Journal of Control Theory and Applications 1 (2006) 32–37

Non-fragile hybrid guaranteed cost control

for a class of uncertain switched linear systems

Rui WANG, Jun ZHAO

(School of Information Science and Engineering, Northeastern University, Shenyang Liaoning 110004, China)

Abstract: This paper focuses on the problem of non-fragile hybrid guaranteed cost control for a class of uncer-

tain switched linear systems. The controller gain to be designed is assumed to have additive gain variations. Based on

multiple-Lyapunov function technique, the design of non-fragile hybrid guaranteed cost controllers is derived to make the

corresponding closed-loop system asymptotically stable for all admissible uncertainties. Furthermore, a convex optimiza-

tion approach with LMIs constraints is introduced to select the optimal non-fragile guaranteed cost controllers. Finally, a

simulation example illustrates the effectiveness of the proposed approach.Keywords: Switched systems; Guaranteed cost control; Non-fragile control; Multiple-Lyapunov function

1 Introduction

In recent years, the switched control systems have beenattracting considerable attention in the control commu-nity [1∼7]. Basically, a switched system belongs to a spe-cial class of hybrid systems, which consist of a family ofcontinuous-time or discrete-time subsystems and a switch-ing law that specifies the switching between them. Suchcontrol systems appear in many applications, such as com-munication networks, switching power converters and manyother fields.

On the other hand, when controlling a real plant, it isalso desirable to design a control system which is not onlyasymptotically stable but also guarantees an adequate levelof performance index. One way to address the robust per-formance problem is the so-called guaranteed cost controlapproach first introduced by Chang and Peng [8]. Basedon the GCC design approach, many significant results havebeen proposed [9∼11]. Although these methods yield con-trollers that are robust with regard to system uncertainty,their robustness with regard to controller uncertainty has notbeen considered. That is to say, these design methods are allbased on a basic assumption that the control will be im-plemented exactly. However, in practice, controllers have acertain degree of errors due to finite word length in any dig-

ital system, the imprecision inherent in analogue systems,actuator degradations, and the requirement for additionaltuning of parameters in the final controller implementation,etc. These errors could destabilize the closed-loop system.In [12], Keel and Bhattacharyya have shown by a numberof examples that the controllers designed by using weightedH∞, μ and l1 synthesis techniques may be very fragile withrespect to errors in the controller coefficients. This brings aproblem: how to design a controller for a given plant suchthat the controller is insensitive to some amount of errorwith respect to its gain, i.e. the controller is non-fragileor resilient. Recently, there have been some efforts to dealwith the non-fragile controller design problem [13,14]. Asto switched systems, because continuous-time and discrete-time dynamics exist simultaneously, the study of the non-fragile control becomes more complicated and difficult. Upto now, to the best of our knowledge, no results of stabi-lization and GCC based on non-fragile control method forswitched systems have been available.

This paper studies the problem of non-fragile hybridguaranteed cost control (NHGCC) for a class of uncertainswitched linear systems. Additive controller gain perturba-tion is considered. By employing multiple-Lyapunov func-tion technique, the design of non-fragile hybrid guaranteed

Received 30 September 2005; revised 14 December 2005.

This work was supported by the National Natural Science Foundation of China (No.60274009, 60574013), and the Natural Science Foundation of

Liaoning Province(No.20032020).

R.WANG et al. / Journal of Control Theory and Applications 1 (2006) 32–37 33

cost controllers is given in terms of linear matrix inequal-ities (LMIs). The upper bound on the guaranteed cost isminimized by solving a convex optimization problem withLMIs constraints.

In this paper, ∗ denotes the symmetric block in one sym-

metric matrix. I denotes the identity matrix of appropriate

dimension. The trace of a matrix is denoted by tr(·).

2 Problem formulation

Consider uncertain switched linear systems described bythe state-space model of the form:

x(t) = (Aσ + ΔAσ)x(t) + (Bσ + ΔBσ)uσ(t)

x(0) = x0, (1)

where x(t) ∈ Rn is the state, σ(t) : [0, +∞) → M =

{1, 2, · · · , m} is a piecewise constant switching signal tobe designed, which usually depends on time t or state x,Aσ, Bσ are known real constant matrices of appropriatedimensions, ui ∈ R

qi is the control input, ΔAσ, ΔBσ arereal-valued matrices representing time-varying parameteruncertainties, and have the following form

[ΔAi, ΔBi] = NiF1i(t)[Ci, Di],

where Ci, Di, Ni are known constant matrices of appropri-ate dimensions, F1i(t) is an unknown matrix function satis-fying

FT1i(t)F1i(t) � I.

For convenience, we adopt the following notations [1] forswitched system (1). Let

Σ = { x0; (i0, t0), (i1, t1), · · · , (in, tn), · · · , |ik ∈ M, k ∈ N} (2)

denote a switching sequence with the initial state x0 and theinitial time t0 , where (ik, tk) means that the ikth subsystemis activated for t ∈ [tk, tk+1).

The cost function associated with switched system (1) isgiven by

J =� +∞

0[x(t)TQx(t) + uT

σ(t)Ruσ(t)]dt, (3)

where Q and R are positive definite weighted matrices.

The difficulty in dealing with fragility is that the con-troller parameters are part of the design. Several mod-ellings of controller uncertainties have been considered tocope with the difficulty. Either the additive uncertainty[13] or multiplicative form [14] enters the system. For thegiven uncertain switched system (1), the controllers actuallyimplemented is

ui(t) = (Ki + ΔKi)x(t), (4)

where Ki is controller gain to be designed, and ΔKi repre-sents controller gain variation. In this paper, the followinggain variation is considered:

ΔKi is of the additive variation:

ΔKi = HiF2iEi, (5)

where FT2i(t)F2i(t) � I, i ∈ M .

Definition 1 For the uncertain switched system (1), ifthere exist a state feedback control law u∗

i for each subsys-tem and a switching law σ(t), and a positive scalar J∗ suchthat for all admissible uncertainties, the closed-loop systemis asymptotically stable and the value of the cost function(3) satisfies J � J∗, then J∗ is said to be a non-fragilehybrid guaranteed cost and u∗

σ(t) is said to be a NHGCClaw.

Remark 1 When σ(t) ≡ i, the switched linear system(1) degenerates into a regular linear system and the NHGCCproblem becomes the standard non-fragile guaranteed costcontrol problem [13].

Remark 2 The NHGCC problem with controller gainvariation is different from the standard guaranteed cost con-trol problem with system uncertainty, because the controllergain perturbation ΔKi not only affects system matrices butalso enters the cost function J as defined in (3), namely,

J =� +∞

0[x(t)TQx(t) + (Ki + ΔKi)TR(Ki + ΔKi)]dt.

The objective of this paper is to design NHGCC law(4) such that switched system (1) satisfies NHGCC for alladmissible uncertainties.

3 Main results

In this section, we present solvability conditions and

design method for NHGCC problem.

Theorem 1 If there exist constants βij (either all non-

negative or all nonpositive), symmetric positive definite

matrices Pi and positive constants ε1i, ε2i, λi, (i ∈ M) such

that the following matrix inequalities⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

Ωi ∗ ∗ ∗ ∗−λiB

Ti Pi −R−1 ∗ ∗ ∗

NTi Pi 0 −ε−1

1i I ∗ ∗Ci − λiDiB

Ti Pi ε2iDiHiH

Ti 0 −ε1iI ∗

HTi BT

i Pi HTi 0 HT

i DTi −ε−1

2i I

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

< 0 (6)

hold, where

34 R.WANG et al. / Journal of Control Theory and Applications 1 (2006) 32–37

Ωi = Ξi +m∑

j=1

βij(Pj − Pi),

Ξi = Q + PiAi + ATi Pi − 2λiPiBiB

Ti Pi + ε−1

2i ETi Ei,

then there exists NHGCC law (4) with control gain vari-

ation ΔKi as given in (5), such that the system (1) satis-

fies NHGCC, where the control gain is Ki = −λiBTi Pi.

When βij � 0, the switching law is given by σ(t) =

arg min{xTPix, i ∈ M}, and the performance upper

bound is

J∗ = min{xT0 Pix0, i ∈ M}; (7)

when βij � 0, the switching law is designed as σ(t) =

arg max{xTPix, i ∈ M}, and the performance upper

bound is

J∗ = max{xT0 Pix0, i ∈ M}. (8)

Proof Without loss of generality, suppose that βij � 0.

Obviously, ∀x ∈ Rn\ {0}, there exists an i ∈ M such that

xT(Pj − Pi)x � 0,∀i ∈ M . Thus, according to the matrix

inequalities (6), we have⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

Ξi ∗ ∗ ∗ ∗−λiB

Ti Pi −R−1 ∗ ∗ ∗

NTi Pi 0 −ε−1

1i I ∗ ∗Ci − λiDiB

Ti Pi ε2iDiHiH

Ti 0 −ε1iI ∗

HTi BT

i Pi HTi 0 HT

i DTi −ε−1

2i I

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

< 0. (9)

For any i ∈ M , let

Ωi = {x ∈ Rn|xT(Pj − Pi)x � 0,∀j ∈ M},

thenm⋃

i=1

Ωi = Rn\{0}. Construct the sets

Ω1 = Ω1, · · · , Ωi = Ωi −i−1⋃j=1

Ωj , · · ·, Ωm = Ωm −m−1⋃j=1

Ωj .

Obviously, we havem⋃

i=1

Ωi = Rn\{0} and Ωi

⋂Ωj =

φ, i = j.

The matrix inequalities (9)are equivalent to the followinginequalities. (Proof see Appendix)⎡

⎣ Σi (Ki + ΔKi)T

Ki + ΔKi −R−1

⎤⎦ < 0, (10)

where

Σi = [(Ai + ΔAi) + (Bi + ΔBi)(Ki + ΔKi)]TPi

+Pi[(Ai + ΔAi) + (Bi + ΔBi)(Ki + ΔKi)] + Q.

By the Schur complement Lemma, (10) ⇔

[(Ai + ΔAi) + (Bi + ΔBi)(Ki + ΔKi)]TPi

+Pi[(Ai + ΔAi) + (Bi + ΔBi)(Ki + ΔKi)] + Q

+(Ki + ΔKi)TR(Ki + ΔKi) < 0. (11)

Define the Lyapunov function candidate as

Vi(x(t)) = xTPix,

where Pi is positive definite matrix satisfying (6). Switchinglaw is designed as follows

σ(t) = arg min{xTPix, i ∈ M}.

When x(t) ∈ Ωi, from (11), the derivative of Vi(x) alongthe trajectory of the closed-loop system (1) is

Vi(x) < −xT[Q + (Ki + ΔKi)TR(Ki + ΔKi)]x

< 0. (12)

According to multiple-Lyapunov function technique, thesystem (1) is asymptotically stable.

In the following, we show that the closed-loop systemsatisfies the performance upper bound. Without loss of gen-erality, suppose xT

0 Pi0x0 = minik∈M

{xT0 Pik

x0}. According to

the switching sequence (2), we obtain

J =� +∞

0[x(t)TQx(t) + uT

σ(t)Ruσ(t)]dt

=m∑

i=1

∞∑j=1

� tij+1

tij[x(t)TQx(t) + uT

i Rui + Vi(x(t))]dt

− ∞∑k=0

� tk+1

tkVik

(x(t))dt

<− ∞∑k=0

� tk+1

tkVik

(x(t))dt

=− (Vi0(x(t1)) − Vi0(x(t0)) + Vi1(x(t2)) + · · ·)= Vi0(x(t0)) = xT

0 Pi0x0 = mini∈M

{xT0 Pix0}.

From Definition 1, we know (7) is a performance upperbound of system (1).

When βij � 0, similar proof can be given with the per-formance upper bound (8).

Remark 3 Pre- and post-multiplying both sides ofinequalities (6) by diag{Xi, I, I, I, I} with Xi = P−1

i ,and applying Schur complement Lemma yield the follow-ing LMIs ⎡

⎣Γi ΨTi

Ψi Υi

⎤⎦ < 0, (13)

R.WANG et al. / Journal of Control Theory and Applications 1 (2006) 32–37 35

where

Γi =⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

Λi ∗ ∗ ∗ ∗−λiB

Ti −R−1 ∗ ∗ ∗

NTi 0 −ε−1

1i I ∗ ∗CiXi − λiDiB

Ti ε2iDiHiH

Ti 0 −ε1iI ∗

HTi BT

i HTi 0 HT

i DTi −ε−1

2i I

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Λi = AiXi + XiATi − 2λiBiB

Ti −

m∑j=1

βijXi ,

Ψi =[Xi EiXi Xi Xi Xi

]T [I 0 0 · · · 0

],

Υi = diag{−Q−1,−ε2iI,−β−1i1 P−1

1 , · · · − β−1i,i−1P

−1i−1,

− β−1i,i+1P

−1i+1, · · · ,−β−1

imP−1m }.

The cost upper bound (7) depends on feasible solution ofLMIs (13). The different feasible solutions of LMIs (13) canresult in different cost upper bounds. Theorem 1 provides asufficient condition for the solution to the NHGCC prob-lem. But it remains unclear as how to optimize the matrixPi in order to achieve the minimal guaranteed cost of theclosed-loop system. This problem is solved in the followingtheorem.

Theorem 2 For the given switched system (1) and per-formance index (3), if the following optimization problems

minX1,X2,··· ,Xm,Mi

tr(Mi)

s. t. i) (14),

ii)

⎡⎣ Mi xT

0

x0 Xi

⎤⎦ > 0, i ∈ M (14)

have solutions (X1, M1), (X2, M2), · · · , (Xm, Mm). Thenthere exists optimal NHGCC law u∗

i (t) = (Ki +ΔKi)x(t),and the minimal cost upper bound is

J∗ = mini∈M

xT0 X−1

i x0,

where Ki = −λiBTi X−1

i .

Proof If (X1, M1), (X2, M2), · · · , (Xm, Mm) aresolutions of the optimization problems (14), then they arealso feasible solutions of constraint condition i). Accordingto Theorem 1 and Remark 3, u∗

i (t) = (Ki + ΔKi)x(t) is aNHGCC law for switched system (1).

In addition, by Schur complement Lemma, constraint

condition ii) is equivalent to Mi > xT0 X−1

i x0. Thus, the

minimization of tr(Mi) implies the minimization of the

tr(xT0 X−1

i x0). Therefore, the minimal cost upper bound is

J∗ = mini∈M

xT0 X−1

i x0. The optimality of the solution of the

optimization problem (14) follows from the convexity of the

objective function and of the constraints. This completes the

proof.

4 Numerical example

Consider the following uncertain switched linear system

x(t) = (Ai + ΔAi)x(t) + (Bi + ΔBi)ui,

x(0) =[−1 2

]T

, i = 1, 2, (15)

where

A1 =

⎡⎣−5 0

−3 1

⎤⎦ , A2 =

⎡⎣ 1 0

2 −2

⎤⎦ , B1 =

⎡⎣ 1 1

−2 −1

⎤⎦ ,

B2 =

⎡⎣−1 2

1 1

⎤⎦ , C1 =

⎡⎣ 0.2 0.5

0.1 0

⎤⎦ , C2 =

⎡⎣ 0 0.2

0.4 0

⎤⎦ ,

Ni =

⎡⎣ 0 0.3

0.5 0

⎤⎦ , Di =

⎡⎣ 0.1 0

0 0.1

⎤⎦ .

We assume that controller gain has additive variation,

namely, ΔKi = HiF2iEi, where Hi = 0.5,

Ei =

⎡⎣ 0.1 0

0 0.1

⎤⎦ , F1i(t) = F2i(t) = sin t, (i = 1, 2).

Consider positive definite weighted matrices

Q = R =

⎡⎣ 0.5 0

0 0.5

⎤⎦ .

It is easy to see that (Ai, Bi) are not controllable, namely,

the two subsystems of system (15) are unstable. Let β12 =

1.5, β21 = 1, λi = 0.5, ε1i = ε2i = 1, (i = 1, 2). Now,

we apply Theorem 1 and Theorem 2 to design the optimal

state feedback controllers. By solving optimization problem

(14), we have

P1opt =

⎡⎣ 1.4780 0.4057

0.4057 1.2206

⎤⎦ , P2opt =

⎡⎣ 19.0822 9.3902

9.3902 5.2374

⎤⎦ .

Let

Ω1 = {x ∈ Rn|xT(P2opt − P1opt)x � 0, x = 0},

Ω2 = {x ∈ Rn|xT(P1opt − P2opt)x � 0, x = 0},

then Ω1

⋃Ω2 = R

n\ {0}. The switching law is designed

by

σ(t) =

{1, x(t) ∈ Ω1,

2, x(t) ∈ Ω2\Ω1.

36 R.WANG et al. / Journal of Control Theory and Applications 1 (2006) 32–37

The optimal state feedback controllers are designed as

u∗i (t) = (Ki+ΔKi)x(t), Ki = −λiB

Ti Piopt, i = 1, 2. It is

easy to see from Fig. 1 that the system (15) is asymptotically

stable under the optimal state feedback controllers. The

performance upper bound is J∗ = 4.7373 without using

optimization problem (14), while it is J∗ = xT0 P2optx0 =

2.4710 after optimization problem (14) is applied.

Fig. 1 The state responses of switched system (15).

5 Conclusions

In this article, based on the multiple-Lyapunov function

technique, we have investigated a design method to the

optimal NHGCC problem for a class of switched linear sys-

tems via state feedback controllers with additive variation.

Furthermore, a convex optimization problem with LMIs

constraints is formulated to design the optimal guaranteed

cost controller which minimizes the guaranteed cost of the

closed-loop uncertain system.

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Appendix

Equivalence of (9) and (10).

Proof (9)⇔26666666666664

Θi + ε−12i ET

i Ei ∗ ∗ ∗ ∗

Ki −R−1 ∗ ∗ ∗

NTi Pi 0 −ε−1

1i I ∗ ∗

Ci + DiKi ε2iDiHiHTi 0 −ε1iI ∗

HTi BT

i Pi HTi 0 HT

i DTi −ε−1

2i I

37777777777775

< 0, (a1)

where

Θi = (Ai + BiKi)TPi + Pi(Ai + BiKi) + Q.

R.WANG et al. / Journal of Control Theory and Applications 1 (2006) 32–37 37

By Schur complement Lemma, (a1) ⇔26666666664

Φi ∗ ∗ ∗

Ki + ΔKi −R−1 ∗ ∗

NTi Pi 0 −ε−1

1i I ∗

Ci + Di(Ki + ΔKi) 0 0 −ε1iI

37777777775

< 0, (a2)

where

Φi = [Ai + Bi(Ki + ΔKi)]TPi + Pi[Ai + Bi(Ki + ΔKi)]

+Q.

According to Schur complement Lemma, (a2) ⇔264

Wi ∗

Ki + ΔKi −R−1

375 < 0 ⇔

264

Vi ∗

Ki + ΔKi −R−1

375 < 0 ⇔

264

Σi (Ki + ΔKi)T

Ki + ΔKi −R−1

375 < 0.

where

Wi = Φi + ε1iPiNiNTi Pi

+ε−11i [Ci + Di(Ki + ΔKi)]

T[Ci + Di(Ki + ΔKi)],

Vi = Φi + ΔATi Pi + PiΔAi

+(Ki + ΔKi)T ΔBT

i Pi + PiΔBi(Ki + ΔKi).

Rui WANG received the B.E. and

M.E. degrees in mathematics in 2001

and 2004, respectively, both from

Bohai University. She is currently a

Ph.D. candidate in Control Theory and

Applications at Northeastern Univer-

sity. Her main research interests include

switched systems, fault-tolerant con-

trol. E-mail: ruiwang01@126.com.

Jun ZHAO received the Ph.D in Con-

trol Theory and Applications in 1991 at

Northeastern University, China. From

1992 to 1993 he was a postdoctoral fel-

low at the same University. Since 1994

he has been with School of Informa-

tion Science and Engineering, North-

eastern University, China, where he is

currently a professor. From February

1998 to February 1999, he was a visiting scholar at the Coordinated Sci-

ence Laboratory, University of Illinois at Urbana-Champaign. He has held

a Research Fellow position at Department of Electronic Engineering, City

University of Hong Kong. His main research interests include switched

systems, nonlinear systems, geometric control theory, and robust control.

E-mail: zhaojun9305@126.com.