Integral Sliding Mode Control for a Class of Uncertain ...

European Journal of Control (2010)1:16–22 # 2010 EUCA DOI:10.3166/EJC.16.16–22

Integral Sliding Mode Control for a Class of Uncertain Switched Nonlinear Systems Jie Lian1,2,, Jun Zhao2,3, Georgi M. Dimirovski4 1

School of Electronic and Information Engineering, Dalian University of Technology, Dalian 116024, P. R. China; Key Laboratory of Integrated Automation of Process Industry, Ministry of Education, Northeastern University, Shenyang 110004, P. R. of China; 3 Department of Information Engineering, Research School of Information Sciences and Engineering, The Australian National University, Canberra ACT 0200, Australia; 4 Department of Computer Engineering, Dogus University, Acibadem, Kadikoy, 34722, Istanbul, Republic of Turkey 2

In this paper, a new integral sliding mode control scheme is proposed for a class of uncertain switched nonlinear systems. First, a nonlinear integral sliding surface is constructed such that the switched system under the sliding mode is exponentially stable for switching laws with an average dwell time switching property. Then, variable structure controllers are designed to guarantee the existence of the sliding mode from the initial time. Robustness of the closed-loop system in the whole state space is obtained. Finally, a numerical example is given to illustrate the effectiveness of the proposed design method. Keywords: Average dwell time, integral sliding mode control, robust control, uncertain switched nonlinear systems

1. Introduction Recently, the study of switched systems has been receiving growing attention mainly because many real-world systems, for example, automotive engine control systems [1], disc drives [23], some robot control systems [9], and the cart-pendulum system [27] can be modeled as switched systems. Several methods have

 Correspondence to: J. Lian, E-mail: [email protected]

been proposed for study of switched systems [8, 13, 19, 26]. A common Lyapunov function for all subsystems guarantees asymptotic stability under arbitrary switching laws [4, 12, 15]. The single Lyapunov function is developed by designing a state-partition-based feedback switching law to stabilize a class of switched linear systems with stable convex combinations [10, 18, 22]. The multiple Lyapunov functions method proposed in [16] is considered as an analysis tool in [2, 24]. It is shown in [14] that based on the dwell time method a switched linear system is exponentially stable if the individual subsystems are exponentially stable. This result is later extended to switching laws with average dwell time [7]. The average dwell time method is considered as one of the most effective methods in the study of switched systems [12]. It can also be applied to the study of switched systems composed of both stable and unstable subsystems [25]. However, the average dwell time method has rarely been applied to study switched systems with uncertainties. In particular, no relevant results for uncertain switched nonlinear systems have been reported so far. On the other hand, sliding mode control is an effective robust control approach [5, 11, 20]. Typically, the complete response of the sliding mode control comprises two phases: reaching phase and sliding phase. We know that in the reaching phase, systems

Received 17 January 2008; Accepted 24 April 2009 Recommended by A. Astolfi, A.J. van der Schaft

17

Integral Sliding Control of Switching Nonlinear Systems

are sensitive to uncertainties and perturbations, which may deteriorate system performance. In order to handle this problem, a new sliding mode is proposed, called integral sliding mode which does not have reaching phase [21]. Having this feature, integral sliding mode control has been widely applied to various systems [3, 6]. Integral sliding mode control is of course expected to be effective for uncertain switched systems. However, so far there have been no results for integral sliding mode control of uncertain switched nonlinear systems. In this paper, the robust integral sliding mode control problem is studied for a class of uncertain switched nonlinear systems. The switched system is composed of two cascade parts: a linear part with mismatched uncertainties and a nonlinear part with matched uncertainties. In order to achieve exponential stability, we present a two-step design strategy. First, we design an integral sliding surface such that the sliding mode is exponentially stabilizable via switching. Then, we design variable structure controllers to guarantee that the trajectory of the system remains on the sliding surface from the initial time. Robustness of the closed-loop system in the whole state space is obtained. Throughout this paper, kk denotes the Euclidean norm for a vector or the matrix-induced norm for a matrix. max ðPÞ and min ðPÞ denote the maximum and minimum eigenvalues of a symmetric matrix P, respectively.

2. Problem Formulation and Preliminaries We consider the following uncertain switched nonlinear system _1 ¼ ðA1 þ A1 Þ1 þ A2 2 ; _2 ¼ f ð2 Þ þ g ð2 Þf½I þ g ð2 ; #Þu þ f ð2 ; #Þg; ð1Þ where  ¼ ½1T ; 2T T is the state with 1 2 Rh and 2 2 Rnh ; ðtÞ : ½0; 1Þ !  ¼ f1; 2; . . . ; lg is the switching signal, which is a piecewise constant function of time, ui 2 Rm is the control input of the ith subsystem, A1i and A2i are constant matrices with appropriate dimensions, fi ð2 Þ and gi ð2 Þ are known nonlinear function vector and matrix, respectively. For some prescribed compact set Y  Rp ; # 2 Y is the vector uncertain variable. fi ð2 ; #Þ and gi ð2 ; #Þ represent the uncertainties satisfying, kfi ð2 ; #Þk  ’i ð2 Þ; kgi ð2 ; #Þk  i < 1; 8ð2 ; #Þ 2 Rnh  Y, where ’i ð2 Þ and i are known nonnegative scalar-valued nonlinear functions and nonnegative constants,

respectively. A1i are system parameter uncertainties that can be represented as A1i ¼ Ei i ðtÞFi ; i 2 ; where Ei ; Fi are constant matrices with appropriate dimensions and i ðtÞ are unknown matrices and satisfy Ti ðtÞi ðtÞ  I. Lemma 1 [17]: Given real matrices R1 and R2 with appropriate dimensions and an unknown matrix ðtÞ with T ðtÞðtÞ  I, there exists  > 0 such that R1 ðtÞR2 þ RT2 T ðtÞRT1  R1 RT1 þ 1 RT2 R2 We will focus on switching signals that satisfy certain average dwell time requirement. Definition 1 [7]: For switching signal  and t >  0, let N ð; tÞ denote the number of discontinuities of  on an interval ð; tÞ. If there exist N0 0; a > 0 such that N ð; tÞ  N0 þ

t a

ð2Þ

holds, then the positive constant a is referred to as average dwell time. As commonly used in the literature, for convenience, we choose N0 ¼ 0 in this paper. The control object is for the uncertain switched nonlinear system (1), determine switching laws with an average dwell time property, a proper nonlinear integral sliding surface and variable structure controllers such that the closed-loop system is globally exponentially stable. We introduce the following assumption. Assumption 1: For each i 2 , the known nominal nonlinear subsystem, _2 ¼ fi ð2 Þ þ gi ð2 Þui is globally asymptotically stabilizable via a nominal controller ui ¼ i ð2 Þ, i.e., there are a collection of positive defined radially unbounded functions fW1 ; W2 ; Wl g and positive numbers i ; 1i ; 2i such that, for each 2 2 Rnh , @Wi ð2 Þ ðfi þ gi i Þ  i k2 k2 ; @2 2

ð3Þ

2

1i k2 k  Wi ð2 Þ  2i k2 k : In this paper, we do not require that all subsystems are stable. Instead, we allow stable subsystems and unstable subsystems to co-exist. Let p be a nonempty ~ p be the complement of p with subset of  and  respect to . We make the following assumption. Assumption 2: There exist a constant 1, positive definite matrices Pi ; i 2 , positive constants

18

J. Lian et al.

"i > 0; i 2 , a nonempty subset p  , positive constants i > 0; i 2 p and negative constants i < 0, ~ p such that the following matrix inequalities hold i2 T T AT1i Pi þ Pi A1i þ "1 i Pi Ei Ei Pi þ "i Fi Fi þ i Pi þ I < 0;

Pi  Pj ;

i; j 2 : ð4Þ

Let Tþ ðt0 ; tÞ denote the total activation time of all ith; i 2 p subsystems during ½t0 ; tÞ; T ðt0 ; tÞ denote ~ p subsystems the total activation time of all jth; j 2   ~ p g; during ½t0 ; tÞ. Define  ¼ minfj < 0; j 2    i . þ ¼ minf^i > 0; i 2 p g, where ^i ¼ min i ; 22i For any given  2 ð0; þ Þ, we choose an arbitrary  2 ð; þ Þ. Motivated by the idea in [25], we propose the switching law satisfying the following condition: Tþ ðt0 ; tÞ    þ : inf  t>t0 T ðt0 ; tÞ   

ð5Þ

Remark 1: The form of A1i ¼ Ei i ðtÞFi is a standard assumption in the robust control literature (see [5, 6, 10] for example). Assumption 2 is very mild. In fact Pi  Pj always hold for some 1, while AT1i Pi þ T T Pi A1i þ "1 i Pi Ei Ei Pi þ "i Fi Fi þ i Pi þ I < 0 are easy to satisfy and this is taken as a typical assumption in the literature ([5, 10]).

Remark 2: From (6), we can see that the sliding surface is designed as integral form and depends on the initial condition 2 ðt0 Þ. Therefore, the trajectory of 2 is driven to remain in sliding surface from the initial time, the reaching phase is avoided and the robustness is improved. Theorem 1: Suppose that Assumptions 1 and 2 hold. The uncertain switched nonlinear system (1) is exponentially stable in the integral sliding surface SðtÞ ¼ 0 for any switching signals  satisfying the condition (5) and the average dwell time ln ^ a a ¼  ð7Þ    2i with ^ ¼ max ; ; i; j 2  . 1j Proof: When 2 evolves in the sliding surface, we have SðtÞ ¼ 0

ð8Þ

Differentiating the sliding function (6) with respect to time, one obtains _ ¼ G_2 ðtÞ  Gðf þ g  Þ SðtÞ

ð9Þ

According to kgi ð2 ; #Þk  i < 1; ðI þ gi ð2 ; #ÞÞ _ ¼ 0 is solved to are invertible. Then, by using (1), SðtÞ obtain the equivalent control uieq uieq ¼ ½I þ gi ð2 ; #Þ1 fi ð2 Þ  fi ð2 ; #Þg; i 2  ð10Þ Substituting (10) into (1) yields the sliding mode

3. Integral Sliding Surface Design According to design of general sliding mode variable structure control, the design procedure is divided into two phases. First, the integral sliding surface is designed, so that the controlled system yields the desired dynamic performance. The second phase is to design the variable structure controllers such that the trajectory of the system remains on the sliding surface from the initial time. In this section, we design a nonlinear integral sliding surface such that the sliding mode is exponentially stable. The nonlinear integral sliding function is defined as follows Z t SðtÞ ¼ G2 ðtÞ  G2 ðt0 Þ  Gðf þ g  Þd; t0

ð6Þ where G 2 RmðnhÞ is a constant matrix and Ggi ; i 2  are invertible. Now, when SðtÞ ¼ 0, we give the stability result for uncertain switched nonlinear system (1).

_1 ¼ ðA1 þ A1 Þ1 þ A2 2 ; _2 ¼ f ð2 Þ þ g ð2 Þ ð2 Þ:

ð11Þ

Define the Lyapunov function candidate as follows VðtÞ ¼ V ð1 ; 2 Þ ¼ 1T P 1 þ KW ð2 Þ

ð12Þ

where K is a positive number to be determined later, it switches in accordance with the piecewise constant switching law . Then, when the i-th subsystem is active, the time derivative of Vi along the trajectory of the switched system (11) is V_ i ¼ 1T ðAT1i Pi þ Pi A1i Þ1 þ 1T ðAT1i Pi þ Pi A1i Þ1 dWi ð2 Þ þ 21T Pi A2i 2 þ K ðfi þ gi i Þ d2 ¼ 1T ðAT1i Pi þ Pi A1i Þ1 þ 1T ðFTi Ti ðtÞETi Pi þ Pi Ei i ðtÞFi Þ1 þ 21T Pi A2i 2 þK

dWi ð2 Þ ðfi þ gi i Þ: d2

19

Integral Sliding Control of Switching Nonlinear Systems

By using Assumption 1 and Lemma 1 with R1 ¼ FTi ; R2 ¼ ETi Pi and  ¼ "i , we have



 eð

T T V_ i  1T ðAT1i Pi þ Pi A1i þ "1 i Pi Ei Ei Pi þ "i Fi Fi Þ1

þ 21T Pi A2i 2  Ki k2 k2 :



VðtÞ  e ðT 

tt0 a

ðt0 ;tÞþTþ ðt0 ;tÞÞþ

lna ^Þðtt0 Þ

ln ^

Vðt0 Þ ðt0 Þ

Vðt0 Þ ðt0 Þ

ð19Þ

 eðtt0 Þ Vðt0 Þ ðt0 Þ:

ð13Þ

Moreover, from 1i k2 k2  Wi ð2 Þ  2i k2 k2 in (3) and (12), it is easy to get

It is easy to show that there exist constants i > 0, i 2  such that

a1 k1 k2 þK1 k2 k2  VðtÞ  a2 k1 k2 þK2 k2 k2 ; ð20Þ

 T   Pi A2i 2   i k1 kk2 k; 1

i2

ð14Þ

Then, V_ i < i 1T Pi 1  k1 k2 þ2 i k1 kk2 k  Ki k2 k2 Ki  i 1T Pi 1  Wi ð2 Þ  ðk1 k  i k2 kÞ2 22i   Ki   i2 k2 k2 : 2 ð15Þ  2    2 i i 0 ; Taking K > max ; i 2  and i ¼ min i ; i 22i i 2 , we get V_ i < 0i Vi ; i 2 

ð16Þ

From Assumption 2, we know that when i 2 p ;   i ~ p ; 0 ¼ > 0, when i 2  0i ¼ ^i ¼ min i ; i 22i   i min i ; ¼ i < 0. 22i From 1i k2 k2  Wi ð2 Þ  2i k2 k2 in (3) and Pi  Pj in (4), we obtain ^ j ðtÞ; 8i; j 2  Vi ðtÞ  V

ð17Þ

For t > 0, let t0 < t1 < < tk ¼ tNðt0 ;tÞ be the switching instants of the switching signal ðtÞ over the interval ðt0 ; tÞ. Therefore, by using (17) we have VðtÞ  Vðtk Þ ðtk Þe  e







þ

¼e



ð21Þ which implies kðtÞk2 

1 d2 VðtÞ  eðtt0 Þ kðt0 Þk2 d1 d1

Thus, sffiffiffiffiffi d2 ðtt0 Þ e 2 kðt0 Þk kðtÞk  d1

ð22Þ

This means that the sliding mode is exponentially stable, which completes the proof. Remark 3: It is observed from the proof of Theorem 1 that we design the sliding surface, such that when the trajectory of 2 remains on the integral sliding surface, the switched system (1) is exponentially stable. Having designed the stable integral sliding surface, we now design variable structure controllers such that the trajectory of 2 is driven to remain on sliding surface from the initial time.

4. Design of Variable Structure Controllers Theorem 2: For the system (1), the sliding function is given as (6). Then the sliding mode can be maintained from the initial time by employing the following variable structure controllers

^ ðtk Þ ðt V kÞ



 ^N ðt0 ;tÞ e

d1 ðk1 k2 þk2 k2 Þ  VðtÞ  d2 ðk1 k2 þk2 k2 Þ;

þ

T ðtk ;tÞ T ðtk ;tÞ

T ðtk ;tÞþ Tþ ðtk ;tÞ

where a1 ¼ min ðPi ; i 2 Þ; a2 ¼ max ðPi ; i 2 Þ; 1 ¼ minf1i ; i 2 g and 2 ¼ maxf2i ; i 2 g. Let d1 ¼ minfa1 ; K1 g and d2 ¼ maxfa2 ; K2 g. We have

T ðt0 ;tÞþ Tþ ðt0 ;tÞ

Vðt0 Þ ðt0 Þ

 T ðt0 ;tÞþ Tþ ðt0 ;tÞþN ðt0 ;tÞ ln ^

Vðt0 Þ ðt0 Þ; ð18Þ

Thus, by exploiting (2), (5) and (7), we obtain

ui ¼ i ð2 Þ 

1 ði ki ð2 Þk þ ’i ð2 Þ þ Þ 1  i

signððGgi ÞT SÞ;

i 2 ; ð23Þ

20

J. Lian et al.

where  > 0 is a design constant to adjust the convergent rate. Proof: We choose a Lyapunov function as 1 Vs ¼ ST S 2

ð24Þ

Differentiating Vs with respect to time along the trajectory of (1) yields

where d1 ¼ minfa1 ; K1i ; i 2 g; d2 ¼ maxfa2 ; K2i ; i 2 g; a1 ¼ min ðPi ; i 2 Þ; a2 ¼ max ðPi ; i 2 Þ;  2  2 i K > max ;i 2  . i

V_ s ¼ ST S_ ¼ ST Ggi ð2 Þf½I þ gi ð2 ; #Þui þ fi ð2 ; #Þ  i ð2 Þg: By substituting (23) into the above equation, one obtains  T _ Vs ¼ S Ggi ð2 Þ 

1 ði ki ð2 Þk 1  i

þ ’i ð2 Þ þ ÞsignððGgi ÞT SÞ 1 þ gi ð2 ; #Þ½i ð2 Þ  ði ki ð2 Þk 1  i

  2i where ^ ¼ max ; ; i; j 2  ;  2 ð0; 0 Þ; 0 ¼    1j i min i ; ;i 2  . 22i Moreover, the state estimate of the system (1) is given by sffiffiffiffiffi d2 0 ðtt0 Þ e 2 ð26Þ kðt0 Þk kðtÞk  d1

5. Example In this section, we present an example to demonstrate the effectiveness of the proposed design method. Consider the following uncertain switched nonlinear system _1 ¼ ðA1 þ A1 Þ1 þ A2 2 ; _2 ¼ f ð2 Þ þ g ð2 Þf½I þ g ð2 Þu þ f ð2 Þg;

o

þ ’i ð2 Þ þ ÞsignððGgi ÞT SÞ þ fi ð2 ; #Þ   i 1  ’i ð2 Þ þ  ki ð2 Þk þ 1  i 1  i 1  i  2  T   i S Ggi  þ i ki ð2 Þk þ ki ð2 Þk 1  i   i i   ST Ggi  ’i ð2 Þ þ þ 1  i 1  i  T      þ ’i ðÞ S Ggi   ST Ggi 

ð27Þ where ðtÞ 2  ¼ f1; 2g;   0:4 1:2 3 A11 ¼ , A12 ¼ 1 0:5 2  A21 ¼

0:1 0:4

 0:2 0:3 , A22 ¼ 0:5 0:1 

Because Ggi are nonsingular and Sðt0 Þ ¼ 0, then, if S 6¼ 0, we can get V_ s < 0. Therefore, the sliding mode is maintained by the controllers (23), from the initial time. This completes the proof. Remark 4: From Theorems 1 and 2, we know that exponential stability of the system (1) is achieved under the controllers (23) and any switching law satisfying average dwell time (7). If p ¼ , then all i > 0; i 2 . We obtain the following corollary. Corollary 1: Suppose that Assumptions 1 and 2 hold. The sliding function is given as (6). The switched nonlinear system (1) under the variable structure controllers (23) is exponentially stable for arbitrary switching law satisfying the average dwell time a a ¼

ln ^ ; 

ð25Þ

f1 ð2 Þ ¼ A1 2 þ

1 , 4 0:5 , 0:4

 0 3 , A1 ¼ 2 0:222 sin ð21 Þ 0:1

0:5 , 2



 0:221 cosð22 Þ 5 1 f2 ð2 Þ ¼ A2 2 þ , A2 ¼ , 0:222 sinð22 Þ 2 3 

 2 1 Þ 0:1 cosð21 , B1 ¼ , g1 ð2 Þ ¼ B1 þ 1 0 2 , , B2 ¼ g2 ð2 Þ ¼ B2 þ 0:1 cosð21 Þ sinð22 Þ 1 



0



g1 ð2 Þ ¼ 0:2cosð21 Þ, g2 ð2 Þ ¼ 0:1sinð22 Þ, 2 f1 ð2 Þ ¼ 0:121 sinð21 22 Þ þ 0:1, 4 2 f2 ð2 Þ ¼ 0:121 cosð22 Þ þ 0:222 sinð21 Þ þ 0:15,

A1i ¼ Ei i ðtÞFi , E1 ¼ ½ 0:1

0:1 T ,

E2 ¼ ½ 0:1 0 T , F1 ¼ ½ 0 0:1 , F2 ¼ ½ 0:1 0 , i ¼ i 2 ½1; 1.

21

Integral Sliding Control of Switching Nonlinear Systems 2 The bounds ’1 ð2 Þ ¼ 0:121 þ 0:1; ’2 ð2 Þ ¼ 4 2 0:121 þ 0:222 þ 0:15; 1 ¼ 0:2 and 2 ¼ 0:1 are known. Choose K1 ¼ ½ 0:0357 0:0357  and K2 ¼ ½ 0:6667 0:0333 , G ¼ ½ 1 1  which makes Ggi ð2 Þ nonsingular. The sliding mode is

_1 ¼ ðA1i þ A1i Þ1 þ A2i 2 ; _2 ¼ ðAi  Bi Ki Þ2 þ Yi ;

ð28Þ

 2 0 0:1 cosð21 Þ  where Y1 ¼ K1 2 0:222 sin2 ð21 Þ 0 and   0 0:221 cosð22 Þ Y2 ¼  K . 0:222 sinð22 Þ 0:1 cosð21 Þ sinð22 Þ 2 2 Define the Lyapunov functions V0i ¼ 2T Qi 2 ;  0:3293 0:0445 i ¼ 1; 2, where Q1 ¼ and Q2 ¼ 0:0445 0:4972  0:3192 0:1522 are the solutions of the inequalities 0:1522 0:3668 

Choose ^ ¼ ;  ¼ 0:1;  ¼ 1:4. By (7), we obtain ln ^ ¼ 1:0963   

ð30Þ

Tþ ðtÞ    ¼ 1:8125: T ðtÞ þ  

ð31Þ

a a ¼ and

Taking  ¼ 2, the variable structure controllers are given as 1 ði kKi 2 k þ ’i ð2 Þ þ 2Þ ui ¼  Ki 2  1  i signððGgi ÞT SÞ; i ¼ 1; 2: ð32Þ The simulation results are depicted in Figs 1 and 2 with the initial state ð0Þ ¼ ½3; 4; 2; 2T .

ðAi  Bi Ki ÞT Qi þ Qi ðAi  Bi Ki Þ þ 0:5I < 0: ð29Þ Then the derivative of V0i along the trajectory of the switched system (28) is V_ 0i ¼ 2T ððAi  Bi Ki ÞT Qi þ Qi ðAi  Bi Ki ÞÞ2 þ YTi Qi 2 þ 2T Qi Yi  max ððAi  Bi Ki ÞT Qi þ Qi ðAi  Bi Ki ÞÞk2 k2 þ 2max ðQi ÞkYi kk2 k  max ððAi  Bi Ki ÞT Qi þ Qi ðAi  Bi Ki ÞÞk2 k2 þ 2max ðQi Þð0:1kKi k þ 0:2Þk2 k2 :

Fig. 1. The state responses of the switched system (27).

Thus, we obtain V_ 01  1:7328k2 k2 ; V_ 02  1:6196k2 k2 : Take  ¼ 1:5; þ ¼ 3; "1 ¼ "2 ¼ 1. Solving (4) gives  1:2339 0:3461 ; P1 ¼ 0:3461 1:8201  0:8487 0:0307 P2 ¼ 0:0307 0:4788 max ðPi Þ ¼ 4:1586; 11 ¼ 0:3; 21 ¼ min ðPj Þ 0:55; 12 ¼ 0:18; 22 ¼ 0:5. These imply that the conditions of Theorem 1 are satisfied. Take ¼ supi;j2

Fig. 2. The switching signal.

22

6. Conclusions We have studied the robust integral sliding mode control problem for a class of uncertain switched cascade nonlinear systems. The nonlinear integral sliding surface has been designed for the switched system. Based on the average dwell time method, we have given the conditions for the existence of the sliding surface and proved exponential stability of the sliding mode. The variable structure controllers have been designed such that the state of the system remains on the integral sliding surface from the initial time. Finally, we have showed the effectiveness of the proposed method by using a numerical example.

Acknowledgment This work was supported in part by the NSF of China under grants 60874024 and 60574013, and Dogus University Fund for Science.

References 1. Balluchi A, Di Benedetto MD, Pinello C, Rossi C, Sangiovanni-Vincentelli A. Hybrid control in automotive applications: the cut-off control. Automatica 1999; 35: 519–535 2. Branicky MS. Multiple Lyapunov functions and other analysis tools for switched and hybrid systems. IEEE Trans Autom Control 1998; 43: 475–482 3. Cao WJ, Xu JX. Nonlinear integral-type sliding surface for both matched and unmatched uncertain systems. IEEE Trans Autom Control 2004; 49: 1355–1360 4. Cheng D, Gao L, Huang J. On quadratic Lyapunov function. IEEE Trans Autom Control 2003; 48: 885–890 5. Choi HH. An LMI-based switching surface design method for a class of mismatched uncertain systems. IEEE Trans Autom Control 2003; 48: 1634–1638 6. Choi HH. LMI-based sliding surface design for integral sliding mode control of mismatched uncertain systems. IEEE Trans Autom Control 2007; 52: 736–741 7. Hespanha JP, Morse AS. Stability of switched systems with average dwell-time. In Proceedings of the 38th IEEE Conference on Decision and Control 1999; pp. 2655–2660 8. Hespanha JP, Morse AS. Switching between stabilizing controllers. Automatica 2002; 38: 351–357 9. Jeon D, Tomizuka M. Learning hybrid force and position control of robot manipulators. IEEE Trans Robot Autom 1993; 9: 423–431

J. Lian et al.

10. Ji Z, Wang L, Xie G, Hao F. Linear matrix inequality approach to quadratic stabilization of switched systems. IEE Proc Control Theory Appl 2004; 151: 289–294 11. Kim KS, Park Y, Oh SH. Designing robust sliding hyperplanes for parametric uncertain systems: a Riccati approach. Automatica 2000; 36: 1041–1048 12. Liberzon D. Switching in Systems and Control. Boston, Birkhauser, 2003 13. Liberzon D, Morse AS. Basic problem in stability and design of switched systems. IEEE Control Syst Mag 1999; 19: 59–70 14. Morse AS. Supervisory control of families of linear setpoint controllers-part I: exact matching. IEEE Trans Autom Control 1996; 41: 1413–1431 15. Narendra KS, Balakrishnan J. A common Lyapunov function for stable LTI systems with com-muting a-matrices. IEEE Trans Autom Control 1994; 39: 2469–2471 16. Peleties P, DeCarlo R. Asymptotic stability of m-switched systems using Lyapunov-like functions. In Proceedings of the 30th American Control Conference 1991; pp. 1679–1684 17. Petersen IR. A stabilization algorithm for a class of uncertain linear systems. Syst Control Lett 1987; 8: 351–357 18. Sun Z. A robust stabilizing law for switching linear systems. Int J Control 2004; 77: 389–398 19. Sun Z, Ge SS, Lee TH. Controllability and reachability criteria for switched linear systems. Automatica 2002; 38: 775–786 20. Utkin VI. Variable structure systems with sliding modes. IEEE Trans Autom Control 1977; 22: 212–222 21. Utkin V, Shi J. Integral sliding mode control in systems operating under uncertainty conditions. In Proceedings of the 35th IEEE Conference on Decision and Control 1996; 4591–4596 22. Xie G, Wang L. Controllability and stabilizability of switched linear-systems. Syst Control Lett 2003; 48: 135–155 23. Yamaguchi T, Numasato H, Hirai H. A modeswitching control for motion control and its application to disk drives: design of optimal mode-switching conditions. IEEE/ASME Trans Mechatronics 1998; 3: 202–209 24. Ye H, Michel AN, Hou L. Stability theory for hybrid dynamical systems. IEEE Trans Autom Control 1998; 43: 461–474 25. Zhai G, Hu B, Yasuda K, Michel AN. Piecewise Lyapunov functions for switched systems with average dwell time. Asian J Control 2000; 2: 192–197 26. Zhao J, Hill DJ. On stability, and L2 -gain and H1

control for switched systems. Automatica 2008; 45: 1220–1232 27. Zhao J, Spong MW. Hybrid control for global stabilization of the cart-pendulum system. Automatica 2001; 37: 1941–1951