Title Stability analysis of markovian jump systems with multiple delay components and polytopic uncertainties Author(s Wang, Q; Du, B; Lam, J; Chen, MZQ Citation Circuits, Systems, And Signal Processing, 212, v 31 n 1, p 143-162 Issued Date 212 URL http://hdlhandlenet/1722/135169 Rights The original publication is available at wwwspringerlinkcom
Circuits Syst Signal Process (212 31:143 162 DOI 117/s34-11-9282- Stability Analysis of Markovian Jump Systems with Multiple Delay Components and Polytopic Uncertainties Qing Wang Baozhu Du James Lam Michael ZQ Chen Received: 19 August 21 / Revised: 22 February 211 / Published online: 26 March 211 The Author(s 211 This article is published with open access at Springerlinkcom Abstract This paper investigates the stability problem of Markovian jump systems with multiple delay components and polytopic uncertainties A new Lyapunov Krasovskii functional is used for the stability analysis of Markovian jump systems with or without polytopic uncertainties Two numerical examples are provided to demonstrate the applicability of the proposed approach Keywords Delay Markovianjumpsystem Stability Polytopic uncertainty This work was supported by National Natural Science Foundation of China under Grant 68432 and by the Hong Kong Research Grants Council under HKU 7137/9E Q Wang ( Department of Electrical and Electronic Engineering, The University of Hong Kong, Pokfulam Road, Hong Kong, Hong Kong e-mail: sunsanqingwang@gmailcom Q Wang School of Information Science and Technology, Sun Yat-sen University, Guangzhou, 51275, China B Du Department of Mathematics, Dalian Maritime University, 11626, Dalian, China e-mail: dubaozhu@gmailcom J Lam MZQ Chen Department of Mechanical Engineering, The University of Hong Kong, Pokfulam Road, Hong Kong, Hong Kong J Lam e-mail: jameslam@hkuhk MZQ Chen e-mail: mzqchen@hkuhk
144 Circuits Syst Signal Process (212 31:143 162 1 Introduction Markovian jump linear systems are hybrid systems with many operation models and each model corresponds to a deterministic dynamic system The switching amongst the system modes is governed by a Markov process This class of systems is often used to model systems whose structures are subject to abrupt changes and their extensive applications have been applied to many physical systems such as manufacturing systems, aircraft control, target tracking, robotics, solar receiver control, power systems, and so on 2, 18 Considerable attention has recently been devoted to the study of Markovian jump linear systems such as controllability 16, 17, stability 5, 1, 37, H control 2, 7, 19, 23, 28, H 2 control 9, H filtering 11, 22, 26, 36, 39, guaranteed cost control 8, and model reduction 3, 42 As time delay is very common in many physical, industrial and engineering systems 1, 24, 25, it is often introduced to model these systems; for example, multiple delays are required to model the RLC circuits with delayed elements 29, networked control systems 14, 2 and nonlinear stochastic systems 1 However, since dynamic systems with time delays are of infinite dimensions, the control of these systems is fairly complicated and the desired performance of the closed-loop systems is difficult to achieve Therefore, the presence of time delays substantially complicates the analytical and theoretical aspects of control system design A survey of recent results developed to analyze the stability of delay systems is given in 38, which includes delay-independent and delay-dependent results Recently, a new simplified and more efficient stability criterion for linear continuous system with multiple components has been used in 13 to improve the delay-dependent stability condition As a result, the study of Markovian jump linear systems with time delay has received attention of many researchers in topics of stability 4, H control 4, 31, 34, and filtering 12, 32, 33 To accurately characterize the physical systems, description of the uncertainty in the system parameters is often needed Two types of uncertainties, norm bounded uncertainties and polytopic ones, are commonly used 6 For jump systems with norm bounded uncertainties, the stability and stabilization 5, 21, 35, 41, H control 15, and robustness H filtering 36 have been addressed The robust stability and H control of Markovian jump systems with constant delay and polytopic uncertainties is studied in 4 In this paper, we are concerned with the robust stability of continuous-time Markovian jump systems with mode-dependent multiple time delays and time-varying polytopic uncertainties The rest of the paper is organized as follows Section 2 describes the continuous-time Markovian jump systems with mode-dependent multiple time delays and polytopic uncertainties, and presents preliminary results The main results are given in Sect 3, which includes the stability and robust stability of continuoustime Markovian jump system with mode-dependent multiple time delays without polytopic uncertainties or with polytopic uncertainties, respectively Two numerical examples are used to illustrate the main results in Sect 4, which is followed by the conclusion in Sect 5 Notation Throughout this paper, the notation X>Y for real symmetric matrices X and Y means that the matrix X Y is positive definite R n is the set of all column
Circuits Syst Signal Process (212 31:143 162 145 vectors with n real entries M T represents the transpose of the matrix M Identity matrices are invariably denoted by I when their dimensions are obvious and otherwise denoted by I n to represent an n n identity matrix, while zero matrices are invariably denoted by The notation col{ } denotes a matrix column with blocks given by the matrices in { } For a matrix M R n m with rank r, the orthogonal complement is defined as a (possibly non-unique m (m r matrix such that MM = and M T M > 2 Problem Formulation Given is a probability space (Ω, F, P where Ω is the sample space, F is the σ - algebra of subsets of the sample space and P is the probability measure defined on F On this probability space (Ω, F, P, we consider the following class of continuoustime Markovian jump systems with mode-dependent multiple time delays and polytopic uncertainties: ( Σ p : ẋ(t = A(r t x(t + A d (r t x t d k (r t, (1 k=1 x(t = ϕ(t, t μ,, where x(t R n is the state, {r t } is a continuous-time homogeneous Markov process with right-continuous trajectories and taking values in a finite set Ñ ={1,,N}, and stationary transition probability matrix Π =π ij i,j Ñ is given by { πij h + o(h i j Pr{r t+h = j r t = i}= 1 + π ii h + o(h i = j where h>, lim h (o(h/h =, and π ij, for i j, is the transition rate from mode i at time t to mode j at time t + h, and π ii = N j=1,j i The d k (r t, k = 1,,r, denote the time delay components in the state and are assumed to satisfy the following conditions: π ij <d k (r t = d ik d ik <, r t = i Ñ, and { μ = max i Ñ k=1 d ik } In (1, ϕ(t is a vector-valued initial continuous function defined on the interval μ, When system Σ d is in mode r t, A(r t and A d (r t are matrix functions of r t
146 Circuits Syst Signal Process (212 31:143 162 For each possible value r t = i, i Ñ, the matrices associated with the ith mode are unknown and assumed to belong to a convex compact set of polytopic type, A(i Ad (i U p (i, where { q U p (i α l (t } q A il A dil, αl (t, α l (t = 1 with A il and A dil being given matrices with appropriate dimensions (2 Definition 1 4 System Σ p is said to be robust stochastically stable if for any A(i and A d (i U p (i, there exists a constant U (ϕ(, r, which is dependent on the initial condition (ϕ(, r and satisfies U (, =, such that E x ( t,ϕ(, r 2 ( dt U ϕ(, r, where E denotes the expectation and x(t,ϕ(, r denotes the solution of system Σ p at time t under the initial conditions ϕ( and r Lemma 1 13 Let Y R n n and the bidiagonal upper triangular block matrix be I n Y I n Y J K (Y = R Kn Kn Y I n S 1 S If Z =J K (Y S R Kn (Kn+m 2, where S = RKn m with S i R n n i=1 S K1 S K (i = 1,,k, then { K K Z = col Y i1 S i, Y i2 S i,,s K,I m } i=2 Lemma 2 27 (Finsler s Lemma Consider real matrices B and M such that B has full row rank and M = M T The following statements are equivalent: 1 There exists a vector x such that x T Mx < and Bx = ; 2 There exists a scalar l R such that 3 The following condition holds: lb T B M>; B T MB <
Circuits Syst Signal Process (212 31:143 162 147 3 Main Results 31 Stability If A(r t and A d (r t are known and do not have polytopic uncertainties, then system Σ p becomes ( Σ: x(t = A(r t x(t + A d (r t x t d k (r t, (3 k=1 x(t = ϕ(t, t μ,, (4 where for each possible value r t = i, i Ñ, hold A i = A(i and A di = A di (i Define { η = max πii }, λ ip = i Ñ p d ik, λ ip = k=1 with λ i = and λ i =, which leads to p d ik, i Ñ, k=1 λ ip λ ip μ, i Ñ, l =, 1,,r (5 Then we have the following theorem by applying the methodology used in 13 Theorem 1 If there exist P i >, i Ñ, Q m >, m = 1,,r, X> and Y> such that Q m+1 Q m, m= 1,,r 1, (6 Γi1 + Γ i2 T T Γ i3 T <, (7 Γ i4 for each i Ñ, then the system Σ is stochastically stable, where Nj=1 π ij P j + X + (1 + ημq 1 +P i A i + A T i P P i A di i Q 2 Q 1 Γ i1 = Q r Q r1 A T di P i Q r R (r+1n (r+1n, Γ i2 = A T i A T di μy A i A di (8 R(r+1n (r+1n, (9
148 Circuits Syst Signal Process (212 31:143 162 Γ i3 = X d 1 i1 Y d 1 i(r1 Y (μ λ i(r1 1 Y R (r+1n (r+1n, (1 π ii d 1 i1 Q 1 Γ i4 = π ii d 1 i(r1 Q r1 Rrn rn, (11 π ii dir 1 Q r { } T = col S i, S i,,s r,i (2r+2n R (3r+2n 2(r+1n, (12 S 1 S 2 S r1 S r = i=1 i=2 I n I n I n I n I n Proof Note that {(x(t, r t, t } is not a Markov process From 4, we define a new process {(x t,r t, t } by x t (s = x(t + s, t μ s t, which leads to conclusion that {(x t,r t, t } is a Markov process with initial state (ϕ(, r Define a new stochastic Lyapunov functional candidate of the system Σ as V(x t,r t = 5 V p (x t,r t, (13 p=1 where V 1 (x t,r t = x T (tp (r t x(t, V 2 (x t,r t = x T (sxx(s ds, V 3 (x t,r t = ẋ T (sy ẋ(sdsdθ, μ t+θ λi(l1 V 4 (x t,r t = x T (sq l x(sds, V 5 (x t,r t = η μ t+θ x T (sq 1 x(sdsdθ
Circuits Syst Signal Process (212 31:143 162 149 Let A denote the weak infinitesimal generator of the random process {x t,r t } Then, for each r t = i, i Ñ, it can be verified that A V 1 (x t,i ( N = x T (t π ij P j x(t + 2x T ( (tp i Ai x(t + A di x(t λ ir, j=1 A V 2 (x t,i = x T (txx(t x T (t μxx(t μ By Jensen s integral inequality and λ i(r1 μ, i Ñ,wehave A V 3 (x t,i = μẋ T (ty ẋ(t ẋ T (sy ẋ(sds = μẋ T (ty ẋ(t r1 λi(l1 λi(r1 ẋ T (sy ẋ(sds ẋ T (sy ẋ(sds (λi(r1 T μẋ T (ty ẋ(t (μ λ i(r1 ẋ(sds 1 Y (λi(r1 ẋ(sds r1 dil 1 (λi(l1 T (λi(l1 ẋ(sds Y ẋ(sds (λi(l1 T μẋ T (ty ẋ(t (μ λ i(r1 ẋ(sds 1 Y (λi(l1 ẋ(sds r1 d 1 il (λi(l1 From the proof of Theorem 31 in 3 and we have V 4 (x t,i= ( x T (sq l x(sds E V 4 (x t+,r t+ (x t,r t = i T (λi(l1 ẋ(sds Y ẋ(sds tλ i(l1 x T (sq l x(sds,
15 Circuits Syst Signal Process (212 31:143 162 = j i E I (rt+ =j ( (+ t x T (sq l x(sds + x T (sq l x(sds (x t,r t = i t+ λ jl ( (+ + E I (rt+ =i x T (sq l x(sds (x t,r t = i t+ λ il ( (+ E x T (sq l x(sds j i I (rt+ =j t + x T (sq l x(sds (x t,r t = i t+ λ j(l1 ( (+ E I (rt+ =i x T (sq l x(sds (x t,r t = i t+ λ i(l1 = W 1 W 2, where W 1 = E j i I (rt+ =j ( (+ t x T (sq l x(sds + x T (sq l x(sds (x t,r t = i t+ λ jl ( (+ + E I (rt+ =i x T (sq l x(sds (x t,r t = i, t+ λ il W 2 = (+ E x T (sq l x(sds j i I (rt+ =j t + x T (sq l x(sds (x t,r t = i t+ λ j(l1 ( (+ + E I (rt+ =i x T (sq l x(sds (x t,r t = i t+ λ i(l1 The first part can be expressed as W 1 = j i E I (rt+ =j ( (+ t x T (sq l x(sds
Circuits Syst Signal Process (212 31:143 162 151 where Note that + x T (sq l x(sds (x t,r t = i t+ λ jl ( (+ + E I (rt+ =i x T (sq l x(sds (x t,r t = i t+ λ il = F + FF + FFF, F = E j i FF = j i E FFF = E I (rt+ =j I (rt+ =j I (rt+ =i ( + x T (sq l x(sds (x t,r t = i, t ( t x T (sq l x(sds (x t,r t = i, t+ λ jl ( (+ x T (sq l x(sds (x t,r t = i t+ λ il F o ( 2, FF = ( P r t+ = j r t = i x T (sq l x(sds j i t+ λ jl = ( πij + o( r ( x T (sq l x(sds, j i t+ λ jl (+ FFF = P r t+ = i r t = i E x T (sq l x(sds (x t,r t = i t+ λ il = ( 1 + π ii + o( r Similarly, W 2 = E j i I (rt+ =j ( (+ t+ λ il x T (sq l x(sds (+ t (+ x T (sq l x(sds (x t,r t = i + ( πij + o( r ( x T (sq l x(sds j i t+ λ j(l1 + ( 1 + π ii + o( r t+ λ j(l1 x T (sq l x(sds
152 Circuits Syst Signal Process (212 31:143 162 Thus we have ( 1 {E V4 (x t+,r t+ (x t,r t = i V 4 (x t,r t = i } ( ( 1 (+ ( = x T (sq l x(sds x T (sq l x(sds t+ λ il (+ + π ii x T (sq l x(sds t+ λ il + ( π ij x T (sq l x(sds j i t+ λ jl ( ( 1 (+ ( x T (sq l x(sds x T (sq l x(sds t+ λ il (+ π ii x T (sq l x(sds t+ λ j(l1 j i π ij Therefore, we obtain ( t+ λ j(l1 x T (sq l x(sds A V 4 (x t,i λi(l1 = A x T (sq l x(sds 1 { = lim E V4 (x t+,r t+ (xt,r t = i V 4 (x t,r t = i } ( = A x T (sq l x(sds A x T (sq l x(sds tλ i(l1 x T (tq l x(t x T (t λ il Q l x(t λ il + N = j=1 π t ij tλil x T (sq l x(sds x T (tq l x(t + x T (t λ i(l1 Q l x(t λ i(l1 N j=1 π t ij tλil x T (sq l x(sds ( = x T (t λ i(l1 Q l x(t λ i(l1 x T (t λ il Q l x(t λ il + N π ij j=1 λj(l1 tλ jl x T (sq l x(sds
Circuits Syst Signal Process (212 31:143 162 153 From (5 and (6 it can be further shown that A V 4 (x t,i = x T (tq 1 x(t x T (t λ ir Q r x(t λ ir r1 + x T (t λ il (Q l Q l+1 x(t λ il N λj(l1 π ij j=1,j i tλ jl λi(l1 π ii x T (sq l x(sds x T (sq l x(sds x T (tq 1 x(t x T (t λ ir Q r x(t λ ir r1 + x T (t λ il (Q l Q l+1 x(t λ il N λj(l1 π ij j=1,j i tλ jl λi(l1 dil 1 π ii x T (sq l x(sds x T (s ds Q l x T (tq 1 x(t x T (t λ ir Q r x(t λ ir r1 + x T (t λ il (Q l Q l+1 x(t λ il N λj(l1 π ij j=1,j i tλ jl λi(l1 dil 1 π ii λi(l1 x T (sq 1 x(sds x T (s ds Q l x T (tq 1 x(t x T (t λ ir Q r x(t λ ir r1 + x T (t λ il (Q l Q l+1 x(t λ il N j=1,j i π ij x T (sq 1 x(sds λi(l1 x(sds x(sds
154 Circuits Syst Signal Process (212 31:143 162 π ii d 1 il λi(l1 x T (s ds Q l = x T (tq 1 x(t x T (t λ ir Q r x(t λ ir r1 π ii + π ii d 1 il x T (t λ il (Q l Q l+1 x(t λ il x T (sq 1 x(sds λi(l1 x T (s ds Q l λi(l1 λi(l1 x T (tq 1 x(t x T (t λ ir Q r x(t λ ir r1 + η x T (t λ il (Q l Q l+1 x(t λ il + π ii x T (sq 1 x(sds d 1 il λi(l1 x T (s ds Q l x(sds λi(l1 and A V 5 (x t,i = ημx T (tq 1 x(t η x T (sq 1 x(sds Then we have A V(x t,i = 5 A V p (x t,i p=1 x(sds x(sds, ( N x T (t π ij P j x(t + 2x T ( (tp i Ai x(t + A di x(t λ ir j=1 + x T (txx(t x T (t μxx(t μ (λi(r1 T + μẋ T (ty ẋ(t (μ λ i(r1 ẋ(sds 1 Y (λi(r1 ẋ(sds r1 (λi(l1 T ( ẋ(sds d il 1 Y (λ i(l1 ẋ(sds
Circuits Syst Signal Process (212 31:143 162 155 which equals + x T (tq 1 x(t x T (t λ ir Q r x(t λ ir r1 + ημx T (tq 1 x(t x T (t λ il (Q l Q l+1 x(t λ il + π ii A V(x t,i ζ T (t (λi(l1 dil 1 x T (s ds Q l (λi(l1 Γi1 + Γ i2 Γ i3 ζ(t, Γ i4 where Γ ij, i Ñ, j = 1, 2, 3, are defined in (8 (11 and ζ(t = x(t x(t λ i1 x(t λ ir x(t μ tλ i1 x(sds tλ i1 ẋ(sds λi(r2 tλ i(r1 From the Newton Leibniz formula we have which is equivalent to Tζ(t= λi(r2 ẋ(sds tλ i(r1 λi(r1 x(sds tλ ir λi(r1 x(t x(t λ i1 ẋ(sds =, tλ i1 λi1 x(t λ i1 x(t λ i2 ẋ(sds =, tλ i2 λi(r2 x(t λ i(r2 x(t λ i(r1 ẋ(sds =, tλ i(r1 x(t λ i(r1 x(t μ In In In In In In In λi(r1 In In In In In x(sds, ẋ(sds T x(sds (14 ẋ(sds =, ζ(t=
156 Circuits Syst Signal Process (212 31:143 162 Therefore, from Lemma 2, AV(x t,i < if ( Γi1 + Γ i2 ζ T (t T T T Γ i3 ζ(t>, Γ i4 which is equivalent to (7 from Lemma 2, where T can be computed from Lemma 1 and given in (12 This completes the proof of Theorem 1 Remark 1 Sufficient condition with reduced conservatism is obtained for the stability analysis of Markovian jump system Σ with mode-dependent multiple time delays The conservatism can be reduced further by introducing more delay components as shown in Remarks 4, 5 and 6 in 13 at the expense of more matrix variables and LMIs in larger dimension in Theorem 1 32 Robust Stability Now we consider the robust stochastic stability of system Σ p Theorem 2 If there exist P i >, i Ñ, Q m >, m = 1,,r, X> and Y> such that Q m+1 Q m, m= 1,,r 1, (15 Γ T T il1 + Γ il2 Γ i3 T <, (16 Γ i4 for each i Ñ, l = 1, 2,,q, then the system Σ p is robust stochastically stable, where Nj=1 π ij P j + X + (1 + ημq 1 + P i A il + A T il P P i A dil i Γ il1 Q 2 Q 1, Q r Q r1 A T dil P i Q r A il Γ il2 A T il A T dil μy A dil Proof From the proof of Theorem 1, it is easy to show that AV(x t,i in (13 satisfies A V(x t,i ( N x T (t π ij P j + X + (1 + ημq 1 x(t j=1
Circuits Syst Signal Process (212 31:143 162 157 + 2x T ( (tp i A(ix(t + Ad (ix(t λ ir x T (t μxx(t μ x T (t λ ir Q r x(t λ ir (λi(r1 T + μẋ T (ty ẋ(t (μ λ i(r1 ẋ(sds 1 Y (λi(r1 ẋ(sds r1 r1 d 1 il + π ii (λi(l1 T (λi(l1 ẋ(sds Y ẋ(sds x T (t λ il (Q l Q l+1 x(t λ il d 1 il λi(l1 x T (s ds Q l λi(l1 x(sds Γ i1 + Γ i2 = ζ T (t Γ i3 ζ(t, (17 Γ i4 where Nj=1 π ij P j + X + (1 + ημq 1 + P i A(i + A T P i A d (i (ip i Γ Q 2 Q 1 i1 =, Q r Q r1 A T d (ip i Q r A(i Γ i2 = A T (i A T d (i μy, A d (i and ζ(t is defined in (14, Γ i3 and Γ i4 are given in (1 (11, and A(i and A d (i are defined in (2 From the proof of Theorem 1, Γ i1 + Γ i2 ζ T (t Γ i3 ζ(t< Γ i4 if and only if Γ i1 + Γ i2 T T Γ i3 T <, Γ i4
158 Circuits Syst Signal Process (212 31:143 162 which, by Schur complement equivalence, is equivalent to T T Γ i1 Γ i3 Γ i4 T T T A(i A d (i μy μy A T (i A T d (i T μy < According to (16 and by Schur complement equivalence again, we have T T Γ il1 Γ i3 Γ i4 T T T A il A dil μy μy A T il AT dil T μy <, which further leads to q α l (t T T Γ il1 Γ i3 Γ i4 T T T A il A dil μy μy A T il AT dil T μy = T T Γ i1 Γ i3 Γ i4 T T T A(i A d (i μy μy A T (i A T d (i T μy < (18
Circuits Syst Signal Process (212 31:143 162 159 By Schur complement equivalence again, it is easy to obtain that (18 holds if and only if (16 is satisfied, which implies A V(x t,i Γ i1 + Γ i2 ζ T (t Γ i3 ζ(t< Γ i4 This completes the proof of Theorem 2 4 Numerical Examples Next, two numerical examples will be given to illustrate the effectiveness of the proposed approach Example 1 The following matrices, 22 A 11 = 1 14 1 A 22 = 2 19 A d21 = 2 1 2 2, A 12 = 2 16, A d11 =, A d22 = 1 2 1 1, 2 1 21, A 21 = 1 9, A d12 =, 1 2 are the system matrices of a Markovian jump system Σ p with mode-dependent delays and polytopic uncertainties in the form of (1 The delays are given by d 11 = 8, d 12 = 26, d 21 = 26, d 22 = 8, and the transition probability matrix is 1 1 M = 2 2 The solutions of (15 and (16aregivenby 11428 58 6554847 4929 X = >, Y = >, 58 43 4929 27 P 1 = 1 3 19823 274 >, P 274 1872 2 = 1 3 2173 246 246 836 Q 1 = 1 3 1346 179 >, Q 179 14 2 = 1 3 12973 179 179 14 So the overall stability bound on delay is the sum of all parts, ie 16 >, >,
16 Circuits Syst Signal Process (212 31:143 162 Example 2 The system matrices of a Markovian jump system Σ p with modedependent delays and polytopic uncertainties in the form of (1aregivenby with 3 A 11 = 3 4 1 A 22 = 1 3 A d21 = 9 7, A 12 =, A d11 =, A d22 = 3 1 1 4 1 1 1 1 9 6 2, A 21 = 4, A d12 =, 8 1 9 d 11 = 122, d 12 = 5, d 21 = 121, d 22 = 51 The transition probability matrix is 3 3 M = 2 2 The solutions of (15 and (16aregivenby 36 165 X = 165 658 173223 878 P 1 = 878 5715 79497 41 Q 1 = 41 388 >, Y = >, P 2 = >, Q 2 = 3397 958 958 36 227518 839 839 4851 79479 38 38 319 >, >,,, > So the overall stability bound on delay is the sum of all parts, ie 172 By the method in 4 without partitioning the delay, the upper bound of the time delay is 57, which shows that our method has improved the result in 4 5 Conclusion We have presented solutions to the stability analysis for Markovian jump systems with multiple delay components and possible polytopic uncertainties A delaydependent sufficient condition in terms of the LMI framework has been obtained by using a new form of stochastic Lyapunov functionals Two examples have been given to demonstrate the effectiveness of the proposed result Open Access This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s and source are credited
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