Nonlinear Analysis: Hybrid Systems 1 (2007) 317–325www.elsevier.com/locate/nahs

Reliable H∞ control for a class of switched nonlinear systems withactuator failuresI

Rui Wanga,∗, Man Liub, Jun Zhaoa

a School of Information Science and Engineering, Northeastern University, Shenyang, 110004, PR Chinab School of Science, Dalian Nationalities University, Dalian, 116600, PR China

Received 13 May 2006; accepted 1 November 2006

Abstract

This paper focuses on the problem of reliable H∞ control for a class of switched nonlinear systems with actuator failures amonga prespecified subset of actuators. We consider the case in which the never failed actuators cannot stabilize the system. The multiple-Lyapunov-function method is exploited to derive a sufficient condition for the switched nonlinear system to be asymptotically stablewith H∞-norm bound. This condition is given in the form of a set of partial differential inequalities. As a special application, ahybrid state feedback strategy is proposed to solve the standard H∞ control problem for non-switched nonlinear systems when nosingle continuous controller is effective.c© 2006 Elsevier Ltd. All rights reserved.

Keywords: Switched systems; Reliable H∞ control; Multiple Lyapunov functions; Actuator failures; LaSalle’s invariance principle

1. Introduction

Recent years have witnessed rapidly growing interest in switched systems which are an important class ofhybrid systems [1–7]. The motivation for studying switched systems is from the fact that many practical systemsare inherently multi-models in the sense that several dynamical subsystems are required to describe their behaviordepending on various changing environmental factors. The widespread applications of switched systems are alsomotivated by increasing performance requirements in control. In the study of stability analysis for switched systems,the multiple-Lyapunov-function approach has been shown to be an effective tool [8–12]. For example, in [11], a hybridnonlinear control methodology is presented by using multiple Lyapunov functions for a class of switched nonlinearsystems with input constraints. On the basis of multiple Lyapunov functions, the problem of H∞ control for switchednonlinear systems is addressed in [12].

On the other hand, owing to the growing demands of system reliability in aerospace and industrial process, thestudy of reliable control has recently attracted considerable attention. Classical H∞ control methods may not providesatisfactory performance because failures of control components often occur in the real world. Therefore, to overcome

I This work was supported by the National Science Foundation of China under Grant 60574013.∗ Corresponding author.

E-mail addresses: ruiwang01@126.com (R. Wang), liuman dl@dlnu.edu.cn (M. Liu), zhaojun@ise.neu.edu.cn (J. Zhao).

1751-570X/$ - see front matter c© 2006 Elsevier Ltd. All rights reserved.doi:10.1016/j.nahs.2006.11.002

318 R. Wang et al. / Nonlinear Analysis: Hybrid Systems 1 (2007) 317–325

this problem, reliable H∞ control has made great progress recently [13–17]. In particular, in [13], a methodology forthe design of a reliable H∞ controller for the case of actuator failures is presented. In [15], the reliable H∞ controlproblem for affine nonlinear systems is solved by using the Hamilton–Jacobi inequality approach for the case ofactuator and sensor failures. However, these reliable H∞ design methods are all based on a basic assumption that thenever failed actuators can stabilize the given system. For the case where actuators suffer “serious failure” — the neverfailed actuators cannot stabilize the given system, these design methods of existing reliable H∞ control do not work.

In this paper, we introduce a switching strategy to solve the reliable H∞ control problem for switched nonlinearsystems with “serious failed” actuators. On the basis of the multiple-Lyapunov-function technique, a sufficientcondition for the switched nonlinear systems to be asymptotically stable with H∞-norm bound is derived for alladmissible actuator failures. Furthermore, as a direct application, a hybrid state feedback strategy is proposed to solvethe standard H∞ control problem for nonlinear systems when no single continuous controller is effective. Due to thecomplexity of switched systems, there are few results on the reliable control for switched systems [18,19]. Comparedwith the existing results on the reliable control for switched systems, the results of this paper have three features.Firstly, we consider the worst case where the never failed actuators cannot stabilize the system, while the existingworks only aimed to address the case where never failed actuators are adequate to stabilize the system. Second, thereliable control problem is solved by using the multiple-Lyapunov-function technique, while the existing literatureused the common Lyapunov function method. Third, the problem of H∞ control is solved in this paper while theexisting works usually considered the problem of stability.

2. Problem formulation

Consider the switched nonlinear systems described by a state-space model of the form

x = fσ (x) + gσ (x)uσ + pσ (x)wσ ,

z =

(hσ

uσ

),

(1)

where σ : R+ → M = 1, 2, . . . , m is the switching signal to be designed, x ∈ Rn is the state, ui =

(ui1, . . . uimi )T

∈ Rmi and wi = (wi1, . . . wiqi )T

∈ Rqi denote the control input and disturbance input of thei-th subsystem respectively, z is the output to be regulated. Further, let fi (x) ∈ Rn, gi (x) = (gi1(x), . . . gimi (x)) ∈

Rn×mi , pi (x) = (pi1(x), . . . piqi (x)) ∈ Rn×qi , hi (x) = (hi1(x), . . . , hi pi (x))T∈ R pi , fi (0) = 0, hi (0) = 0, i =

1, 2, . . . , m.We adopt the following notation from [8]. A switching sequence is expressed by

Σ = x0; (i0, t0), (i1, t1), . . . , (i j , t j ), . . . , |i j ∈ M, j ∈ N

in which t0 is the initial time, x0 is the initial state, (ik, jk) means that the ik-th subsystem is activated for t ∈ [tk, tk+1).Therefore, when t ∈ [tk, tk+1), the trajectory of the switched system (1) is produced by the ik-th subsystem. For anyj ∈ M ,

Σt ( j) = [t j1 , t j1+1), [t j2 , t j2+1), . . . [t jn , t jn+1) . . . , σ (t) = j, t jk ≤ t < t jk+1, k ∈ N (2)

denotes the sequence of switching times of the j-th subsystem, in which the j-th subsystem is switched on at t jk andswitched off at t jk+1.

We classify actuators of the i-th subsystem into two groups. One is a set of actuators susceptible to failures,denoted by Θi ⊆ 1, 2, . . . , mi , i ∈ M . The other is a set of actuators robust to failures, denoted by Θi ⊆

1, 2, . . . , mi − Θi , i ∈ M . For ωi ⊆ Θi , introduce the decomposition

gi (x) = gωi (x) + gωi (x),

where

gωi (x) = (δωi (1)gi1(x), δωi (2)gi2(x), . . . , δωi (mi )gimi (x))

R. Wang et al. / Nonlinear Analysis: Hybrid Systems 1 (2007) 317–325 319

with δωi defined by

δωi (k) =

1, k ∈ ωi0, k 6∈ ωi .

When actuator failures occur corresponding to ωi ⊆ Θi the resulting i-th subsystem can be described by

x = fi (x) + gωi (x)uωi + pi (x)wi ,

zω =

(hi (x)

uωi

).

(3)

The following inequalities are obvious and will be used in the sequel:

gωi (x)gTωi

(x) ≤ gΘi (x)gTΘi

(x), gΘi(x)gT

Θi(x) ≤ gωi (x)gT

ωi(x).

Now, the reliable H∞ control problem for the switched system (1) is stated as follows:Let a constant γ > 0 be given. For actuator failures corresponding to any ωi ⊆ Θi , find continuous state feedback

controllers ui = ui (x) for all subsystems, and a switching law i = σ(t) such that:

(1) The closed-loop system of system (1) is asymptotically stable when wi = 0.(2) The output z satisfies ‖z‖2 ≤ γ ‖wi‖2 under the zero initial condition.

Remark 1. The reliable H∞ control problem for a switched nonlinear system is solved by Theorem 1. WhenM = 1, switched system (1) degenerates into a regular nonlinear system and the H∞ control problem becomesthe standard reliable H∞ control problem for nonlinear systems [15].

Definition ([20]). A system

x = f (x),

y = h(x)

is called asymptotically zero state detectable if for any ε > 0, there exists δ > 0 such that when ‖y(t + s)‖ < δ holdsfor some t ≥ 0,∆ > 0 and 0 ≤ s ≤ ∆, we have ‖x(t)‖ < ε.

Remark 2. This asymptotically zero state detectability is a weaker version of small-time norm observability [21].

Remark 3. In the existing standard reliable control problem, ( f, gΘ ) is assumed to be a stabilizable pair. This strongassumption is no longer needed here for switched systems. In fact, if ( f j , gΘ j

) is a stabilizable pair for some j ∈ M ,then we can fix σ(t) ≡ j and design a state feedback controller for the j-th subsystem which makes the system (1)stabilizable and thus the problem becomes trivial.

3. Main results

This section gives a condition for the reliable H∞ control problem to be solvable, and designs continuouscontrollers for subsystems and a switching law.

Theorem 1. Let a constant γ > 0 be given. Suppose that:

(1) The pair fi , hi is asymptotically zero state detectable.(2) There exist functions βi j (x), i, j ∈ M, (either all nonnegative or all nonpositive) and radially unbounded positive

definite smooth functions Vi (x) with Vi (0) = 0, i ∈ M, satisfying the partial differential inequalities

∂Vi

∂xfi +

14

∂Vi

∂x

(1γ 2 pi pT

i − gΘigTΘi

)∂T Vi

∂x+ hT

i hi +

m∑j=1

βi j (Vi − V j ) ≤ 0, i ∈ M. (4)

320 R. Wang et al. / Nonlinear Analysis: Hybrid Systems 1 (2007) 317–325

Then, the state feedback reliable controllers

ui = ui (x) = −12

gTi (x)

∂T Vi

∂x(x), i = 1, 2, . . . m (5)

and the switching law σ = σ(x) solve the reliable H∞ control problem.

Proof. Consider actuator failures corresponding to any ωi ⊆ Θi . Since the control input ui (x) is applied to the plantonly through normal actuators, we have

ui = uωi (x) = −12

gTωi

(x)∂T Vi

∂x(x).

Without loss of generality, suppose βi j ≥ 0. For any fixed x ∈ Rn , if Vi (x)−V j (x) ≥ 0, for some i ∈ M and ∀ j ∈ M ,(4) gives

∂Vi

∂xfi +

14

∂Vi

∂x

(1γ 2 pi pT

i − gΘigTΘi

)∂T Vi

∂x+ hT

i hi ≤ 0. (6)

Obviously, for any x ∈ Rn\ 0, there exists an i ∈ M such that Vi (x) − V j (x) ≥ 0, ∀ j ∈ M . For any i ∈ M , let

Ωi = x ∈ Rn|Vi (x) − V j (x) ≥ 0, ∀ j ∈ M, (7)

then⋃m

i=1 Ωi = Rn\ 0. Construct the sets

Ω1 = Ω1, . . . , Ωi = Ωi −

i−1⋃j=1

Ω j , . . . , Ωm = Ωm −

m−1⋃j=1

Ω j .

Obviously, it holds thatm⋃

i=1

Ωi = Rn\ 0, and Ωi

⋂Ω j = φ, i 6= j.

Design the switching law

σ(t) = i, when x(t) ∈ Ωi , i ∈ M. (8)

When x(t) ∈ Ωi , the time derivative of Vi (x(t)) along the trajectory of the system (3) is given by

Vi (x(t)) =∂Vi

∂x( fi + gωi uωi + piwi )

=∂Vi

∂x( fi + piwi + gi ui − gωi uωi )

≤∂Vi

∂x( fi + piwi + gi ui ) +

14

∂Vi

∂xgωi g

Tωi

∂T Vi

∂x+ uT

ωiuωi

≤∂Vi

∂x( fi + piwi + gi ui ) +

14

∂Vi

∂xgΘi g

TΘi

∂T Vi

∂x+ uT

ωiuωi

=∂Vi

∂x( fi + piwi ) +

∥∥∥∥ui +12

gTi

∂T Vi

∂x

∥∥∥∥2

− uTωi

uωi −14

∂Vi

∂xgΘi

gTΘi

∂T Vi

∂x. (9)

When wi = 0, substituting (5) into (9) and noticing (6), we have

Vi (x(t)) ≤∂Vi

∂xfi − uT

ωiuωi −

14

∂Vi

∂xgΘi

gTΘi

∂T Vi

∂x

≤ −1

4γ 2∂Vi

∂xpi pT

i∂T Vi

∂x− hT

i hi − uTωi

uωi

≤ 0.

R. Wang et al. / Nonlinear Analysis: Hybrid Systems 1 (2007) 317–325 321

Thus, according to the asymptotically zero state detectability, the closed-loop system (1) and (5) is asymptoticallystable by using the method in [20].

In the following, we show that the overall L2-gain from wi to zω is less than or equal to γ . We suppose x(0) = 0,and without loss of generality, assume that the first subsystem (σ = 1) is activated at the initial time, i.e. tk1 = t0 = 0.Now we introduce

J =

∫ T

0(‖zω(t)‖2

− γ 2‖wi (t)‖2) dt.

According to the switching sequence (2), when T ∈ [tk, tk+1), we have

J ≤

k−1∑j=0

(∫ t j+1

t j

(‖hi j (t)‖2+ ‖uωi j

(t)‖2− γ 2

‖wi j (t)‖2+ Vi j (t)) dt − (Vi j (x(t j+1)) − Vi j (x(t j )))

)

+

∫ T

tk(‖hi j (t)‖

2+ ‖uωi j

(t)‖2− γ 2

‖wi j (t)‖2+ Vi j (t)) dt − (Vik (x(T )) − Vik (x(tk))).

Note that

Vi j (t) + ‖hi j (t)‖2+ ‖uωi j

(t)‖2− γ 2

‖wi j (t)‖2

≤∂Vi j

∂x( fi j + pi j wi j + gi j ui j ) +

14

∂Vi j

∂xgΘi j

gΘTi j

∂T Vi j

∂x+ ‖hi j (t)‖

2+ ‖ui j

(t)‖2− γ 2

‖wi j (t)‖2

=∂Vi j

∂xfi j +

14γ 2

∂Vi j

∂xpi j pT

i j

∂T Vi j

∂x−

14

∂Vi j

∂xgΘi j

gTΘi j

∂T Vi j

∂x+ hT

i jhi j

+

∥∥∥∥∥ui j +12

gTi j

∂T Vi j

∂x

∥∥∥∥∥2

−

∥∥∥∥∥γwi j −1

2γpT

i j

∂T Vi j

∂x

∥∥∥∥∥2

. (10)

Substituting (5) into (10) gives rise to

Vi j (t) + ‖hi j (t)‖2+ ‖uωi j

(t)‖2− γ 2

‖wi j (t)‖2

≤ −γ 2

∥∥∥∥∥wi j −1γ 2 pT

i j

∂T Vi j

∂x

∥∥∥∥∥2

≤ 0.

Therefore,

J ≤ −

k−1∑j=0

(Vi j (x(t j+1)) − Vi j (x(t j ))) − (Vik (x(T )) − Vik (x(tk)))

= −(Vi0(x(t1)) − Vi0(x(t0)) + Vi1(x(t2)) − Vi1(x(t1)) + · · · + Vik−1(x(tk))

− Vik−1(x(tk−1))) − Vik (x(T )) + Vik (x(tk)).

In view of Vσ(tk−1)(tk) = Vσ(tk )(tk), we have

J ≤ Vi0(x(t0)) − Vik (x(T ))

= −Vik (x(T ))

≤ 0.

This completes the proof.

Remark 4. For the switched linear system

x = Ai x + Bi u + Diw,

z = Ci x,

322 R. Wang et al. / Nonlinear Analysis: Hybrid Systems 1 (2007) 317–325

(4) turns out to be the matrix inequalities

Pi Ai + ATi Pi + Pi (γ

−2 Di DTi − ε−1 BΘi

BTΘi

)Pi + CTi Ci +

m∑j=1

βi j (Pi − Pj ) < 0, i ∈ M,

where Pi are positive definite matrices, βi j are constants either all nonnegative or all nonpositive. In particular, ifm = 1, the Riccati inequality follows.

4. Hybrid reliable H∞ control for nonlinear systems

In engineering, a continuous reliable H∞ controller for a nonlinear system may not exist or may be sometimestoo complex to implement. Thus, in some control problems, control actions are decided by switching between finitecandidate controllers. In this section, we try to use this methodology to solve the standard reliable H∞ control problemfor nonlinear systems.

Consider the following nonlinear system:

x = f (x) + g(x)u + p(x)w,

z =

(h(x)

u

),

(11)

where x ∈ Rn is the state, u and w denote the control input and disturbance input respectively, z is the outputto be regulated, f (x) ∈ Rn, g(x) = (g1(x), . . . gm(x)) ∈ Rn×m, p(x) = (p1(x), . . . pq(x)) ∈ Rn×q , h(x) =

(h1(x), . . . h p(x))T∈ R p, f (0) = 0, h(0) = 0.

Suppose that we have a finite number of candidate controllers for system (11). When actuator failures occur, noneof the individual controllers stabilizes the system. In particular, we consider the following class of candidate statefeedback controllers:

ui = ui (x) = −12

gT (x)∂T Vi

∂x(x), i ∈ M, (12)

where Vi will be specified later.

Theorem 2. Let a constant γ > 0 be given. Suppose that:

(1) The pair f, h is asymptotically zero state detectable.(2) There exist functions βi j (x), i, j ∈ M (either all nonnegative or all nonpositive), and radially unbounded, positive

smooth functions Vi (x) with Vi (0) = 0, i ∈ M, satisfying the partial differential inequalities

∂Vi

∂xf +

14

∂Vi

∂x

(1γ 2 ppT

− gΘ gTΘ

)∂T Vi

∂x+ hT h +

m∑j=1

βi j (Vi − V j ) ≤ 0, i ∈ M. (13)

Then, for actuator failures corresponding to any ωi ⊆ Θi , the state feedback reliable controllers (12) with theswitching law (8) solve the reliable H∞ control problem.

Proof. Substituting (12) into the system (11) results in a switched nonlinear system. Then, applying the Theorem 1gives the result.

Next, we give a sufficient condition guaranteeing (13) by extending the results [22] to switched systems.

Proposition 1. Suppose p = gΘ s + q and there exist a positive definite and radially unbounded function Wi (x)

satisfying the following assumptions:

(i) ∂Wi∂x f (x) ≤ −αi (x), αi (x) > 0,

(ii) 0 < β < γ 2, 0 < η <γ 2

‖s‖2 − 1,

(iii) hT h +1ε

∑mj=1 βi j

∫ Wi0 Ki (t) dt ≤ αi Ki (Wi ),

(iv) Ki (Wi )(∂Wi∂x qqT ∂T Wi

∂x ) ≤ βαi (1 +1η)−1,

R. Wang et al. / Nonlinear Analysis: Hybrid Systems 1 (2007) 317–325 323

(v) Vi (Wi ) =1ε

∫ Wi0 Ki (t) dt , where 1

2 (1 −

√1 −

β

γ 2 ) < ε < 12 (1 +

√1 −

β

γ 2 ).

Then, the partial differential inequalities (13) are solvable and the function Vi (Wi ) is a solution of inequalities (13).

Proof. From assumptions (i)–(v), we have

∂Vi

∂xf +

14

∂Vi

∂x

(1γ 2 ppT

− gΘ gTΘ

)∂T Vi

∂x+ hT h +

m∑j=1

βi j (Vi − V j )

=1ε

Ki∂Wi

∂xf +

14γ 2ε2 K 2

i∂Wi

∂x(gΘ s + q)(gΘ s + q)T ∂T Wi

∂x−

K 2i

4ε2∂Wi

∂xgΘ gT

Θ

∂T Wi

∂x

+ hT h +1ε

m∑j=1

βi j

∫ Wi

0Ki (t)dt −

1ε

m∑j=1

βi j

∫ W j

0K j (t) dt

≤ −1εαi Ki +

14γ 2ε2 K 2

i∂Wi

∂x(gΘ s + q)(gΘ s + q)T ∂T Wi

∂x−

14ε2 K 2

i∂Wi

∂xgΘ gT

Θ

∂T Wi

∂x+ αi Ki

= −1εαi Ki −

14ε2 K 2

i∂Wi

∂xgΘ

(1 −

1γ 2 ssT

)gTΘ

∂T Wi

∂x+

12γ 2ε2 K 2

i∂Wi

∂xgΘ sqT ∂T Wi

∂x

+1

4γ 2ε2 K 2i∂Wi

∂xqqT ∂T Wi

∂x+ αi Ki

≤ −1εαi Ki −

14ε2 K 2

i∂Wi

∂xgΘ

(1 −

1 + η

γ 2 ssT)

gTΘ

∂T Wi

∂x+

14γ 2ε2 K 2

i

(1 +

1η

)∂Wi

∂xqqT ∂T Wi

∂x+ αi Ki

≤ −1εαi Ki +

14γ 2ε2 K 2

i

(1 +

1η

)∂Wi

∂xqqT ∂T Wi

∂x+ αi Ki

≤ −1εαi Ki +

β

4γ 2ε2 αi Ki + αi Ki

=−4γ 2ε + 4γ 2ε2

+ β

4γ 2ε2 αi Ki .

From (v), we know ε2− ε +

β

4γ 2 < 0, which is equivalent to

−4γ 2ε + 4γ 2ε2+ β

4γ 2ε2 < 0.

Thus, the function Vi (Wi ) is a solution of inequalities (13).

Remark 5. The partial differential inequalities (13) are much easier to satisfy than the Hamilton–Jacobi inequalitybecause the term

∑mj=1 βi j (Vi − V j ) is added, which may change sign when x varies. In particular, if m = 1, (13)

degenerates into the Hamilton–Jacobi inequality.

5. Example

In this section, we present an example to illustrate the effectiveness of the proposed design method. Consider thefollowing nonlinear switched system:

x = fi (x) + gi (x)ui + pi (x)wi

z =

(hiui

), i = 1, 2,

(14)

where

f1(x) = −3x3, p1(x) = x, h1(x) = h2(x) = x2, g1(x) = (3 x2),

f2(x) = −3x3+ x, p2(x) = 1, g2(x) = (x2 2), Θ1 = 1, Θ2 = 2.

324 R. Wang et al. / Nonlinear Analysis: Hybrid Systems 1 (2007) 317–325

It is easy to check that fi , hi is detectable, but ( fi , gΘi) is not a stabilizable pair. We will show that the reliable H∞

control problem is solvable via switching between subsystems. We choose

V1(x) = x2, V2(x) = x4, x ∈ Rn .

Both V1 and V2 are globally positive definite and V1(0) = V2(0) = 0.Let γ = 1, β1(x) = 3x2, β2(x) = 5x2; then

∂V1

∂xf1 +

14

∂V1

∂x

(1γ 2 p1 pT

1 − gΘ1gTΘ1

)∂T V1

∂x+ hT

1 h1 + β1(V1 − V2)

= 2x(−3x3) + x2(x2− x4) + x4

+ 3x2(x2− x4)

= −x4− 4x6

≤ 0

and

∂V2

∂xf2 +

14

∂V2

∂x

(1γ 2 p2 pT

2 − gΘ2gTΘ2

)∂T V2

∂x+ hT

2 h2 + β2(V2 − V1)

= 4x3(−x3+ x) + 4x6(1 − x4) + x4

+ 5x2(x4− x2)

= −3x6− 4x10

≤ 0.

Let

Ω1 = x ∈ Rn|V1(x) − V2(x) ≥ 0, x 6= 0;

Ω2 = x ∈ Rn|V2(x) − V1(x) ≥ 0, x 6= 0,

then Ω1 ∪ Ω2 = Rn\ 0. The switching law is designed by

σ(t) =

1 x(t) ∈ Ω1,

2 x(t) ∈ Ω2 \ Ω1.

So the controllers

u1 = −12

gT1 (x)

∂T V1

∂x(x) =

(−3x−x3

), u2 = −

12

gT2 (x)

∂T V2

∂x(x) =

(−2x5

−4x3

)with the switching law σ(t) solve the reliable H∞ control problem with γ = 1.

6. Conclusions

We have considered the problem of reliable H∞ control for switched nonlinear systems. In particular, attentionis concentrated on actuators suffering “serious failures”, which has not been considered in previous reliable work.On the basis of switching strategy, we design controllers and a switching law to solve the problem of reliable H∞

control. Moreover, a hybrid state feedback strategy is proposed to solve the standard H∞ control problem for nonlinearsystems when no single continuous controller is effective.

References

[1] D. Liberzon, Switching in Systems and Control, Birkhauser, Boston, 2003.[2] D.Z. Cheng, L. Guo, Y.D. Lin, Y. Wang, Stabilization of switched linear systems, IEEE Transactions on Automatic Control 50 (5) (2005)

661–666.[3] J. Zhao, G.M. Dimirovski, Quadratic stability of a class of switched nonlinear systems, IEEE Transactions on Automatic Control 49 (4) (2004)

574–578.[4] Z.D. Sun, S.S. Ge, Switched Linear Systems—Control and Design, Springer-Verlag, New York, 2004.[5] J. Zhao, M.W. Spong, Hybrid control for global stabilization of the cart–pendulum system, Automatica 37 (12) (2001) 1941–1951.

R. Wang et al. / Nonlinear Analysis: Hybrid Systems 1 (2007) 317–325 325

[6] G.M. Xie, L. Wang, Necessary and sufficient conditions for controllability and observability of switched impulsive control systems, IEEETransactions on Automatic Control 49 (6) (2004) 960–966.

[7] G.S. Zhai, B. Hu, K. Yasuda, A.N. Michel, Disturbance attenuation properties of time-controlled switched systems, Journal of the FranklinInstitute 338 (7) (2001) 765–779.

[8] M.S. Branicky, Multiple Lyapunov functions and other analysis tools for switched and hybrid systems, IEEE Transactions on AutomaticControl 43 (4) (1998) 475–482.

[9] J.P. Hespanha, Uniform stability of switched linear systems: Extensions of LaSalle’s invariance principle, IEEE Transactions on AutomaticControl 49 (4) (2004) 470–482.

[10] H. Ishii, T. Basar, R. Tempo. Synthesis of switching rules for switched linear systems through randomized algorithms, in: Proceedings of theIEEE Conference on Decision and Control, vol. 5, 2003, pp. 4788–4793.

[11] N.H. El-Farra, P. Mhaskar, P.D. Christofides, Output feedback control of switched nonlinear systems using multiple Lyapunov functions,Systems and Control Letters 54 (12) (2005) 1163–1182.

[12] J. Zhao, D.J. Hill, Hybrid H∞ control based on multiple Lyapunov functions, in: 6th IFAC Symposium on Nonlinear Control Systems,NOLCOS 2004, vol. 2, 2004, pp. 567–572.

[13] C.J. Seo, B.K. Kim, Robust and reliable H∞ control for linear systems with parameter uncertainty and actuator failure, Automatica 32 (3)(1996) 465–467.

[14] Z.D. Wang, H. Qiao, H∞ reliable control of uncertain linear state delayed systems, Journal of Dynamical and Control Systems 10 (1) (2004)55–76.

[15] G.H. Yang, J. Lam, J.L. Wang, Reliable H∞ control for affine nonlinear systems, IEEE Transactions on Automatic Control 43 (8) (1998)1112–1117.

[16] J.C. Wang, H.H. Shao, Delay-dependent robust and reliable H∞ control for uncertain time-delay systems with actuator failures, Journal ofthe Franklin Institute 337 (6) (2000) 781–791.

[17] D. Yue, J. Lam, D.W.C. Ho, Reliable H∞ control of uncertain descriptor systems with multiple time delays, IEE Proceedings—ControlTheory and Applications 50 (6) (2003) 557–564.

[18] M.A. Demetriou, Adaptive reorganization of switched systems with faulty actuators, in: Proc. of the 40rd IEEE conference on Decision andControl, Orlando, FL, 2001, pp. 1879–1884.

[19] R. Wang, J. Zhao, Robust fault-tolerant control for a class of switched nonlinear systems in lower triangular form, Asian Journal of Control 9(1) (2006).

[20] J. Zhao, D.J. Hill, Dissipativity theory for switched systems, in: Proc. of the 44rd IEEE conference on Decision and Control, Seville, Spain,2005, pp. 7003–7008.

[21] J.P. Hespanha, D. Liberzon, A. Angeli, E.D. Sontag, Nonlinear norm observability notions and stability of switched systems, IEEETransactions on Automatic Control 50 (2) (2005) 154–168.

[22] W.Z. Su, C.E. de Souza, L.H. Xie, H∞ control for asymptotically stable nonlinear systems, IEEE Transactions on Automatic Control 44 (5)(1999) 989–993.