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Delay-dependent H∞ control for singular Markovian jump systems with time delay
Nonlinear Analysis: Hybrid Systems 8 (2013) 1–12
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Delay-dependent H∞ control for singular Markovian jump systems with time delay Junru Wang a , Huijiao Wang a,∗ , Anke Xue b , Renquan Lu b a
Institute of Automation, College of Mechanical Engineering and Automation, Zhejiang Sci-Tech University, Hangzhou 310018, PR China
b
Institute of Information & Control, Hangzhou Dianzi University, Hangzhou 310018, PR China
article
info
Article history: Received 23 May 2012 Accepted 28 August 2012 Keywords: Singular Markovian jump system H∞ control Time delay Delay subinterval decomposition approach
abstract This paper deals with the problem of delay-dependent H∞ control for singular Markovian jump systems with time delay. Based on the delay subinterval decomposition approach, a new Lyapunov–Krasovskii functional is proposed to develop the new delay-dependent bounded real lemma (BRL), which ensures the considered system to be regular, impulsefree and stochastically stable with given H∞ performance index γ . Based on this new BRL, the explicit expression of the desired controller gains is also presented by solving a set of strict LMIs. Some numerical examples are given to show the effectiveness and less conservativeness of the proposed methods. © 2012 Elsevier Ltd. All rights reserved.
1. Introduction Markovian jump systems, as a special class of stochastic hybrid systems, have the advantage of better describing physical systems with abrupt variations. Applications of this kind of systems can be found in manufacturing systems [1], networked control systems [2], economics systems [3,4], air intake system [5], and other practical systems. Recently, a lot of attention has been paid to the stability analysis and controller synthesis for Markovian jump systems [6–13]. On the other hand, singular systems have been extensively studied in the past years due to the fact that singular systems better describe physical systems than state-space ones. When singular systems experience abrupt changes in their structures, it is natural to model them as singular Markovian jump systems [14–18]. Time delays always exist in many dynamical systems and delays are sources of poor stability and performance of a system. For example, it is a very common phenomenon that feedback current is time delay in the DC motor in position control servomechanisms [14]. When time delay is considered, the results in [14] cannot be applied to analyze the dynamical model of the DC motor. The singular Markovian jump systems with time delay may also have applications in other practical systems. So studies of the stability criteria and the performance for singular Markovian jump systems with delays are of theoretical and practical importance. During the recent years, much attention has been devoted to the study of singular Markovian jump systems with time delay. The criteria can be generally classified into two categories: delay-independent [19] and delay-dependent ones [20–22]. Generally, the delay-dependent case is less conservative than delay-independent ones, especially when the delay is comparatively small. So more attention has been paid to delay-dependent criteria. Various methods have been proposed to obtain delay-dependent criteria, such as properly chosen Lyapunov–Krasovskii functionals [23], the slack variable method [11,24], the delay decomposition approach [25] etc. Among these, the delay decomposition approach provides much less conservative delay-dependent stability criteria than the other techniques. The idea of delay partitioning has appeared in [26]. As mentioned in [26], some free-weighting matrices (or slack variable) may be redundant and
∗
Corresponding author. E-mail addresses: [email protected] (J. Wang), [email protected] (H. Wang), [email protected] (A. Xue), [email protected] (R. Lu).
1751-570X/$ – see front matter © 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.nahs.2012.08.003
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J. Wang et al. / Nonlinear Analysis: Hybrid Systems 8 (2013) 1–12
they will increase the computational burden in the case of stability analysis for deterministic delay systems by constant Lyapunov–Krasovskii functionals. The H∞ control problem for time-delay singular systems was investigated in [27–30]. For Markovian jump singular systems, since stability, regularity, impulse elimination and system switch should be considered at the same time, it is difficult to discuss Markovian jump singular systems. The robust stability and H∞ control problems for discrete Markovian jump systems with mode-dependent time-delay were discussed in [31]. Based on the LMI approach, the H∞ control [23], H∞ filtering [32] and dissipative control [33] for singular Markovian jump systems with time delay were discussed. For singular Markovian jump systems with time-varying delays, using the free-weighting-matrix approach, stability and stabilization [22], delay-depend robust control [24], and H∞ filtering [34] were discussed. To the best of our knowledge, the delay-dependent H∞ control problem for singular Markovian jump systems with time-delay has not yet been fully investigated. There has room for improvement. Motivated by the idea of delay partitioning, this paper deals with the problem of delay-dependent H∞ control for singular Markovian jump systems with time-delay. The purpose is to design the state feedback controller such that the resulted closeloop system is regular, impulse-free and stochastically stable with H∞ performance index γ . Based on the delay subinterval decomposition approach, in which the interval [t − d, t ] is decomposed into [t − d, t − kd] and [t − kd, t ](0 < k < 1) and each subinterval has different Lyapunov weighted matrices, a new Lyapunov–Krasovskii functional for singular Markovian jump systems is presented and a novel delay-dependent bounded real lemma (BRL) is established via LMIs. Based on this new BRL, the delay-dependent H∞ control problem is solved and a strict LMI-based design method of the desired controller is proposed. The effectiveness of the method is illustrated by some numerical examples. Notations. Through this paper, the superscripts ‘‘T ’’ and ‘‘−1’’ stand for the transpose of a matrix and the inverse of a matrix; Rn denotes n-dimensional Euclidean space; Rn×m is the set of all real matrices with m rows and n columns; ∥ · ∥ stands for the Euclidean norm for a vector. E {·} denotes the expectation operator. For a symmetric matrix, ∗ denotes the matrix entries implied by symmetry. 2. Problem formulation and preliminaries Consider the following singular Markovian jump time-delay system
E x˙ (t ) = A(rt )x(t ) + Ad (rt )x(t − d) + B(rt )u(t ) + Bω (rt )ω(t ) z (t ) = C (rt )x(t ) + D(rt )u(t ) x(t ) = φ(t ), t ∈ [−d¯ , 0]
(1)
where x(t ) ∈ Rn is the state vector, u(t ) ∈ Rm is the control input, z (t ) ∈ Rq is the system output, ω(t ) ∈ Rp is the deterministic disturbance input which belongs to L2 [0, ∞). Here L2 [0, ∞) stands for the space of square integrable vector functions ¯ φ(·) is the initial condition defined on over the interval [0, ∞). d is an unknown but constant delay satisfying 0 ≤ d ≤ d. [−d¯ , 0]. The matrix E ∈ Rn×n may be singular and we assume that rank E = r ≤ n. {rt , t ≥ 0} is a continuous-time discretestate Markov process with right continuous trajectory values in a finite set S = {1, 2, . . . , s}. A(rt ), Ad (rt )B(rt ), Bω (rt ), C (rt ) and D(rt ) are known real constant matrices with appropriate dimensions for each rt ∈ S . The transition probability matrix Π = {πij } (i, j ∈ S ) given by
P{rt +h = j|rt = i} =
πij h + o(h), i ̸= j. 1 + πij h + o(h), i = j
o(h)
where h > 0 and limh→0 h = 0, πij ≥ 0, for j ̸= i, is the transition rate from mode i at time t to mode j at time t + h and πii = − sj=1,j̸=i πij . For the singular Markovian jump time-delay system
E x˙ (t ) = A(rt )x(t ) + Ad (rt )x(t − d), x(t ) = φ(t ), t ∈ [−d¯ , 0]
(2)
we have the following definition. Definition 1 ([17,32]). (1) For a given scalar d¯ > 0, the nominal singular Markovian jump time-delay system (2) is said to be regular and impulsefree for any constant time delay d satisfying 0 ≤ d ≤ d¯ if the pairs (E , A(rt )) and (E , A(rt ) + Ad (rt )) are regular and impulse-free for every rt ∈ S . (2) The singular Markovian jump time-delay system (2) is said to be stochastically stable, if there exists a scalar M (r0 , φ(·)) such that T
lim E
T →∞
∥x(t )∥ dt |r0 , x(s) = φ(s) ≤ M (r0 , φ(·)). 2
0
(3)
J. Wang et al. / Nonlinear Analysis: Hybrid Systems 8 (2013) 1–12
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(3) The singular Markovian jump time-delay system (2) is said to be stochastically admissible, if it is regular, impulse-free and stochastically stable. Remark 1. The regularity and absence of impulse of the pair (E , A(rt )) for every rt ∈ S ensures the system (2) with time delay d ̸= 0 to be regular and impulse free, while the fact that for every rt ∈ S , the pair (E , A(rt ) + Ad (rt )) is regular and impulse free ensures the system (2) with time delay d = 0 to be regular and impulse free. The unforced singular Markovian jump time-delay system of system (1) is as follows:
E x˙ (t ) = A(rt )x(t ) + Ad (rt )x(t − d) + Bω (rt )ω(t ), z (t ) = C (rt )x(t ) x(t ) = φ(t ), t ∈ [−d¯ , 0].
(4)
Definition 2. Given a scalar γ > 0, the singular Markovian jump time-delay system (4) is said to be stochastically admissible with H∞ performance γ , if the system with ω(t ) ≡ 0 is stochastically admissible and under zero initial condition, satisfies ∞
z T (t )z (t )dt
E
≤ γ2
∞
0
ωT (t )ω(t )dt
(5)
0
for any non-zero ω(t ) ∈ L2 [0, ∞). The H∞ control problem to be addressed here is formulated as follows: for given scalar γ > 0 and the singular Markovian jump time-delay system (1), design feedback control law u(t ) = K (rt )x(t )
(6)
such that the resulting closed-loop system is stochastically admissible with H∞ performance γ . For notational simplicity, in this paper, when r (t ) = i, i ∈ S , a matrix M (rt ) will be denoted by Mi ; for example, A(rt ) is denoted by Ai , Ad (rt ) by Adi and so on. The following lemmas will play an important role in the subsequent sections. Lemma 1 ([35]). For any constant matrix X ∈ Rn×n , X = X T > 0, scalar r > 0, and vector function x˙ : [−r , 0] → Rn such that the following integration is well defined, then
0
x˙ T (t + s)X x˙ (t + s)ds ≤ xT (t )
−r
xT ( t − r )
−X
−r
X
X −X
x(t ) . x(t − r )
(7)
Lemma 2 ([32]). The singular Markovian jump system E x˙ (t ) = Ai x(t )
(8)
is stochastically admissible if and only if there exist symmetric positive-definite matrices Pi and matrices Si such that for every i ∈ S, s
πij E T Pj E + E T Pi Ai + Si RT Ai + ATi Pi E + ATi RSiT < 0
(9)
j =1
where R ∈ Rn×(n−r ) is any matrix with full column rank and satisfies E T R = 0. 3. Main results In this section, the H∞ control problem will be investigated for the singular Markovian jump time-delay system (1) in terms of the delay subinterval decomposition approach. To this end, we first provide the following delay-dependent bounded real lemma (BRL) for (4), which will play a key role in the derivation of our main results. 3.1. Delay-dependent bounded real lemma Theorem 1. For given scalar d¯ > 0, γ > 0 and 0 < k < 1, the singular Markovian jump time-delay system (4) is stochastically ¯ if there exist symmetric positive-definite admissible with H∞ performance γ for any constant time-delay d satisfying 0 ≤ d ≤ d,
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J. Wang et al. / Nonlinear Analysis: Hybrid Systems 8 (2013) 1–12
matrices Pi , Qi , U , V , Q , Z , W and matrices Si such that for every i ∈ S ,
Γ1i ∗ ∗ ∗ ∗ ∗
Γ4i Γ2i ∗
Γ6i Γ5i Γ3i
Γ7i
∗
∗
0 0
¯ Ti Z dA 0 ¯ Tdi Z dA
¯ Ti W dA 0 ¯ Tdi W dA
∗
−γ 2 I
¯ Tωi Z dB
¯ Tωi W dB
∗
∗
∗
−
∗
∗
∗
∗
∗
∗
∗
∗
Z
0
k
−
W 1−k
∗
CiT 0 0
0
<0 0
(10a)
0
−I
s
πij Qj ≤ Q
(10b)
j =1
where s
Γ1i =
¯ − πij E T Pj E + E T Pi Ai + Si RT Ai + ATi Pi E + ATi RSiT + Qi + U + dQ
j =1
E T ZE
Γ2i = −U + V −
k
E T WE
Γ3i = −Qi − V − E T ZE
Γ4i =
k
,
−
1−k
Γ5i =
E T WE 1−k
,
,
E T ZE k
,
Γ6i = (E T Pi + Si RT )Adi
Γ7i = (E T Pi + Si RT )Bωi
E T WE 1−k
and R ∈ Rn×(n−r ) is any matrix with full column satisfying E T R = 0. Proof. We first show the singular Markovian jump time-delay system (4) is regular and impulse-free. Since rank E = r ≤ n, there must exist two invertible matrices G and H ∈ Rn×n such that
¯E = GEH = Ir
0
0 . 0
(11)
Then, R can be parameterized as T
R=G
0
¯ Φ
¯ ∈ R(n−r )×(n−r ) is any nonsingular matrix. where Φ Similar to (11), we define A¯ i = GAi H =
A¯ i,11 A¯ i,21
A¯ i,12 A¯ i,22
P¯ P¯ i = G−T Pi G−1 = ¯ i,11 Pi,21
S¯i = H T Si =
S¯i,11 S¯i,21
P¯ i,12 P¯ i,22
(12)
for every i ∈ S . Pre-multiplying and post-multiplying Γ1i < 0 by H T and H respectively, we have
¯ S¯iT,21 + S¯i,21 Φ ¯ T A¯ i,22 < 0 A¯ Ti,22 Φ which implies A¯ i,22 is nonsingular for every i ∈ S and thus the pair (E , Ai ) is regular and impulse-free for every i ∈ S . By (10a), it is easy to see that
Γ1i ∗ ∗
Γ4i Γ2i ∗
Γ6i Γ5i < 0. Γ3i
(13)
J. Wang et al. / Nonlinear Analysis: Hybrid Systems 8 (2013) 1–12
I Pre-multiplying and post-multiplying on both sides of (13) by
I
I I 0
0 0
0 I
I and
0 I 0
I I
5
0 0 I
, we have
Γ1i + Γ2i + Γ3i + Γ4i + Γ5i + Γ6i + Γ4iT + Γ5iT + Γ6iT < 0. Hence, s
πij E T Pj E + (E T Pi + Si RT )(Ai + Adi ) + (Ai + Adi )T (Pi E + RSiT ) < 0.
(14)
j =1
According to Lemma 2, (14) implies that the pair (E , Ai + Adi ) is regular and impulse-free for every i ∈ S . Thus, by Definition 1, ¯ the system (4) is regular and impulse-free for any constant time-delay d satisfying 0 ≤ d ≤ d. Next, we will show the stochastic stability of the system (4). For t ≥ d, define the following stochastic Lyapunov candidate for the system (4) with ω(t ) ≡ 0, 5
V (xt , rt , t ) =
Vm (xt , rt , t )
(15)
m=1
where V1 (xt , rt , t ) = xT (t )E T P (rt )Ex(t ), V2 (xt , rt , t ) =
t
xT (s)Q (rt )x(s)ds, t −d
V3 (xt , rt , t ) =
t
xT (s)Ux(s)ds +
t −kd
V4 (xt , rt , t ) = d¯
0
−d
t
−kd
V5 (xt , rt , t ) =
xT (s)Vx(s)ds, t −d
0
t −kd
x˙ T (s)E T ZE x˙ (s)dsdθ + d¯
−kd
t +θ
−d
t
x˙ T (s)E T WE x˙ (s)dsdθ , t +θ
t
xT (s)Qx(s)dsdθ , t +θ
symmetric positive-definite matrices Pi , Qi , U , V , Q , Z , W for every i ∈ S are matrices to be determined. Let A be the weak infinitesimal generator of the random process {(xt , rt ), t ≥ 0}. Then, for every i ∈ S and t ≥ d, we have A V (xt , i, t ) ≤ 2xT (t )(E T Pi + Si RT )E x˙ (t ) + xT (t )
s
¯ )x(t ) πij E T Pj E x(t ) + xT (t )(Qi + U + dQ
j =1
− x (t − d)(Qi + V )x(t − d) + T
t
x (s) T
s
t −d
πij Qj x(s)ds − xT (t − kd)(U − V )x(t − kd)
j=1
+ kd¯ 2 x˙ T (t )E T ZE x˙ (t ) + (1 − k)d¯ 2 x˙ T (t )E T WE x˙ (t ) t t −kd − d¯ x˙ T (s)E T ZE x˙ (s)ds − d¯ x˙ T (s)E T WE x˙ (s)ds − t −kd
t −d
t
xT (s)Qx(s)ds. t −d
From (10b), it is clear that
t
x (s) T
t −d
s
πij Qj x(s)ds ≤
t
xT (s)Qx(s)ds.
t −d
j =1
It follows from Lemma 1 that d¯
t
T
x(t ) x˙ (s)E ZE x˙ (s)ds ≤ x ( t − kd) k t −kd T
T
1
−E T ZE E T ZE
E T ZE −E T ZE
x(t ) x(t − kd)
and d¯
t −kd
x˙ T (s)E T WE x˙ (s)ds ≤ t −d
x(t − kd) 1 − k x(t − d) 1
T
−E T WE E T WE
E T WE −E T WE
x(t − kd) . x(t − d)
Noting E T R = 0, we have A V (xt , i, t ) ≤ ηT (t )Θi η(t )
(16)
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J. Wang et al. / Nonlinear Analysis: Hybrid Systems 8 (2013) 1–12
where ηT (t ) = xT (t )
xT (t − kd)
Γ4i Γ2i ∗
Xi
Θi =
∗ ∗
Yi
xT (t − d)
Γ5i
(17)
Zi
with Xi = Γ1i + ATi [kd¯ 2 E T ZE + (1 − k)d¯ 2 E T WE ]Ai Yi = Γ6i + ATi [kd¯ 2 E T ZE + (1 − k)d¯ 2 E T WE ]Adi Zi = Γ3i + ATdi [kd¯ 2 E T ZE + (1 − k)d¯ 2 E T WE ]Adi .
Using (10a), it is easy to see that there exists a scalar λ > 0 such that for every i ∈ S A V (xt , i, t ) ≤ −λ∥x(t )∥2 .
Therefore, by Dynkin’s formula, we get for any t ≥ d
E V (xt , i, t ) − E V (xd , rd , d) ≤ −λE
t
∥x(s)∥2 ds d
which yields t
∥x(s)∥2 ds ≤ λ−1 E V (xd , rd , d).
E
(18)
d
Following the similar line to that of [32], it is clear that there exists a scalar β such that t
∥x(s)∥2 ds = E
E
d
0
∥x(s)∥2 ds + E
t
0
d
∥x(s)∥2 ds ≤ β E ∥φ∥2d¯ .
According to Definition 1, the system (4) with ω(t ) ≡ 0 is stochastically stable for any constant time delay d satisfying ¯ 0 ≤ d ≤ d. In the following, we consider the stochastic Lyapunov candidate (15) and the following index for the system (4): Jz ω (t ) = E
t
[z (s) z (s) − γ ω(s) ω(s)]ds . T
2
T
0
Under zero initial condition, it is easy to see that t
[z (s)T z (s) − γ 2 ω(s)T ω(s) + A V (xs , i, s)]ds 0 t T ≤E ς (s)Ξi ς (s)ds
Jz ω (t ) = E
0
where ς T (t ) = x (t )
T
Xi + CiT Ci
Ξi =
∗ ∗ ∗
xT (t − kd)
Γ4i Γ2i ∗ ∗
Yi
Γ5i Zi
∗
xT (t − d)
Γ7i
ωT (t )
0 . 0 2 −γ I
(19)
Hence, by Schur complement and from (10a), we have Jz ω (t ) < 0, for all t > 0. Therefore, for any non-zero ω(t ) ∈ L2 [0, ∞), (5) holds. Then according to Definition 2, the singular Markovian jump time-delay system (4) is stochastically admissible with H∞ performance γ . This completes the proof. Remark 2. We can divide [0, d¯ ] into N segments [0, N1 d¯ ], [ N1 d¯ , N2 d¯ ], . . . , [ NN−1 d¯ , d¯ ], the correspondent stability results can also be derived for singular Markovian jump systems. It is also called the delay decomposition approach or the delay partitioning method, which is widely used in time-delay systems derived [36,37], Markovian jump systems [25], singular systems [38], T –S fuzzy systems [39] and stochastic neural networks [40], etc.
J. Wang et al. / Nonlinear Analysis: Hybrid Systems 8 (2013) 1–12
7
3.2. Delay-dependent H∞ controller Theorem 2. For given scalar d¯ > 0, γ > 0 and 0 < k < 1, the singular Markovian jump time-delay system (1) is stochastically ¯ if there exist symmetric positive-definite admissible with H∞ performance γ for any constant time-delay d satisfying 0 ≤ d ≤ d, matrices Pi , Qi , U , V , Q , Z , W and matrices Si , Ti , Fi such that for every i ∈ S , EZE T
Υ1i
Υ2i
∗
−FiT − Fi
k 0
∗
∗
Υ7i
∗ ∗
∗ ∗
∗ ∗
∗
∗
∗
FiT ATdi
Υ5i
0
0
FiT ATdi
Υ5i
¯ dZ
¯ dW
0
0
0
0
0
0 0 Z
0 0
0 0
0
0
EWE T 1−k
Υ9i ∗
−γ 2 I
∗
∗
−
k
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
s
−
Bωi
∗
<0
0
W 1−k
0
(20a)
−I
πij Qj ≤ Q
(20b)
j=1
where
Υ1i =
s
¯ − πij EPj E T + Ai Fi + FiT ATi + Bi Ti + TiT BTi + Qi + U + dQ
j=1
EZE T k
,
Υ2i = EPi + Si RT − FiT + Ai Fi + Bi Ti , Υ5i = FiT CiT + TiT DTi , Υ7i = −U + V − Υ9i = −Qi − V −
EZE T k
−
EWE T 1−k
EWE T 1−k
,
,
and R ∈ Rn×(n−r ) is any matrix with full column satisfying ER = 0. Moreover, the controller gain can be given by Ki = Ti Fi−1 .
(21)
Proof. Substituting the state feedback controller (6) to system (1) yields the following closed-loop system
E x˙ (t ) = (Ai + Bi Ki )x(t ) + Adi x(t − d) + Bωi ω(t ) z (t ) = (Ci + Di Ki )x(t ).
(22)
Following the similar philosophy as that in [41], we represent the system (1) to the following equivalent form
E¯ x˙¯ (t ) = A¯ i x(t ) + A¯ di x(t − d) + B¯ ωi ω(t ) z (t ) = C¯ i x(t )
where y(t ) = E x˙ (t ), x¯ (t ) =
x(t ) y(t )
, E¯ =
Ei 0
(23)
0 0
, A¯ i =
0 Ai + Bi Ki
I
, A¯ di =
0 Adi
0 0
, B¯ ωi =
0 Bωi
, C¯ i = Ci + Di Ki 0 . Then, by Theorem 1, it is easy to see that the system (22) is stochastically admissible with H∞ performance γ for any constant ¯ if there exist symmetric positive-definite matrices P¯i , Q¯ i , U¯ , V¯ , Q¯ , Z¯ , W ¯ and matrices S¯i time-delay d satisfying 0 ≤ d ≤ d, satisfying (10) for every i ∈ S . As a particular case, we set ¯Pi = Pi 0 , ¯Pj = Pj 0 , ¯R = R 0 , 0 εI 0 εI 0 Fi S I Qi 0 Q 0 S¯i = i , Q¯ i = , Q¯ = , 0 I 0 εI 0 εI U 0 V 0 Z 0 ¯ = W 0 U¯ = , V¯ = , Z¯ = , W 0 εI 0 εI 0 εI 0 εI −I
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J. Wang et al. / Nonlinear Analysis: Hybrid Systems 8 (2013) 1–12
where Pi , Fi ∈ Rn×n are nonsingular matrices with Pi symmetric and positive-definite, R ∈ Rn×(n−r ) satisfies E T R = 0 and rank R = n − r , ε > 0. It is obvious that E¯ T R¯ = 0 and R¯ ∈ R2n×(2n−r ) is with full column rank. Letting ε → 0+ , we have
Φ 1i ∗ ∗ ∗ ∗ ∗ ∗
E T ZE
Φ2i
−FiT − Fi
k 0
∗
Φ7i
∗ ∗
∗ ∗
∗
∗
∗
Φ10i
FiT Adi
FiT Bωi
0
0
FiT Adi
FiT Bωi
¯ dZ
¯ dW
0
0
0
0
0
0 0 Z
0 0
0 0
0
0
E T WE 1−k
Φ9i ∗
−γ 2 I
∗
∗
−
k
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
−
W 1−k
∗
<0
0
0
(24a)
−I
s
πij Qj ≤ Q
(24b)
j =1
where
Φ1i =
s
¯ − πij E T Pj E + FiT (Ai + Bi Ki ) + (Ai + Bi Ki )T Fi + Qi + U + dQ
E T ZE
j=1
k
,
Φ2i = E T Pi + Si RT − FiT + (Ai + Bi Ki )T Fi , Φ7i = −U + V − Φ9i = −Qi − V −
E T ZE k
−
E T WE 1−k
,
E T WE 1−k
,
Φ10i = (Ci + Di Ki )T
and R ∈ Rn×(n−r ) is any matrix with full column satisfying E T R = 0. Since det(sE −(Ai +Bi Ki )) = det(sE T −(Ai +Bi Ki )T ), the pairs (E , (Ai +Bi Ki )) are regular, impulse-free if and only if the pairs T (E , (Ai + Bi Ki )T ) are regular, impulse-free for every i ∈ S . Moreover, since the solution of det(sE − (Ai + Bi Ki ) − e−ds Adi ) = 0 is the same as that of det(sE T − (Ai + Bi Ki )T − e−ds ATdi ) = 0 and the
∥G(s)∥∞ = sup σmax {(Ci + Di Ki )(jωE − (Ai + Bi Ki ) − e−djω Adi )−1 Bωi } ω∈[0,∞)
is equal to
∥H (s)∥∞ = sup σmax {BTωi (jωE T − (Ai + Bi Ki )T − e−djω ATdi )−1 (Ci + Di Ki )T } ω∈[0,∞)
as long as the regularity, absence of impulses and stability with H∞ performance are the only concern, system (22) is equivalent to the system E T x˙ (t ) = (Ai + Bi Ki )T x(t ) + ATdi x(t − d) + (Ci + Di Ki )T ω(t ) z (t ) = BTωi x(t ).
(25)
Hence, replacing E , (Ai + Bi Ki ), Adi , Bωi and (Ci + Di Ki ) in (24) with E T , (Ai + Bi Ki )T , ATdi , (Ci + Di Ki )T and BTωi , respectively, and setting Ki = Ti Fi−1 yields (20). Remark 3. When γ is given, Theorem 2 provides a method of designing the H∞ controller for singular Markovian jump time-delay system (1). On the other hand, when γ is unknown, a minimum bound of γ can be obtained by solving the following LMI-based optimization problem min γ s.t. LMI (20a), (20b) withPi > 0, Qi > 0, U , V , Q , Z , W > 0 for i ∈ S which will be shown in Examples 1 and 3.
(26)
J. Wang et al. / Nonlinear Analysis: Hybrid Systems 8 (2013) 1–12
9
3.3. Delay-dependent robust H∞ controller Consider the following uncertain singular Markovian jump time-delay system
E x˙ (t ) = (Ai (t ) + 1Ai (t ))x(t ) + (Adi (t ) + 1Adi (t ))x(t − d) + (Bi (t ) + 1Bi (t ))u(t ) + Bω (rt )ω(t ) z (t ) = C (rt )x(t ) + D(rt )u(t ).
(27)
The admissible parameter uncertainties are assumed to be norm-bounded, which satisfy
1Ai (t )
1Adi (t )
1Bi (t ) = Mi ∆i (t ) N1i
N2i
N3i
(28)
with Mi , N1i , N2i and N3i are known real constant matrices with appropriate dimensions and ∆i satisfy
∆Ti ∆i
≤ I , ∀t ≥ 0.
Remark 4. By the routine process to deal with the norm-bounded uncertainties in [42], the results in this paper can all be extended to solve the problem of delay-dependent robust H∞ control. Theorem 3. For given scalar d¯ > 0, γ > 0 and 0 < k < 1, the uncertain singular Markovian jump time-delay system (27) is ¯ if there exist symmetric stochastically admissible with H∞ performance γ for any constant time-delay d satisfying 0 ≤ d ≤ d, positive-definite matrices Pi , Qi , U , V , Q , Z , W and matrices Si , Ti , Fi and scalars αi > 0, βi > 0 such that for every i ∈ S , (20b) and
Θ 1i ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗
Υ2i
EZE T
Θ2i
k 0
∗
Υ7i
∗ ∗
∗ ∗
∗
∗
FiT ATdi
Υ5i
0
0
Bωi
Θ2i
T FiT N2i
FiT ATdi
Υ5i
¯ dZ
¯ dW
0
Θ2i
T FiT N2i
0
0
0
0
0
0
0
0 0 Z
0 0
0 0
0 0
0 0
0
0
0
0
0
0
0
−I ∗ ∗
0
0 0
EWE T 1−k
Θ3i ∗
−γ 2 I
∗
∗
−
k
∗
∗
∗
∗
∗
∗ ∗ ∗
∗ ∗ ∗
∗ ∗ ∗
∗ ∗ ∗
∗ ∗ ∗
−
W 1−k
∗ ∗ ∗
−αi I ∗
<0
−βi I
hold, where Θ1i = Υ1i + α , Θ2i = − − Fi , Θ3i = Υ9i + β , Θ2i = (N1i Fi + N3i Ti )T , and Υ1i , Υ2i , Υ5i , Υ7i , Υ9i , n×(n−r ) R∈R are given in Theorem 2. Moreover, the controller gain can be given by T i Mi Mi
FiT
T i Mi Mi
Ki = Ti Fi−1 .
(29)
4. Numerical examples In this section, some numerical examples are presented to show the usefulness and effectiveness of the results developed in this paper. Example 1. Consider the Markovian jump time-delay system (4) with E = I, two modes and the following parameters [32]:
−3.5 0.8 −2.5 0.3 , A2 = −0.6 −3.3 1.4 −0.1 −0.9 −1.3 −2.8 0.5 Ad1 = , Ad2 = −0.7 −2.1 −0.8 −1.0 0.5 0.3 Bω1 = , Bω2 = 0.4 0.2 C1 = 0.1 0.3 , C2 = 0.2 0.15
A1 =
that is, the singular Markovian jump time-delay system (4) reduces to a regular Markovian jump time-delay system. First, we consider the singular time-delay system (4) with ω(t ) = 0 and suppose π22 = −0.8. Table 1 presents the comparison results with various π11 , which show that the delay-dependent stochastic stability condition in Theorem 1 gives less conservative results than those in [32,43–45]. Next, we consider the Markovian jump time-delay system (4) and choose π11 = −0.2 and π22 = −0.8. Table 2 gives the comparison results on minimum allowed γ for various d¯ by different methods. It is clear that the results of Theorem 1 are
10
J. Wang et al. / Nonlinear Analysis: Hybrid Systems 8 (2013) 1–12 Table 1 Comparisons of maximum allowed d¯ for Example 1.
π11
−0.4
−0.55
−0.7
−0.85
−1.00
[43–45] [32] Theorem 1(k = 0.1) Theorem 1(k = 0.3) Theorem 1(k = 0.5)
0.5044 0.6078 0.6078 0.6220 0.6322
0.5025 0.5894 0.5894 0.6020 0.6120
0.5010 0.5768 0.5770 0.5889 0.5981
0.4998 0.5675 0.5682 0.5796 0.5881
0.4987 0.5603 0.5617 0.5726 0.5805
Table 2 Comparisons of minimum allowed γ for Example 1. d¯
0.2
0.3
0.4
0.5
0.6
[43,44] [32] Theorem 1(k = 0.1) Theorem 1(k = 0.3) Theorem 1(k = 0.5)
0.0642 0.0620 0.0447 0.0469 0.0487
0.0971 0.0888 0.0566 0.0553 0.0562
0.2060 0.1465 0.0798 0.0744 0.0699
3.2465 0.2544 0.1452 0.1360 0.1248
– 0.6938 0.4802 0.3966 0.3519
Table 3 Comparisons of maximum allowed d¯ for Example 2.
π11
0.60
0.55
0.50
0.45
0.40
0.35
[21] Theorem 1 (k = 0.1) Theorem 1 (k = 0.3) Theorem 1 (k = 0.5)
1.0808 1.1362 1.15 1.1588
1.0842 1.1360 1.1499 1.1587
1.0878 1.1358 1.1498 1.1587
1.0918 1.1356 1.1497 1.1587
1.0960 1.1354 1.1496 1.1587
1.1006 1.1352 1.1496 1.1587
better than those in [32,43–45]. In particular, when d¯ = 0.6, the methods of [43,44] fail, the minimum allowed γ = 0.6938 in [32], whereas, the minimum allowed γ = 0.4802 (k = 0.1), γ = 0.3966 (k = 0.3), γ = 0.3519 (k = 0.5) in this paper. Example 2. Consider the singular Markovian jump time-delay system (2) with two modes, that is, S = {1, 2}. The mode −π11 π11 switching is governed by the rate matrix 0.3 −0.3 . The system parameters are described as follows [21]: 0.4972 A1 = 0
Ad1 =
0.5121 A2 = 0
0 , −0.9541
−1.010 0
1.5415 , 0.5449
Ad2 =
0 , −0.7215
−0.8521
1 E= 0
0 , 0
1.9721 . 0.4321
0
When π11 = 0.45, according to Theorem 1, it can be shown that the system is stochastically admissible for any constant time delay d satisfying 0 < d < 1.1587. Table 3 presents the maximum allowed time-delay d¯ for different π11 > 0. It is clear that the results of Theorem 1 are less conservative than those in [21]. But the result of [20] fails to determine the stochastic stability of the above system. From Examples 1 and 2, we conclude that the results in this paper are less conservative than some existing studies. Example 3. Consider the singular Markovian jump time-delay system (1) with the following parameters:
−3 1 0 −2.5 0.5 −0.1 −2.5 −4 , −3.5 0.3 A1 = 0.3 A2 = 0.1 −0.1 0.3 −3.8 −0.1 1 −2 −0.1669 0.0802 1.682 0.1053 −0.1948 0.0755 Ad1 = −0.8162 −0.9373 0.5936 , Ad2 = −0.1586 2.0941 0.6357 0.7902 0.8709 −0.5266 1 0 0 1 −0.6 E= 0 1 0 , Bω 1 = 0 , Bω2 = 0.5 ,
0
B1 =
0
1.5 1.0 , 1.0
C2 = 0
0
1
1
1.0 0.5 , 1.5
B2 = 0.6 ,
0
D1 = 0.1,
C1 = 0.5
−0.1
D2 = 0.3.
1
−0.6855 −0.2684 −1.1883
J. Wang et al. / Nonlinear Analysis: Hybrid Systems 8 (2013) 1–12
The mode switching is governed by the rate matrix
−0.8 0.3
0.8 −0.3 . Using Theorem 1 and letting k
11
= 0.5, it can be shown that
the system is stochastically admissible for any constant time delay d satisfying 0 < d < 2.4295. As an example, we assume d¯ = 0.8, the minimum attenuation level γmin obtained from Theorem 2 is γmin = 0.8635 and the delay-dependent state feedback controller has the following gains: K1 = −2.6889
−1.6278
K2 = −0.2408
−1.0242
−3.5553 , −2.1675 .
Through this example, we found that our results are effective. 5. Conclusion Delay-dependent H∞ control for singular Markovian jump systems with time-delay has been discussed in this paper. Based on the delay subinterval decomposition method, new bounded real lemma for unforced singular Markovian jump systems has been established. The explicit expression of the desired controller gain is also presented. All the results reported in this paper are formulated in terms of strict LMIs, which can be readily solved using standard numerical software. Some numerical examples are provided to show the effectiveness of the proposed methods. Furthermore, the delay subinterval decomposition method used in this paper can also be extended to deal with the problem of stabilization, H∞ filtering, L2 − L∞ filtering, state estimation for full and partial knowledge of transition probabilities of Markovian jump systems, and so on. Acknowledgments The second author’s work was supported by the National Natural Science Foundation of People’s Republic of China under Grant 61104094, the Natural Science Foundation of Zhejiang Province under Grant Y1080690 and Y2110690, and the Science Technology Department of Zhejiang Province under Grant 2012C21012. References [1] E.K. Boukas, Q. Zhang, G. Yin, Robust production and maintenance planning in stochastic manufacturing systems, IEEE Trans. Automat. Control 40 (1998) 1098–1102. [2] M. Huang, S. Dey, Stability of Kalman filtering with Markovian packet losses, Automatica 43 (2007) 598–607. [3] L.J. Mirman, O.F. Morand, K.L. Reffett, A qualitative approach to Markovian equilibrium in infinite horizon economies with capital, J. Econom. Theory 139 (2008) 75–98. [4] R.C. Tsaur, A fuzzy time series-Markov chain model with an application to forecast the exchange rate between the Taiwan and US dollar, Int. J. Innovative Comput. Inform. Control 8 (2012) 4931–4942. [5] Q. Ahmed, A. 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Contents lists available at SciVerse ScienceDirect
Nonlinear Analysis: Hybrid Systems journal homepage: www.elsevier.com/locate/nahs
Delay-dependent H∞ control for singular Markovian jump systems with time delay Junru Wang a , Huijiao Wang a,∗ , Anke Xue b , Renquan Lu b a
Institute of Automation, College of Mechanical Engineering and Automation, Zhejiang Sci-Tech University, Hangzhou 310018, PR China
b
Institute of Information & Control, Hangzhou Dianzi University, Hangzhou 310018, PR China
article
info
Article history: Received 23 May 2012 Accepted 28 August 2012 Keywords: Singular Markovian jump system H∞ control Time delay Delay subinterval decomposition approach
abstract This paper deals with the problem of delay-dependent H∞ control for singular Markovian jump systems with time delay. Based on the delay subinterval decomposition approach, a new Lyapunov–Krasovskii functional is proposed to develop the new delay-dependent bounded real lemma (BRL), which ensures the considered system to be regular, impulsefree and stochastically stable with given H∞ performance index γ . Based on this new BRL, the explicit expression of the desired controller gains is also presented by solving a set of strict LMIs. Some numerical examples are given to show the effectiveness and less conservativeness of the proposed methods. © 2012 Elsevier Ltd. All rights reserved.
1. Introduction Markovian jump systems, as a special class of stochastic hybrid systems, have the advantage of better describing physical systems with abrupt variations. Applications of this kind of systems can be found in manufacturing systems [1], networked control systems [2], economics systems [3,4], air intake system [5], and other practical systems. Recently, a lot of attention has been paid to the stability analysis and controller synthesis for Markovian jump systems [6–13]. On the other hand, singular systems have been extensively studied in the past years due to the fact that singular systems better describe physical systems than state-space ones. When singular systems experience abrupt changes in their structures, it is natural to model them as singular Markovian jump systems [14–18]. Time delays always exist in many dynamical systems and delays are sources of poor stability and performance of a system. For example, it is a very common phenomenon that feedback current is time delay in the DC motor in position control servomechanisms [14]. When time delay is considered, the results in [14] cannot be applied to analyze the dynamical model of the DC motor. The singular Markovian jump systems with time delay may also have applications in other practical systems. So studies of the stability criteria and the performance for singular Markovian jump systems with delays are of theoretical and practical importance. During the recent years, much attention has been devoted to the study of singular Markovian jump systems with time delay. The criteria can be generally classified into two categories: delay-independent [19] and delay-dependent ones [20–22]. Generally, the delay-dependent case is less conservative than delay-independent ones, especially when the delay is comparatively small. So more attention has been paid to delay-dependent criteria. Various methods have been proposed to obtain delay-dependent criteria, such as properly chosen Lyapunov–Krasovskii functionals [23], the slack variable method [11,24], the delay decomposition approach [25] etc. Among these, the delay decomposition approach provides much less conservative delay-dependent stability criteria than the other techniques. The idea of delay partitioning has appeared in [26]. As mentioned in [26], some free-weighting matrices (or slack variable) may be redundant and
∗
Corresponding author. E-mail addresses: [email protected] (J. Wang), [email protected] (H. Wang), [email protected] (A. Xue), [email protected] (R. Lu).
1751-570X/$ – see front matter © 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.nahs.2012.08.003
2
J. Wang et al. / Nonlinear Analysis: Hybrid Systems 8 (2013) 1–12
they will increase the computational burden in the case of stability analysis for deterministic delay systems by constant Lyapunov–Krasovskii functionals. The H∞ control problem for time-delay singular systems was investigated in [27–30]. For Markovian jump singular systems, since stability, regularity, impulse elimination and system switch should be considered at the same time, it is difficult to discuss Markovian jump singular systems. The robust stability and H∞ control problems for discrete Markovian jump systems with mode-dependent time-delay were discussed in [31]. Based on the LMI approach, the H∞ control [23], H∞ filtering [32] and dissipative control [33] for singular Markovian jump systems with time delay were discussed. For singular Markovian jump systems with time-varying delays, using the free-weighting-matrix approach, stability and stabilization [22], delay-depend robust control [24], and H∞ filtering [34] were discussed. To the best of our knowledge, the delay-dependent H∞ control problem for singular Markovian jump systems with time-delay has not yet been fully investigated. There has room for improvement. Motivated by the idea of delay partitioning, this paper deals with the problem of delay-dependent H∞ control for singular Markovian jump systems with time-delay. The purpose is to design the state feedback controller such that the resulted closeloop system is regular, impulse-free and stochastically stable with H∞ performance index γ . Based on the delay subinterval decomposition approach, in which the interval [t − d, t ] is decomposed into [t − d, t − kd] and [t − kd, t ](0 < k < 1) and each subinterval has different Lyapunov weighted matrices, a new Lyapunov–Krasovskii functional for singular Markovian jump systems is presented and a novel delay-dependent bounded real lemma (BRL) is established via LMIs. Based on this new BRL, the delay-dependent H∞ control problem is solved and a strict LMI-based design method of the desired controller is proposed. The effectiveness of the method is illustrated by some numerical examples. Notations. Through this paper, the superscripts ‘‘T ’’ and ‘‘−1’’ stand for the transpose of a matrix and the inverse of a matrix; Rn denotes n-dimensional Euclidean space; Rn×m is the set of all real matrices with m rows and n columns; ∥ · ∥ stands for the Euclidean norm for a vector. E {·} denotes the expectation operator. For a symmetric matrix, ∗ denotes the matrix entries implied by symmetry. 2. Problem formulation and preliminaries Consider the following singular Markovian jump time-delay system
E x˙ (t ) = A(rt )x(t ) + Ad (rt )x(t − d) + B(rt )u(t ) + Bω (rt )ω(t ) z (t ) = C (rt )x(t ) + D(rt )u(t ) x(t ) = φ(t ), t ∈ [−d¯ , 0]
(1)
where x(t ) ∈ Rn is the state vector, u(t ) ∈ Rm is the control input, z (t ) ∈ Rq is the system output, ω(t ) ∈ Rp is the deterministic disturbance input which belongs to L2 [0, ∞). Here L2 [0, ∞) stands for the space of square integrable vector functions ¯ φ(·) is the initial condition defined on over the interval [0, ∞). d is an unknown but constant delay satisfying 0 ≤ d ≤ d. [−d¯ , 0]. The matrix E ∈ Rn×n may be singular and we assume that rank E = r ≤ n. {rt , t ≥ 0} is a continuous-time discretestate Markov process with right continuous trajectory values in a finite set S = {1, 2, . . . , s}. A(rt ), Ad (rt )B(rt ), Bω (rt ), C (rt ) and D(rt ) are known real constant matrices with appropriate dimensions for each rt ∈ S . The transition probability matrix Π = {πij } (i, j ∈ S ) given by
P{rt +h = j|rt = i} =
πij h + o(h), i ̸= j. 1 + πij h + o(h), i = j
o(h)
where h > 0 and limh→0 h = 0, πij ≥ 0, for j ̸= i, is the transition rate from mode i at time t to mode j at time t + h and πii = − sj=1,j̸=i πij . For the singular Markovian jump time-delay system
E x˙ (t ) = A(rt )x(t ) + Ad (rt )x(t − d), x(t ) = φ(t ), t ∈ [−d¯ , 0]
(2)
we have the following definition. Definition 1 ([17,32]). (1) For a given scalar d¯ > 0, the nominal singular Markovian jump time-delay system (2) is said to be regular and impulsefree for any constant time delay d satisfying 0 ≤ d ≤ d¯ if the pairs (E , A(rt )) and (E , A(rt ) + Ad (rt )) are regular and impulse-free for every rt ∈ S . (2) The singular Markovian jump time-delay system (2) is said to be stochastically stable, if there exists a scalar M (r0 , φ(·)) such that T
lim E
T →∞
∥x(t )∥ dt |r0 , x(s) = φ(s) ≤ M (r0 , φ(·)). 2
0
(3)
J. Wang et al. / Nonlinear Analysis: Hybrid Systems 8 (2013) 1–12
3
(3) The singular Markovian jump time-delay system (2) is said to be stochastically admissible, if it is regular, impulse-free and stochastically stable. Remark 1. The regularity and absence of impulse of the pair (E , A(rt )) for every rt ∈ S ensures the system (2) with time delay d ̸= 0 to be regular and impulse free, while the fact that for every rt ∈ S , the pair (E , A(rt ) + Ad (rt )) is regular and impulse free ensures the system (2) with time delay d = 0 to be regular and impulse free. The unforced singular Markovian jump time-delay system of system (1) is as follows:
E x˙ (t ) = A(rt )x(t ) + Ad (rt )x(t − d) + Bω (rt )ω(t ), z (t ) = C (rt )x(t ) x(t ) = φ(t ), t ∈ [−d¯ , 0].
(4)
Definition 2. Given a scalar γ > 0, the singular Markovian jump time-delay system (4) is said to be stochastically admissible with H∞ performance γ , if the system with ω(t ) ≡ 0 is stochastically admissible and under zero initial condition, satisfies ∞
z T (t )z (t )dt
E
≤ γ2
∞
0
ωT (t )ω(t )dt
(5)
0
for any non-zero ω(t ) ∈ L2 [0, ∞). The H∞ control problem to be addressed here is formulated as follows: for given scalar γ > 0 and the singular Markovian jump time-delay system (1), design feedback control law u(t ) = K (rt )x(t )
(6)
such that the resulting closed-loop system is stochastically admissible with H∞ performance γ . For notational simplicity, in this paper, when r (t ) = i, i ∈ S , a matrix M (rt ) will be denoted by Mi ; for example, A(rt ) is denoted by Ai , Ad (rt ) by Adi and so on. The following lemmas will play an important role in the subsequent sections. Lemma 1 ([35]). For any constant matrix X ∈ Rn×n , X = X T > 0, scalar r > 0, and vector function x˙ : [−r , 0] → Rn such that the following integration is well defined, then
0
x˙ T (t + s)X x˙ (t + s)ds ≤ xT (t )
−r
xT ( t − r )
−X
−r
X
X −X
x(t ) . x(t − r )
(7)
Lemma 2 ([32]). The singular Markovian jump system E x˙ (t ) = Ai x(t )
(8)
is stochastically admissible if and only if there exist symmetric positive-definite matrices Pi and matrices Si such that for every i ∈ S, s
πij E T Pj E + E T Pi Ai + Si RT Ai + ATi Pi E + ATi RSiT < 0
(9)
j =1
where R ∈ Rn×(n−r ) is any matrix with full column rank and satisfies E T R = 0. 3. Main results In this section, the H∞ control problem will be investigated for the singular Markovian jump time-delay system (1) in terms of the delay subinterval decomposition approach. To this end, we first provide the following delay-dependent bounded real lemma (BRL) for (4), which will play a key role in the derivation of our main results. 3.1. Delay-dependent bounded real lemma Theorem 1. For given scalar d¯ > 0, γ > 0 and 0 < k < 1, the singular Markovian jump time-delay system (4) is stochastically ¯ if there exist symmetric positive-definite admissible with H∞ performance γ for any constant time-delay d satisfying 0 ≤ d ≤ d,
4
J. Wang et al. / Nonlinear Analysis: Hybrid Systems 8 (2013) 1–12
matrices Pi , Qi , U , V , Q , Z , W and matrices Si such that for every i ∈ S ,
Γ1i ∗ ∗ ∗ ∗ ∗
Γ4i Γ2i ∗
Γ6i Γ5i Γ3i
Γ7i
∗
∗
0 0
¯ Ti Z dA 0 ¯ Tdi Z dA
¯ Ti W dA 0 ¯ Tdi W dA
∗
−γ 2 I
¯ Tωi Z dB
¯ Tωi W dB
∗
∗
∗
−
∗
∗
∗
∗
∗
∗
∗
∗
Z
0
k
−
W 1−k
∗
CiT 0 0
0
<0 0
(10a)
0
−I
s
πij Qj ≤ Q
(10b)
j =1
where s
Γ1i =
¯ − πij E T Pj E + E T Pi Ai + Si RT Ai + ATi Pi E + ATi RSiT + Qi + U + dQ
j =1
E T ZE
Γ2i = −U + V −
k
E T WE
Γ3i = −Qi − V − E T ZE
Γ4i =
k
,
−
1−k
Γ5i =
E T WE 1−k
,
,
E T ZE k
,
Γ6i = (E T Pi + Si RT )Adi
Γ7i = (E T Pi + Si RT )Bωi
E T WE 1−k
and R ∈ Rn×(n−r ) is any matrix with full column satisfying E T R = 0. Proof. We first show the singular Markovian jump time-delay system (4) is regular and impulse-free. Since rank E = r ≤ n, there must exist two invertible matrices G and H ∈ Rn×n such that
¯E = GEH = Ir
0
0 . 0
(11)
Then, R can be parameterized as T
R=G
0
¯ Φ
¯ ∈ R(n−r )×(n−r ) is any nonsingular matrix. where Φ Similar to (11), we define A¯ i = GAi H =
A¯ i,11 A¯ i,21
A¯ i,12 A¯ i,22
P¯ P¯ i = G−T Pi G−1 = ¯ i,11 Pi,21
S¯i = H T Si =
S¯i,11 S¯i,21
P¯ i,12 P¯ i,22
(12)
for every i ∈ S . Pre-multiplying and post-multiplying Γ1i < 0 by H T and H respectively, we have
¯ S¯iT,21 + S¯i,21 Φ ¯ T A¯ i,22 < 0 A¯ Ti,22 Φ which implies A¯ i,22 is nonsingular for every i ∈ S and thus the pair (E , Ai ) is regular and impulse-free for every i ∈ S . By (10a), it is easy to see that
Γ1i ∗ ∗
Γ4i Γ2i ∗
Γ6i Γ5i < 0. Γ3i
(13)
J. Wang et al. / Nonlinear Analysis: Hybrid Systems 8 (2013) 1–12
I Pre-multiplying and post-multiplying on both sides of (13) by
I
I I 0
0 0
0 I
I and
0 I 0
I I
5
0 0 I
, we have
Γ1i + Γ2i + Γ3i + Γ4i + Γ5i + Γ6i + Γ4iT + Γ5iT + Γ6iT < 0. Hence, s
πij E T Pj E + (E T Pi + Si RT )(Ai + Adi ) + (Ai + Adi )T (Pi E + RSiT ) < 0.
(14)
j =1
According to Lemma 2, (14) implies that the pair (E , Ai + Adi ) is regular and impulse-free for every i ∈ S . Thus, by Definition 1, ¯ the system (4) is regular and impulse-free for any constant time-delay d satisfying 0 ≤ d ≤ d. Next, we will show the stochastic stability of the system (4). For t ≥ d, define the following stochastic Lyapunov candidate for the system (4) with ω(t ) ≡ 0, 5
V (xt , rt , t ) =
Vm (xt , rt , t )
(15)
m=1
where V1 (xt , rt , t ) = xT (t )E T P (rt )Ex(t ), V2 (xt , rt , t ) =
t
xT (s)Q (rt )x(s)ds, t −d
V3 (xt , rt , t ) =
t
xT (s)Ux(s)ds +
t −kd
V4 (xt , rt , t ) = d¯
0
−d
t
−kd
V5 (xt , rt , t ) =
xT (s)Vx(s)ds, t −d
0
t −kd
x˙ T (s)E T ZE x˙ (s)dsdθ + d¯
−kd
t +θ
−d
t
x˙ T (s)E T WE x˙ (s)dsdθ , t +θ
t
xT (s)Qx(s)dsdθ , t +θ
symmetric positive-definite matrices Pi , Qi , U , V , Q , Z , W for every i ∈ S are matrices to be determined. Let A be the weak infinitesimal generator of the random process {(xt , rt ), t ≥ 0}. Then, for every i ∈ S and t ≥ d, we have A V (xt , i, t ) ≤ 2xT (t )(E T Pi + Si RT )E x˙ (t ) + xT (t )
s
¯ )x(t ) πij E T Pj E x(t ) + xT (t )(Qi + U + dQ
j =1
− x (t − d)(Qi + V )x(t − d) + T
t
x (s) T
s
t −d
πij Qj x(s)ds − xT (t − kd)(U − V )x(t − kd)
j=1
+ kd¯ 2 x˙ T (t )E T ZE x˙ (t ) + (1 − k)d¯ 2 x˙ T (t )E T WE x˙ (t ) t t −kd − d¯ x˙ T (s)E T ZE x˙ (s)ds − d¯ x˙ T (s)E T WE x˙ (s)ds − t −kd
t −d
t
xT (s)Qx(s)ds. t −d
From (10b), it is clear that
t
x (s) T
t −d
s
πij Qj x(s)ds ≤
t
xT (s)Qx(s)ds.
t −d
j =1
It follows from Lemma 1 that d¯
t
T
x(t ) x˙ (s)E ZE x˙ (s)ds ≤ x ( t − kd) k t −kd T
T
1
−E T ZE E T ZE
E T ZE −E T ZE
x(t ) x(t − kd)
and d¯
t −kd
x˙ T (s)E T WE x˙ (s)ds ≤ t −d
x(t − kd) 1 − k x(t − d) 1
T
−E T WE E T WE
E T WE −E T WE
x(t − kd) . x(t − d)
Noting E T R = 0, we have A V (xt , i, t ) ≤ ηT (t )Θi η(t )
(16)
6
J. Wang et al. / Nonlinear Analysis: Hybrid Systems 8 (2013) 1–12
where ηT (t ) = xT (t )
xT (t − kd)
Γ4i Γ2i ∗
Xi
Θi =
∗ ∗
Yi
xT (t − d)
Γ5i
(17)
Zi
with Xi = Γ1i + ATi [kd¯ 2 E T ZE + (1 − k)d¯ 2 E T WE ]Ai Yi = Γ6i + ATi [kd¯ 2 E T ZE + (1 − k)d¯ 2 E T WE ]Adi Zi = Γ3i + ATdi [kd¯ 2 E T ZE + (1 − k)d¯ 2 E T WE ]Adi .
Using (10a), it is easy to see that there exists a scalar λ > 0 such that for every i ∈ S A V (xt , i, t ) ≤ −λ∥x(t )∥2 .
Therefore, by Dynkin’s formula, we get for any t ≥ d
E V (xt , i, t ) − E V (xd , rd , d) ≤ −λE
t
∥x(s)∥2 ds d
which yields t
∥x(s)∥2 ds ≤ λ−1 E V (xd , rd , d).
E
(18)
d
Following the similar line to that of [32], it is clear that there exists a scalar β such that t
∥x(s)∥2 ds = E
E
d
0
∥x(s)∥2 ds + E
t
0
d
∥x(s)∥2 ds ≤ β E ∥φ∥2d¯ .
According to Definition 1, the system (4) with ω(t ) ≡ 0 is stochastically stable for any constant time delay d satisfying ¯ 0 ≤ d ≤ d. In the following, we consider the stochastic Lyapunov candidate (15) and the following index for the system (4): Jz ω (t ) = E
t
[z (s) z (s) − γ ω(s) ω(s)]ds . T
2
T
0
Under zero initial condition, it is easy to see that t
[z (s)T z (s) − γ 2 ω(s)T ω(s) + A V (xs , i, s)]ds 0 t T ≤E ς (s)Ξi ς (s)ds
Jz ω (t ) = E
0
where ς T (t ) = x (t )
T
Xi + CiT Ci
Ξi =
∗ ∗ ∗
xT (t − kd)
Γ4i Γ2i ∗ ∗
Yi
Γ5i Zi
∗
xT (t − d)
Γ7i
ωT (t )
0 . 0 2 −γ I
(19)
Hence, by Schur complement and from (10a), we have Jz ω (t ) < 0, for all t > 0. Therefore, for any non-zero ω(t ) ∈ L2 [0, ∞), (5) holds. Then according to Definition 2, the singular Markovian jump time-delay system (4) is stochastically admissible with H∞ performance γ . This completes the proof. Remark 2. We can divide [0, d¯ ] into N segments [0, N1 d¯ ], [ N1 d¯ , N2 d¯ ], . . . , [ NN−1 d¯ , d¯ ], the correspondent stability results can also be derived for singular Markovian jump systems. It is also called the delay decomposition approach or the delay partitioning method, which is widely used in time-delay systems derived [36,37], Markovian jump systems [25], singular systems [38], T –S fuzzy systems [39] and stochastic neural networks [40], etc.
J. Wang et al. / Nonlinear Analysis: Hybrid Systems 8 (2013) 1–12
7
3.2. Delay-dependent H∞ controller Theorem 2. For given scalar d¯ > 0, γ > 0 and 0 < k < 1, the singular Markovian jump time-delay system (1) is stochastically ¯ if there exist symmetric positive-definite admissible with H∞ performance γ for any constant time-delay d satisfying 0 ≤ d ≤ d, matrices Pi , Qi , U , V , Q , Z , W and matrices Si , Ti , Fi such that for every i ∈ S , EZE T
Υ1i
Υ2i
∗
−FiT − Fi
k 0
∗
∗
Υ7i
∗ ∗
∗ ∗
∗ ∗
∗
∗
∗
FiT ATdi
Υ5i
0
0
FiT ATdi
Υ5i
¯ dZ
¯ dW
0
0
0
0
0
0 0 Z
0 0
0 0
0
0
EWE T 1−k
Υ9i ∗
−γ 2 I
∗
∗
−
k
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
s
−
Bωi
∗
<0
0
W 1−k
0
(20a)
−I
πij Qj ≤ Q
(20b)
j=1
where
Υ1i =
s
¯ − πij EPj E T + Ai Fi + FiT ATi + Bi Ti + TiT BTi + Qi + U + dQ
j=1
EZE T k
,
Υ2i = EPi + Si RT − FiT + Ai Fi + Bi Ti , Υ5i = FiT CiT + TiT DTi , Υ7i = −U + V − Υ9i = −Qi − V −
EZE T k
−
EWE T 1−k
EWE T 1−k
,
,
and R ∈ Rn×(n−r ) is any matrix with full column satisfying ER = 0. Moreover, the controller gain can be given by Ki = Ti Fi−1 .
(21)
Proof. Substituting the state feedback controller (6) to system (1) yields the following closed-loop system
E x˙ (t ) = (Ai + Bi Ki )x(t ) + Adi x(t − d) + Bωi ω(t ) z (t ) = (Ci + Di Ki )x(t ).
(22)
Following the similar philosophy as that in [41], we represent the system (1) to the following equivalent form
E¯ x˙¯ (t ) = A¯ i x(t ) + A¯ di x(t − d) + B¯ ωi ω(t ) z (t ) = C¯ i x(t )
where y(t ) = E x˙ (t ), x¯ (t ) =
x(t ) y(t )
, E¯ =
Ei 0
(23)
0 0
, A¯ i =
0 Ai + Bi Ki
I
, A¯ di =
0 Adi
0 0
, B¯ ωi =
0 Bωi
, C¯ i = Ci + Di Ki 0 . Then, by Theorem 1, it is easy to see that the system (22) is stochastically admissible with H∞ performance γ for any constant ¯ if there exist symmetric positive-definite matrices P¯i , Q¯ i , U¯ , V¯ , Q¯ , Z¯ , W ¯ and matrices S¯i time-delay d satisfying 0 ≤ d ≤ d, satisfying (10) for every i ∈ S . As a particular case, we set ¯Pi = Pi 0 , ¯Pj = Pj 0 , ¯R = R 0 , 0 εI 0 εI 0 Fi S I Qi 0 Q 0 S¯i = i , Q¯ i = , Q¯ = , 0 I 0 εI 0 εI U 0 V 0 Z 0 ¯ = W 0 U¯ = , V¯ = , Z¯ = , W 0 εI 0 εI 0 εI 0 εI −I
8
J. Wang et al. / Nonlinear Analysis: Hybrid Systems 8 (2013) 1–12
where Pi , Fi ∈ Rn×n are nonsingular matrices with Pi symmetric and positive-definite, R ∈ Rn×(n−r ) satisfies E T R = 0 and rank R = n − r , ε > 0. It is obvious that E¯ T R¯ = 0 and R¯ ∈ R2n×(2n−r ) is with full column rank. Letting ε → 0+ , we have
Φ 1i ∗ ∗ ∗ ∗ ∗ ∗
E T ZE
Φ2i
−FiT − Fi
k 0
∗
Φ7i
∗ ∗
∗ ∗
∗
∗
∗
Φ10i
FiT Adi
FiT Bωi
0
0
FiT Adi
FiT Bωi
¯ dZ
¯ dW
0
0
0
0
0
0 0 Z
0 0
0 0
0
0
E T WE 1−k
Φ9i ∗
−γ 2 I
∗
∗
−
k
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
−
W 1−k
∗
<0
0
0
(24a)
−I
s
πij Qj ≤ Q
(24b)
j =1
where
Φ1i =
s
¯ − πij E T Pj E + FiT (Ai + Bi Ki ) + (Ai + Bi Ki )T Fi + Qi + U + dQ
E T ZE
j=1
k
,
Φ2i = E T Pi + Si RT − FiT + (Ai + Bi Ki )T Fi , Φ7i = −U + V − Φ9i = −Qi − V −
E T ZE k
−
E T WE 1−k
,
E T WE 1−k
,
Φ10i = (Ci + Di Ki )T
and R ∈ Rn×(n−r ) is any matrix with full column satisfying E T R = 0. Since det(sE −(Ai +Bi Ki )) = det(sE T −(Ai +Bi Ki )T ), the pairs (E , (Ai +Bi Ki )) are regular, impulse-free if and only if the pairs T (E , (Ai + Bi Ki )T ) are regular, impulse-free for every i ∈ S . Moreover, since the solution of det(sE − (Ai + Bi Ki ) − e−ds Adi ) = 0 is the same as that of det(sE T − (Ai + Bi Ki )T − e−ds ATdi ) = 0 and the
∥G(s)∥∞ = sup σmax {(Ci + Di Ki )(jωE − (Ai + Bi Ki ) − e−djω Adi )−1 Bωi } ω∈[0,∞)
is equal to
∥H (s)∥∞ = sup σmax {BTωi (jωE T − (Ai + Bi Ki )T − e−djω ATdi )−1 (Ci + Di Ki )T } ω∈[0,∞)
as long as the regularity, absence of impulses and stability with H∞ performance are the only concern, system (22) is equivalent to the system E T x˙ (t ) = (Ai + Bi Ki )T x(t ) + ATdi x(t − d) + (Ci + Di Ki )T ω(t ) z (t ) = BTωi x(t ).
(25)
Hence, replacing E , (Ai + Bi Ki ), Adi , Bωi and (Ci + Di Ki ) in (24) with E T , (Ai + Bi Ki )T , ATdi , (Ci + Di Ki )T and BTωi , respectively, and setting Ki = Ti Fi−1 yields (20). Remark 3. When γ is given, Theorem 2 provides a method of designing the H∞ controller for singular Markovian jump time-delay system (1). On the other hand, when γ is unknown, a minimum bound of γ can be obtained by solving the following LMI-based optimization problem min γ s.t. LMI (20a), (20b) withPi > 0, Qi > 0, U , V , Q , Z , W > 0 for i ∈ S which will be shown in Examples 1 and 3.
(26)
J. Wang et al. / Nonlinear Analysis: Hybrid Systems 8 (2013) 1–12
9
3.3. Delay-dependent robust H∞ controller Consider the following uncertain singular Markovian jump time-delay system
E x˙ (t ) = (Ai (t ) + 1Ai (t ))x(t ) + (Adi (t ) + 1Adi (t ))x(t − d) + (Bi (t ) + 1Bi (t ))u(t ) + Bω (rt )ω(t ) z (t ) = C (rt )x(t ) + D(rt )u(t ).
(27)
The admissible parameter uncertainties are assumed to be norm-bounded, which satisfy
1Ai (t )
1Adi (t )
1Bi (t ) = Mi ∆i (t ) N1i
N2i
N3i
(28)
with Mi , N1i , N2i and N3i are known real constant matrices with appropriate dimensions and ∆i satisfy
∆Ti ∆i
≤ I , ∀t ≥ 0.
Remark 4. By the routine process to deal with the norm-bounded uncertainties in [42], the results in this paper can all be extended to solve the problem of delay-dependent robust H∞ control. Theorem 3. For given scalar d¯ > 0, γ > 0 and 0 < k < 1, the uncertain singular Markovian jump time-delay system (27) is ¯ if there exist symmetric stochastically admissible with H∞ performance γ for any constant time-delay d satisfying 0 ≤ d ≤ d, positive-definite matrices Pi , Qi , U , V , Q , Z , W and matrices Si , Ti , Fi and scalars αi > 0, βi > 0 such that for every i ∈ S , (20b) and
Θ 1i ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗
Υ2i
EZE T
Θ2i
k 0
∗
Υ7i
∗ ∗
∗ ∗
∗
∗
FiT ATdi
Υ5i
0
0
Bωi
Θ2i
T FiT N2i
FiT ATdi
Υ5i
¯ dZ
¯ dW
0
Θ2i
T FiT N2i
0
0
0
0
0
0
0
0 0 Z
0 0
0 0
0 0
0 0
0
0
0
0
0
0
0
−I ∗ ∗
0
0 0
EWE T 1−k
Θ3i ∗
−γ 2 I
∗
∗
−
k
∗
∗
∗
∗
∗
∗ ∗ ∗
∗ ∗ ∗
∗ ∗ ∗
∗ ∗ ∗
∗ ∗ ∗
−
W 1−k
∗ ∗ ∗
−αi I ∗
<0
−βi I
hold, where Θ1i = Υ1i + α , Θ2i = − − Fi , Θ3i = Υ9i + β , Θ2i = (N1i Fi + N3i Ti )T , and Υ1i , Υ2i , Υ5i , Υ7i , Υ9i , n×(n−r ) R∈R are given in Theorem 2. Moreover, the controller gain can be given by T i Mi Mi
FiT
T i Mi Mi
Ki = Ti Fi−1 .
(29)
4. Numerical examples In this section, some numerical examples are presented to show the usefulness and effectiveness of the results developed in this paper. Example 1. Consider the Markovian jump time-delay system (4) with E = I, two modes and the following parameters [32]:
−3.5 0.8 −2.5 0.3 , A2 = −0.6 −3.3 1.4 −0.1 −0.9 −1.3 −2.8 0.5 Ad1 = , Ad2 = −0.7 −2.1 −0.8 −1.0 0.5 0.3 Bω1 = , Bω2 = 0.4 0.2 C1 = 0.1 0.3 , C2 = 0.2 0.15
A1 =
that is, the singular Markovian jump time-delay system (4) reduces to a regular Markovian jump time-delay system. First, we consider the singular time-delay system (4) with ω(t ) = 0 and suppose π22 = −0.8. Table 1 presents the comparison results with various π11 , which show that the delay-dependent stochastic stability condition in Theorem 1 gives less conservative results than those in [32,43–45]. Next, we consider the Markovian jump time-delay system (4) and choose π11 = −0.2 and π22 = −0.8. Table 2 gives the comparison results on minimum allowed γ for various d¯ by different methods. It is clear that the results of Theorem 1 are
10
J. Wang et al. / Nonlinear Analysis: Hybrid Systems 8 (2013) 1–12 Table 1 Comparisons of maximum allowed d¯ for Example 1.
π11
−0.4
−0.55
−0.7
−0.85
−1.00
[43–45] [32] Theorem 1(k = 0.1) Theorem 1(k = 0.3) Theorem 1(k = 0.5)
0.5044 0.6078 0.6078 0.6220 0.6322
0.5025 0.5894 0.5894 0.6020 0.6120
0.5010 0.5768 0.5770 0.5889 0.5981
0.4998 0.5675 0.5682 0.5796 0.5881
0.4987 0.5603 0.5617 0.5726 0.5805
Table 2 Comparisons of minimum allowed γ for Example 1. d¯
0.2
0.3
0.4
0.5
0.6
[43,44] [32] Theorem 1(k = 0.1) Theorem 1(k = 0.3) Theorem 1(k = 0.5)
0.0642 0.0620 0.0447 0.0469 0.0487
0.0971 0.0888 0.0566 0.0553 0.0562
0.2060 0.1465 0.0798 0.0744 0.0699
3.2465 0.2544 0.1452 0.1360 0.1248
– 0.6938 0.4802 0.3966 0.3519
Table 3 Comparisons of maximum allowed d¯ for Example 2.
π11
0.60
0.55
0.50
0.45
0.40
0.35
[21] Theorem 1 (k = 0.1) Theorem 1 (k = 0.3) Theorem 1 (k = 0.5)
1.0808 1.1362 1.15 1.1588
1.0842 1.1360 1.1499 1.1587
1.0878 1.1358 1.1498 1.1587
1.0918 1.1356 1.1497 1.1587
1.0960 1.1354 1.1496 1.1587
1.1006 1.1352 1.1496 1.1587
better than those in [32,43–45]. In particular, when d¯ = 0.6, the methods of [43,44] fail, the minimum allowed γ = 0.6938 in [32], whereas, the minimum allowed γ = 0.4802 (k = 0.1), γ = 0.3966 (k = 0.3), γ = 0.3519 (k = 0.5) in this paper. Example 2. Consider the singular Markovian jump time-delay system (2) with two modes, that is, S = {1, 2}. The mode −π11 π11 switching is governed by the rate matrix 0.3 −0.3 . The system parameters are described as follows [21]: 0.4972 A1 = 0
Ad1 =
0.5121 A2 = 0
0 , −0.9541
−1.010 0
1.5415 , 0.5449
Ad2 =
0 , −0.7215
−0.8521
1 E= 0
0 , 0
1.9721 . 0.4321
0
When π11 = 0.45, according to Theorem 1, it can be shown that the system is stochastically admissible for any constant time delay d satisfying 0 < d < 1.1587. Table 3 presents the maximum allowed time-delay d¯ for different π11 > 0. It is clear that the results of Theorem 1 are less conservative than those in [21]. But the result of [20] fails to determine the stochastic stability of the above system. From Examples 1 and 2, we conclude that the results in this paper are less conservative than some existing studies. Example 3. Consider the singular Markovian jump time-delay system (1) with the following parameters:
−3 1 0 −2.5 0.5 −0.1 −2.5 −4 , −3.5 0.3 A1 = 0.3 A2 = 0.1 −0.1 0.3 −3.8 −0.1 1 −2 −0.1669 0.0802 1.682 0.1053 −0.1948 0.0755 Ad1 = −0.8162 −0.9373 0.5936 , Ad2 = −0.1586 2.0941 0.6357 0.7902 0.8709 −0.5266 1 0 0 1 −0.6 E= 0 1 0 , Bω 1 = 0 , Bω2 = 0.5 ,
0
B1 =
0
1.5 1.0 , 1.0
C2 = 0
0
1
1
1.0 0.5 , 1.5
B2 = 0.6 ,
0
D1 = 0.1,
C1 = 0.5
−0.1
D2 = 0.3.
1
−0.6855 −0.2684 −1.1883
J. Wang et al. / Nonlinear Analysis: Hybrid Systems 8 (2013) 1–12
The mode switching is governed by the rate matrix
−0.8 0.3
0.8 −0.3 . Using Theorem 1 and letting k
11
= 0.5, it can be shown that
the system is stochastically admissible for any constant time delay d satisfying 0 < d < 2.4295. As an example, we assume d¯ = 0.8, the minimum attenuation level γmin obtained from Theorem 2 is γmin = 0.8635 and the delay-dependent state feedback controller has the following gains: K1 = −2.6889
−1.6278
K2 = −0.2408
−1.0242
−3.5553 , −2.1675 .
Through this example, we found that our results are effective. 5. Conclusion Delay-dependent H∞ control for singular Markovian jump systems with time-delay has been discussed in this paper. Based on the delay subinterval decomposition method, new bounded real lemma for unforced singular Markovian jump systems has been established. The explicit expression of the desired controller gain is also presented. All the results reported in this paper are formulated in terms of strict LMIs, which can be readily solved using standard numerical software. Some numerical examples are provided to show the effectiveness of the proposed methods. Furthermore, the delay subinterval decomposition method used in this paper can also be extended to deal with the problem of stabilization, H∞ filtering, L2 − L∞ filtering, state estimation for full and partial knowledge of transition probabilities of Markovian jump systems, and so on. 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