptotically zero, i.e. lim�→0+G1y(0)=0, where G1y(0) is given in (23).

5.3. Existence of the stabilizing decentralized mismatch compensator parameters

The 2DOF state/output feedback topology, analogous to its 1DOF counterpart, is shown inFigure 16. The offline controller subsystem in this figure without controller uncertainty can be



Figure 16. 2DOF state/output feedback topology.

Figure 17. Reduced offline subsystem in 2DOF state/output feedback topology.

reduced to the diagram in Figure 17. It is easily seen from comparisons between Figures 17 and 7,and between (5) and (24) that the transfer function from online controller output u′ to offlinecontroller output u in Figure 7 is identical to that in Figure 17 if the matrices A, B, and C inthe former are substituted by A, B1 and C , respectively, in the latter. This fact, combined withthe result on the steady-state identity of G2 for 2DOF LQ bumpless transfer topology given inCorollary 11, yields the following existence result for controllers with the input dimension no lessthan the output one.

Corollary 14Suppose the assumptions in either Theorem 2 or 3 with matrices A, B, and C therein replaced byA, B1, and C of the 2DOF controller in (20), respectively, are satisfied. Then, there exist a set ofconstants kui , i=1, . . . ,n, and a family of MIMO integral controllers KI in (8) with parameters0<ki sgn(ki )�|kui |, i=1, . . . ,n, such that for each KI in the family the topology in Figure 17 isrobustly stable with respect to uncertainty in both G2 and KI . Moreover, the values of ku1 , . . . ,k

un

can be found by the convex optimization procedure proposed in [9].Similarly, when the offline controller input dimension is less than its output one, the results

described in Theorems 9 and 10 extend in a straightforward manner to 2DOF case.

Corollary 15In the state/output feedback topology in Figure 16, stabilizing mismatch compensators in the formof K (s)=K ′(s)/s, where K ′(s) is a real rational transfer matrix such that K (s) is proper, existonly if the offline controller input dimension associated with the reference signal is no less thanits output one, i.e. dim(B1)�dim(C) in (20).



Corollary 16Suppose the offline controller input dimension is less than its output one. Then, there exists amismatch compensator in the form of (19) such that the offline controller subsystem in Figure 17is stable.

6. CONCLUSION

The recent results on the steady-state structure of both 1DOF and 2DOF LQ bumpless transfertopologies are summarized. These results, combined with the steady-state diagonal dominancecondition proposed, establish the existence of the robustly stabilizing decentralized mismatchcompensator parameters in the 1DOF and 2DOF state/output feedback topologies required forbumpless transfer under controller uncertainty. Moreover, such parameters can be easily obtainedusing the convex optimization algorithm previously proposed. As a result, a broadly applicabletransfer operator synthesis technique is obtained, which permits easily implementable steady-statebumpless transfer under controller uncertainty.

ACKNOWLEDGEMENTS

The authors would like to thank the Associate Editor Professor Henrik Niemann and the anonymousreviewers for their insightful comments and illuminating suggestions resulting in a substantial restructuringand improvement of the original submission.

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Decentralized compensation of controller uncertainty in the steady-state bumpless transfer under the state/output feedback