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Robust H∞ stabilization for uncertain switched impulsive control systems with state delay: An LMI approach
Nonlinear Analysis: Hybrid Systems 2 (2008) 1287–1300
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Robust H∞ stabilization for uncertain switched impulsive control systems with state delay: An LMI approach Guangdeng Zong a,b,∗ , Shengyuan Xu b , Yuqiang Wu a a
Research Institute of Automation, Qufu Normal University, Qufu, Shandong 273165, People’s Republic of China
b
School of Automation, Nanjing University of Science & Technology, Nanjing 210094, People’s Republic of China
article
info
Article history: Received 3 February 2007 Accepted 19 September 2008 Keywords: Switched linear systems Impulse effects Robust asymptotic stability H∞ stabilization Parameter uncertainties
a b s t r a c t This paper deals with the problem of robust H∞ state feedback stabilization for uncertain switched linear systems with state delay. The system under consideration involves time delay in the state, parameter uncertainties and nonlinear uncertainties. The parameter uncertainties are norm-bounded time-varying uncertainties which enter all the state matrices. The nonlinear uncertainties meet with the linear growth condition. In addition, the impulsive behavior is introduced into the given switched system, which results a novel class of hybrid and switched systems called switched impulsive control systems. Using the switched Lyapunov function approach, some sufficient conditions are developed to ensure the globally robust asymptotic stability and robust H∞ disturbance attenuation performance in terms of certain linear matrix inequalities (LMIs). Not only the robustly stabilizing state feedback H∞ controller and impulsive controller, but also the stabilizing switching law can be constructed by using the corresponding feasible solution to the LMIs. Finally, the effectiveness of the algorithms is illustrated with an example. Crown Copyright © 2008 Published by Elsevier Ltd. All rights reserved.
1. Introduction In recent years, there has been growing interest in stability analysis and controller design for hybrid and switched systems, see e.g., [1–4] for some review. By a switched system, we mean a hybrid dynamical system consisting of a family of continuous-time or discrete-time subsystems and a rule that orchestrates the switching among them. The motivation for studying hybrid and switched systems mainly comes from the fact that they provide a natural and convenient unified framework for mathematical modeling of many physical phenomena and practical applications. Typical examples include autonomous transmission systems, computer disc driver, room temperature control, power electronics, chaos generators, to name a few. Hybrid control has been proved to achieve better performance in many aspects, for instance, on achieving stability and improving transient response. Many researchers have devoted their study to hybrid systems and/or hybrid control. Lots of valuable results in the stability analysis and stabilization for linear or nonlinear hybrid and switched systems were established, see [1–11,17–20] and the references cited therein. As is well known, many practical systems in physics, biology, engineering, and information science exhibit impulsive dynamical behaviors due to abrupt changes at certain instants during the dynamical process [12–14]. Although hybrid and switched system is an important model for dealing with many complex physical processes, it does not cover above dynamical process when the impulse effects appear at the switching points because state jump usually occurs under such circumstances. Here, we just call these hybrid and switched control systems with impulse effects as switched impulsive
∗ Corresponding author at: Research Institute of Automation, Qufu Normal University, Qufu, Shandong 273165, People’s Republic of China. Tel.: +86 537 4455717; fax: +86 537 4455717. E-mail addresses: [email protected] (G. Zong), [email protected] (S. Xu), [email protected] (Y. Wu). 1751-570X/$ – see front matter Crown Copyright © 2008 Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.nahs.2008.09.018
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control systems. Due to the existence of the states jump, these new class of hybrid systems will not be well described by using the traditional pure continuous or pure discrete modes [13]. Therefore, it is theoretically important and necessary to study switched impulsive control systems. To the best of our knowledge, there are very few results [11–14] on such hybrid dynamical systems and the corresponding control problems reported. In this note, we consider the problem of the robust H∞ state feedback stabilization for a class of uncertain switched impulsive control systems with state delay. The system under consideration involves time delay in the state, parameter uncertainties and nonlinear uncertainties. The parameter uncertainties are norm-bounded time-varying uncertainties which enter all the state matrices. The nonlinear uncertainties meet with the linear growth condition. The purpose of this paper is to find some criteria for robust asymptotic stability and robust H∞ disturbance attenuation performance as well as design robustly state feedback H∞ controller, impulsive controller and stabilizing switching law for the above systems. Using the switched Lyapunov function approach [17], some sufficient conditions are provided. All the results are given in terms of LMI form, which are easy to testify by utilizing Matlab tool-box [15]. A numerical example is presented to demonstrate the effectiveness of the algorithm. This note is organized as follows. Section 2 formulates the problem and states some preliminary results. Sections 3 and 4 presents some criteria for the robust asymptotic stability and H∞ disturbance attenuation performance, and designs the H∞ state feedback controller and impulsive controller. A numerical example illustrating our design procedures and the effectiveness is provided in Section 5. Finally, in Section 6, concluding remarks end the paper. Notations: We use standard notations throughout this paper. Rn denotes the n-dimensional Euclidean space, Rm×n is the set of all real m × n matrices. Given any two symmetric real matrices A and B, A > B refers to the fact that A − B is positive definite. AT is the transpose of matrix A. The identity matrix of order m is denoted as Im (or simply I if no confusion arises). L2 [0, ∞) denotes the space of square integrable functions on [0, ∞) and k · k2 stands for the usual L2 [0, ∞)-norm. The symbol ∗ will be used in some matrix expressions to induce a symmetric structure. 2. Problem formulation Consider the class of uncertain switched impulsive control systems with state delay described by
x˙ (t ) = Aσ (t ) + 1Aσ (t ) (t ) x(t ) + Adσ (t ) + 1Adσ (t ) (t ) x(t − d) + B1σ (t ) + 1B1σ (t ) (t ) u(t ) + fσ (t ) (t , x) + B2σ (t ) ω(t ), t ∈ (tk , tk+1 ] 1x = Eσ (tk ) x(tk ) + u(tk ), t = tk z (t ) = Cσ (t ) x(t ) + Dσ (t ) u(t ) x(t ) = 0, t ∈ [−d, t0 ], t0 = 0
(1)
where x(t ) ∈ Rn is the state, u(t ) ∈ Rm is the control input, u(tk ) ∈ Rn is the impulsive control at tk , ω(t ) ∈ Rq is the external 1
disturbance which belongs to L2 [0, ∞), z (t ) ∈ Rp is the controlled output. σ (t ) : [0, ∞) → {1, 2, . . . , N } = N is the switching signal which is a piecewise constant function, σ (t ) = i implies that the i-th subsystem or mode is activated. d > 0 is the time-delay which is a positive constant number. 1x(t ) = x(t + )−x(t − ), limh→0+ x(t −h) = x(t − ), limh→0+ x(t +h) = x(t + ), tk (k = 1, 2, . . .) are the impulsive jumping points or switching points and t0 < t1 < · · · < tk < · · · , limk→∞ t (k) = ∞. Without loss of generality, it is assumed that x(tk ) = x(tk− ) = limh→0+ x(tk − h), i.e. the solution x(t ) to system (1) is left continuous at tk . Ai , Adi , B1i , B2i , Ei , Ci , and Di are known real constant matrices with appropriate dimensions. 1Ai (t ), 1Adi (t ) and 1B1i (t ) are time-varying parameter uncertainties assumed to be of the form
1Ai (t )
1Adi (t )
1B1i (t ) = Mi Γi (t ) NAi
Ndi
NBi
(2)
where Mi , NAi , Ndi and NBi are constant matrices, and Γi (t ) is the uncertain matrix satisfying
ΓiT (t )Γi (t ) ≤ I
(3)
which reflects the uncertainties of the system matrices. 1Ai (t ), 1Adi (t ) and 1B1i (t ) are said to be admissible if both of (2) and (3) hold. There are several reasons for assuming that the system uncertainties have the structure given in (2) and (3), see Reference [16] for some details. fi (t , x) : Rn → Rn is a nonlinear vector valued function, which denotes the nonlinear uncertainty and satisfying
kfi (t , x)k ≤ kFi xk. Assumption 1. DTi [Ci , Di ] = [0, I ], k = 1, 2, . . . . Remark 1. Assumption 1 is commonly used in robust H∞ control which does not lose any generality.
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Remark 2. Introduce the following switched system
x˙ (t ) = Aσ (t ) + 1Aσ (t ) (t ) x(t ) + Adσ (t ) + 1Adσ (t ) (t ) x(t − d) + v(x, t ) + fσ (t ) (t , x) + B2σ (t ) ω(t ) z (t ) = C x(t ) + D ω(t ) x(t ) = 0,σ (t ) t ∈ [−d,σt(t )], t = 0. 0 0
(5)
And construct the hybrid controller v(x, t ) = u1 + u2 for system (5) as follows u1 =
∞ X
B1σ (t ) + 1B1σ (t ) (t ) u(t )lk (t ),
u2 =
k=1
∞ X (Eσ (t ) x(t ) + u(t ))δ(t − tk )
(6)
k=1
where δ(·) is the Dirac impulse, lk (t ) = 1 as tk < t ≤ tk+1 , otherwise, lk (t ) = 0 with discontinuity points 0 = t0 < t1 < t2 < · · · < tk < · · · , limk→∞ tk = ∞. Then system (5) under the hybrid controller (6) results in switched impulsive control system (1). For more details, please refer to Reference [13]. From this point, we can see that switched impulsive control system (1) is a more general class of hybrid and switched system model. Throughout this paper, we shall use the following definition. Definition 1. Switched state feedback H∞ stabilization problem: given any disturbance attenuation level γ > 0 if there exist switched state feedback H∞ controller u(t ) = Kσ (t ) x(t ), t ∈ [tk−1 , tk ) and impulsive controller u(tk ) = K¯ σ (tk ) x(tk ), t = tk such that the closed-loop switched system satisfies (i) with ω(t ) ≡ 0, system (1) is globally asymptotically stable for all admissible uncertainties; (ii) with zero-initial condition x(0) = 0, kz (t )k2 ≤ γ kω(t )k2 for all admissible uncertainties and all nonzero ω(t ) ∈ L2 [0, ∞). In this paper, we shall develop some stabilizable sufficient conditions and design stabilizing switched state feedback controller and impulsive controller for the uncertain switched impulsive control systems (1)–(3). Different from the usual switched control systems, in order to stabilize the uncertain switched impulsive control system (1), we not only need to design the switched state feedback controller u(t ) = Ki x(t ), t ∈ [tk−1 , tk ), but also need to design the impulsive controller u(tk ) = K¯i x(tk ), t = tk . Therefore, the stabilization problem of system (1) is more complex. In order to study the above robust H∞ stabilization problem, we introduce the following lemma. Lemma 1. For any matrix P , D ∈ Rn×n and n-dimensional vectors x, ω, the following inequality holds for any positive constant ε>0 xT P T Dω + ωT DT Px ≤ εωT ω +
1
ε
xT P T DDT Px.
(7)
Proof. The above inequality easily follows by the fact that
1 T √ 2
0 ≤ √ D Px − εω
ε 2 T √ √ 1 T 1 = √ D Px − εω √ DT Px − εω ε ε =
1
ε
xT P T DDT Px + εωT ω − xT P T Dω − ωT DT Px.
(8)
3. Analysis of autonomous switched impulsive systems In this section, hinging on the switched Lyapunov function approach, we firstly study the problem of the asymptotic stability and H∞ performance for the following autonomous switched impulsive system (setting u(t ) = 0, u(tk ) = 0 for system (1)).
˙ x(t ) = Aσ (t ) + 1Aσ (t ) (t ) x(t ) + Adσ (t ) + 1Adσ (t ) (t ) x(t − d) + fσ (t ) (t , x) + B2σ (t ) ω(t ), 1x = Eσ (tk ) x(tk ), t = tk z (t ) = Cσ (t ) x(t ) x(t ) = 0, t ∈ [−d, t0 ], t0 = 0.
t ∈ (tk , tk+1 ] (9)
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Theorem 1. With ω(t ) ≡ 0, system (9) is globally asymptotically stable if there exit symmetric positive definite matrices Xi , Ti and positive numbers λi , εi (i ∈ N ) such that the following LMIs hold
αi ∗ ∗ ∗ ∗
Adi Tj −Tj
Xi FiT 0 0 −λi I
Xi 0 −Ti
∗ ∗ ∗
∗ ∗
T Xi NAi T Tj Ndi 0 < 0, 0 −εi I
∗
(∀ i, j ∈ N )
(10)
(∀ i, l ∈ N , i 6= l).
(11)
4
αi = Xi ATi + Ai Xi + λi I + εi Mi MiT and
−X l ∗ ∗
Xl (I + El )T −Xi
∗
Xl EiT 0 < 0, −Ti
In this case, a stabilizing switching law is defined as
σ (t ) = i,
t > tk σ (tk ) = j, if Sj − (I + Ej )T Si (I + Ej ) < 0.
(j 6= i, i, j ∈ N¯ )
(12)
4 4 Proof. Denote Pi = Xi−1 , Si = Ti−1 . Choose the candidate Lyapunov energy function as
V (t ) = x (t )Pσ (t ) x(t ) + T
t
Z
xT (τ )Sσ (τ ) x(τ )dτ ,
σ (t ) ∈ N .
(13)
t −d
Assume that system (9) is in the ith mode for t ∈ (tk , tk+1 ], k = 0, 1, 2, 3, . . .. Moreover, suppose that the system switched to ith mode from the lth mode at t = tk and that (t − d) ∈ (tk−1 , tk ]. Then for t ∈ (tk , tk + d] V (t ) = xT (t )Pi x(t ) +
Z
tk
xT (τ )Sl x(τ )dτ +
t −d
Z
t
xT (τ )Si x(τ )dτ
(14)
tk
while for t ∈ (tk + d, tk+1 ] V (t ) = xT (t )Pi x(t ) +
Z
t
xT (τ )Si x(τ )dτ .
(15)
t −d
Therefore, due to the time delay, for the first part of the time interval when the system is in the ith mode, the Lyapunov energy function V (t ) depends on both the ith and lth subsystems. Note that if tk + d > tk+1 , then the case in (14) will cover the entire interval (tk , tk+1 ], i.e. (14) will apply for t ∈ (tk , tk+1 ]. More generally, if (t − d) ∈ (tk−m , tk−m+1 ] and mode θj−1 is active on (tk−j , tk−j+1 ], j = 1, 2, . . . , m, then for t ∈ (tk , tk + d] (or t ∈ (tk , tk+1 ], if tk + d > tk+1 ) V (t ) = xT (t )Pi x(t ) +
Z
tk−m+1 t −d
m−1
xT (τ )Sθm−1 x(τ )dτ +
XZ j =1
tk−j+1
xT (τ )Sθj−1 x(τ )dτ +
tk−j
Z
t
xT (τ )Si x(τ )dτ
(16)
tk
while for t ∈ (tk + d, tk+1 ], V (t ) is given by (15). The reason that we discuss the Lyapunov energy function as above is that the state x(t ) jumps at the switching points tk due to the impulsive effects, which makes the integrals become hard to tackle. Now we are in a position to compute the derivative of V (t ) along the trajectory of system (9). Since the expression for V (t ) depends on whether d is greater than or less than tk+1 − tk . Henceforth, we divide the rest of the proof into two cases. Case 1: tk+1 − tk ≤ d. In this case, V (t ) can be expressed by (16). Thus, for t ∈ (tk , tk+1 ], we have dV (t ) dt
= 2xT (t )Pi x˙ (t ) + xT (tk−m+1 )Sθm−1 x(tk−m+1 ) − xT (t − d)Sθm−1 x(t − d) m−1
+
X
xT (tk−j+1 )Sθj−1 x(tk−j+1 ) − xT (tk+−j )Sθj−1 x(tk+−j ) + xT (t )Si x(t ) − xT (tk+ )Si x(tk+ ).
j =1
(17)
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Since 1x(tq ) = x(tq+ ) − x(tq ) = Eσ (tq ) x(tq ) at the switching instant tq , we have x(tq+ ) = (I + Eσ (tq ) )x(tq )
(18)
for any nonnegative integers q = 0, 1, 2, . . . . Combing (18) and (17) gives dV (t ) dt
m−1
= 2xT (t )Pi x˙ (t ) + xT (tk−m+1 )Sθm−1 x(tk−m+1 ) − xT (t − d)Sθm−1 x(t − d) +
X
xT (tk−j+1 )Sθj−1 x(tk−j+1 )
j =1 m−1
−
X
xT (tk−j )(I + Eθj )T Sθj−1 (I + Eθj )x(tk−j ) + xT (t )Si x(t ) − xT (tk )(I + Eθ0 )T Si (I + Eθ0 )x(tk ).
(19)
j=1
After some manipulations, (19) can be written as dV (t ) dt
= 2xT (t )Pi x˙ (t ) − xT (t − d)Sθm−1 x(t − d) + xT (tk−m+1 )Sθm−1 x(tk−m+1 ) − xT (tk−m+1 )(I + Eθm−1 )T Sθm−2 (I + Eθm−1 )x(tk−m+1 ) + · · · − xT (tk−1 )(I + Eθ1 )T Sθ0 (I + Eθ1 )x(tk−1 ) + xT (tk )Sθ0 x(tk ) − xT (tk )(I + Eθ0 )T Si (I + Eθ0 )x(tk ) + xT Si x(t ). (20)
According to the definition of switching law (12), we obtain xT (tk−m+1 )[Sθm−1 − (I + Eθm−1 )T Sθm−2 (I + Eθm−1 )]x(tk−m+1 ) ≤ 0
.. . xT (tk )[Sθ0 − (I + Eθ0 )T Si (I + Eθ0 )]x(tk ) ≤ 0
.
(21)
Substituting (21) into (20) yields dV (t ) dt
≤ 2xT (t )Pi x˙ (t ) − xT (t − d)Sθm−1 x(t − d) + xT Si x(t ).
For the sake of convenience, let j = θm−1 . Then we have dV (t ) dt
≤ 2xT (t )Pi x˙ (t ) − xT (t − d)Sj x(t − d) + xT Si x(t ).
(22)
Furthermore, when ω(t ) ≡ 0, we get dV (t ) dt
= 2xT (t )Pi x˙ (t ) + xT (t )Si x(t ) − xT (t − d)Sj x(t − d) = xT (t ) A¯ Ti Pi + Pi A¯ i + Si x(t ) + 2xT (t − d)A¯ Tdi Pi x(t ) + 2fiT (t , x)Pi x(t ) − xT (t − d)Sj x(t − d)
(23)
with A¯ i = Ai + 1Ai , A¯ di = Adi + 1Adi . By Lemma 1, we have 2fiT (t , x)Pi x(t ) ≤
1
λi
xT (t )fiT (t , x)fi (t , x) + λi xT (t )Pi Pi x(t )
≤ xT (t )
1
FiT Fi + λi Pi Pi x(t )
λi
(24)
where λi > 0 is an arbitrary constant number. Combining (24) and (23) leads to dV (t ) dt
1 T T ¯ ¯ ≤ x (t ) Ai Pi + Pi Ai + Si + Fi Fi + λi Pi Pi x(t ) + 2xT (t − d)A¯ Tdi Pi x(t ) − xT (t − d)Sj x(t − d) λi T x( t ) x(t ) = Σij x(t − d) x(t − d) T
(25)
where 1
" 4
Σij =
A¯ Ti Pi + Pi A¯ i + Si +
λi
A¯ Tdi Pi
FiT Fi + λi Pi Pi
Pi A¯ di
−Sj
# .
(26)
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Denote 1
" 4
Yij =
ATi Pi + Pi Ai + Si +
λi
FiT Fi + λi Pi Pi
Pi Adi
ATdi Pi
# .
(27)
−Sj
Then from Lemma 1, we get
Σij = Yij +
≤ Yi + εi
Pi Mi Γi (t ) NAi 0
Pi Mi 0
Pi M i 0
ATdi Pi
+
1
εi
T +
1
εi
Pi Adi +
y1i
=
NdiT NAi
Ndi + NAi
−S j +
NAi
1
εi
1
εi
Ndi
T NAi Ndi
NdiT Ndi
T
Ndi
ΓiT (t )
T
NAi
Ndi
Pi Mi 0
T
.
(28)
with y1i = + Pi Ai + Si + + λi Pi Pi + εi Pi Mi MiT Pi + ε1i NAiT NAi . Multiplying both sides of (28) by diag(Xi , Ti ) gives 1
ATi Pi
Tj ATdi
F TF λi i i
Adi Tj +
y2i
+
1
εi
Tj NdiT NAi Xi
1
εi
1
−T j +
εi
T Xi NAi Ndi Tj
Tj NdiT Ndi Tj
<0
(29)
where y2i = Xi ATi + Ai Xi + Xi Si Xi +
1
λi
Xi FiT Fi Xi + λi I + εi Mi MiT +
1
εi
T Xi NAi NAi Xi .
It is apparent that (29) is equivalent to inequality (10) by Schur complement. Case 2: tk+1 − tk > d In this case, we need to consider two subintervals: (tk , tk + d] and (tk + d, tk+1 ]. For t ∈ (tk , tk + d], V (t ) is given by (16) and the analysis is same as for Case 1. For t ∈ (tk + d, tk+1 ], V (t ) is given by (15). Then, under the assumption that ω(t ) ≡ 0, we get dV (t ) dt
= 2xT (t )Pi x˙ (t ) + xT (t )Si x(t ) − xT (t − d)Si x(t − d) = xT (t ) A¯ Ti Pi + Pi A¯ i + Si x(t ) + 2xT (t − d)A¯ Tdi Pi x(t ) + 2fiT (t , x)Pi x(t ) − xT (t − d)Si x(t − d)
where A¯ i = Ai + 1Ai , A¯ di = Adi + 1Adi . Following the proof of Case 1, we conclude that
Ti ATdi
Adi Ti +
y2i
+
1
εi
Ti NdiT NAi Xi
−Ti +
1
εi
1
εi
T Xi NAi Ndi Ti
Ti NdiT Ndi Ti
dV (t ) dt
(30)
< 0 if
< 0.
(31)
Obviously, by Schur complement, inequality (31) is equivalent to LMI (10) in the case of j = i. Combining Case 1 and Case 2, we draw the conclusion that when ω(t ) = 0, the time derivative of V (t )(t ∈ (tk , tk+1 ]) is negative along the solution of (9) if LMI (10) holds. According to the multiple Lyapunov function theory [7], V (t ) should decrease along the switching instants to ensure the asymptotical stability. In what follows, we will check this property. Just as stated previously, it is assumed that system (9) switched to ith mode from the lth mode at t = tk . That is to say, σ (t ) = l, t ∈ (tk−1 , tk ] and σ (t ) = i, t ∈ (tk , tk+1 ]. From (9) and (13), we derive
(tk+ ) = x(tk ) + 1x(tk ) =Z [I + El ]x(tk )
V (tk ) = xT (tk )Pl x(tk ) +
tk
tk − d
(32)
xT (τ )Sσ (τ ) x(τ )dτ
V (tk+ ) = xT (tk+ )Pσ (t + ) x(tk+ ) + k
Z
+ tk
+ tk − d
xT (τ )Sσ (τ ) x(τ )dτ
= xT (tk )(I + El )T × Pi (I + El )x(tk ) +
Z
tk tk − d
xT (τ )Sσ (τ ) x(τ )dτ + xT (tk )EiT Si Ei x(tk ).
(33)
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Therefore, V (tk+ ) − V (tk− ) = V (tk+ ) − V (tk ) = xT (tk )Θli x(tk )
(34)
where Θli = (I + El ) Pi (I + El ) − Pl + It shows by Schur complement that Θli < 0 is equivalent to EiT Si Ei .
T
(I + El )T −Pi−1 ∗
−P l ∗ ∗
EiT 0 < 0. −Si−1
(35)
Multiplying both sides of (35) by diag(Pl−1 , I , I ) and paying attention to that Xl = Pl−1 , Ti = Si−1 , l ∈ N, we arrive at the inequality (11). This shows that V (tk+ ) − V (tk− ) = V (tk+ ) − V (tk ) < 0
(36)
if the LMI (11) holds, i.e. V (t ) decreases along the switching instants. Therefore, by the multiple Lyapunov functions theory [7], we conclude that the trivial solution of the time-delay switched impulsive system (9) is globally asymptotically stable. The proof of Theorem 1 is complete. Remark 3. The switched Lyapunov function (13) is a particular quadratic Lyapunov function making use of the switching nature of system (9), specifically, it has the same switching signal as switched system (9). Therefore, the Lyapunov function (12) is actually a switching signal dependent Lyapunov function. Compared with the common Lyapunov function approach (i.e. Pi = P > 0, Si = S > 0 is common for all the subsystems), the switched Lyapunov function approach is less conservation, see reference [17] for some details. Now let us study the disturbance attenuation property of switched system (9). Theorem 2. For system (9), design the switching law as (12), then with zero-initial condition x(0) = 0, kz (t )k2 ≤ γ kω(t )k2 for all nonzero ω(t ) ∈ L2 [0, ∞) and admissible uncertainties, if there exit symmetric positive definite matrices Xi , Ti and positive numbers λi , εi (i ∈ N ) solving the following LMIs
αi + γ −2 B2i BT2i ∗ ∗ ∗ ∗ ∗
Adi Tj −Tj
Xi 0 −Ti
∗ ∗ ∗ ∗
∗ ∗ ∗
Xi FiT 0 0 −λi I
∗ ∗
T Xi NAi T Tj Ndi 0 0 −εi I
∗
Xi CiT 0 0 < 0, 0 0 −I
(∀i, j ∈ N )
(37)
4
αi = Xi ATi + Ai Xi + λi I + εi Mi MiT and
−X l ∗ ∗
Xl (I + El )T −Xi
∗
Xl EiT 0 < 0, −Ti
(∀ i, l ∈ N , i 6= l).
(38)
Moreover, switched impulsive system (9) is globally asymptotically stable when ω(t ) ≡ 0. Proof. It easily follows that inequality (37) holds implies inequality (10) holds. Hence, by Theorem 1, system (9) is globally asymptotically stable when ω(t ) ≡ 0, and the switched Lyapunov function V (t ) is given as (13). Secondly, suppose that x(0) = 0 and introduce the performance index T
Z
(z T (t )z (t ) − γ 2 ωT (t )ω(t ))dt .
J =
(39)
0
For any given T > 0, suppose that T ∈ (tk , tk+1 ], then we have T
Z
z T (t )z (t ) − γ 2 ωT (t )ω(t ) + V˙ (t ) dt −
J = 0
T
V˙ (x(t ))dt = 0
V˙ (t )dt . 0
Bearing V (0) = 0 in mind, we obtain from (36) that
Z
T
Z
k−1 Z X m=0
tm+1 tm
V˙ (t )dt +
Z
T
V˙ (t )dt tk
(40)
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= V (t1 ) − V (0) + V (t2 ) − V (t1+ ) + · · · + V (T ) − V (tk+ ) = V (T ) +
k X
+ V (tm ) − V (tm ) ≥ V (T ) (≥ 0).
(41)
m=1
From (37) and Schur complement, we have
π i ∗
Adi Tj +
−Tj +
1
T Xi NAi Ndi Tj
εi
1
εi
Tj NdiT Ndi Tj
<0
(42)
T NAi Xi + Xi CiT Ci Xi + γ −2 B2i BT2i . where πi = Xi ATi + Ai Xi + λi I + εi Mi MiT + Xi Si Xi + λ1 Xi FiT Fi Xi + ε1 Xi NAi i i Multiplying both sides of (42) by diag(Pi , Sj ) yields
π0 i ∗ with πi = 0
Pi Adi +
−Sj + ATi Pi
1
εi
1
εi
T NAi Ndi
NdiT Ndi
<0
(43)
+ Pi Ai + λi Pi Pi + εi Pi Mi MiT Pi + Si +
1
T Fi + ε1 NAi NAi + CiT Ci + γ −2 Pi B2i BT2i Pi . i in two different cases due to the existence of time delay d.
T
F λi i
dV (t ) dt
By the proof of Theorem 1, we should compute Case 1: tk+1 − tk ≤ d. In this case, V (t ) is given by (16). Under the switching law (12), and following the deducing process (17)–(22), we obtain dV (t ) dt
≤ 2xT (t )Pi x˙ (t ) − xT (t − d)Sj x(t − d) + xT Si x(t ).
(44)
Furthermore, by the use of (43) z T (t )z (t ) − γ 2 ωT (t )ω(t ) +
dV (t )
= z T (t )z (t ) − γ 2 ωT (t )ω(t ) + 2xT (t )Pi x˙ (t ) + xT Si x(t ) − xT (t − d)Sj x(t − d) 1 ≤ xT (t ) A¯ Ti Pi + Pi A¯ i + Si + λi Pi Pi + FiT Fi + γ −2 Pi B2i BT2i Pi + CiT Ci x(t ) λi T + 2xT (t − d)A¯ Tdi Pi x(t ) − xT (t − d)Sj x(t − d) − γ −1 BT2i Pi x(t ) − γ ω(t ) γ −1 BT2i Pi x(t ) − γ ω(t ) 1 ≤ xT (t ) A¯ Ti Pi + Pi A¯ i + Si + λi Pi Pi + FiT Fi + γ −2 Pi B2i BT2i Pi + CiT Ci x(t ) λi + 2xT (t − d)A¯ di Pi x(t ) − xT (t − d)Sj x(t − d) T π 0 Pi Adi + 1 N T Ndi Ai x(t ) x(t ) i ε i ≤ (45) x(t − d) < 0. 1 x(t − d) ∗ −Sj + NdiT Ndi εi Case 2: tk+1 − tk > d. As declared in Case 2 in Theorem 1, we need to consider two subintervals: (tk , tk + d] and (tk + d, tk+1 ]. For t ∈ (tk , tk + d], V (t ) is given by (16) and we can get the same analysis as (45). For t ∈ (tk + d, tk+1 ], V (t ) is given by (15). We can similarly dt
obtain z T (t )z (t ) − γ 2 ωT (t )ω(t ) +
dV (t ) dt
= z T (t )z (t ) − γ 2 ωT (t )ω(t ) + 2xT (t )Pi x˙ (t ) + xT Si x(t ) − xT (t − d)Si x(t − d) T π 0 x(t ) i ≤ x(t − d) ∗
Thus z (t )z (t ) − γ ω (t )ω(t ) + T
π0 i ∗
2
Pi Adi +
−Si +
T
1
εi
1
εi
T NAi Ndi
NdiT Ndi
dV (t ) dt
Pi Adi +
−Si +
1
εi
1
εi
T NAi Ndi
NdiT Ndi
x( t ) x(t − d) .
(46)
< 0 is equivalent to
<0
which is exactly the case of j = i in (43).
(47)
G. Zong et al. / Nonlinear Analysis: Hybrid Systems 2 (2008) 1287–1300
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Therefore, both Case 1 and Case 2 indicate that z T (t )z (t ) − γ 2 ωT (t )ω(t ) +
dV (t ) dt
<0
(48)
regardless of d. Substituting (41) and (48) into (40) gives rise to T
Z
z T (t )z (t ) − γ 2 ωT (t )ω(t ) dt < 0
J = 0
and ∞
Z
z T (t )z (t ) − γ 2 ωT (t )ω(t ) dt = lim
T →∞
0
T
Z
z T (t )z (t ) − γ 2 ωT (t )ω(t ) dt < 0
(49)
0
for any nonzero disturbance ω(t ) ∈ L2 [0, ∞) and all admissible uncertainties. Therefore, we have kz (t )k2 ≤ γ kω(t )k2 . The proof of Theorem 2 is complete. 4. Controller design In this section, we shall study the robust H∞ stabilization problem proposed in Definition 1 and design the corresponding state feedback stabilizer u(t ) = Kσ (t ) x(t ), t ∈ (tk−1 , tk ] and impulsive controller u(tk ) = K¯ σ (tk ) x(tk ), t = tk . The closed-loop switched impulsive system is given as below
x˙ (t ) = A˜ σ (t ) + 1A˜ σ (t ) (t ) x(t ) + A¯ di x(t − d) + fσ (t ) (t , x) + B2σ (t ) ω(t ), 1x = E˜ σ (tk ) x(tk ), t = tk z (t ) = C˜ σ (t ) x(t ) x(t ) = 0, t = t0 = 0
t ∈ (tk−1 , tk ] (50)
where A˜ i = Ai + B1i Ki , 1A˜ i (t ) = Mi Γi (t )(NAi + NBi Ki ), A¯ di = Adi + 1Adi (t ), E˜ i = Ei + K¯ i , C˜ i = Ci + Di Ki . Theorem 3. Given any constant γ > 0, the switched state feedback H∞ stabilization problem is feasible if there exist symmetric positive definite matrices Pi , Si and positive numbers λi , εi (i ∈ N ) solving the following LMIs
− Si + Pl < 0,
(i, l ∈ N )
(51)
and
ϑ i ∗ ∗ ∗ ∗ ∗
ηi −Sj ∗
Pi 0 1
−
λi
∗
∗
∗ ∗
∗ ∗
I
Pi Mi 0
T NAi NdiT
0
0
1
− I εi ∗ ∗
Pi B2i 0
0
< 0, 0
0
−εi I ∗
( i, j ∈ N )
(52)
0
−γ 2 I
where 1
4
4
1
ϑi = ATi Pi + Pi Ai + Si + CiT Ci + FiT Fi , ψi2 = I + NBiT NBi λi εi 1 1 T 4 T ηi = Pi Adi − Pi B1i + NAi NBi ψi−2 NBi Ndi . εi εi Moreover, a switched state feedback H∞ stabilizing controller and an impulsive controller are respectively given by u(t ) = Ki x(t ), u(tk ) = K¯ i x(tk ),
Ki = −ψi−2
BT1i Pi +
1
εi
T NBi NAi
, t ∈ (tk−1 , tk ]
K¯ i = −(I + Ei ), t = tk .
(53) (54)
And the stabilizing switching law is constructed as
σ (t ) = i,
t > tk σ (tk ) = j, if Sj − (I + Ej )T Si (I + Ej ) < 0.
(j 6= i, i, j ∈ N¯ )
(55)
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Proof. From (52) and Schur complement, for any [x(t ), x(t − d)]T 6= 0, we have
T ϑ1i x(t ) x( t − d ) ∗
1
ηi +
εi
T NAi Ndi
1
−Sj +
εi
NdiT Ndi
x( t ) x(t − d) < 0
(56)
T NAi + γ −2 Pi B2i BT2i Pi . where ϑ1i = ϑi + λi Pi Pi + εi Pi Mi MiT Pi + ε1 NAi i For any given switching signal σ (t ), Choose the candidate switched Lyapunov function as (13). According to Definition 1, we shall firstly study the stability of the closed-loop system (50), and then verify its H∞ disturbance attenuation performance.
A: Stability analysis and controller design By Theorem 1, we ought to calculate the derivative of V (t ) under two different cases: tk+1 − tk ≤ d and tk+1 − tk > d. Here, in order to avoid repeating some similar manipulations, we merely go on with the former case. The analysis of the latter case is in a similar way. When tk+1 − tk ≤ d, the Lyapunov energy function is displayed as (16) and its time derivative satisfies (22) under the switching law (55). Through some manipulations, with ω(t ) = 0, we have dV (t )
= x˙ T (t )Pi x(t ) + xT (t )Pi x˙ (t ) + xT Si x(t ) − xT (t − d)Sj x(t − d) 1 ≤ xT (t ) Aˆ Ti Pi + Pi Aˆ i + Si + FiT Fi + λi Pi Pi x(t ) + 2xT (t − d)A¯ Tdi Pi x(t ) − xT (t − d)Sj x(t − d) λi T x(t ) x(t ) ˆi Σ = x( t − d ) x(t − d)
dt
(57)
where Aˆ i = A˜ i + 1A˜ i (t ), and 1
"
Aˆ Ti Pi + Pi Aˆ i + Si +
ˆi = Σ
λi
FiT Fi + λi Pi Pi
A¯ Tdi Pi
Pi A¯ di
# .
(58)
−Sj
Denote 1
" 4
Yˆi =
A˜ Ti Pi + Pi A˜ i + Si +
λi
FiT Fi + λi Pi Pi
Pi Adi
ATdi Pi
# .
(59)
−S j
Then, following the proof of (20), we get
ˆ i = Yˆi + Pi Mi Γi (t ) (NAi + NBi Ki ) Σ 0
yˆ 1i
≤
∗
Pi Adi +
1
εi
(NAi + NBi Ki )T Ndi
−Sj +
1
εi
NdiT Ndi
Ndi + (NAi + NBi Ki )
Ndi
T
ΓiT (t )
Pi Mi 0
T
(60)
where εi > 0 is a positive constant number, and yˆ 1i = ATi Pi + Pi Ai + Si +
1
λi
FiT Fi + λi Pi Pi + εi Pi Mi MiT Pi +
1
εi
T NAi NAi
T 1 1 − KiT Ki − BT1i Pi + NBiT NAi ψi−2 BT1i Pi + NBiT NAi εi εi T 1 T 1 T −2 T −2 T 2 + Ki + ψi (B1i Pi + NBi NAi ) ψi Ki + ψi (B1i Pi + NBi NAi ) . εi εi
(61)
T Define the control gain Ki as Ki = −ψi−2 (BT1i Pi + ε1 NBi NAi ), then we have i
yˆ 1i ≤ ATi Pi + Pi Ai + Si +
1
λi
FiT Fi + λi Pi Pi + εi Pi Mi MiT Pi +
1
εi
T NAi NAi .
(62)
G. Zong et al. / Nonlinear Analysis: Hybrid Systems 2 (2008) 1287–1300
1297
Combining (56), (60) and (62) gives dV (t )
<0
dt
(63)
for any [x(t ), x(t − d)] 6= 0. In order to testify the robust asymptotic stability, the following condition has to be satisfied by Theorem 1 V (tk+ ) − V (tk− ) = V (tk+ ) − V (tk ) < 0.
(64)
From (35) and (36), (64) holds if
(I + E˜ l )T −Pi−1 ∗
−P l ∗ ∗
E˜ iT 0 < 0. −Si−1
(65)
Take K¯ l = −(I + El ), then (I + E˜ l ) = 0 and the above inequality (64) holds naturally by (51). Therefore the closed-loop switched impulsive system (50) is globally asymptotically stable. B: H∞ performance analysis Define T
Z
(z T (t )z (t ) − γ 2 ωT (t )ω(t ))dt .
J = 0
Since x(0) = 0, from (41) and (64), we directly get T
Z
J ≤
z (t )z (t ) − γ ω (t )ω(t ) + T
2
T
0
dV (t ) dt
dt
(66)
for any given 0 < T ∈ (tk , tk+1 ]. dV (t ) Since dt has different forms for different time delay d, we ought to compute z T (t )z (t ) − γ 2 ωT (t )ω(t ) + cases. Case 1: tk+1 − tk ≤ d. In this case, V (t ) is given by (16). Under the switching law (55), it follows that dV (t )
≤ 2xT (t )Pi x˙ (t ) − xT (t − d)Sj x(t − d) + xT Si x(t ).
dt
dV (t ) dt
in all the
(67)
Furthermore, z T (t )z (t ) − γ 2 ωT (t )ω(t ) +
dV (t ) dt
1 ≤ xT (t ) Aˆ Ti Pi + Pi Aˆ i + Si + λi Pi Pi + FiT Fi + CiT Ci + KiT Ki + γ −2 Pi B2i BT2i Pi x(t ) λi
+ 2x (t − d)A¯ Tdi Pi x(t ) − xT (t − d)Sj x(t − d) − (γ −1 BT2i Pi x(t ) − γ ω(t ))T (γ −1 BT2i Pi x(t ) − γ ω(t )) 1 T T T T −2 T ˆ ˆ ≤ x (t ) Ai Pi + Pi Ai + Si + λi Pi Pi + Fi Fi + Ci Ci + γ Pi B2i B2i Pi x(t ) λi T T T + 2x (t − d)A¯ di Pi x(t ) − x (t − d)Sj x(t − d) T x(t ) x(t ) ≤ Ωij x( t − d ) x(t − d) T
(68)
where
µ i Ωij = ∗
Pi Adi +
1
εi
(NAi + NBi Ki )T Ndi
−Sj +
1
εi
NdiT Ndi
(69)
and
µi = ATi Pi + Pi Ai + Si +
1
F T Fi + λi Pi Pi + CiT Ci + γ −2 Pi B2i BT2i Pi + εi Pi Mi MiT Pi +
λi i T
1 ψi−2 BT1i Pi + NBiT NAi εi εi T 1 1 + Ki + ψi−2 BT1i Pi + NBiT NAi ψi2 Ki + ψi−2 BT1i Pi + NBiT NAi . εi εi −
BT1i Pi +
1
1
εi
T NAi NAi
T NBi NAi
(70)
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T Substituting Ki = −ψi−2 (BT1i Pi + ε1 NBi NAi ) from (53) into (69) and (70) gives rise to i
µi ≤ ATi Pi + Pi Ai + Si + Pi Adi +
1
εi
1
λi
FiT Fi + λi Pi Pi + CiT Ci + γ −2 Pi B2i BT2i Pi + εi Pi Mi MiT Pi +
(NAi + NBi Ki )T Ndi = ηi +
1
εi
1
εi
T NAi NAi = ϑ1i
T NAi Ndi .
(71) (72)
Furthermore,
Ωij < 0. That is to say z T (t )z (t ) − γ 2 ωT (t )ω(t ) +
dV (t ) dt
< 0.
(73)
Case 2: tk+1 − tk > d. In this case, we need to consider two subintervals: (tk , tk + d] and (tk + d, tk+1 ]. For t ∈ (tk , tk + d], V (t ) is given by (16) and we can get the same result as (73) by repeating the process from (67)–(72). For t ∈ (tk + d, tk+1 ], V (t ) is given by (15), and we have z T (t )z (t ) − γ 2 ωT (t )ω(t ) +
dV (t ) dt
= z T (t )z (t ) − γ 2 ωT (t )ω(t ) + 2xT (t )Pi x˙ (t ) + xT Si x(t ) − xT (t − d)Si x(t − d) T
x(t ) x(t − d)
≤
Ωii
x(t ) <0 x( t − d )
(74)
where Ωii is equal to Ωij in the case of j = i in (69), which is negative by the above deduction. Combining (73) with (74) gives z T (t )z (t ) − γ 2 ωT (t )ω(t ) +
dV (t ) dt
<0
(75)
irrespective of the size of time delay d. Substituting (75) into (66) yields ∞
Z
(z T (t )z (t ) − γ 2 ωT (t )ω(t ))dt = lim 0
T →∞
T
Z
(z T (t )z (t ) − γ 2 ωT (t )ω(t ))dt < 0
(76)
0
for any nonzero disturbance ω(t ) ∈ L2 [0, ∞) and all admissible uncertainties, which implies that kz (t )k2 ≤ γ kω(t )k2 . The proof of Theorem 3 is complete. 5. An illustrative example In this section, we give an illustrative example to show the effectiveness of the proposed algorithms. Consider the following control systems with N = 2 and the parameters as follows
" # " # −12.8 0 1.8 −4.4 0 8.01 1.2 −0.9 0.6 , 0 A1 = A2 = 2.63 −5.5 ; 1.2 −0.56 −2.5 0 0.2 −7.02 " # " # 0.02 0.86 0.8 −4.4 0 8.01 0 −0.1 0.51 , 0 Ad1 = Ad2 = 2.63 −5.5 ; 0 0.12 −0.22 0 0.2 −7.02 " # " # " # 0.02 0.02 0.10 0.02 1.00 0.04 0 0.10 , 0 0.10 , 0 B11 = B12 = M1 = 0.01 ; 0.63 0 0.08 0 0.01 0 " # " # 2.00 0.40 0.1 0.04 0.1 0 0 ; M2 = 0.10 , NA1 = 0.03 0.01 0.20 0 0.5 0 0 " # " # 0.02 0.04 0.10 0.09 0.01 0.01 0 0 0 0 NA2 = 0.03 0.10 , Nd1 = ; 0.03 0 0 0.02 0 0.04
G. Zong et al. / Nonlinear Analysis: Hybrid Systems 2 (2008) 1287–1300
" Nd2 =
0.05 0 0
0 0 0.02
0.01 0 , 0.01
#
0.4100 0 , −1.2100
" B21 =
0.04 C2 = 0
#
0.01 0.05
" NB1 = 0.01 0 0.04
" E1 =
0.1 0.2 0
"
0 , 0.01
B22 =
0.01 0.06 0.01
0 0 , 0.02
0 0 .3 , 0.05
0.01 0.02 0.10
0 0 ; 0.10
0.02 0
0.01 0.03
0 ; 0.01
"
#
0.05 0 , 0.08
C1 =
#
0.43 0 0.40
" E2 =
1299
NB2 =
#
0.10 0.15 0
#
0 0.30 . 0.15
#
Choose γ = 0.2, ε1 = 0.5, ε2 = 0.4, λ1 = 0.4, λ2 = 0.2. Solving the LMIs (38) and (39) in Pi , Si , we obtain the following feasible solution 0.1114 0.0062 0.0113
" P1 =
0.2055 −0.0143 0.0217
" S1 =
0.0062 0.0595 0.0017
0.0113 0.0017 , 0.0750
−0.0143 0.0751 −0.0143
#
0.0497 −0.0056 −0.0056
" P2 =
0.0217 −0.0143 , 0.1003
#
" S2 =
−0.0056 0.0397 0.0397
0.2055 −0.0143 0.0217
0.0413 −0.0036 −0.0036
−0.0143 0.0751 −0.0143
#
0.0217 −0.0143 . 0.1003
#
After some manipulations, we can see that P1 , P2 and S1 , S2 are symmetric positive definite matrices. Then by Theorem 3, we know the given uncertain switched impulsive control system is stabilizable with H∞ disturbance attenuation level γ = 0.2. According to (40) and (41), the robust switched state feedback H∞ controller and impulsive controller gain can be given as follows
−0.0355 −0.0020 −0.6308 , −0.0400 −0.1000 0.0001 " # −1.0100 −0.1000 0 ¯K1 = 0 −1.2000 −0.3000 , −0.0400 0 −1.0500
K1 =
−0.2084 −0.0052 −0.1618 −0.0465 −0.1992 0.0006 " # −1.4300 −0.1000 0 ¯K2 = 0 −1.1500 −0.3000 . −0.4000 0 −1.1500
K2 =
6. Conclusions In this paper, we have investigated the problem of robust H∞ stabilization for a class of uncertain switched impulsive control systems with state delay. By the use of the switched Lyapunov function approach, some sufficient conditions have been developed to ensure the robust globally asymptotic stability and H∞ disturbance attenuation performance. These conditions are presented in terms of LMI form, which are easy to testify and compute utilizing Matlab tool-box. A suitable robust H∞ stabilizing state feedback controller, an impulsive controller and switching law have also been given. When compared with the common Lyapunov function based methods, the conservatism of the proposed approach is highly reduced. The main contribute of this paper lies in providing a practically useful and easily executable investigation algorithm for uncertain switched impulsive control systems. Acknowledgments This work was supported by the National Natural Science Foundation of China (No. 60674027), Specialized Research Fund for the Doctoral Program of Higher Education (No. 20050446001), China Postdoctoral Science Foundation(No. 20070410336), Postdoctoral Foundation of Jiangsu Province (No. 0602042B) and Scientific Research Foundation of Qufu Normal University. References [1] D. Liberzon, A.S. Morse, Basic problems in stability and design of switched systems, IEEE Control Systems Magazine 19 (1999) 59–70. [2] R. Decarlo, M.S. Branicky, B. Lennartson, Perspective and results on the stability and stabilizability of hybrid systems, Proceedings of the IEEE 88 (2000) 1069–1082. [3] D. Liberzon, Switching in Systems and Control, Birkhauser, Boston, MA, 2003. [4] Z.D. Sun, S.S. Ge, Switched Linear Systems-control and Design, Springer Verlag, New York, 2004. [5] Z.G. Li, Y.C. Soh, C.Y. Wen, Robust stability of quasiperiodic hybrid dynamic uncertain systems, IEEE Transactions on Automatic Control 46 (2001) 107–111. [6] Z.G. Li, Y.C. Soh, C.Y. Wen, Robust stability for a class of hybrid nonlinear systems, IEEE Transactions on Automatic Control 47 (2002) 897–903. [7] M.S. Branicky, Multiple Lyapunov functions and other analysis tools for switched and hybrid systems, IEEE Transactions on Automatic Control 43 (1998) 60–66. [8] D.Z. Cheng, G. Lei, Y.D. Lin, Y. Wang, Stabilization of switched systems, IEEE Transactions on Automatic Control 50 (2005) 661–666. [9] G.M. Xie, L. Wang, Necessary and sufficient conditions for controllability and observability of switched impulsive control systems, IEEE Transactions on Automatic Control 49 (2004) 960–966.
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Nonlinear Analysis: Hybrid Systems journal homepage: www.elsevier.com/locate/nahs
Robust H∞ stabilization for uncertain switched impulsive control systems with state delay: An LMI approach Guangdeng Zong a,b,∗ , Shengyuan Xu b , Yuqiang Wu a a
Research Institute of Automation, Qufu Normal University, Qufu, Shandong 273165, People’s Republic of China
b
School of Automation, Nanjing University of Science & Technology, Nanjing 210094, People’s Republic of China
article
info
Article history: Received 3 February 2007 Accepted 19 September 2008 Keywords: Switched linear systems Impulse effects Robust asymptotic stability H∞ stabilization Parameter uncertainties
a b s t r a c t This paper deals with the problem of robust H∞ state feedback stabilization for uncertain switched linear systems with state delay. The system under consideration involves time delay in the state, parameter uncertainties and nonlinear uncertainties. The parameter uncertainties are norm-bounded time-varying uncertainties which enter all the state matrices. The nonlinear uncertainties meet with the linear growth condition. In addition, the impulsive behavior is introduced into the given switched system, which results a novel class of hybrid and switched systems called switched impulsive control systems. Using the switched Lyapunov function approach, some sufficient conditions are developed to ensure the globally robust asymptotic stability and robust H∞ disturbance attenuation performance in terms of certain linear matrix inequalities (LMIs). Not only the robustly stabilizing state feedback H∞ controller and impulsive controller, but also the stabilizing switching law can be constructed by using the corresponding feasible solution to the LMIs. Finally, the effectiveness of the algorithms is illustrated with an example. Crown Copyright © 2008 Published by Elsevier Ltd. All rights reserved.
1. Introduction In recent years, there has been growing interest in stability analysis and controller design for hybrid and switched systems, see e.g., [1–4] for some review. By a switched system, we mean a hybrid dynamical system consisting of a family of continuous-time or discrete-time subsystems and a rule that orchestrates the switching among them. The motivation for studying hybrid and switched systems mainly comes from the fact that they provide a natural and convenient unified framework for mathematical modeling of many physical phenomena and practical applications. Typical examples include autonomous transmission systems, computer disc driver, room temperature control, power electronics, chaos generators, to name a few. Hybrid control has been proved to achieve better performance in many aspects, for instance, on achieving stability and improving transient response. Many researchers have devoted their study to hybrid systems and/or hybrid control. Lots of valuable results in the stability analysis and stabilization for linear or nonlinear hybrid and switched systems were established, see [1–11,17–20] and the references cited therein. As is well known, many practical systems in physics, biology, engineering, and information science exhibit impulsive dynamical behaviors due to abrupt changes at certain instants during the dynamical process [12–14]. Although hybrid and switched system is an important model for dealing with many complex physical processes, it does not cover above dynamical process when the impulse effects appear at the switching points because state jump usually occurs under such circumstances. Here, we just call these hybrid and switched control systems with impulse effects as switched impulsive
∗ Corresponding author at: Research Institute of Automation, Qufu Normal University, Qufu, Shandong 273165, People’s Republic of China. Tel.: +86 537 4455717; fax: +86 537 4455717. E-mail addresses: [email protected] (G. Zong), [email protected] (S. Xu), [email protected] (Y. Wu). 1751-570X/$ – see front matter Crown Copyright © 2008 Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.nahs.2008.09.018
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control systems. Due to the existence of the states jump, these new class of hybrid systems will not be well described by using the traditional pure continuous or pure discrete modes [13]. Therefore, it is theoretically important and necessary to study switched impulsive control systems. To the best of our knowledge, there are very few results [11–14] on such hybrid dynamical systems and the corresponding control problems reported. In this note, we consider the problem of the robust H∞ state feedback stabilization for a class of uncertain switched impulsive control systems with state delay. The system under consideration involves time delay in the state, parameter uncertainties and nonlinear uncertainties. The parameter uncertainties are norm-bounded time-varying uncertainties which enter all the state matrices. The nonlinear uncertainties meet with the linear growth condition. The purpose of this paper is to find some criteria for robust asymptotic stability and robust H∞ disturbance attenuation performance as well as design robustly state feedback H∞ controller, impulsive controller and stabilizing switching law for the above systems. Using the switched Lyapunov function approach [17], some sufficient conditions are provided. All the results are given in terms of LMI form, which are easy to testify by utilizing Matlab tool-box [15]. A numerical example is presented to demonstrate the effectiveness of the algorithm. This note is organized as follows. Section 2 formulates the problem and states some preliminary results. Sections 3 and 4 presents some criteria for the robust asymptotic stability and H∞ disturbance attenuation performance, and designs the H∞ state feedback controller and impulsive controller. A numerical example illustrating our design procedures and the effectiveness is provided in Section 5. Finally, in Section 6, concluding remarks end the paper. Notations: We use standard notations throughout this paper. Rn denotes the n-dimensional Euclidean space, Rm×n is the set of all real m × n matrices. Given any two symmetric real matrices A and B, A > B refers to the fact that A − B is positive definite. AT is the transpose of matrix A. The identity matrix of order m is denoted as Im (or simply I if no confusion arises). L2 [0, ∞) denotes the space of square integrable functions on [0, ∞) and k · k2 stands for the usual L2 [0, ∞)-norm. The symbol ∗ will be used in some matrix expressions to induce a symmetric structure. 2. Problem formulation Consider the class of uncertain switched impulsive control systems with state delay described by
x˙ (t ) = Aσ (t ) + 1Aσ (t ) (t ) x(t ) + Adσ (t ) + 1Adσ (t ) (t ) x(t − d) + B1σ (t ) + 1B1σ (t ) (t ) u(t ) + fσ (t ) (t , x) + B2σ (t ) ω(t ), t ∈ (tk , tk+1 ] 1x = Eσ (tk ) x(tk ) + u(tk ), t = tk z (t ) = Cσ (t ) x(t ) + Dσ (t ) u(t ) x(t ) = 0, t ∈ [−d, t0 ], t0 = 0
(1)
where x(t ) ∈ Rn is the state, u(t ) ∈ Rm is the control input, u(tk ) ∈ Rn is the impulsive control at tk , ω(t ) ∈ Rq is the external 1
disturbance which belongs to L2 [0, ∞), z (t ) ∈ Rp is the controlled output. σ (t ) : [0, ∞) → {1, 2, . . . , N } = N is the switching signal which is a piecewise constant function, σ (t ) = i implies that the i-th subsystem or mode is activated. d > 0 is the time-delay which is a positive constant number. 1x(t ) = x(t + )−x(t − ), limh→0+ x(t −h) = x(t − ), limh→0+ x(t +h) = x(t + ), tk (k = 1, 2, . . .) are the impulsive jumping points or switching points and t0 < t1 < · · · < tk < · · · , limk→∞ t (k) = ∞. Without loss of generality, it is assumed that x(tk ) = x(tk− ) = limh→0+ x(tk − h), i.e. the solution x(t ) to system (1) is left continuous at tk . Ai , Adi , B1i , B2i , Ei , Ci , and Di are known real constant matrices with appropriate dimensions. 1Ai (t ), 1Adi (t ) and 1B1i (t ) are time-varying parameter uncertainties assumed to be of the form
1Ai (t )
1Adi (t )
1B1i (t ) = Mi Γi (t ) NAi
Ndi
NBi
(2)
where Mi , NAi , Ndi and NBi are constant matrices, and Γi (t ) is the uncertain matrix satisfying
ΓiT (t )Γi (t ) ≤ I
(3)
which reflects the uncertainties of the system matrices. 1Ai (t ), 1Adi (t ) and 1B1i (t ) are said to be admissible if both of (2) and (3) hold. There are several reasons for assuming that the system uncertainties have the structure given in (2) and (3), see Reference [16] for some details. fi (t , x) : Rn → Rn is a nonlinear vector valued function, which denotes the nonlinear uncertainty and satisfying
kfi (t , x)k ≤ kFi xk. Assumption 1. DTi [Ci , Di ] = [0, I ], k = 1, 2, . . . . Remark 1. Assumption 1 is commonly used in robust H∞ control which does not lose any generality.
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Remark 2. Introduce the following switched system
x˙ (t ) = Aσ (t ) + 1Aσ (t ) (t ) x(t ) + Adσ (t ) + 1Adσ (t ) (t ) x(t − d) + v(x, t ) + fσ (t ) (t , x) + B2σ (t ) ω(t ) z (t ) = C x(t ) + D ω(t ) x(t ) = 0,σ (t ) t ∈ [−d,σt(t )], t = 0. 0 0
(5)
And construct the hybrid controller v(x, t ) = u1 + u2 for system (5) as follows u1 =
∞ X
B1σ (t ) + 1B1σ (t ) (t ) u(t )lk (t ),
u2 =
k=1
∞ X (Eσ (t ) x(t ) + u(t ))δ(t − tk )
(6)
k=1
where δ(·) is the Dirac impulse, lk (t ) = 1 as tk < t ≤ tk+1 , otherwise, lk (t ) = 0 with discontinuity points 0 = t0 < t1 < t2 < · · · < tk < · · · , limk→∞ tk = ∞. Then system (5) under the hybrid controller (6) results in switched impulsive control system (1). For more details, please refer to Reference [13]. From this point, we can see that switched impulsive control system (1) is a more general class of hybrid and switched system model. Throughout this paper, we shall use the following definition. Definition 1. Switched state feedback H∞ stabilization problem: given any disturbance attenuation level γ > 0 if there exist switched state feedback H∞ controller u(t ) = Kσ (t ) x(t ), t ∈ [tk−1 , tk ) and impulsive controller u(tk ) = K¯ σ (tk ) x(tk ), t = tk such that the closed-loop switched system satisfies (i) with ω(t ) ≡ 0, system (1) is globally asymptotically stable for all admissible uncertainties; (ii) with zero-initial condition x(0) = 0, kz (t )k2 ≤ γ kω(t )k2 for all admissible uncertainties and all nonzero ω(t ) ∈ L2 [0, ∞). In this paper, we shall develop some stabilizable sufficient conditions and design stabilizing switched state feedback controller and impulsive controller for the uncertain switched impulsive control systems (1)–(3). Different from the usual switched control systems, in order to stabilize the uncertain switched impulsive control system (1), we not only need to design the switched state feedback controller u(t ) = Ki x(t ), t ∈ [tk−1 , tk ), but also need to design the impulsive controller u(tk ) = K¯i x(tk ), t = tk . Therefore, the stabilization problem of system (1) is more complex. In order to study the above robust H∞ stabilization problem, we introduce the following lemma. Lemma 1. For any matrix P , D ∈ Rn×n and n-dimensional vectors x, ω, the following inequality holds for any positive constant ε>0 xT P T Dω + ωT DT Px ≤ εωT ω +
1
ε
xT P T DDT Px.
(7)
Proof. The above inequality easily follows by the fact that
1 T √ 2
0 ≤ √ D Px − εω
ε 2 T √ √ 1 T 1 = √ D Px − εω √ DT Px − εω ε ε =
1
ε
xT P T DDT Px + εωT ω − xT P T Dω − ωT DT Px.
(8)
3. Analysis of autonomous switched impulsive systems In this section, hinging on the switched Lyapunov function approach, we firstly study the problem of the asymptotic stability and H∞ performance for the following autonomous switched impulsive system (setting u(t ) = 0, u(tk ) = 0 for system (1)).
˙ x(t ) = Aσ (t ) + 1Aσ (t ) (t ) x(t ) + Adσ (t ) + 1Adσ (t ) (t ) x(t − d) + fσ (t ) (t , x) + B2σ (t ) ω(t ), 1x = Eσ (tk ) x(tk ), t = tk z (t ) = Cσ (t ) x(t ) x(t ) = 0, t ∈ [−d, t0 ], t0 = 0.
t ∈ (tk , tk+1 ] (9)
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Theorem 1. With ω(t ) ≡ 0, system (9) is globally asymptotically stable if there exit symmetric positive definite matrices Xi , Ti and positive numbers λi , εi (i ∈ N ) such that the following LMIs hold
αi ∗ ∗ ∗ ∗
Adi Tj −Tj
Xi FiT 0 0 −λi I
Xi 0 −Ti
∗ ∗ ∗
∗ ∗
T Xi NAi T Tj Ndi 0 < 0, 0 −εi I
∗
(∀ i, j ∈ N )
(10)
(∀ i, l ∈ N , i 6= l).
(11)
4
αi = Xi ATi + Ai Xi + λi I + εi Mi MiT and
−X l ∗ ∗
Xl (I + El )T −Xi
∗
Xl EiT 0 < 0, −Ti
In this case, a stabilizing switching law is defined as
σ (t ) = i,
t > tk σ (tk ) = j, if Sj − (I + Ej )T Si (I + Ej ) < 0.
(j 6= i, i, j ∈ N¯ )
(12)
4 4 Proof. Denote Pi = Xi−1 , Si = Ti−1 . Choose the candidate Lyapunov energy function as
V (t ) = x (t )Pσ (t ) x(t ) + T
t
Z
xT (τ )Sσ (τ ) x(τ )dτ ,
σ (t ) ∈ N .
(13)
t −d
Assume that system (9) is in the ith mode for t ∈ (tk , tk+1 ], k = 0, 1, 2, 3, . . .. Moreover, suppose that the system switched to ith mode from the lth mode at t = tk and that (t − d) ∈ (tk−1 , tk ]. Then for t ∈ (tk , tk + d] V (t ) = xT (t )Pi x(t ) +
Z
tk
xT (τ )Sl x(τ )dτ +
t −d
Z
t
xT (τ )Si x(τ )dτ
(14)
tk
while for t ∈ (tk + d, tk+1 ] V (t ) = xT (t )Pi x(t ) +
Z
t
xT (τ )Si x(τ )dτ .
(15)
t −d
Therefore, due to the time delay, for the first part of the time interval when the system is in the ith mode, the Lyapunov energy function V (t ) depends on both the ith and lth subsystems. Note that if tk + d > tk+1 , then the case in (14) will cover the entire interval (tk , tk+1 ], i.e. (14) will apply for t ∈ (tk , tk+1 ]. More generally, if (t − d) ∈ (tk−m , tk−m+1 ] and mode θj−1 is active on (tk−j , tk−j+1 ], j = 1, 2, . . . , m, then for t ∈ (tk , tk + d] (or t ∈ (tk , tk+1 ], if tk + d > tk+1 ) V (t ) = xT (t )Pi x(t ) +
Z
tk−m+1 t −d
m−1
xT (τ )Sθm−1 x(τ )dτ +
XZ j =1
tk−j+1
xT (τ )Sθj−1 x(τ )dτ +
tk−j
Z
t
xT (τ )Si x(τ )dτ
(16)
tk
while for t ∈ (tk + d, tk+1 ], V (t ) is given by (15). The reason that we discuss the Lyapunov energy function as above is that the state x(t ) jumps at the switching points tk due to the impulsive effects, which makes the integrals become hard to tackle. Now we are in a position to compute the derivative of V (t ) along the trajectory of system (9). Since the expression for V (t ) depends on whether d is greater than or less than tk+1 − tk . Henceforth, we divide the rest of the proof into two cases. Case 1: tk+1 − tk ≤ d. In this case, V (t ) can be expressed by (16). Thus, for t ∈ (tk , tk+1 ], we have dV (t ) dt
= 2xT (t )Pi x˙ (t ) + xT (tk−m+1 )Sθm−1 x(tk−m+1 ) − xT (t − d)Sθm−1 x(t − d) m−1
+
X
xT (tk−j+1 )Sθj−1 x(tk−j+1 ) − xT (tk+−j )Sθj−1 x(tk+−j ) + xT (t )Si x(t ) − xT (tk+ )Si x(tk+ ).
j =1
(17)
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Since 1x(tq ) = x(tq+ ) − x(tq ) = Eσ (tq ) x(tq ) at the switching instant tq , we have x(tq+ ) = (I + Eσ (tq ) )x(tq )
(18)
for any nonnegative integers q = 0, 1, 2, . . . . Combing (18) and (17) gives dV (t ) dt
m−1
= 2xT (t )Pi x˙ (t ) + xT (tk−m+1 )Sθm−1 x(tk−m+1 ) − xT (t − d)Sθm−1 x(t − d) +
X
xT (tk−j+1 )Sθj−1 x(tk−j+1 )
j =1 m−1
−
X
xT (tk−j )(I + Eθj )T Sθj−1 (I + Eθj )x(tk−j ) + xT (t )Si x(t ) − xT (tk )(I + Eθ0 )T Si (I + Eθ0 )x(tk ).
(19)
j=1
After some manipulations, (19) can be written as dV (t ) dt
= 2xT (t )Pi x˙ (t ) − xT (t − d)Sθm−1 x(t − d) + xT (tk−m+1 )Sθm−1 x(tk−m+1 ) − xT (tk−m+1 )(I + Eθm−1 )T Sθm−2 (I + Eθm−1 )x(tk−m+1 ) + · · · − xT (tk−1 )(I + Eθ1 )T Sθ0 (I + Eθ1 )x(tk−1 ) + xT (tk )Sθ0 x(tk ) − xT (tk )(I + Eθ0 )T Si (I + Eθ0 )x(tk ) + xT Si x(t ). (20)
According to the definition of switching law (12), we obtain xT (tk−m+1 )[Sθm−1 − (I + Eθm−1 )T Sθm−2 (I + Eθm−1 )]x(tk−m+1 ) ≤ 0
.. . xT (tk )[Sθ0 − (I + Eθ0 )T Si (I + Eθ0 )]x(tk ) ≤ 0
.
(21)
Substituting (21) into (20) yields dV (t ) dt
≤ 2xT (t )Pi x˙ (t ) − xT (t − d)Sθm−1 x(t − d) + xT Si x(t ).
For the sake of convenience, let j = θm−1 . Then we have dV (t ) dt
≤ 2xT (t )Pi x˙ (t ) − xT (t − d)Sj x(t − d) + xT Si x(t ).
(22)
Furthermore, when ω(t ) ≡ 0, we get dV (t ) dt
= 2xT (t )Pi x˙ (t ) + xT (t )Si x(t ) − xT (t − d)Sj x(t − d) = xT (t ) A¯ Ti Pi + Pi A¯ i + Si x(t ) + 2xT (t − d)A¯ Tdi Pi x(t ) + 2fiT (t , x)Pi x(t ) − xT (t − d)Sj x(t − d)
(23)
with A¯ i = Ai + 1Ai , A¯ di = Adi + 1Adi . By Lemma 1, we have 2fiT (t , x)Pi x(t ) ≤
1
λi
xT (t )fiT (t , x)fi (t , x) + λi xT (t )Pi Pi x(t )
≤ xT (t )
1
FiT Fi + λi Pi Pi x(t )
λi
(24)
where λi > 0 is an arbitrary constant number. Combining (24) and (23) leads to dV (t ) dt
1 T T ¯ ¯ ≤ x (t ) Ai Pi + Pi Ai + Si + Fi Fi + λi Pi Pi x(t ) + 2xT (t − d)A¯ Tdi Pi x(t ) − xT (t − d)Sj x(t − d) λi T x( t ) x(t ) = Σij x(t − d) x(t − d) T
(25)
where 1
" 4
Σij =
A¯ Ti Pi + Pi A¯ i + Si +
λi
A¯ Tdi Pi
FiT Fi + λi Pi Pi
Pi A¯ di
−Sj
# .
(26)
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Denote 1
" 4
Yij =
ATi Pi + Pi Ai + Si +
λi
FiT Fi + λi Pi Pi
Pi Adi
ATdi Pi
# .
(27)
−Sj
Then from Lemma 1, we get
Σij = Yij +
≤ Yi + εi
Pi Mi Γi (t ) NAi 0
Pi Mi 0
Pi M i 0
ATdi Pi
+
1
εi
T +
1
εi
Pi Adi +
y1i
=
NdiT NAi
Ndi + NAi
−S j +
NAi
1
εi
1
εi
Ndi
T NAi Ndi
NdiT Ndi
T
Ndi
ΓiT (t )
T
NAi
Ndi
Pi Mi 0
T
.
(28)
with y1i = + Pi Ai + Si + + λi Pi Pi + εi Pi Mi MiT Pi + ε1i NAiT NAi . Multiplying both sides of (28) by diag(Xi , Ti ) gives 1
ATi Pi
Tj ATdi
F TF λi i i
Adi Tj +
y2i
+
1
εi
Tj NdiT NAi Xi
1
εi
1
−T j +
εi
T Xi NAi Ndi Tj
Tj NdiT Ndi Tj
<0
(29)
where y2i = Xi ATi + Ai Xi + Xi Si Xi +
1
λi
Xi FiT Fi Xi + λi I + εi Mi MiT +
1
εi
T Xi NAi NAi Xi .
It is apparent that (29) is equivalent to inequality (10) by Schur complement. Case 2: tk+1 − tk > d In this case, we need to consider two subintervals: (tk , tk + d] and (tk + d, tk+1 ]. For t ∈ (tk , tk + d], V (t ) is given by (16) and the analysis is same as for Case 1. For t ∈ (tk + d, tk+1 ], V (t ) is given by (15). Then, under the assumption that ω(t ) ≡ 0, we get dV (t ) dt
= 2xT (t )Pi x˙ (t ) + xT (t )Si x(t ) − xT (t − d)Si x(t − d) = xT (t ) A¯ Ti Pi + Pi A¯ i + Si x(t ) + 2xT (t − d)A¯ Tdi Pi x(t ) + 2fiT (t , x)Pi x(t ) − xT (t − d)Si x(t − d)
where A¯ i = Ai + 1Ai , A¯ di = Adi + 1Adi . Following the proof of Case 1, we conclude that
Ti ATdi
Adi Ti +
y2i
+
1
εi
Ti NdiT NAi Xi
−Ti +
1
εi
1
εi
T Xi NAi Ndi Ti
Ti NdiT Ndi Ti
dV (t ) dt
(30)
< 0 if
< 0.
(31)
Obviously, by Schur complement, inequality (31) is equivalent to LMI (10) in the case of j = i. Combining Case 1 and Case 2, we draw the conclusion that when ω(t ) = 0, the time derivative of V (t )(t ∈ (tk , tk+1 ]) is negative along the solution of (9) if LMI (10) holds. According to the multiple Lyapunov function theory [7], V (t ) should decrease along the switching instants to ensure the asymptotical stability. In what follows, we will check this property. Just as stated previously, it is assumed that system (9) switched to ith mode from the lth mode at t = tk . That is to say, σ (t ) = l, t ∈ (tk−1 , tk ] and σ (t ) = i, t ∈ (tk , tk+1 ]. From (9) and (13), we derive
(tk+ ) = x(tk ) + 1x(tk ) =Z [I + El ]x(tk )
V (tk ) = xT (tk )Pl x(tk ) +
tk
tk − d
(32)
xT (τ )Sσ (τ ) x(τ )dτ
V (tk+ ) = xT (tk+ )Pσ (t + ) x(tk+ ) + k
Z
+ tk
+ tk − d
xT (τ )Sσ (τ ) x(τ )dτ
= xT (tk )(I + El )T × Pi (I + El )x(tk ) +
Z
tk tk − d
xT (τ )Sσ (τ ) x(τ )dτ + xT (tk )EiT Si Ei x(tk ).
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Therefore, V (tk+ ) − V (tk− ) = V (tk+ ) − V (tk ) = xT (tk )Θli x(tk )
(34)
where Θli = (I + El ) Pi (I + El ) − Pl + It shows by Schur complement that Θli < 0 is equivalent to EiT Si Ei .
T
(I + El )T −Pi−1 ∗
−P l ∗ ∗
EiT 0 < 0. −Si−1
(35)
Multiplying both sides of (35) by diag(Pl−1 , I , I ) and paying attention to that Xl = Pl−1 , Ti = Si−1 , l ∈ N, we arrive at the inequality (11). This shows that V (tk+ ) − V (tk− ) = V (tk+ ) − V (tk ) < 0
(36)
if the LMI (11) holds, i.e. V (t ) decreases along the switching instants. Therefore, by the multiple Lyapunov functions theory [7], we conclude that the trivial solution of the time-delay switched impulsive system (9) is globally asymptotically stable. The proof of Theorem 1 is complete. Remark 3. The switched Lyapunov function (13) is a particular quadratic Lyapunov function making use of the switching nature of system (9), specifically, it has the same switching signal as switched system (9). Therefore, the Lyapunov function (12) is actually a switching signal dependent Lyapunov function. Compared with the common Lyapunov function approach (i.e. Pi = P > 0, Si = S > 0 is common for all the subsystems), the switched Lyapunov function approach is less conservation, see reference [17] for some details. Now let us study the disturbance attenuation property of switched system (9). Theorem 2. For system (9), design the switching law as (12), then with zero-initial condition x(0) = 0, kz (t )k2 ≤ γ kω(t )k2 for all nonzero ω(t ) ∈ L2 [0, ∞) and admissible uncertainties, if there exit symmetric positive definite matrices Xi , Ti and positive numbers λi , εi (i ∈ N ) solving the following LMIs
αi + γ −2 B2i BT2i ∗ ∗ ∗ ∗ ∗
Adi Tj −Tj
Xi 0 −Ti
∗ ∗ ∗ ∗
∗ ∗ ∗
Xi FiT 0 0 −λi I
∗ ∗
T Xi NAi T Tj Ndi 0 0 −εi I
∗
Xi CiT 0 0 < 0, 0 0 −I
(∀i, j ∈ N )
(37)
4
αi = Xi ATi + Ai Xi + λi I + εi Mi MiT and
−X l ∗ ∗
Xl (I + El )T −Xi
∗
Xl EiT 0 < 0, −Ti
(∀ i, l ∈ N , i 6= l).
(38)
Moreover, switched impulsive system (9) is globally asymptotically stable when ω(t ) ≡ 0. Proof. It easily follows that inequality (37) holds implies inequality (10) holds. Hence, by Theorem 1, system (9) is globally asymptotically stable when ω(t ) ≡ 0, and the switched Lyapunov function V (t ) is given as (13). Secondly, suppose that x(0) = 0 and introduce the performance index T
Z
(z T (t )z (t ) − γ 2 ωT (t )ω(t ))dt .
J =
(39)
0
For any given T > 0, suppose that T ∈ (tk , tk+1 ], then we have T
Z
z T (t )z (t ) − γ 2 ωT (t )ω(t ) + V˙ (t ) dt −
J = 0
T
V˙ (x(t ))dt = 0
V˙ (t )dt . 0
Bearing V (0) = 0 in mind, we obtain from (36) that
Z
T
Z
k−1 Z X m=0
tm+1 tm
V˙ (t )dt +
Z
T
V˙ (t )dt tk
(40)
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= V (t1 ) − V (0) + V (t2 ) − V (t1+ ) + · · · + V (T ) − V (tk+ ) = V (T ) +
k X
+ V (tm ) − V (tm ) ≥ V (T ) (≥ 0).
(41)
m=1
From (37) and Schur complement, we have
π i ∗
Adi Tj +
−Tj +
1
T Xi NAi Ndi Tj
εi
1
εi
Tj NdiT Ndi Tj
<0
(42)
T NAi Xi + Xi CiT Ci Xi + γ −2 B2i BT2i . where πi = Xi ATi + Ai Xi + λi I + εi Mi MiT + Xi Si Xi + λ1 Xi FiT Fi Xi + ε1 Xi NAi i i Multiplying both sides of (42) by diag(Pi , Sj ) yields
π0 i ∗ with πi = 0
Pi Adi +
−Sj + ATi Pi
1
εi
1
εi
T NAi Ndi
NdiT Ndi
<0
(43)
+ Pi Ai + λi Pi Pi + εi Pi Mi MiT Pi + Si +
1
T Fi + ε1 NAi NAi + CiT Ci + γ −2 Pi B2i BT2i Pi . i in two different cases due to the existence of time delay d.
T
F λi i
dV (t ) dt
By the proof of Theorem 1, we should compute Case 1: tk+1 − tk ≤ d. In this case, V (t ) is given by (16). Under the switching law (12), and following the deducing process (17)–(22), we obtain dV (t ) dt
≤ 2xT (t )Pi x˙ (t ) − xT (t − d)Sj x(t − d) + xT Si x(t ).
(44)
Furthermore, by the use of (43) z T (t )z (t ) − γ 2 ωT (t )ω(t ) +
dV (t )
= z T (t )z (t ) − γ 2 ωT (t )ω(t ) + 2xT (t )Pi x˙ (t ) + xT Si x(t ) − xT (t − d)Sj x(t − d) 1 ≤ xT (t ) A¯ Ti Pi + Pi A¯ i + Si + λi Pi Pi + FiT Fi + γ −2 Pi B2i BT2i Pi + CiT Ci x(t ) λi T + 2xT (t − d)A¯ Tdi Pi x(t ) − xT (t − d)Sj x(t − d) − γ −1 BT2i Pi x(t ) − γ ω(t ) γ −1 BT2i Pi x(t ) − γ ω(t ) 1 ≤ xT (t ) A¯ Ti Pi + Pi A¯ i + Si + λi Pi Pi + FiT Fi + γ −2 Pi B2i BT2i Pi + CiT Ci x(t ) λi + 2xT (t − d)A¯ di Pi x(t ) − xT (t − d)Sj x(t − d) T π 0 Pi Adi + 1 N T Ndi Ai x(t ) x(t ) i ε i ≤ (45) x(t − d) < 0. 1 x(t − d) ∗ −Sj + NdiT Ndi εi Case 2: tk+1 − tk > d. As declared in Case 2 in Theorem 1, we need to consider two subintervals: (tk , tk + d] and (tk + d, tk+1 ]. For t ∈ (tk , tk + d], V (t ) is given by (16) and we can get the same analysis as (45). For t ∈ (tk + d, tk+1 ], V (t ) is given by (15). We can similarly dt
obtain z T (t )z (t ) − γ 2 ωT (t )ω(t ) +
dV (t ) dt
= z T (t )z (t ) − γ 2 ωT (t )ω(t ) + 2xT (t )Pi x˙ (t ) + xT Si x(t ) − xT (t − d)Si x(t − d) T π 0 x(t ) i ≤ x(t − d) ∗
Thus z (t )z (t ) − γ ω (t )ω(t ) + T
π0 i ∗
2
Pi Adi +
−Si +
T
1
εi
1
εi
T NAi Ndi
NdiT Ndi
dV (t ) dt
Pi Adi +
−Si +
1
εi
1
εi
T NAi Ndi
NdiT Ndi
x( t ) x(t − d) .
(46)
< 0 is equivalent to
<0
which is exactly the case of j = i in (43).
(47)
G. Zong et al. / Nonlinear Analysis: Hybrid Systems 2 (2008) 1287–1300
1295
Therefore, both Case 1 and Case 2 indicate that z T (t )z (t ) − γ 2 ωT (t )ω(t ) +
dV (t ) dt
<0
(48)
regardless of d. Substituting (41) and (48) into (40) gives rise to T
Z
z T (t )z (t ) − γ 2 ωT (t )ω(t ) dt < 0
J = 0
and ∞
Z
z T (t )z (t ) − γ 2 ωT (t )ω(t ) dt = lim
T →∞
0
T
Z
z T (t )z (t ) − γ 2 ωT (t )ω(t ) dt < 0
(49)
0
for any nonzero disturbance ω(t ) ∈ L2 [0, ∞) and all admissible uncertainties. Therefore, we have kz (t )k2 ≤ γ kω(t )k2 . The proof of Theorem 2 is complete. 4. Controller design In this section, we shall study the robust H∞ stabilization problem proposed in Definition 1 and design the corresponding state feedback stabilizer u(t ) = Kσ (t ) x(t ), t ∈ (tk−1 , tk ] and impulsive controller u(tk ) = K¯ σ (tk ) x(tk ), t = tk . The closed-loop switched impulsive system is given as below
x˙ (t ) = A˜ σ (t ) + 1A˜ σ (t ) (t ) x(t ) + A¯ di x(t − d) + fσ (t ) (t , x) + B2σ (t ) ω(t ), 1x = E˜ σ (tk ) x(tk ), t = tk z (t ) = C˜ σ (t ) x(t ) x(t ) = 0, t = t0 = 0
t ∈ (tk−1 , tk ] (50)
where A˜ i = Ai + B1i Ki , 1A˜ i (t ) = Mi Γi (t )(NAi + NBi Ki ), A¯ di = Adi + 1Adi (t ), E˜ i = Ei + K¯ i , C˜ i = Ci + Di Ki . Theorem 3. Given any constant γ > 0, the switched state feedback H∞ stabilization problem is feasible if there exist symmetric positive definite matrices Pi , Si and positive numbers λi , εi (i ∈ N ) solving the following LMIs
− Si + Pl < 0,
(i, l ∈ N )
(51)
and
ϑ i ∗ ∗ ∗ ∗ ∗
ηi −Sj ∗
Pi 0 1
−
λi
∗
∗
∗ ∗
∗ ∗
I
Pi Mi 0
T NAi NdiT
0
0
1
− I εi ∗ ∗
Pi B2i 0
0
< 0, 0
0
−εi I ∗
( i, j ∈ N )
(52)
0
−γ 2 I
where 1
4
4
1
ϑi = ATi Pi + Pi Ai + Si + CiT Ci + FiT Fi , ψi2 = I + NBiT NBi λi εi 1 1 T 4 T ηi = Pi Adi − Pi B1i + NAi NBi ψi−2 NBi Ndi . εi εi Moreover, a switched state feedback H∞ stabilizing controller and an impulsive controller are respectively given by u(t ) = Ki x(t ), u(tk ) = K¯ i x(tk ),
Ki = −ψi−2
BT1i Pi +
1
εi
T NBi NAi
, t ∈ (tk−1 , tk ]
K¯ i = −(I + Ei ), t = tk .
(53) (54)
And the stabilizing switching law is constructed as
σ (t ) = i,
t > tk σ (tk ) = j, if Sj − (I + Ej )T Si (I + Ej ) < 0.
(j 6= i, i, j ∈ N¯ )
(55)
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G. Zong et al. / Nonlinear Analysis: Hybrid Systems 2 (2008) 1287–1300
Proof. From (52) and Schur complement, for any [x(t ), x(t − d)]T 6= 0, we have
T ϑ1i x(t ) x( t − d ) ∗
1
ηi +
εi
T NAi Ndi
1
−Sj +
εi
NdiT Ndi
x( t ) x(t − d) < 0
(56)
T NAi + γ −2 Pi B2i BT2i Pi . where ϑ1i = ϑi + λi Pi Pi + εi Pi Mi MiT Pi + ε1 NAi i For any given switching signal σ (t ), Choose the candidate switched Lyapunov function as (13). According to Definition 1, we shall firstly study the stability of the closed-loop system (50), and then verify its H∞ disturbance attenuation performance.
A: Stability analysis and controller design By Theorem 1, we ought to calculate the derivative of V (t ) under two different cases: tk+1 − tk ≤ d and tk+1 − tk > d. Here, in order to avoid repeating some similar manipulations, we merely go on with the former case. The analysis of the latter case is in a similar way. When tk+1 − tk ≤ d, the Lyapunov energy function is displayed as (16) and its time derivative satisfies (22) under the switching law (55). Through some manipulations, with ω(t ) = 0, we have dV (t )
= x˙ T (t )Pi x(t ) + xT (t )Pi x˙ (t ) + xT Si x(t ) − xT (t − d)Sj x(t − d) 1 ≤ xT (t ) Aˆ Ti Pi + Pi Aˆ i + Si + FiT Fi + λi Pi Pi x(t ) + 2xT (t − d)A¯ Tdi Pi x(t ) − xT (t − d)Sj x(t − d) λi T x(t ) x(t ) ˆi Σ = x( t − d ) x(t − d)
dt
(57)
where Aˆ i = A˜ i + 1A˜ i (t ), and 1
"
Aˆ Ti Pi + Pi Aˆ i + Si +
ˆi = Σ
λi
FiT Fi + λi Pi Pi
A¯ Tdi Pi
Pi A¯ di
# .
(58)
−Sj
Denote 1
" 4
Yˆi =
A˜ Ti Pi + Pi A˜ i + Si +
λi
FiT Fi + λi Pi Pi
Pi Adi
ATdi Pi
# .
(59)
−S j
Then, following the proof of (20), we get
ˆ i = Yˆi + Pi Mi Γi (t ) (NAi + NBi Ki ) Σ 0
yˆ 1i
≤
∗
Pi Adi +
1
εi
(NAi + NBi Ki )T Ndi
−Sj +
1
εi
NdiT Ndi
Ndi + (NAi + NBi Ki )
Ndi
T
ΓiT (t )
Pi Mi 0
T
(60)
where εi > 0 is a positive constant number, and yˆ 1i = ATi Pi + Pi Ai + Si +
1
λi
FiT Fi + λi Pi Pi + εi Pi Mi MiT Pi +
1
εi
T NAi NAi
T 1 1 − KiT Ki − BT1i Pi + NBiT NAi ψi−2 BT1i Pi + NBiT NAi εi εi T 1 T 1 T −2 T −2 T 2 + Ki + ψi (B1i Pi + NBi NAi ) ψi Ki + ψi (B1i Pi + NBi NAi ) . εi εi
(61)
T Define the control gain Ki as Ki = −ψi−2 (BT1i Pi + ε1 NBi NAi ), then we have i
yˆ 1i ≤ ATi Pi + Pi Ai + Si +
1
λi
FiT Fi + λi Pi Pi + εi Pi Mi MiT Pi +
1
εi
T NAi NAi .
(62)
G. Zong et al. / Nonlinear Analysis: Hybrid Systems 2 (2008) 1287–1300
1297
Combining (56), (60) and (62) gives dV (t )
<0
dt
(63)
for any [x(t ), x(t − d)] 6= 0. In order to testify the robust asymptotic stability, the following condition has to be satisfied by Theorem 1 V (tk+ ) − V (tk− ) = V (tk+ ) − V (tk ) < 0.
(64)
From (35) and (36), (64) holds if
(I + E˜ l )T −Pi−1 ∗
−P l ∗ ∗
E˜ iT 0 < 0. −Si−1
(65)
Take K¯ l = −(I + El ), then (I + E˜ l ) = 0 and the above inequality (64) holds naturally by (51). Therefore the closed-loop switched impulsive system (50) is globally asymptotically stable. B: H∞ performance analysis Define T
Z
(z T (t )z (t ) − γ 2 ωT (t )ω(t ))dt .
J = 0
Since x(0) = 0, from (41) and (64), we directly get T
Z
J ≤
z (t )z (t ) − γ ω (t )ω(t ) + T
2
T
0
dV (t ) dt
dt
(66)
for any given 0 < T ∈ (tk , tk+1 ]. dV (t ) Since dt has different forms for different time delay d, we ought to compute z T (t )z (t ) − γ 2 ωT (t )ω(t ) + cases. Case 1: tk+1 − tk ≤ d. In this case, V (t ) is given by (16). Under the switching law (55), it follows that dV (t )
≤ 2xT (t )Pi x˙ (t ) − xT (t − d)Sj x(t − d) + xT Si x(t ).
dt
dV (t ) dt
in all the
(67)
Furthermore, z T (t )z (t ) − γ 2 ωT (t )ω(t ) +
dV (t ) dt
1 ≤ xT (t ) Aˆ Ti Pi + Pi Aˆ i + Si + λi Pi Pi + FiT Fi + CiT Ci + KiT Ki + γ −2 Pi B2i BT2i Pi x(t ) λi
+ 2x (t − d)A¯ Tdi Pi x(t ) − xT (t − d)Sj x(t − d) − (γ −1 BT2i Pi x(t ) − γ ω(t ))T (γ −1 BT2i Pi x(t ) − γ ω(t )) 1 T T T T −2 T ˆ ˆ ≤ x (t ) Ai Pi + Pi Ai + Si + λi Pi Pi + Fi Fi + Ci Ci + γ Pi B2i B2i Pi x(t ) λi T T T + 2x (t − d)A¯ di Pi x(t ) − x (t − d)Sj x(t − d) T x(t ) x(t ) ≤ Ωij x( t − d ) x(t − d) T
(68)
where
µ i Ωij = ∗
Pi Adi +
1
εi
(NAi + NBi Ki )T Ndi
−Sj +
1
εi
NdiT Ndi
(69)
and
µi = ATi Pi + Pi Ai + Si +
1
F T Fi + λi Pi Pi + CiT Ci + γ −2 Pi B2i BT2i Pi + εi Pi Mi MiT Pi +
λi i T
1 ψi−2 BT1i Pi + NBiT NAi εi εi T 1 1 + Ki + ψi−2 BT1i Pi + NBiT NAi ψi2 Ki + ψi−2 BT1i Pi + NBiT NAi . εi εi −
BT1i Pi +
1
1
εi
T NAi NAi
T NBi NAi
(70)
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G. Zong et al. / Nonlinear Analysis: Hybrid Systems 2 (2008) 1287–1300
T Substituting Ki = −ψi−2 (BT1i Pi + ε1 NBi NAi ) from (53) into (69) and (70) gives rise to i
µi ≤ ATi Pi + Pi Ai + Si + Pi Adi +
1
εi
1
λi
FiT Fi + λi Pi Pi + CiT Ci + γ −2 Pi B2i BT2i Pi + εi Pi Mi MiT Pi +
(NAi + NBi Ki )T Ndi = ηi +
1
εi
1
εi
T NAi NAi = ϑ1i
T NAi Ndi .
(71) (72)
Furthermore,
Ωij < 0. That is to say z T (t )z (t ) − γ 2 ωT (t )ω(t ) +
dV (t ) dt
< 0.
(73)
Case 2: tk+1 − tk > d. In this case, we need to consider two subintervals: (tk , tk + d] and (tk + d, tk+1 ]. For t ∈ (tk , tk + d], V (t ) is given by (16) and we can get the same result as (73) by repeating the process from (67)–(72). For t ∈ (tk + d, tk+1 ], V (t ) is given by (15), and we have z T (t )z (t ) − γ 2 ωT (t )ω(t ) +
dV (t ) dt
= z T (t )z (t ) − γ 2 ωT (t )ω(t ) + 2xT (t )Pi x˙ (t ) + xT Si x(t ) − xT (t − d)Si x(t − d) T
x(t ) x(t − d)
≤
Ωii
x(t ) <0 x( t − d )
(74)
where Ωii is equal to Ωij in the case of j = i in (69), which is negative by the above deduction. Combining (73) with (74) gives z T (t )z (t ) − γ 2 ωT (t )ω(t ) +
dV (t ) dt
<0
(75)
irrespective of the size of time delay d. Substituting (75) into (66) yields ∞
Z
(z T (t )z (t ) − γ 2 ωT (t )ω(t ))dt = lim 0
T →∞
T
Z
(z T (t )z (t ) − γ 2 ωT (t )ω(t ))dt < 0
(76)
0
for any nonzero disturbance ω(t ) ∈ L2 [0, ∞) and all admissible uncertainties, which implies that kz (t )k2 ≤ γ kω(t )k2 . The proof of Theorem 3 is complete. 5. An illustrative example In this section, we give an illustrative example to show the effectiveness of the proposed algorithms. Consider the following control systems with N = 2 and the parameters as follows
" # " # −12.8 0 1.8 −4.4 0 8.01 1.2 −0.9 0.6 , 0 A1 = A2 = 2.63 −5.5 ; 1.2 −0.56 −2.5 0 0.2 −7.02 " # " # 0.02 0.86 0.8 −4.4 0 8.01 0 −0.1 0.51 , 0 Ad1 = Ad2 = 2.63 −5.5 ; 0 0.12 −0.22 0 0.2 −7.02 " # " # " # 0.02 0.02 0.10 0.02 1.00 0.04 0 0.10 , 0 0.10 , 0 B11 = B12 = M1 = 0.01 ; 0.63 0 0.08 0 0.01 0 " # " # 2.00 0.40 0.1 0.04 0.1 0 0 ; M2 = 0.10 , NA1 = 0.03 0.01 0.20 0 0.5 0 0 " # " # 0.02 0.04 0.10 0.09 0.01 0.01 0 0 0 0 NA2 = 0.03 0.10 , Nd1 = ; 0.03 0 0 0.02 0 0.04
G. Zong et al. / Nonlinear Analysis: Hybrid Systems 2 (2008) 1287–1300
" Nd2 =
0.05 0 0
0 0 0.02
0.01 0 , 0.01
#
0.4100 0 , −1.2100
" B21 =
0.04 C2 = 0
#
0.01 0.05
" NB1 = 0.01 0 0.04
" E1 =
0.1 0.2 0
"
0 , 0.01
B22 =
0.01 0.06 0.01
0 0 , 0.02
0 0 .3 , 0.05
0.01 0.02 0.10
0 0 ; 0.10
0.02 0
0.01 0.03
0 ; 0.01
"
#
0.05 0 , 0.08
C1 =
#
0.43 0 0.40
" E2 =
1299
NB2 =
#
0.10 0.15 0
#
0 0.30 . 0.15
#
Choose γ = 0.2, ε1 = 0.5, ε2 = 0.4, λ1 = 0.4, λ2 = 0.2. Solving the LMIs (38) and (39) in Pi , Si , we obtain the following feasible solution 0.1114 0.0062 0.0113
" P1 =
0.2055 −0.0143 0.0217
" S1 =
0.0062 0.0595 0.0017
0.0113 0.0017 , 0.0750
−0.0143 0.0751 −0.0143
#
0.0497 −0.0056 −0.0056
" P2 =
0.0217 −0.0143 , 0.1003
#
" S2 =
−0.0056 0.0397 0.0397
0.2055 −0.0143 0.0217
0.0413 −0.0036 −0.0036
−0.0143 0.0751 −0.0143
#
0.0217 −0.0143 . 0.1003
#
After some manipulations, we can see that P1 , P2 and S1 , S2 are symmetric positive definite matrices. Then by Theorem 3, we know the given uncertain switched impulsive control system is stabilizable with H∞ disturbance attenuation level γ = 0.2. According to (40) and (41), the robust switched state feedback H∞ controller and impulsive controller gain can be given as follows
−0.0355 −0.0020 −0.6308 , −0.0400 −0.1000 0.0001 " # −1.0100 −0.1000 0 ¯K1 = 0 −1.2000 −0.3000 , −0.0400 0 −1.0500
K1 =
−0.2084 −0.0052 −0.1618 −0.0465 −0.1992 0.0006 " # −1.4300 −0.1000 0 ¯K2 = 0 −1.1500 −0.3000 . −0.4000 0 −1.1500
K2 =
6. Conclusions In this paper, we have investigated the problem of robust H∞ stabilization for a class of uncertain switched impulsive control systems with state delay. By the use of the switched Lyapunov function approach, some sufficient conditions have been developed to ensure the robust globally asymptotic stability and H∞ disturbance attenuation performance. These conditions are presented in terms of LMI form, which are easy to testify and compute utilizing Matlab tool-box. A suitable robust H∞ stabilizing state feedback controller, an impulsive controller and switching law have also been given. When compared with the common Lyapunov function based methods, the conservatism of the proposed approach is highly reduced. The main contribute of this paper lies in providing a practically useful and easily executable investigation algorithm for uncertain switched impulsive control systems. Acknowledgments This work was supported by the National Natural Science Foundation of China (No. 60674027), Specialized Research Fund for the Doctoral Program of Higher Education (No. 20050446001), China Postdoctoral Science Foundation(No. 20070410336), Postdoctoral Foundation of Jiangsu Province (No. 0602042B) and Scientific Research Foundation of Qufu Normal University. References [1] D. Liberzon, A.S. Morse, Basic problems in stability and design of switched systems, IEEE Control Systems Magazine 19 (1999) 59–70. [2] R. Decarlo, M.S. Branicky, B. Lennartson, Perspective and results on the stability and stabilizability of hybrid systems, Proceedings of the IEEE 88 (2000) 1069–1082. [3] D. Liberzon, Switching in Systems and Control, Birkhauser, Boston, MA, 2003. [4] Z.D. Sun, S.S. Ge, Switched Linear Systems-control and Design, Springer Verlag, New York, 2004. [5] Z.G. Li, Y.C. Soh, C.Y. Wen, Robust stability of quasiperiodic hybrid dynamic uncertain systems, IEEE Transactions on Automatic Control 46 (2001) 107–111. [6] Z.G. Li, Y.C. Soh, C.Y. Wen, Robust stability for a class of hybrid nonlinear systems, IEEE Transactions on Automatic Control 47 (2002) 897–903. [7] M.S. Branicky, Multiple Lyapunov functions and other analysis tools for switched and hybrid systems, IEEE Transactions on Automatic Control 43 (1998) 60–66. [8] D.Z. Cheng, G. Lei, Y.D. Lin, Y. Wang, Stabilization of switched systems, IEEE Transactions on Automatic Control 50 (2005) 661–666. [9] G.M. Xie, L. Wang, Necessary and sufficient conditions for controllability and observability of switched impulsive control systems, IEEE Transactions on Automatic Control 49 (2004) 960–966.
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