Stabilization of planar discrete-time switched systems: Switched Lyapunov functional approach

Nonlinear Analysis: Hybrid Systems 2 (2008) 1062–1068

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Nonlinear Analysis: Hybrid Systems journal homepage: www.elsevier.com/locate/nahs

Stabilization of planar discrete-time switched systems: Switched Lyapunov functional approach Yuan Gong Sun a,b,∗ , Long Wang b a

School of Mathematical Science, Qufu Normal University, Qufu 273165, China

b

Center for Systems and Control, College of Engineering, Peking University, Beijing 100871, China

article

a b s t r a c t

info

Article history: Received 5 August 2006 Accepted 4 September 2008

In this paper, we investigate the problem of stabilization for single-input planar discretetime switched systems by establishing necessary and/or sufficient conditions for the existence of switched quadratic Lyapunov functions of the closed-loop system. The results given in terms of a series of matrix inequalities generalize those results in our recent paper [Y.G. Sun, L. Wang, G. Xie, Necessary and sufficient conditions for stabilization of discretetime planar switched systems, Nonlinear Anal.: Theory and Methods 65 (2006) 1039–1049] and clearly describe the set of switched quadratic Lyapunov functions for the system. © 2008 Published by Elsevier Ltd

Keywords: Discrete-time switched system Stabilization Switched quadratic Lyapunov function Necessary and sufficient condition

1. Introduction By a switched system, we mean a hybrid dynamical system which consists of a family of continuous-time subsystems and a rule that orchestrates the switching between them. While the continuous dynamics of subsystems govern the local behavior of the system, the discrete dynamics of switching mechanisms determine the global performance of the system. In this way, switched systems provide a unifying formulation for conventional dynamic systems, intelligent systems, and complex systems. In the last decade, switched systems have been investigated by many authors (see [2,4,10,17–20,24,25] and the references therein). This problem is not only theoretically interesting but also practically important. Many real-world systems such as chemical processes, transportation systems, computer controlled systems, power systems and communication industries can be modeled as switched systems (see [7,8,16,21–23]). For continuous-time switched linear systems, one way to solve the stabilization problem is to find a common quadratic Lyapunov function (CQLF) for the closed-loop systems. Finding a CQLF for a switched linear system is still an open problem, even though several progresses have been made [1,9,11–15]. Recently, Cheng [3] studied the single-input planar continuoustime switched control system x˙ (t ) = Aσ (x,t ) x(t ) + bσ (x,t ) u(t ),

(1)

where σ (x, t ): R × [0, ∞) → N = {1, 2, . . . , N } is an arbitrary mapping unless elsewhere specified, N ≥ 1 is an integer. Cheng established a necessary and sufficient condition for system (1) to be stabilizable (i.e., there exist a positive-definite matrix P, a set of state feedback controls ui = Ki x such that Ai + bi Ki , i = 1, 2, . . . , N, share a CQLF, xT Px). In this paper, we will consider the following single-input planar discrete-time switched system 2

xk+1 =

N X i=1

∗

σi (k)Ai xk +

N X

σi (k)bi uk ,

i =1

Corresponding author at: School of Mathematical Science, Qufu Normal University, Qufu, 273165, China. E-mail address: [email protected] (Y.G. Sun).

1751-570X/$ – see front matter © 2008 Published by Elsevier Ltd doi:10.1016/j.nahs.2008.09.004

(2)

Y.G. Sun, L. Wang / Nonlinear Analysis: Hybrid Systems 2 (2008) 1062–1068

1063

where σi (k) : {0, 1, 2, . . . , } → {0, 1}, i=1 σi (k) = 1 for all k ∈ {0, 1, 2, . . . , }. The vector σ (k) := [σ1 (k), σ2 (k), . . . , σN (k)] is called a switching signal vector, which specifies which switching model will be activated at the discrete time k. For example, σi (k0 ) = 1, i ∈ N , means that the ith switching model (Ai , bi ) is activated at time k0 . Finding a common quadratic Lyapunov function for the closed-loop system of (2) is somewhat similar to that for the continuous-time case. This issue has been studied in our recent paper [18]. However, for the discrete-time switched system PN T (2), we can introduce the so-called switched quadratic Lyapunov function (SQLF) V (k) = i=1 σi (k)xk Pi xk defined in [5] instead of a CQLF, where Pi > 0 are positive-definite matrices. It is easy to see that the SQLF reduces to the CQLF when all Pi , i ∈ N , are the same. The main contribution of this paper is to establish necessary and/or sufficient conditions for the existence of SQLFs of the closed-loop system of system (2). Such conditions guarantee stabilization of system (2) under an arbitrary switching signal. These conditions given in terms of a series of matrix inequalities are only dependent on the system parameter (Ai , bi ) and generalize the main results in our recent paper [18]. We use standard notations throughout this paper. M T is the transpose of the matrix M. M > 0 (M < 0) means that M is positive definite (negative definite).

PN

2. Main results First, we give two definitions to formulate the problem concerned in this section. Definition 1 ([5]). The following quadratic Lyapunov function V (k, xk ) :=

N X

σi (k)xTk Pi xk

(3)

i=1

is called a SQLF of the discrete-time switched system xk+1 = i=1 σi (k)Ai xk , if there exist Pi > 0, i ∈ N , such that ∆V (k, xk ) = V (xk+1 ) − V (xk ) is negative definite along the solution of the system.

PN

Remark 1. Here, we say V (k, xk ) defined by (3) is negative definite if ∆V (k, 0) = 0, ∀ k ≥ 0, and ∆V (k, xk ) ≤ −γ (kxk k), ∀ k ≥ 0, ∀xk ∈ Rn , where the function γ is continuous, strictly increasing, zero at zero. Based on the analysis in [5], we PN know that the discrete-time switched system xk+1 = i=1 σi (k)Ai xk is asymptotically stable if it shares a SQLF. Thus, in order to study the problem of stabilization of system (2), it is sufficient to study necessary and/or sufficient conditions for the existence of SQLF of the closed-loop system (2). On the other hand, according to the direct computation we also have PN that the function V (k, xk ) defined by (3) is negative definite along the solution of the system xk+1 = i=1 σi (k)Ai xk if and only if

 xTk 

N X

!T σi (k)Ai

i=1

N X

! σi (k + 1)Pi

i =1

N X

! σi (k)Ai −

i =1

N X

 σi (k)Pi  xk < 0.

(4)

i=1

(4) holds for any switching signal vector σ (k) if and only if ATl Pj Al − Pl < 0,

∀ (l, j) ∈ N × N .

Definition 2. System (2) is said to be stabilizable for any switching signal if there exist Pi > 0, a set of state feedback controls u = Ki x, i ∈ N , such that the closed-loop system of (2) has a SQLF defined by (3), or equivalently, there exist Pi > 0, i ∈ N , such that

(Ai + bi Ki )T Pj (Ai + bi Ki ) − Pi < 0,

∀ (i, j) ∈ N × N .

(5)

In what follows, for the sake of convenience we assume N = 2. That is, system (2) has only two subsystems. The general case can be similarly discussed. For N = 2, condition (5) contains four linear matrix inequalities. The following lemma shows that we only need to solve two of them if the following assumption is satisfied: (A1) All the subsystems of (2) are reachable. Lemma 1. Assume that (A1) holds. System (2) is stabilizable for any switching signal vector σ (k) = [σ1 (k), σ2 (k)] satisfying σ (k0 ) = σ (k0 + 1) for some k0 ≥ 0. Proof. According to the definition of the switching signal vector, we know that the first or the second subsystem of (2) is activated at times k0 and k0 + 1. Without loss of generality, we assume the first subsystem (A1 , b1 ) is activated at times k0 and k0 + 1. Since (A1 , b1 ) is reachable, we can choose a feedback matrix K1 ∈ R1×2 such that eigenvalues λi (i = 1, 2) of A1 + b1 K1 satisfy λ1 = λ2 = 0, Let T1 = [Ab1 , b1 ], then we have T1−1 (A1 + b1 K1 )T1 =



0 0



1 . 0

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Y.G. Sun, L. Wang / Nonlinear Analysis: Hybrid Systems 2 (2008) 1062–1068

It yields

(A1 + b1 K1 ) = T1 2





0 0

1 0

0 0



1 −1 T = 0. 0 1

Therefore, xk0 +2 = (A1 + b1 K1 )2 xk0 = 0. This implies that there exist state feedback matrices K1 , K2 ∈ R1×2 such that xk = 0 for k ≥ k0 + 2, i.e., system (2) is stabilizable. The proof of Lemma 1 is complete.  Now, if a switching signal vector σ (k) = [σ1 (k), σ2 (k)] satisfies σ (k0 ) = σ (k0 + 1) for some k0 ≥ 0, we can ignore it. We only need to consider those switching signals satisfying σ (k) 6= σ (k + 1) for k ≥ 0. Thus, for the case when N = 2 and (A1) holds, (5) reduces to

(A1 + b1 K1 )T P2 (A1 + b1 K1 ) − P1 < 0, (A2 + b2 K2 )T P1 (A2 + b2 K2 ) − P2 < 0.



(6)

Lemma 2. Given two positive-definite matrices

 M =

m1 m2

m2 m3



n1 n2

N =

n2 n3





> 0,

> 0.

There exists a feedback matrix K = (k1 , k2 ) such that A˜ T M A˜ − N < 0,

(7)

if and only if det M m3

<

det N n1

,

where A˜

 =

A + bK =

 :=

0

1

α



β

0 a1

1 a2



  +

0 (k1 k2 ) 1

.

Proof. According to a simple computation, we can easily obtain (7) holds if and only if D(α, β) :=

m 3 α 2 − n1 m2 α + m3 αβ − n2

m2 α + m3 αβ − n2 m1 + 2m2 β + m3 β 2 − n3





< 0.

(8)

Now for (8) to hold, it is necessary and sufficient to have m3 α < n1 and det D(α, β) > 0. (Necessity.) Assume that there is a feedback matrix K = (k1 , k2 ) such that (7) holds, then we have m3 α 2 < n1 and 2

det D(α, β) = (m3 α 2 − n1 )(m1 + 2m2 β + m3 β 2 − n3 ) − (m2 α + m3 αβ − n2 )2

= −m3 n1 β 2 + 2(m3 n2 α − m2 n1 )β + (m3 α 2 − n1 )(m1 − n3 ) − (m2 α − n2 )2 > 0. Thus, when

β=

n2 n1

α−

m2 m3

,

we have that det D(α, β) gets its maximum



Dmax = (m3 α 2 − n1 ) m1 −

m22 m3 m22



+

m3 n22 n21 n22

  2 m 3 n2 2 α 2 − n3 − α − n2

= (m3 α 2 − n1 ) m1 − + − n3 m3 n1   det M det N = (m3 α 2 − n1 ) − m3

> 0.

n1

n1



Y.G. Sun, L. Wang / Nonlinear Analysis: Hybrid Systems 2 (2008) 1062–1068

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Since m3 α 2 < n1 , we have that det M m3

<

det N n1

,

which proves the necessity. (Sufficiency.) We can simply choose any α satisfying m3 α 2 < n1 and let

β=

n2 n1

α−

m2

.

m3

Then D(α, β) = (m3 α 2 − n1 ) The sufficiency is proved.



det M m3

−

det N n1



> 0.



Remark 2. According to the proof of Lemma 2, for simplicity we may let α = 0 and β = −m2 /m3 such that (7) holds. Hence, the feedback matrix can be chosen as K = [−a1 , −a2 − m2 /m3 ]. Now, let us consider the single-input planar discrete-time switched system (2). For each switching model we denote the state transformation matrix, which converts it to the canonical controllable form, by Ti for i = 1, 2. Based on Lemma 2 we have the following necessary and sufficient condition for stabilization of system (2). Theorem 1. Assume that (A1) holds. The single-input planar discrete-time system (2) is stabilizable if and only if there exist P1 > 0 and P2 > 0 such that

 det P2 det P1    e T T P T eT < e T T P T eT , 2 1 2 1 2 1 1 1 1 1 det P2  det P1  < ,  T T T T e2 T2 P1 T2 e2

(9)

e1 T2 P2 T2 e1

where e1 = (1, 0) and e2 = (0, 1). Proof. The single-input planar discrete-time system (2) is stabilizable, if and only if there exist Pi > 0, a set of state feedback controls u = Ki x, i = 1, 2 such that



(A1 + b1 K1 )T P2 (A1 + b1 K1 ) − P1 < 0, (A2 + b2 K2 )T P1 (A2 + b2 K2 ) − P2 < 0,

which is equivalent to



A˜ T1 T1T P2 T1 A˜ 1 − T1T P1 T1 < 0, A˜ T2 T2T P1 T2 A˜ 2 − T2T P2 T2 < 0,

(10)

where A˜ i = Ti−1 (Ai + bi Ki )Ti has the form A˜ i =



0

αi

1



βi

,

i = 1, 2.

It means that there exist T1T P2 T1 > 0, T1T P1 T1 > 0, T2T P1 T2 > 0, T2T P2 T2 > 0, and state feedback K˜ i = Ki Ti such that (10) holds. By Lemma 2, we have that this is equivalent to

 det(T1T P2 T1 ) det(T1T P1 T1 )     e T T P T eT < e T T P T eT , 2 1 2 1 2 1 1 1 1 1  det(T2T P1 T2 ) det(T2T P2 T2 )   < ,  T T T T e2 T2 P1 T2 e2

e1 T2 P2 T2 e1

which is equivalent to (9). This completes the proof of Theorem 1.



By Remark 2, we can easily construct a stabilizing control for system (2). We state it formally as the following step-by-step algorithm.

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Y.G. Sun, L. Wang / Nonlinear Analysis: Hybrid Systems 2 (2008) 1062–1068

Algorithm 2. The following step-by-step algorithm presents a stabilizing control for system (2): Step 1. Determine the state transformation matrices Ti for i = 1, 2 such that in the coordinate frame x = Ti y the ith switching model is in the Brunovsky canonical form. That is,



0 ai1

Ti−1 Ai Ti = Ti−1 bi =

1 ai2



,

 

0 . 1

The parameters ai1 and ai2 are



ai1 ai2



= (bi , Ai bi )−1 A2i bi ,

and the unique state transformation matrix Ti = (Ai bi − ai2 bi , bi ). Step 2. Use (9) to find positive-definite matrices P1 and P2 . Step 3. Calculate M1 = T1T P2 T1 and M2 = T2T P1 T2 . Then the feedback controls K˜ i for the Brunovsky canonical switching system can be constructed as



K˜ i = −ai1 , −ai2 −

eT1 Mi e2



eT2 Mi e2

,

i = 1, 2,

where e1 and e2 are the same as in Theorem 1. Step 4. Back to the original coordinate frame x, the controls should be Ki = K˜ i Ti−1 , i = 1, 2. When det P1 = det P2 , we have the following corollary: Corollary 3. Assume that (A1) holds. The single-input planar discrete-time system (2) is stabilizable if there exist P1 > 0 and P2 > 0 such that det P1 = det P2 , and

(

e2 T1T P2 T1 eT2 > e1 T1T P1 T1 eT1 ,

(11)

e2 T2T P1 T2 eT2 > e1 T2T P2 T2 eT1 , where e1 and e2 are the same as in Theorem 1. When P1 = P2 , we have the following simple corollary which is the same as Theorem 1 in [18]:

Corollary 4. Assume that (A1) holds. The single-input planar discrete-time system (2) is stabilizable if there exists P > 0 such that e2 TiT PTi eT2 > e1 TiT PTi eT1 ,

i = 1, 2,

(12)

where e1 and e2 are the same as in Theorem 1. 3. A numerical example In this section, we will work out a numerical example to illustrate our main results. Consider the following discrete-time switched system 2 X

xk+1 =

σi (k)Ai xk +

i=1

2 X

σi (k)bi uk ,

i =1

where

 A1 =

−4 9

 A2 =

−2 2

 −2



 −1

, b1 = ; 2    −4.5 2 , b2 = . 3 −1 5

According to a simple computation, we can get

 T1 =

 T2 =

1 −1

 −1 2

−1.5

2

2

−1

, 

(13)

Y.G. Sun, L. Wang / Nonlinear Analysis: Hybrid Systems 2 (2008) 1062–1068

1067

such that A¯ 1 = T1−1 A1 T1 = A¯ 2 = T2−1 A2 T2 =



0 2



0 −3



1 , 1

T1−1 b1 =



1 , 1

 

T2−1 b2 =

0 , 1

 

0 . 1

Step 1 has been done. Now, go to Step 2. By using MatLab’s LMI Control Toolbox (c.f. [6]) to solve (12), we have

 P =

98.5108 14.9204

14.9204 . 41.1660



(14)

Step 3 yields M1 =

T1T PT1



109.8360 −136.0816



296.7909 −295.8022

=

M2 = T2T PT2 =

 −136.0816 , 203.4932  −295.8022 . 375.5276

Thus, K˜ 1 = [−2, − 0.3312],

K˜ 2 = [3, − 0.2123].

(15)

By Step 4, we obtain the feedback law as u = Ki x, i = 1, 2, where K1 = [−4.3312, − 2.3312],

K2 = [1.0302, 2.2726].

Now let us check whether P in (14) is indeed a common quadratic Lyapunov function of the closed-loop system of (13) under the feedback control law (15). Since

−79.6764 (A1 + b1 K1 ) P (A1 + b1 K1 ) − P = 3.9140  −57.6863 (A2 + b2 K2 )T P (A2 + b2 K2 ) − P = 15.6979 T



3.9140 −22.3316



< 0,

15.6979 < 0. −18.2022



The verification is completed. 4. Conclusion In this paper, we establish necessary and/or sufficient conditions for the existence of switched quadratic Lyapunov functions, which guarantee stabilization of the planar discrete-time switched system under an arbitrary switching signal. By solving a set of matrix inequalities, we get the designed switched Lyapunov function and stabilizing controls. The general problem of stabilization for a discrete-time switched system of dimension greater than two is left for future study. Acknowledgements This work was supported by NSFC (60704039, 60674050 and 60528007), National 973 Program (2002CB312200), National 863 Program (2006AA04Z258) and 11-5 project (A2120061303). References [1] S. Boyd, L.E. Ghaoui, E. Feron, Linear matrix inequalities in system and control theory, in: SIAM Studies in Application Mathematics, SIAM, Philadelphia, 1994. [2] M.S. Branicky, Multiple Lyapunov functions and other analysis tools for switched hybrid systems, IEEE Trans. Automat. Control 43 (1998) 475–482. [3] D. Cheng, Stabilization of planar switched systems, Syst. Control Lett. 51 (2004) 79–88. [4] W.P. Dayawansa, C.F. Martin, A converse Lyapunov theorem for a class of dynamical systems which undergo switching, IEEE Trans. Automat. Control 44 (1999) 751–760. [5] J. Daafouz, P. Riedinger, C. Iung, Stability analysis and control synthesis for switched systems: A switched Lyapunov function approach, IEEE Trans. Atuomat. Control 47 (2002) 1883–1887. [6] P. Gahinet, A. Nemirovski, A. Laub, M. Chilali, LMI Control Toolbox User’s Guide, The Math Works, Natick, MA, 1995. [7] D. Jeon, M. Tomizuka, Learning hybrid force and position control of robot manipulators, IEEE Trans. Robot. Autom. 9 (1996) 423–431. [8] Y.D. Ji, H.J. Chizeck, Jump linear quadratic Gaussian control: steady-state solution and testable conditions, Control Theory Adv. Technol. 6 (1990) 289–319. [9] D. Liberzon, J.P. Hespanha, A.S. Morse, Stability of switched systems: a Lie-algebraic condition, Syst. Control Lett. 37 (1999) 117–122. [10] D. Liberzon, A.S. Morse, Basic problems on stability and design of switched systems, IEEE Control Syst. Mag. 34 (1999) 1914–1946. [11] T. Ooba, Y. Funahashi, Two conditions concerning common QLFs for linear systems, IEEE Trans. Automa. Control 42 (1997) 719–721. [12] T. Ooba, Y. Funahashi, Stability robustness for linear state space models–a Lyapunov mapping approach, Syst. Control Lett. 29 (1997) 191–196.

1068

Y.G. Sun, L. Wang / Nonlinear Analysis: Hybrid Systems 2 (2008) 1062–1068

[13] R.N. Shorten, K.S. Narendra, A sufficient condition for the existence of a common Lyapunov function for two second-order linear systems, in: Proceedings of the 36th Conference on Decision and Control, San Diego, CA, 1997, pp. 3521–3522. [14] R.N. Shorten, K.S. Narendra, Necessary and sufficient condition for the existence of a common quadratic Lyapunov function for M-stable linear secondorder systems, in: Proceedings of 2000 ACC, Chicago, IL, 2000, pp. 359–363. [15] R.N. Shorten, K.S. Narendra, On the existence of a common Lyapunov function for linear stable switched systems, in; Proceedings of the 10th Yale Workshop on Adaptive and Learning Systems, Tampa, FL, 1998, pp. 130–140. [16] H. Sira-Ranirez, Nonlinear P-I controller design for switch mode DC-to-DC power converts, IEEE Trans. Circuits Systems 38 (1991) 410–417. [17] Z. Sun, D. Zheng, On reachability and stabilization of switched linear systems, IEEE Trans. Automat. Control 46 (2001) 291–295. [18] Y.G. Sun, L. Wang, G. Xie, Necessary and sufficient conditions for stabilization of discrete-time planar switched systems, Nonlinear Anal.: Theory and Methods 65 (2006) 1039–1049. [19] Y.G. Sun, L. Wang, G. Xie, Delay-dependent robust stability and stabilization for discrete-time switched systems with mode-dependent time-varying delays, Appl. Math. Comput. 180 (2006) 428–435. [20] Y.G. Sun, L. Wang, G. Xie, Delay-dependent robust stability and H∞ control for uncertain discrete-time switched systems with mode-dependent time delays, Appl. Math. Comput. 187 (2007) 1228–1237. [21] P.P. Varaiya, Smart car on small roads: problems of control, IEEE Trans. Automat. Control 38 (1993) 195–207. [22] S.M. Williams, R.G. Hoft, Adaptive frequency domain control of PM switched power line conditioner, IEEE Trans. Power Electron. 6 (1991) 665–670. [23] F. Xue, L. Guo, Necessary and sufficient conditions for adaptive stabilization of jump linear systems, Comm. Inform. Syst. 1 (2001) 205–224. [24] G. Xie, L. Wang, Controllability and stabilizability of switched linear systems, Syst. Control Lett. 48 (2003) 135–155. [25] G. Xie, D. Zheng, L. Wang, Controllability of switched linear systems, IEEE Trans. Automat. Control 47 (2002) 1401–1405.