Robust non-fragile stabilisation and H ∞ control for uncertain discrete-time stochastic systems with Markovian jump parameters and time-varying delay

Abstract

In this paper, the robust non-fragile stabilisation and H_∞ control problem is investigated for a class of uncertain discrete-time stochastic systems with Markovian jumping parameters and time-varying delay. The parameter uncertainties are supposed to be time-varying as well as norm-bounded. The aim of the robust non-fragile stabilisation problem is to design a non-fragile state feedback controller which guarantees the robust stability of the closed-loop system for all admissible uncertainties. At the same time, in addition to the robust stability requirement, a prescribed H_∞ performance level is required to be achieved for the robust H_∞ control problem. By Lyapunov stability theory, delay-segment-dependent conditions for the solvability of these problems are formulated in terms of linear matrix inequality technique. Finally, numerical examples are shown to demonstrate the usefulness and applicability of the proposed design method.

Keywords

parameter uncertainty discrete-time stochastic system Markovian jumping parameter time-varying delay LMI

Introduction

Over the past ten years, robust H_∞ control as a dual of the robust H_∞ filter, has received extensive consideration. It is known that there exists inherent uncertainties and time delays in the implementation of a designed controller, which may cause poor performance such as oscillation and instability. Therefore, a great number of results on the H_∞ control problem for uncertain systems or delayed systems have been reported (Balasubramaniam and Senthilkumar, 2013; Chen et al., 2011; Chen and Tu, 2012; Chen et al., 2008; Gao and Wu, 2013; Ran and Zhang, 2011; Sakthivel et al., 2012; Senthilkumar and Balasubramaniam, 2012; Su et al., 2011; Wang and Shen, 2011; Xu and Chen, 2002, 2005; Xu et al., 2004, 2006; Yang et al., 2010; Zhang and Xu, 2009a; Zhang et al., 2009b). It should be noted that a reserved assumption in the course of controller implementation is that the control parameters will be implemented exactly. However, it turns out that controllers can be sensitive to the uncertainties in the controller coefficients (Keel and Bhatacharyya, 1997). Therefore, it is of great significance to design a controller which is insensitive to the errors with respect to its gain, i.e. the designed controller is non-fragile or resilient. This general topic has been widely considered in (Chen et al., 2011; Chen and Tu, 2012; Chen et al., 2008; Senthilkumar and Balasubramaniam, 2012; Su et al., 2011; Wang and Shen, 2011; Xu et al., 2006; Zhang et al., 2009b) for the continuous-time case and (Ran and Zhang, 2011) for the discrete-time case. For example, delay-dependent non-fragile robust stabilization and H_∞ control of uncertain stochastic systems with time-varying delay and nonlinearity is studied in Zhang et al. (2009b), where a method for designing a non-fragile controller for a nonlinear system is proposed. The results in Zhang et al. (2009b) have been further improved (Wang and Shen, 2011). The non-fragile observer-based stabilisation and H_∞ control problems for a neutral stochastic hybrid system with time-varying delay is considered (Chen et al., 2011).

On the other hand, the Markovian jump system has been an attractive subject of research recently since it is appropriate to model the behavior of practical systems; including manufacturing systems, load prediction of power systems, maneuvered target tracking, robotics and premium adjustment of bonus-malus systems. It is well known that these systems can be modelled by a Markov chain taking values in a finite set. Recently, considerable attention has been devoted to the study of robust H_∞ control problems concerning the Markovian jumping system. For instance, via sliding mode design method, the H_∞ control for stochastic systems with Markovian switching and time-varying delay is considered (Gao and Wu, 2013). By the free-weighting method, non-fragile robust stabilisation and H_∞ control for uncertain stochastic time delay systems with Markovian jump parameters and nonlinear disturbances is investigated (Senthilkumar and Balasubramaniam, 2012). The robust H_∞ control problem for uncertain discrete-time stochastic bilinear systems with Markovian switching is discussed (Xu and Chen, 2005). The passivity-based control problem for stochastic jumping systems with mode-dependent round-trip time-varying delays and norm-bounded parametric uncertainties is studied (Shen et al., 2012). The problem of delay-dependent robust stabilisation for uncertain discrete-time fuzzy Markovian jump systems with mode-dependent time-varying delays is considered (Zhang et al., 2011).

The robust non-fragile H_∞ control problem has not been fully investigated, especially for uncertain discrete-time stochastic systems. Robust H_∞ control for uncertain discrete stochastic time-delay systems is studied, in which the controller gain uncertainties are not taken into account (Xu et al., 2004; Xu and Chen, 2005). The delay-independent results of non-fragile H_∞ control for the discrete-time Markovian jump system is obtained in a deterministic setting rather than a stochastic setting (Ran et al., 2011). The non-fragile H_∞ control problem is discussed in the context of continuous stochastic delayed systems with Markovian jumping parameters (Senthilkumar and Balasubramaniam, 2012). In fact, this paper stems from the following motivations. Firstly, the results in Xu et al. (2004) and Xu and Chen (2005) are not about non-fragile H_∞ control for discrete-time stochastic systems. Secondly, the results in Ran and Zhang (2011) and Senthilkumar and Balasubramaniam (2012) are not about non-fragile control for discrete-time stochastic systems. To the best of our knowledge, no results on robust non-fragile H_∞ control for discrete-time stochastic delayed systems with Markovian jumping parameters have been available in the literature to date. Considering all the factors aforementioned, the non-fragile stabilisation and H_∞ control for uncertain discrete-time stochastic systems with Markovian jumping parameters is still open and challenging, which is also the motivation for this work.

In this paper, we deal with the problem of robust non-fragile stabilisation and the H_∞ control problem for a class of uncertain discrete-time stochastic systems with Markovian jumping parameters and time-varying delay. By employing an appropriate Lyapunov functional, delay-segment-dependent conditions are formulated in terms of a linear matrix inequality (LMI) such that the closed-loop system is not only robustly, stochastically stable but also the required H_∞ performance level is satisfied for all admissible uncertainties. Finally, numerical examples are given to show the effectiveness of the proposed method.

Notations: $R^{n}$ denotes the n-dimensional Euclidean space, $R^{n \times m}$ is the set of n × m real matrices. I is the identity matrix. |⋅| denotes the Euclidean norm of vectors and ||·|| denotes the spectral norm of matrices. N denotes the set of all natural numbers, i.e. N = {0,1,2,…}. $(Ω, F, {F_{k}}_{k \geq 0}, P)$ is a complete probability space with a filtration ${F_{k}}_{k \geq 0}$ satisfying the usual conditions. M^T stands for the transpose of the matrix M. For symmetric matrices X and Y, the notation X > Y (respectively X ≥ Y) means that X − Y is positive definite (respectively, positive semi-definite). * denotes a block that is readily inferred by symmetry. E{·} stands for the mathematical expectation operator with respect to the given probability measure $P$ .

Problem description

Consider the following uncertain discrete-time stochastic systems with Markovian jumping parameters and time-varying delay.

\begin{matrix} Σ : x (k + 1) = & A (r_{k}, k) x (k) + A_{1} (r_{k}, k) x (k - d (k)) + A_{2} (r_{k}) u (k) \\ + A_{3} (r_{k}, k) v (k) \\ + [D (r_{k}, k) x (k) + D_{1} (r_{k}, k) x (k - d (k)) \\ + D_{2} (r_{k}) u (k) + D_{3} (r_{k}, k) v (k)] w (k) \end{matrix}

(1)

z (k) = C (r_{k}) x (k) + C_{1} (r_{k}) x (k - d (k)) + C_{2} (r_{k}) u (k)

(2)

where $x (k) \in R^{n}$ is the state vector, $u (k) \in R^{p}$ is the control input, $v (k) \in R^{q}$ is the disturbance input in L₂[0, ∞), $z (k) \in R^{m}$ is the controlled output. A(r_k , k), A₁(r_k , k), A₂(r_k ), A₃(r_k , k), D(r_k , k), D₁(r_k , k), D₂(r_k ), D₃(r_k , k), C(r_k ), C₁(r_k ), C₂(r_k ) are the matrix functions of the random jumping process r_k , where r_k is a finite system mode and takes discrete values in a given finite set $S = 1, 2, \dots, N$ . For simplicity, in the sequel, when $r_{k} = i \in S$ , A(r_k , k), A₁(r_k , k), A₂(r_k ), A₃(r_k , k), D(r_k , k), D₁(r_k , k), D₂(r_k ), D₃(r_k , k), C(r_k ), C₁(r_k ), C₂(r_k ), are denoted by A_i (k), A_1i(k), A_2i, A_3i(k) D_i (k), D_1i(k), D_2i, D_3i(k), C_i , C_1i, C_2i, respectively. w(k) is a scalar Brownian motion defined on the complete probability space $(Ω, F, {F_{k}}_{k \geq 0}, P)$ with

E {w (k)} = 0, E {w^{2} (k)} = σ, E {w (i) w (j)} = 0 (i \neq j)

(3)

where the scalar σ > 0, and d(k) represents the time-varying delay satisfying

0 \leq d_{m} \leq d (k) \leq d_{M}, k = 0, 1, 2, \dots

(4)

The transition probability matrix Π = (π_ij )_N×N is given by

P_{r} (r_{k + 1} = j | r_{k} = i) = π_{ij}, \forall i, j \in S

(5)

where π_ij $(\forall i, j \in S)$ is the transition rate from mode i to mode j and satisfies $\sum_{j = 1}^{N} π_{ij} = 1, \forall i, j \in S$ . In system (1),

\begin{matrix} A_{i} (k) = A_{i} + Δ A_{i} (k), A_{1 i} (k) = A_{1 i} + Δ A_{1 i} (k), \\ A_{3 i} (k) = A_{3 i} + Δ A_{3 i} (k) \\ D_{i} (k) = D_{i} + Δ D_{i} (k), D_{1 i} (k) = D_{1 i} + Δ D_{1 i} (k), \\ D_{3 i} (k) = D_{3 i} + Δ D_{3 i} (k) \end{matrix}

(6)

where A_i , A_1i, A_2i, A_3i, D_i , D_1i, D_2i, D_3i, C_i , C_1i, C_2i are known real constant matrices. Norm-bounded parameter uncertainties ΔA_i (k), ΔA_1i(k), ΔA_3i(k), ΔD_i (k), ΔD_1i(k), ΔD_3i(k) are assumed to satisfy

\begin{matrix} [\begin{matrix} Δ A_{i} (k) & Δ A_{1 i} (k) & Δ A_{3 i} (k) & Δ D_{i} (k) & Δ D_{1 i} (k) & Δ D_{3 i} (k) \end{matrix}] \\ = M_{1 i} F_{i} (k) [\begin{matrix} N_{1 i} & N_{2 i} & N_{3 i} & N_{4 i} & N_{5 i} & N_{6 i} \end{matrix}] \end{matrix}

(7)

where M_1i, N_1i, N_2i, N_3i, N_4i, N_5i, N_6i are known real constant matrices. F_i (·) : $R \to R^{k \times l}$ is an unknown time-varying matrix function satisfying $F_{i}^{T} (k) F_{i} (k) \leq I, \forall k$ .

The parameter uncertainties ΔA_i (k), ΔA_1i(k), ΔA_3i(k), ΔD_i (k), ΔD_1i(k) and ΔD_3i(k) are said to be admissible if both (6) and (7) hold.

The following definitions are used throughout this paper.

Definition 1. (Xu et al., 2004; Xu and Chen, 2005) The uncertain discrete-time system Σ is said to be robustly stochastically stable if there exists a scalar $\tilde{M} (x_{0}, r_{0}) > 0$ such that

lim_{N \to \infty} E {\sum_{k = 0}^{N} | x (k) |^{2} | x_{0}, r_{0}} \leq \tilde{M} (x_{0}, r_{0})

(8)

for all admissible uncertainties when u(k) = 0 and v(k) = 0.

Definition 2. (Xu et al., 2004; Xu and Chen, 2005) Given a scalar γ > 0, the system Σ with u(k) = 0 is said to be robustly stochastically stable with disturbance attenuation level γ if it is robustly stochastically stable, and under zero initial conditions the following inequity holds

| | z (k) | |_{2} < γ | | v (k) | |_{2}

(9)

for all nonzero v(k) ∈ L₂[0, ∞).

This paper considers the problem of robust non-fragile stabilisation and H_∞ control for an uncertain discrete-time stochastic system Σ. The main attention is paid to the design of a memoryless non-fragile state feedback controller

u (k) = K_{i} (k) x (k) = [K_{i} + Δ K_{i} (k)] x (k)

(10)

such that the closed-loop system is robustly stochastically stable, where K_i is the controller gain and ΔK_i (k) satisfies

Δ K_{i} (k) = M_{2 i} F_{1 i} (k) N_{7 i}

(11)

where M_2i and N_7i are known constant matrices. The time-varying uncertain matrix function F_1i(k) satisfies $F_{1 i}^{T} (k) F_{1 i} (k) \leq I, \forall k$ .

Before ending this section, the following Lemma is provided which is essential in establishing our main results.

Lemma 1. (Xu and Chen, 2002) Matrices D and E are real matrices of appropriate dimensions with F^TF ≤ I. For any vectors X, Y ∈ R ⁿ and any scalar ϵ > 0, the following inequality holds

2 X^{T} DFEY \leq ϵ^{- 1} X^{T} {DD}^{T} X + ϵ Y^{T} {EE}^{T} Y

Robust non-fragile stabilization

In this section, we shall establish the design of non-fragile feedback controller for uncertain discrete-time stochastic system (1). According to Definition 1, first, we provide a sufficient condition for robust stochastic stability of system (1) with u(k) = 0 and v(k) = 0 in the following theorem.

Theorem 1. The system (1) with u(k) = 0 and v(k) = 0 is robustly stochastically stable for all admissible uncertainties satisfying (7), if there exist matrices P_i > 0, Q₁ > 0, Q₂ > 0, Q₃ > 0, R₁ > 0, R₂ > 0, T_1i, T_2i, S_1i, S_2i, U_1i, U_2i and scalars ε_1i > 0, $ε_{2 i} > 0 (i \in S)$ such that the following matrix inequalities hold:

[\begin{matrix} Θ_{i} & ϕ_{i} \\ * & - ε_{i} I \end{matrix}] < 0

(12)

Where

\begin{matrix} ϕ_{i} = [\begin{matrix} ϕ_{1 i}^{T} & ϕ_{2 i}^{T} & ε_{1 i} ϕ_{3 i} & ε_{2 i} ϕ_{4 i} \end{matrix}] \\ ε_{i} = diag {- ε_{1 i} I, - ε_{2 i} I, - ε_{1 i} I, - ε_{2 i} I} \\ Θ_{i} = [\begin{matrix} Θ_{11 i} & Θ_{12 i} & U_{1 i} & - T_{1 i} & Θ_{1 i} \\ * & Θ_{22 i} & U_{2 i} & - T_{2 i} & Θ_{2 i} \\ * & * & - Q_{2} & 0 & 0 \\ * & * & * & - Q_{3} & 0 \\ * & * & * & * & Θ_{5 i} \end{matrix}] \\ Θ_{11 i} = (\bar{d} + 1) Q_{1} + Q_{2} + Q_{3} + S_{1 i} + S_{1 i}^{T} - P_{i} \\ Θ_{12 i} = S_{2 i}^{T} - S_{1 i} + T_{1 i} - U_{1 i} \end{matrix}

\begin{matrix} Θ_{22 i} = - Q_{1} - S_{2 i} - S_{2 i}^{T} + T_{2 i} + T_{2 i}^{T} - U_{2 i} - U_{2 i}^{T} \\ Θ_{1 i} = [\begin{matrix} Θ_{15 i} & Θ_{16 i} & Θ_{17 i} \end{matrix}] \\ Θ_{2 i} = [\begin{matrix} Θ_{25 i} & Θ_{26 i} & Θ_{27 i} \end{matrix}] \\ Θ_{5 i} = diag {Θ_{55 i}, Θ_{55 i}, Θ_{77 i}} \\ Θ_{15 i} = [\begin{matrix} A_{i}^{T} \sqrt{π_{i 1}} P_{1} & \dots & A_{i}^{T} \sqrt{π_{iN}} P_{N} \end{matrix}] \\ Θ_{25 i} = [\begin{matrix} A_{1 i}^{T} \sqrt{π_{i 1}} P_{1} & \dots & A_{1 i}^{T} \sqrt{π_{iN}} P_{N} \end{matrix}] \\ Θ_{55 i} = diag {- P_{1}, \dots, - P_{N}} \\ Θ_{16 i} = [\begin{matrix} D_{i}^{T} \sqrt{σ π_{i 1}} P_{1} & \dots & D_{i}^{T} \sqrt{σ π_{iN}} P_{N} \end{matrix}] \\ Θ_{26 i} = [\begin{matrix} D_{1 i}^{T} \sqrt{σ π_{i 1}} P_{1} & \dots & D_{1 i}^{T} \sqrt{σ π_{iN}} P_{N} \end{matrix}] \\ Θ_{17 i} = [\begin{matrix} \sqrt{d_{M}} (A_{i}^{T} - I) R_{1} & \sqrt{\bar{d}} (A_{i}^{T} - I) R_{2} & \sqrt{d_{M} σ} D_{i}^{T} R_{1} & \sqrt{\bar{d} σ} D_{i}^{T} R_{2} & \sqrt{d_{M}} S_{1 i} & \sqrt{\bar{d}} T_{1 i} & \sqrt{\bar{d}} U_{1 i} \end{matrix}] \\ Θ_{27 i} = [\begin{matrix} \sqrt{d_{M}} A_{1 i}^{T} R_{1} & \sqrt{\bar{d}} A_{1 i}^{T} R_{2} & \sqrt{d_{M} σ} D_{1 i}^{T} R_{1} & \sqrt{\bar{d} σ} D_{1 i}^{T} R_{2} & \sqrt{d_{M}} S_{2 i} & \sqrt{\bar{d}} T_{2 i} & \sqrt{\bar{d}} U_{2 i} \end{matrix}] \\ Θ_{77 i} = diag {- R_{1}, - R_{2}, - R_{1}, - R_{2}, - R_{1}, - (R_{1} + R_{2}), - R_{2}} \\ φ_{1 i} = [N_{1 i} N_{2 i} 0_{2} 0 \dots 0 0 \dots 0 0_{7}] \\ φ_{2 i} = [\sqrt{σ} N_{4 i} \sqrt{σ} N_{5 i} 0_{2} 0 \dots 0 0 \dots 0 0_{7}] \\ φ_{3 i} = {[0_{4} \sqrt{π_{i 1}} M_{1 i}^{T} P_{1} \dots \sqrt{π_{iN}} M_{1 i}^{T} P_{N} 0 \dots 0 \sqrt{d_{M}} M_{1 i}^{T} R_{1} \sqrt{\bar{d}} M_{1 i}^{T} R_{2} 0_{5}]}^{T} \\ φ_{4 i} = {[0_{4} 0 \dots 0 \sqrt{π_{i 1}} M_{1 i}^{T} P_{1} \dots \sqrt{π_{iN}} M_{1 i}^{T} P_{N} 0_{2} \sqrt{d_{M}} M_{1 i}^{T} R_{1} \sqrt{\bar{d}} M_{1 i}^{T} R_{2} 0_{3}]}^{T} \\ \bar{d} = d_{M} - d_{m} \end{matrix}

Proof: For convenience, the following notations are adopted:

\begin{matrix} ξ (k) = {[\begin{matrix} x^{T} (k) & x^{T} (k - d (k)) & x^{T} (k - d_{m}) & x^{T} (k - d_{M}) \end{matrix}]}^{T} \\ Γ_{1} = [\begin{matrix} A_{i} (k) & A_{1 i} (k) & 0 & 0 \end{matrix}] \\ Γ_{2} = [\begin{matrix} D_{i} (k) & D_{1 i} (k) & 0 & 0 \end{matrix}] \\ Γ_{3} = [\begin{matrix} A_{i} (k) - I & A_{1 i} (k) & 0 & 0 \end{matrix}] \end{matrix}

Choose the following Lyapunov functional

V (k) = \sum_{l = 1}^{5} V_{l} (k)

(13)

where

\begin{matrix} V_{1} (k) = x^{T} (k) P_{i} x (k) \\ V_{2} (k) = \sum_{i = k - d (k)}^{k - 1} x^{T} (i) Q_{1} x (i) \\ V_{3} (k) = \sum_{j = k - d_{M} + 1}^{k - d_{m}} \sum_{i = j}^{k - 1} x^{T} (i) Q_{1} x (i) \\ V_{4} (k) = \sum_{l = d - d_{m}}^{k - 1} x^{T} (l) Q_{2} x (l) + \sum_{l = d - d_{M}}^{k - 1} x^{T} (l) Q_{3} x (l) \\ V_{5} (k) = \sum_{j = - d_{M} + 1}^{0} \sum_{l = k + j - 1}^{k - 1} η^{T} (l) R_{1} η (l) \\ + \sum_{l = - d_{M} + 1}^{- d_{m}} \sum_{l = k + j - 1}^{k - 1} η^{T} (l) R_{2} η (l) \end{matrix}

where η(l) = x(l + 1) − x(l), then we have

\begin{matrix} x (k + 1) = Γ_{1} ξ (k) + Γ_{2} ξ (k) w (k) \\ η (k) = Γ_{3} ξ (k) + Γ_{2} ξ (k) w (k) \end{matrix}

Calculating the difference of V(k) with u(k) = 0, v(k) = 0 and taking mathematical expectation, we have

E {Δ V (k)} = E {\sum_{l = 1}^{5} Δ V_{l} (k)}

(14)

where

\begin{matrix} E {Δ V_{1} (k)} & = E {V_{1} (x (k + 1), r_{k + 1}) | x (k), r_{k} = i} - V_{1} (x (k), r_{k} = i) \\ = E {ξ^{T} (k) Γ_{1}^{T} (\sum_{j = 1}^{N} π_{ij} P_{j}) Γ_{1} ξ (k) \\ + σ ξ^{T} (k) Γ_{2}^{T} (\sum_{j = 1}^{N} π_{ij} P_{j}) Γ_{2} ξ (k) - x^{T} (k) P_{i} x (k)} \end{matrix}

(15)

\begin{matrix} E {Δ V_{2} (k)} & = E {x^{T} (k) Q_{1} x (k) - x^{T} (k - d (k)) Q_{1} x (k - d (k)) \\ + \sum_{i = k + 1 - d (k + 1)}^{k - 1} x^{T} (i) Q_{1} x (i) - \sum_{i = k - d (k) + 1}^{k - 1} x^{T} (i) Q_{1} x (i)} \\ \leq E {x^{T} (k) Q_{1} x (k) - x^{T} (k - d (k)) Q_{1} x (k - d (k)) + \sum_{i = k - d_{M} + 1}^{k - d_{m}} x^{T} (i) Q_{1} x (i)} \end{matrix}

(16)

\begin{matrix} E {Δ V_{3} (k)} & = E {\sum_{j = k - d_{M} + 1}^{k - d_{m}} (x^{T} (k) Q_{1} x (k) - x^{T} (j) Q_{1} x (j))} \\ = E {(d_{M} - d_{m}) x^{T} (k) Q_{1} x (k) - \sum_{i = k - d_{M} + 1}^{k - d_{m}} x^{T} (i) Q_{1} x (i)} \end{matrix}

(17)

\begin{matrix} E {Δ V_{4} (k)} = & E {x^{T} (k) (Q_{2} + Q_{3}) x (k) \\ - x^{T} (k - d_{m}) Q_{2} x (k - d_{m}) - x^{T} (k - d_{M}) Q_{3} x (k - d_{M})} \end{matrix}

(18)

\begin{matrix} E {Δ V_{5} (k)} & = E {d_{M} η^{T} (k) R_{1} η (k) - \sum_{j = k - d_{M}}^{k - 1} η^{T} (j) R_{1} η (j) + \bar{d} η^{T} (k) R_{2} η (k) - \sum_{j = k - d_{M}}^{k - d_{m} - 1} η^{T} (j) R_{2} η (j)} \\ = E {ξ^{T} (k) (d_{M} Γ_{3}^{T} R_{1} Γ_{3} + d_{M} σ Γ_{2}^{T} R_{1} Γ_{2} + \bar{d} Γ_{3}^{T} R_{2} Γ_{3} + \bar{d} σ Γ_{2}^{T} R_{2} Γ_{2}) ξ (k) \\ - \sum_{l = k - d (k)}^{k - 1} η^{T} (l) R_{1} η (l) - \sum_{l = k - d_{M}}^{k - d (k) - 1} η^{T} (l) (R_{1} + R_{2}) η (l) - \sum_{l = k - d (k)}^{k - d_{m} - 1} η^{T} (l) R_{2} η (l)} \end{matrix}

(19)

Then, for arbitrary matrices S_i , T_i and U_i with appropriate dimensions, the following equations hold

2 ξ^{T} (k) S_{i} [x (k) - x (k - d (k)) - \sum_{l = k - d (k)}^{k - 1} η (l)] = 0

(20)

2 ξ^{T} (k) T_{i} [x (k - d (k)) - x (k - d_{M}) - \sum_{l = k - d_{M}}^{k - d (k) - 1} η (l)] = 0

(21)

2 ξ^{T} (k) U_{i} [x (k - d_{m}) - x (k - d (k)) - \sum_{l = k - d (k)}^{k - d_{m} - 1} η (l)] = 0

(22)

where

S_{i} = {[\begin{matrix} S_{1 i}^{T} & S_{2 i}^{T} & 0 & 0 \end{matrix}]}^{T}, T_{i} = {[\begin{matrix} T_{1 i}^{T} & T_{2 i}^{T} & 0 & 0 \end{matrix}]}^{T}, U_{i} = {[\begin{matrix} U_{1 i}^{T} & U_{2 i}^{T} & 0 & 0 \end{matrix}]}^{T}

Combining (15)–(19), and considering the left sides of (20)–(22), we have

\begin{matrix} E {V (e (k + 1), r_{k + 1}) | x (k), r_{k} = i} - V (x (k), r_{k} = i) \\ \leq E {ξ^{T} (k) ({\bar{Θ}}_{i} + Γ_{i}) ξ (k) - \sum_{l = k - d (k)}^{k - 1} [ξ^{T} (k) S_{i} + η^{T} (l) R_{1}] R_{1}^{- 1} [S_{i}^{T} ξ (k) + R_{1} η (l)] \\ - \sum_{l = k - d_{M}}^{k - d (k) - 1} [ξ^{T} (k) T_{i} + η^{T} (l) R_{1}] {(R_{1} + R_{2})}^{- 1} [T_{i}^{T} ξ (k) + R_{1} η (l)] \\ - \sum_{l = k - d (k)}^{k - d_{m} - 1} [ξ^{T} (k) U_{i} + η^{T} (l) R_{2}] R_{2}^{- 1} [U_{i}^{T} ξ (k) + R_{2} η (l)]} \\ \leq E {ξ^{T} (k) ({\bar{Θ}}_{i} + Γ_{i}) ξ (k)} \end{matrix}

(23)

where

\begin{matrix} {\bar{Θ}}_{i} = [\begin{matrix} Θ_{11 i} & Θ_{12 i} & U_{1 i} & - T_{1 i} \\ * & Θ_{22 i} & U_{2 i} & - T_{2 i} \\ * & * & - Q_{2} & 0 \\ * & * & * & - Q_{3} \end{matrix}] \\ Γ_{i} = d_{M} Γ_{3}^{T} R_{1} Γ_{3} + d_{M} σ Γ_{2}^{T} R_{1} Γ_{2} + \bar{d} Γ_{3}^{T} R_{2} Γ_{3} + \bar{d} σ Γ_{2}^{T} R_{2} Γ_{2} \\ + d_{M} S_{i} R_{1}^{- 1} S_{i}^{T} + \bar{d} T_{i} {(R_{1} + R_{2})}^{- 1} T_{i}^{T} + \bar{d} U_{i} R_{2}^{- 1} U_{i}^{T} \\ + Γ_{1}^{T} (\sum_{j = 1}^{N} π_{ij} P_{j}) Γ_{1} + σ Γ_{2}^{T} (\sum_{j = 1}^{N} π_{ij} P_{j}) Γ_{2} \end{matrix}

It remains to show that ${\bar{Θ}}_{i} + Γ_{i} < 0$ . According to Schur complement, ${\bar{Θ}}_{i} + Γ_{i} < 0$ is equivalent to

[\begin{matrix} Θ_{11 i} & Θ_{12 i} & U_{1 i} & - T_{1 i} & {\bar{Θ}}_{1 i} \\ * & Θ_{22 i} & U_{2 i} & - T_{2 i} & {\bar{Θ}}_{2 i} \\ * & * & - Q_{2} & 0 & 0 \\ * & * & * & - Q_{3} & 0 \\ * & * & * & * & Θ_{5 i} \end{matrix}] < 0

(24)

where

\begin{matrix} {\bar{Θ}}_{1 i} = [\begin{matrix} {\bar{Θ}}_{15 i} & {\bar{Θ}}_{16 i} & {\bar{Θ}}_{17 i} \end{matrix}] \\ {\bar{Θ}}_{2 i} = [\begin{matrix} {\bar{Θ}}_{25 i} & {\bar{Θ}}_{26 i} & {\bar{Θ}}_{27 i} \end{matrix}] \\ Θ_{5 i} = diag {Θ_{55 i}, Θ_{55 i}, Θ_{77 i}} \\ {\bar{Θ}}_{15 i} = [\begin{matrix} A_{i}^{T} (k) \sqrt{π_{i 1}} P_{1} & \dots & A_{i}^{T} (k) \sqrt{π_{iN}} P_{N} \end{matrix}] \\ {\bar{Θ}}_{25 i} = [\begin{matrix} A_{1 i}^{T} (k) \sqrt{π_{i 1}} P_{1} & \dots & A_{1 i}^{T} (k) \sqrt{π_{iN}} P_{N} \end{matrix}] \\ {\bar{Θ}}_{16 i} = [\begin{matrix} D_{i}^{T} (k) \sqrt{σ π_{i 1}} P_{1} & \dots & D_{i}^{T} (k) \sqrt{σ π_{iN}} P_{N} \end{matrix}] \\ {\bar{Θ}}_{26 i} = [\begin{matrix} D_{1 i}^{T} (k) \sqrt{σ π_{i 1}} P_{1} & \dots & D_{1 i}^{T} (k) \sqrt{σ π_{iN}} P_{N} \end{matrix}] \\ {\bar{Θ}}_{17 i} = [\begin{matrix} \sqrt{d_{M}} (A_{i}^{T} (k) - I) R_{1} & \sqrt{\bar{d}} (A_{i}^{T} (k) - I) R_{2} \end{matrix} \\ \begin{matrix} \sqrt{d_{M} σ} D_{i}^{T} (k) R_{1} & \sqrt{\bar{d} σ} D_{i}^{T} (k) R_{2} & \sqrt{d_{M}} S_{1 i} & \sqrt{\bar{d}} T_{1 i} & \sqrt{\bar{d}} U_{1 i} \end{matrix}] \\ {\bar{Θ}}_{27 i} = [\begin{matrix} \sqrt{d_{M}} A_{1 i}^{T} (k) R_{1} & \sqrt{\bar{d}} A_{1 i}^{T} (k) R_{2} \end{matrix} \\ \begin{matrix} \sqrt{d_{M} σ} D_{1 i}^{T} (k) R_{1} & \sqrt{\bar{d} σ} D_{1 i}^{T} (k) R_{2} & \sqrt{d_{M}} S_{2 i} & \sqrt{\bar{d}} T_{2 i} & \sqrt{\bar{d}} U_{2 i} \end{matrix}] \end{matrix}

On the basis of (7) and Lemma 1, (24) is equal to

{\bar{Θ}}_{i} + ε_{1 i}^{- 1} ϕ_{1 i}^{T} ϕ_{1 i} + ε_{2 i}^{- 1} ϕ_{2 i}^{T} ϕ_{2 i} + ε_{1 i} ϕ_{3 i} ϕ_{3 i}^{T} + ε_{2 i} ϕ_{4 i} ϕ_{4 i}^{T} < 0

(25)

By Schur complement, (25) is equivalent to (12), which guarantees ${\bar{Θ}}_{i} + Γ_{i} < 0$ . Obviously, there exists a scalar α > 0 such that ${\bar{Θ}}_{i} + Γ_{i} + α I < 0$ . From (23), it is easy to obtain

E {V (e (k + 1), r_{k + 1}) | e (k), r_{k} = i} - V (e (k), r_{k} = i) < - α | x (k) |^{2}

(26)

Then follow the same lines of the proof of Theorem 1 (Xu et al., 2004), according to Definition 1, we can obtain that system (1) with u(k) = 0 and v(k) = 0 is robustly stochastically stable. This completes the proof.

Next, we shall present the robust non-fragile stabilization problem for system (1).

Theorem 2. System (1) with v(k) = 0 is robustly stochastically stable for all admissible uncertainties satisfying (7) and (11), if there exist matrices X_i > 0, Y_i , ${\tilde{Q}}_{1} > 0$ , ${\tilde{Q}}_{2} > 0$ , ${\tilde{Q}}_{3} > 0$ , Z₁ > 0, Z₂ > 0, ${\tilde{T}}_{1 i}$ , ${\tilde{T}}_{2 i}$ , ${\tilde{S}}_{1 i}$ , ${\tilde{S}}_{2 i}$ , ${\tilde{U}}_{1 i}$ , ${\tilde{U}}_{2 i}$ , L and scalars ε_1i > 0, ε_2i > 0, ε_3i > 0 $(i \in S)$ such that the following LMIs hold

[\begin{matrix} Φ_{i} & {\tilde{ϕ}}_{i} \\ * & - {\tilde{ε}}_{i} I \end{matrix}] < 0

(27)

Where

\begin{matrix} Φ_{i} = [\begin{matrix} Φ_{11 i} & Φ_{12 i} & {\tilde{U}}_{1 i} & - {\tilde{T}}_{1 i} & Φ_{1 i} \\ * & Φ_{22 i} & {\tilde{U}}_{2 i} & - {\tilde{T}}_{2 i} & Φ_{2 i} \\ * & * & - Q_{2} & 0 & 0 \\ * & * & * & - Q_{3} & 0 \\ * & * & * & * & Φ_{5 i} \end{matrix}] \\ {\tilde{υ}}_{i} = [\begin{matrix} {\tilde{υ}}_{1 i}^{T} & {\tilde{υ}}_{2 i}^{T} & ε_{1 i} {\tilde{υ}}_{3 i} & ε_{2 i} {\tilde{υ}}_{4 i} & {\tilde{υ}}_{5 i}^{T} & ε_{3 i} {\tilde{υ}}_{6 i} \end{matrix}] \\ {\tilde{ε}}_{i} = diag {- ε_{1 i} I, - ε_{2 i} I, - ε_{1 i} I, - ε_{2 i} I, - ε_{3 i} I, - ε_{3 i} I} \\ Φ_{1 i} = [\begin{matrix} Φ_{15 i} & Φ_{16 i} & Φ_{17 i} \end{matrix}] \\ Φ_{2 i} = [\begin{matrix} Φ_{25 i} & Φ_{26 i} & Φ_{27 i} \end{matrix}] \\ Φ_{5 i} = diag {Φ_{55 i}, Φ_{55 i}, Φ_{77 i}} \\ Φ_{11 i} = (\bar{d} + 1) {\tilde{Q}}_{1} + {\tilde{Q}}_{2} + {\tilde{Q}}_{3} + {\tilde{S}}_{1 i} + {\tilde{S}}_{1 i}^{T} + X_{i} - L - L^{T} \\ Φ_{12 i} = {\tilde{S}}_{2 i}^{T} - {\tilde{S}}_{1 i} + {\tilde{T}}_{1 i} - {\tilde{U}}_{1 i} \\ Φ_{22 i} = - {\tilde{Q}}_{1} - {\tilde{S}}_{2 i} - {\tilde{S}}_{2 i}^{T} + {\tilde{T}}_{2 i} + {\tilde{T}}_{2 i}^{T} - {\tilde{U}}_{2 i} - {\tilde{U}}_{2 i}^{T} \\ Φ_{15 i} = [\sqrt{π_{i 1}} (L^{T} A_{i}^{T} + Y_{i}^{T} A_{2 i}^{T}) \dots \sqrt{π_{iN}} (L^{T} A_{i}^{T} + Y_{i}^{T} A_{2 i}^{T})] \\ Φ_{25 i} = [\sqrt{π_{i 1}} L^{T} A_{1 i}^{T} \dots \sqrt{π_{iN}} L^{T} A_{1 i}^{T}] \\ Φ_{55 i} = diag {- X_{1}, \dots, - X_{N}} \\ Φ_{16 i} = [\sqrt{σ π_{i 1}} (L^{T} D_{i}^{T} + Y_{i}^{T} D_{2 i}^{T}) \dots \sqrt{σ π_{iN}} (L^{T} D_{i}^{T} + Y_{i}^{T} D_{2 i}^{T})] \\ Φ_{26 i} = [\sqrt{σ π_{i 1}} L^{T} D_{1 i}^{T} \dots \sqrt{σ π_{iN}} L^{T} D_{1 i}^{T}] \\ Φ_{17 i} = [\sqrt{d_{M}} (L^{T} A_{i}^{T} + Y_{i}^{T} A_{2 i}^{T} - L^{T}) \sqrt{\bar{d}} (L^{T} A_{i}^{T} + Y_{i}^{T} A_{2 i}^{T} - L^{T}) \\ \sqrt{d_{M} σ} (L^{T} D_{i}^{T} + Y_{i}^{T} D_{2 i}^{T}) \sqrt{\bar{d} σ} (L^{T} D_{i}^{T} + Y_{i}^{T} D_{2 i}^{T}) \sqrt{d_{M}} {\tilde{S}}_{1 i} \sqrt{\bar{d}} {\tilde{T}}_{1 i} \sqrt{\bar{d}} {\tilde{U}}_{1 i}] \\ Φ_{27 i} = [\sqrt{d_{M}} L^{T} A_{1 i}^{T} \sqrt{\bar{d}} L^{T} A_{1 i}^{T} \sqrt{d_{M} σ} L^{T} D_{1 i}^{T} \sqrt{\bar{d} σ} L^{T} D_{1 i}^{T} \sqrt{d_{M}} {\tilde{S}}_{2 i} \sqrt{\bar{d}} {\tilde{T}}_{2 i} \sqrt{\bar{d}} {\tilde{U}}_{2 i}] \\ Φ_{77 i} = diag {Z_{1} - L - L^{T}, Z_{2} - L - L^{T}, Z_{1} - L - L^{T}, Z_{2} - L - L^{T}, - Z_{1}, - (Z_{1} + Z_{2}), - Z_{2}} \\ {\tilde{υ}}_{1 i} = [N_{1 i} L N_{2 i} L 0_{2} 0 \dots 0 0 \dots 0 0_{7}] \\ {\tilde{υ}}_{2 i} = [\sqrt{σ} N_{4 i} L \sqrt{σ} N_{5 i} L 0_{2} 0 \dots 0 0 \dots 0 0_{7}] \\ {\tilde{υ}}_{3 i} = {[0_{4} \sqrt{π_{i 1}} M_{1 i}^{T} \dots \sqrt{π_{iN}} M_{1 i}^{T} 0 \dots 0 \sqrt{d_{M}} M_{1 i}^{T} \sqrt{\bar{d}} M_{1 i}^{T} 0_{5}]}^{T} \\ {\tilde{υ}}_{4 i} = {[0_{4} 0 \dots 0 \sqrt{π_{i 1}} M_{1 i}^{T} \dots \sqrt{π_{iN}} M_{1 i}^{T} 0_{2} \sqrt{d_{M}} M_{1 i}^{T} \sqrt{\bar{d}} M_{1 i}^{T} 0_{3}]}^{T} \\ {\tilde{υ}}_{5 i} = [N_{7 i} L 0_{3} 0 \dots 0 0 \dots 0 0_{7}] \\ {\tilde{υ}}_{6 i} = [0_{4} \sqrt{π_{i 1}} M_{2 i}^{T} A_{2 i}^{T} \dots \sqrt{π_{iN}} M_{2 i}^{T} A_{2 i}^{T} \sqrt{σ π_{i 1}} M_{2 i}^{T} D_{2 i}^{T} \dots \sqrt{σ π_{iN}} M_{2 i}^{T} D_{2 i}^{T} \\ {\sqrt{d_{M}} M_{2 i}^{T} A_{2 i}^{T} \sqrt{\bar{d}} M_{2 i}^{T} A_{2 i}^{T} \sqrt{σ d_{M}} M_{2 i}^{T} D_{2 i}^{T} \sqrt{σ \bar{d}} M_{2 i}^{T} D_{2 i}^{T} 0_{3}]}^{T} \end{matrix}

Furthermore, a desired non-fragile controller in the form of (10) with the state-feedback gains is given as follows

K_{i} = Y_{i} L^{- 1}, i = 1, 2, \dots, N

(28)

Proof Substituting the non-fragile controller (10) into (24) with v(k) = 0, then (24) is equal to

\begin{matrix} {\bar{Θ}}_{i} + ε_{1 i}^{- 1} ϕ_{1 i}^{T} ϕ_{1 i} + ε_{2 i}^{- 1} ϕ_{2 i}^{T} ϕ_{2 i} + ε_{1 i} ϕ_{3 i} ϕ_{3 i}^{T} + ε_{2 i} ϕ_{4 i} ϕ_{4 i}^{T} \\ + ε_{3 i}^{- 1} ϕ_{5 i}^{T} ϕ_{5 i} + ε_{3 i} ϕ_{6 i}^{T} ϕ_{6 i} < 0 \end{matrix}

(29)

where

\begin{matrix} ϕ_{5} & = [N_{7} 0_{3} 0 \dots 0 0 \dots 0 0_{7}] \\ ϕ_{6} & = [0_{4} \sqrt{π_{i 1}} M_{2 i}^{T} A_{2 i}^{T} P_{1} \dots \sqrt{π_{iN}} M_{2 i}^{T} A_{2 i}^{T} P_{N} \sqrt{σ π_{i 1}} M_{2 i}^{T} D_{2 i}^{T} P_{1} \dots \sqrt{σ π_{iN}} M_{2 i}^{T} D_{2 i}^{T} P_{N} \\ {\sqrt{d_{M}} M_{2 i}^{T} A_{2 i}^{T} R_{1} \sqrt{\bar{d}} M_{2 i}^{T} A_{2 i}^{T} R_{2} \sqrt{σ d_{M}} M_{2 i}^{T} D_{2 i}^{T} R_{1} \sqrt{σ \bar{d}} M_{2 i}^{T} D_{2 i}^{T} R_{2} 0_{3}]}^{T} \end{matrix}

By Schur complement, we have

[\begin{matrix} {\overset{⌣}{Θ}}_{i} & {\overset{⌣}{ϕ}}_{i} \\ * & - {\tilde{ε}}_{i} I \end{matrix}] < 0

(30)

where

\begin{matrix} {\overset{⌣}{Θ}}_{i} = [\begin{matrix} Θ_{11 i} & Θ_{12 i} & U_{1} & - T_{1} & {\overset{⌣}{Θ}}_{1 i} \\ * & Θ_{22 i} & U_{2} & - T_{2} & Θ_{2 i} \\ * & * & - Q_{2} & 0 & 0 \\ * & * & * & - Q_{3} & 0 \\ * & * & * & * & Θ_{5 i} \end{matrix}] \\ {\overset{⌣}{ϕ}}_{i} = [\begin{matrix} ϕ_{1 i}^{T} & ϕ_{2 i}^{T} & ε_{1 i} ϕ_{3 i} & ε_{2 i} ϕ_{4 i} & ϕ_{5 i}^{T} & ε_{3 i} ϕ_{6 i} \end{matrix}] \\ {\overset{⌣}{Θ}}_{1 i} = [\begin{matrix} {\overset{⌣}{Θ}}_{15 i} & {\overset{⌣}{Θ}}_{16 i} & {\overset{⌣}{Θ}}_{17 i} \end{matrix}] \\ {\overset{⌣}{Θ}}_{15 i} = [{(A_{i} + A_{2 i} K_{i})}^{T} \sqrt{π_{i 1}} P_{1} \dots {(A_{i} + A_{2 i} K_{i})}^{T} \sqrt{π_{iN}} P_{N}] \\ {\overset{⌣}{Θ}}_{16 i} = [{(D_{i} + D_{2 i} K_{i})}^{T} \sqrt{σ π_{i 1}} P_{1} \dots {(D_{i} + D_{2 i} K_{i})}^{T} \sqrt{σ π_{iN}} P_{N}] \\ {\overset{⌣}{Θ}}_{17 i} = [\sqrt{d_{M}} {(A_{i} + A_{2 i} K_{i} - I)}^{T} R_{1} \sqrt{\bar{d}} {(A_{i} + A_{2 i} K_{i} - I)}^{T} R_{2} \\ \sqrt{d_{M} σ} {(D_{i} + D_{2 i} K_{i})}^{T} R_{1} \sqrt{\bar{d} σ} {(D_{i} + D_{2 i} K_{i})}^{T} R_{2} \\ \sqrt{d_{M}} S_{1 i} \sqrt{\bar{d}} T_{1 i} \sqrt{\bar{d}} U_{1 i}] \end{matrix}

Now, by denoting

\begin{matrix} P_{i} = X_{i}^{- 1}, Q_{1} = L^{- T} {\tilde{Q}}_{1} L^{- 1}, Q_{2} = L^{- T} {\tilde{Q}}_{2} L^{- 1}, Q_{3} = L^{- T} {\tilde{Q}}_{3} L^{- 1} \\ S_{1} = L^{- T} {\tilde{S}}_{1} L^{- 1}, S_{2} = L^{- T} {\tilde{S}}_{2} L^{- 1}, T_{1} = L^{- T} {\tilde{T}}_{1} L^{- 1}, T_{2} = L^{- T} {\tilde{T}}_{2} L^{- 1} \\ U_{1} = L^{- T} {\tilde{U}}_{1} L^{- 1}, U_{2} = L^{- T} {\tilde{U}}_{2} L^{- 1}, R_{1} = L^{- T} {\tilde{Z}}_{1} L^{- 1}, R_{2} = L^{- T} {\tilde{Z}}_{2} L^{- 1} \end{matrix}

with

V = diag {L_{4}, X_{1} \dots X_{N}, X_{1} \dots X_{N}, R_{1}^{- 1}, R_{2}^{- 1}, R_{1}^{- 1}, R_{2}^{- 1}, L_{3}, I_{7}}

Pre- and post-multiplying the aforementioned inequality (30) by V^T and V, respectively. For each $i \in S$

{(X_{i} - L)}^{T} X_{i}^{- 1} (X_{i} - L) = X_{i} - L - L^{T} + L^{T} X_{i}^{- 1} L \geq 0

Then we have

- L^{T} X_{i}^{- 1} L \leq X_{i} - L - L^{T}

Similarly,

- L^{T} Z_{1}^{- 1} L \leq Z_{1} - L - L^{T}, - L^{T} Z_{2}^{- 1} L \leq Z_{2} - L - L^{T}

By setting Y_i = LK_i , it can be easily seen that the resulting LMIs are equivalent to those of equation (27).

Remark 1. In this paper, inspired by the works in Zhang et al. (2011), free-weighting matrices were introduced. The problem of delay-dependent robust stabilisation for uncertain discrete-time fuzzy Markovian jump systems with mode-dependent time-varying delays is considered in Zhang et al. (2011). By using the same method, the newly obtained delay-dependent results are less conservative than the existing ones in Xu et al. (2004).

Robust non-fragile H_∞ control

In order to solve the robust non-fragile H_∞ control problem for system Σ, we first give a solution to the H_∞ performance analysis for systems Σ with u(k) = 0.

Theorem 3. For a given γ > 0, the systems Σ with u(k) = 0 are robustly stochastically stable for all admissible uncertainties satisfying (7), if there exist matrices P_i > 0, Q₁ > 0, Q₂ > 0, Q₃ > 0, R₁ > 0, R₂ > 0, T_1i, T_2i, S_1i, S_2i, U_1i, U_2i and scalars ε_1i > 0, ε_2i > 0 $(i \in S)$ such that the following matrix inequalities hold

[\begin{matrix} {\hat{Θ}}_{i} & {\hat{ϕ}}_{5}^{T} & {\hat{ϕ}}_{i} \\ * & - I & 0 \\ * & * & - ε_{i} I \end{matrix}] < 0

(31)

where

\begin{matrix} {\hat{Θ}}_{i} = [\begin{matrix} Θ_{11 i} & Θ_{12 i} & U_{1 i} & - T_{1 i} & 0 & Θ_{1 i} \\ * & Θ_{22 i} & U_{2 i} & - T_{2 i} & 0 & Θ_{2 i} \\ * & * & - Q_{2} & 0 & 0 & 0 \\ * & * & * & - Q_{3} & 0 & 0 \\ * & * & * & * & - γ^{2} I & Θ_{3 i} \\ * & * & * & * & * & Θ_{5 i} \end{matrix}] \\ {\hat{φ}}_{i} = [\begin{matrix} {\hat{φ}}_{1 i}^{T} & {\hat{φ}}_{2 i}^{T} & ε_{1 i} {\hat{φ}}_{3 i} & ε_{2 i} {\hat{φ}}_{4 i} \end{matrix}] \\ Θ_{3 i} = [\begin{matrix} Θ_{35 i} & Θ_{36 i} & Θ_{37 i} \end{matrix}] \\ Θ_{35 i} = [A_{3 i}^{T} \sqrt{π_{i 1}} P_{1} \dots A_{3 i}^{T} \sqrt{π_{iN}} P_{N}] \\ Θ_{36 i} = [D_{3 i}^{T} \sqrt{σ π_{i 1}} P_{1} \dots D_{3 i}^{T} \sqrt{σ π_{iN}} P_{N}] \\ Θ_{37 i} = [\sqrt{d_{M}} A_{3 i}^{T} R_{1} \sqrt{\bar{d}} A_{3 i}^{T} R_{2} \sqrt{d_{M} σ} D_{3 i}^{T} R_{1} \sqrt{\bar{d} σ} D_{3 i}^{T} R_{2} 0_{3}] \\ {\hat{φ}}_{1 i} = [N_{1 i} N_{2 i} 0_{2} N_{3 i} 0 \dots 0 0 \dots 0 0_{7}] \\ {\hat{φ}}_{2 i} = [\sqrt{σ} N_{4 i} \sqrt{σ} N_{5 i} 0_{2} \sqrt{σ} N_{6 i} 0 \dots 0 0 \dots 0 0_{7}] \\ {\hat{φ}}_{3 i} = {[0_{5} \sqrt{π_{i 1}} M_{1 i}^{T} P_{1} \dots \sqrt{π_{iN}} M_{1 i}^{T} P_{N} 0 \dots 0 \sqrt{d_{M}} M_{1 i}^{T} R_{1} \sqrt{\bar{d}} M_{1 i}^{T} R_{2} 0_{5}]}^{T} \\ {\hat{φ}}_{4 i} = {[0_{5} 0 \dots 0 \sqrt{π_{i 1}} M_{1 i}^{T} P_{1} \dots \sqrt{π_{iN}} M_{1 i}^{T} P_{N} 0_{2} \sqrt{d_{M}} M_{1 i}^{T} R_{1} \sqrt{\bar{d}} M_{1 i}^{T} R_{2} 0_{3}]}^{T} \\ {\hat{φ}}_{5 i} = [C_{i} C_{1 i} 0_{3} 0 \dots 0 0 \dots 0 0_{7}] \end{matrix}

Proof. From (31), it is clear that (12) holds, which implies that the systems Σ with u(k) = 0 are robustly stochastically stable. Then, under zero initial conditions, we will prove that the system Σ with u(k) = 0 satisfies ∥z(k)∥₂ < γ∥v(k)∥₂ for all nonzero v(k) ∈ L₂[0, ∞).

In the following, we suppose the zero initial condition and define

J = E {\sum_{k = 0}^{N} (| z (k) |^{2} - γ^{2} | v (k) |^{2})}

(32)

Choosing the Lyapunov-Krasovskii functional candidate V (k) as (13), we have

\begin{matrix} J & = & E {\sum_{k = 0}^{N} (| z (k) |^{2} + E [V_{k + 1} (x (k + 1), r_{k + 1}) | x_{k}, r_{k}] - V_{k} (x (k), r_{k}) - γ^{2} | v (k) |^{2})} \\ - E {V_{N + 1} (x (N + 1), r_{N + 1})} \\ \leq & E {\sum_{k = 0}^{N} (| z (k) |^{2} + E [V_{k + 1} (x (k + 1), r_{k + 1}) | x_{k}, r_{k}] - V_{k} (x (k), r_{k}) - γ^{2} | v (k) |^{2})} \\ = & E {ζ^{T} (k) ({\tilde{Θ}}_{i} + {\tilde{Γ}}_{i}) ζ (k)} \end{matrix}

(33)

where

\begin{matrix} {\tilde{Θ}}_{i} = [\begin{matrix} Θ_{11 i} & Θ_{12 i} & U_{1 i} & - T_{1 i} & 0 \\ * & Θ_{22 i} & U_{2 i} & - T_{2 i} & 0 \\ * & * & - Q_{2} & 0 & 0 \\ * & * & * & - Q_{3} & 0 \\ * & * & * & * & - γ^{2} I \end{matrix}] \\ {\tilde{Γ}}_{i} = {\tilde{Γ}}_{1}^{T} (\sum_{j = 1}^{N} π_{ij} P_{j}) {\tilde{Γ}}_{1} + σ {\tilde{Γ}}_{2}^{T} (\sum_{j = 1}^{N} π_{ij} P_{j}) {\tilde{Γ}}_{2} + d_{M} {\tilde{Γ}}_{3}^{T} R_{1} {\tilde{Γ}}_{3} + d_{M} σ {\tilde{Γ}}_{2}^{T} R_{1} {\tilde{Γ}}_{2} + \bar{d} {\tilde{Γ}}_{3}^{T} R_{2} {\tilde{Γ}}_{3} \\ + \bar{d} σ {\tilde{Γ}}_{2}^{T} R_{2} {\tilde{Γ}}_{2} + d_{M} {\tilde{S}}_{i} R_{1}^{- 1} {\tilde{S}}_{i}^{T} + \bar{d} {\tilde{T}}_{i} {(R_{1} + R_{2})}^{- 1} {\tilde{T}}_{i}^{T} + \bar{d} {\tilde{U}}_{i} R_{2}^{- 1} {\tilde{U}}_{i}^{T} + {\tilde{Γ}}_{4}^{T} {\tilde{Γ}}_{4} \\ ζ (k) = {[\begin{matrix} x^{T} (k) & x^{T} (k - d (k)) & x^{T} (k - d_{m}) & x^{T} (k - d_{M}) & v^{T} (k) \end{matrix}]}^{T} \\ {\tilde{Γ}}_{1} = [\begin{matrix} A_{i} (k) & A_{1 i} (k) & 0 & 0 & A_{3 i} (k) \end{matrix}] \\ {\tilde{Γ}}_{2} = [\begin{matrix} D_{i} (k) & D_{1 i} (k) & 0 & 0 & D_{3 i} (k) \end{matrix}] \\ {\tilde{Γ}}_{3} = [\begin{matrix} A_{i} (k) - I & A_{1 i} (k) & 0 & 0 & A_{3 i} (k) \end{matrix}] \\ {\tilde{Γ}}_{4} = [\begin{matrix} C_{i} & C_{1 i} & 0 & 0 & 0 \end{matrix}] \\ {\tilde{S}}_{i} = {[\begin{matrix} S_{1 i}^{T} & S_{2 i}^{T} & 0 & 0 & 0 \end{matrix}]}^{T}, {\tilde{T}}_{i} = {[\begin{matrix} T_{1 i}^{T} & T_{2 i}^{T} & 0 & 0 & 0 \end{matrix}]}^{T}, {\tilde{U}}_{i} = {[\begin{matrix} U_{1 i}^{T} & U_{2 i}^{T} & 0 & 0 & 0 \end{matrix}]}^{T} \end{matrix}

where $Θ_{11 i}, Θ_{12 i}, Θ_{22 i}, Θ_{1 i}, Θ_{2 i}, Θ_{5 i}, {\tilde{ε}}_{i}$ are defined in Theorem 1. Then Theorem 3 can be easily proved by following the method of the proof of Theorem 1.

Next, based on Theorem 3, we are in a position to present the result on the robust non-fragile H_∞ control problem for system Σ.

Theorem 4. For a given γ > 0, the systems Σ are robustly stochastically stable for all admissible uncertainties satisfying (7) and (11), if there exist matrices X_i > 0, Y_i , ${\tilde{Q}}_{1} > 0$ , ${\tilde{Q}}_{2} > 0$ , ${\tilde{Q}}_{3} > 0$ , Z₁ > 0, Z₂ > 0, ${\tilde{T}}_{1 i}$ , ${\tilde{T}}_{2 i}$ , ${\tilde{S}}_{1 i}$ , ${\tilde{S}}_{2 i}$ , ${\tilde{U}}_{1 i}$ , ${\tilde{U}}_{2 i}$ , L and scalars ε_1i > 0, ε_2i > 0, ε_3i > 0 $(i \in S)$ such that the following LMIs hold

[\begin{matrix} {\bar{Φ}}_{i} & {\bar{ϕ}}_{7 i}^{T} & {\bar{ϕ}}_{i} \\ * & - I & {\bar{ϕ}}_{8 i} \\ * & * & - {\tilde{ε}}_{i} I \end{matrix}] < 0

(34)

where

\begin{matrix} {\bar{Φ}}_{i} = [\begin{matrix} Φ_{11 i} & Φ_{12 i} & {\tilde{U}}_{1 i} & - {\tilde{T}}_{1 i} & 0 & Φ_{1 i} \\ * & Φ_{22 i} & {\tilde{U}}_{2 i} & - {\tilde{T}}_{2 i} & 0 & Φ_{2 i} \\ * & * & - Q_{2} & 0 & 0 & 0 \\ * & * & * & - Q_{3} & 0 & 0 \\ * & * & * & * & - γ^{2} I & Φ_{3 i} \\ * & * & * & * & * & Φ_{5 i} \end{matrix}] \\ {\bar{ϕ}}_{i} = [\begin{matrix} {\bar{ϕ}}_{1 i}^{T} & {\bar{ϕ}}_{2 i}^{T} & ε_{1 i} {\bar{ϕ}}_{3 i} & ε_{2 i} {\bar{ϕ}}_{4 i} & {\bar{ϕ}}_{5 i}^{T} & ε_{3 i} {\bar{ϕ}}_{6 i} \end{matrix}] \\ Φ_{3 i} = [\begin{matrix} Φ_{35 i} & Φ_{36 i} & Φ_{37 i} \end{matrix}] \\ Φ_{35 i} = [\begin{matrix} \sqrt{π_{i 1}} A_{3 i}^{T} & \dots & \sqrt{π_{iN}} A_{3 i}^{T} \end{matrix}] \\ Φ_{36 i} = [\begin{matrix} \sqrt{σ π_{i 1}} D_{3 i}^{T} & \dots & \sqrt{σ π_{iN}} D_{3 i}^{T} \end{matrix}] \\ Φ_{37 i} = [\begin{matrix} \sqrt{d_{M}} A_{3 i}^{T} & \sqrt{\bar{d}} A_{3 i}^{T} & \sqrt{d_{M} σ} D_{3 i}^{T} & \sqrt{\bar{d} σ} D_{3 i}^{T} 0_{3} \end{matrix}] \\ {\bar{ϕ}}_{1 i} = [N_{1 i} L N_{2 i} L 0_{2} N_{3 i} L 0 \dots 0 0 \dots 0 0_{7}] \\ {\bar{ϕ}}_{2 i} = [\sqrt{σ} N_{4 i} L \sqrt{σ} N_{5 i} L 0_{2} \sqrt{σ} N_{6 i} L 0 \dots 0 0 \dots 0 0_{7}] \\ {\bar{ϕ}}_{3 i} = {[0_{5} \sqrt{π_{i 1}} M_{1 i}^{T} \dots \sqrt{π_{iN}} M_{1 i}^{T} 0 \dots 0 \sqrt{d_{M}} M_{1 i}^{T} \sqrt{\bar{d}} M_{1 i}^{T} 0_{5}]}^{T} \\ {\bar{ϕ}}_{4 i} = {[0_{5} 0 \dots 0 \sqrt{π_{i 1}} M_{1 i}^{T} \dots \sqrt{π_{iN}} M_{1 i}^{T} 0_{2} \sqrt{d_{M}} M_{1 i}^{T} \sqrt{\bar{d}} M_{1 i}^{T} 0_{3}]}^{T} \\ {\bar{ϕ}}_{5 i} = [N_{7 i} L 0_{4} 0 \dots 0 0 \dots 0 0_{7}] \\ {\bar{ϕ}}_{6 i} = [0_{5} \sqrt{π_{i 1}} M_{2 i}^{T} A_{2 i}^{T} \dots \sqrt{π_{iN}} M_{2 i}^{T} A_{2 i}^{T} \sqrt{σ π_{i 1}} M_{2 i}^{T} D_{2 i}^{T} \dots \\ \sqrt{σ π_{iN}} M_{2 i}^{T} D_{2 i}^{T} \sqrt{d_{M}} M_{2 i}^{T} A_{2 i}^{T} \sqrt{\bar{d}} M_{2 i}^{T} A_{2 i}^{T} \sqrt{σ d_{M}} M_{2 i}^{T} D_{2 i}^{T} \\ \sqrt{σ \bar{d}} M_{2 i}^{T} D_{2 i}^{T} 0_{3}] T \\ {\bar{ϕ}}_{7 i} = [C_{i} L + C_{2 i} Y_{i} C_{1 i} L 0_{3} 0 \dots 0 0 \dots 0 0_{7}] \\ {\bar{ϕ}}_{8 i} = [0 0 0 0 0 ε_{3 i} C_{2 i} M_{2 i}] \end{matrix}

where $Φ_{11 i}, Φ_{12 i}, Φ_{22 i}, Φ_{1 i}, Φ_{2 i}, Φ_{5 i}, {\tilde{ε}}_{i}$ are defined in Theorem 2. Furthermore, a desired non-fragile H_∞ controller in the form of (10) with the state-feedback gains is given as follows

K_{i} = Y_{i} L^{- 1}, i = 1, 2, \dots, N

(35)

Proof. Substituting the non-fragile controller (10) into system (24), equation (34) can be formulated by following the same method used in the proof of Theorem 2 and Theorem 3.

Remark 2. In the following, in order to show that our result is less conservative than the one studied in Xu et al. (2004), we consider the case that there are no jumping parameters in system Σ, then the system Σ reduces to the following form

\begin{matrix} x (k + 1) = & A (k) x (k) + A_{1} (k) x (k - d (k)) + A_{2} u (k) + A_{3} (k) v (k) \\ + [D (k) x (k) + D_{1} (k) x (k - d (k)) + D_{2} u (k) \\ + D_{3} (k) v (k)] w (k) \end{matrix}

(36)

z (k) = Cx (k) + C_{1} x (k - d (k)) + C_{2} u (k)

(37)

where

\begin{matrix} A (k) = A + Δ A (k), A_{1} (k) = A_{1} + Δ A_{1} (k), A_{3} (k) = A_{3} + Δ A_{3} (k) \\ D (k) = D + Δ D (k), D_{1} (k) = D_{1} + Δ D_{1} (k), D_{3} (k) = D_{3} + Δ D_{3} (k) \end{matrix}

with

\begin{matrix} [\begin{matrix} Δ A (k) & Δ A_{1} (k) & Δ A_{3} (k) & Δ D (k) & Δ D_{1} (k) & Δ D_{3} (k) \end{matrix}] \\ = M_{1} F (k) [\begin{matrix} N_{1} & N_{2} & N_{3} & N_{4} & N_{5} & N_{6} \end{matrix}] \end{matrix}

(38)

F(k)(·): $R \to R^{k \times l}$ is an unknown time-varying matrix function satisfying $F^{T} (k) F (k) \leq I, \forall k$ . In such a case, the robust H_∞ controller is chosen as

u (k) = Kx (k)

(39)

The following corollary gives the result for the robust H_∞ control problem for system (43) and (44), which is a consequence of Theorem 4 and hence the proofs are omitted.

Corollary 1. For a given γ > 0, the systems (36) and (37) are robustly stochastically stable for all admissible uncertainties satisfying (41), if there exist matrices X > 0, Y, ${\tilde{Q}}_{1} > 0$ , ${\tilde{Q}}_{2} > 0$ , ${\tilde{Q}}_{3} > 0$ , Z₁ > 0, Z₂ > 0, ${\tilde{T}}_{1}$ , ${\tilde{T}}_{2}$ , ${\tilde{S}}_{1}$ , ${\tilde{S}}_{2}$ , ${\tilde{U}}_{1}$ , ${\tilde{U}}_{2}$ , L and scalars ε₁ > 0, ε₂ > 0 such that the following LMI holds

[\begin{matrix} Φ & ϕ_{6}^{T} & ϕ \\ * & - I & 0 \\ * & * & - ε I \end{matrix}] < 0

(40)

where

\begin{matrix} ϕ = [\begin{matrix} ϕ_{1}^{T} & ϕ_{2}^{T} & ε_{1} ϕ_{3} & ε_{2} ϕ_{4} \end{matrix}] \\ ϕ_{1} = [\begin{matrix} N_{1} L + N_{3} L & N_{2} L 0_{12} \end{matrix}] \\ ϕ_{2} = [\begin{matrix} N_{4} L + N_{6} L & N_{5} L 0_{12} \end{matrix}] \\ ϕ_{3} = {[0_{5} M_{1}^{T} 0 \sqrt{d_{M}} M_{1}^{T} \sqrt{\bar{d}} M_{1}^{T} 0_{5}]}^{T} \\ ϕ_{4} = {[0_{6} M_{1}^{T} 0_{2} \sqrt{d_{M}} M_{1}^{T} \sqrt{\bar{d}} M_{1}^{T} 0_{3}]}^{T} \\ ϕ_{6} = [\begin{matrix} CL + C_{2} Y & C_{1} L 0_{12} \end{matrix}] \\ ε = diag {- ε_{1} I, - ε_{2} I, - ε_{1} I, - ε_{2} I} \end{matrix}

\begin{matrix} Φ = [\begin{matrix} Φ_{11} & Φ_{12} & {\tilde{U}}_{1} & - {\tilde{T}}_{1} & 0 & Φ_{1} \\ * & Φ_{22} & {\tilde{U}}_{2} & - {\tilde{T}}_{2} & 0 & Φ_{2} \\ * & * & - Q_{2} & 0 & 0 & 0 \\ * & * & * & - Q_{3} & 0 & 0 \\ * & * & * & * & - γ^{2} I & Φ_{3} \\ * & * & * & * & * & Φ_{5} \end{matrix}] \\ Φ_{1} = [\begin{matrix} Φ_{15} & Φ_{16} & Φ_{17} \end{matrix}] \\ Φ_{2} = [\begin{matrix} Φ_{25} & Φ_{26} & Φ_{27} \end{matrix}] \\ Φ_{3} = [\begin{matrix} Φ_{35} & Φ_{36} & Φ_{37} \end{matrix}] \\ Φ_{5} = diag {Φ_{55}, Φ_{66}, Φ_{77}} \\ Φ_{11} = (\bar{d} + 1) {\tilde{Q}}_{1} + {\tilde{Q}}_{2} + {\tilde{Q}}_{3} + {\tilde{S}}_{1} + {\tilde{S}}_{1}^{T} + X - L - L^{T} \\ Φ_{12} = {\tilde{S}}_{2}^{T} - {\tilde{S}}_{1} + {\tilde{T}}_{1} - {\tilde{U}}_{1} \\ Φ_{22} = - {\tilde{Q}}_{1} - {\tilde{S}}_{2} - {\tilde{S}}_{2}^{T} + {\tilde{T}}_{2} + {\tilde{T}}_{2}^{T} - {\tilde{U}}_{2} - {\tilde{U}}_{2}^{T} \\ Φ_{15} = [\begin{matrix} L^{T} A^{T} + Y^{T} A_{2}^{T} & L^{T} D^{T} + Y^{T} D_{2}^{T} \end{matrix}] \\ Φ_{25} = [\begin{matrix} L^{T} A_{1}^{T} & L^{T} D_{1}^{T} \end{matrix}], Φ_{35} = [\begin{matrix} A_{3}^{T} & D_{3}^{T} \end{matrix}] \end{matrix}

\begin{matrix} Φ_{16} = [\begin{matrix} \sqrt{d_{M}} (L^{T} A^{T} + Y^{T} A_{2}^{T} - L^{T}) & \sqrt{\bar{d}} (L^{T} A^{T} + Y^{T} A_{2}^{T} - L^{T}) \end{matrix}] \\ Φ_{26} = [\begin{matrix} \sqrt{d_{M}} L^{T} A_{1}^{T} & \sqrt{\bar{d}} L^{T} A_{1}^{T} \end{matrix}] \\ Φ_{36} = [\begin{matrix} \sqrt{d_{M}} A_{3}^{T} & \sqrt{\bar{d}} A_{3}^{T} \end{matrix}] \\ Φ_{17} = [\begin{matrix} \sqrt{d_{M} σ} (L^{T} D^{T} + Y^{T} D_{2}^{T}) & \sqrt{\bar{d} σ} (L^{T} D^{T} + Y^{T} D_{2}^{T}) & \sqrt{d_{M}} {\tilde{S}}_{1} & \sqrt{\bar{d}} {\tilde{T}}_{1} & \sqrt{\bar{d}} {\tilde{U}}_{1} \end{matrix}] \\ Φ_{27} = [\begin{matrix} \sqrt{d_{M} σ} L^{T} D_{1}^{T} & \sqrt{\bar{d} σ} L^{T} D_{1}^{T} & \sqrt{d_{M}} {\tilde{S}}_{2} & \sqrt{\bar{d}} {\tilde{T}}_{2} & \sqrt{\bar{d}} {\tilde{U}}_{2} \end{matrix}] \\ Φ_{37} = [\begin{matrix} \sqrt{d_{M} σ} D_{3}^{T} & \sqrt{\bar{d} σ} D_{3}^{T} & 0_{3} \end{matrix}] \\ Φ_{55} = diag {- X, - X} \\ Φ_{66} = diag {Z_{1} - L - L^{T}, Z_{2} - L - L^{T}} \\ Φ_{77} = diag {Z_{1} - L - L^{T}, Z_{2} - L - L^{T}, - Z_{1}, - (Z_{1} + Z_{2}), - Z_{2}} \end{matrix}

Furthermore, a desired H_∞ controller in the form of (39) with the state-feedback gains is given as follows

K = {YL}^{- 1}

(41)

Remark 3. The results obtained in Theorem 4 and Corollary 1 are formulated in terms of LMIs. Hence, the controller gain matrix K_i (or K) can be effectively acquired by solving LMIs, which can be facilitated readily by resorting to the Matlab LMI Control Toolbox. At the same time the controller performance γ can also be optimised. For example, the H_∞ performance index γ described in Corollary 1 can be optimised by a convex optimisation algorithm:

Algorithm. $\min_{X, Y, {\tilde{Q}}_{1}, {\tilde{Q}}_{2}, {\tilde{Q}}_{3}, Z_{1}, Z_{2}, {\tilde{T}}_{1}, {\tilde{T}}_{2}, {\tilde{S}}_{1}, {\tilde{S}}_{2}, {\tilde{U}}_{1}, {\tilde{U}}_{2}, L, ε_{1}, ε_{2}} γ^{2}$ subject to LMI (40).

Numerical examples

Example 1. Consider system (36) and (37) with the matrices (Xu et al., 2004)

\begin{matrix} A = [\begin{matrix} - 1 & 0.5 \\ 0.2 & 0.3 \end{matrix}], A_{1} = [\begin{matrix} 0.2 & - 0.1 \\ 0 & 0.1 \end{matrix}], \\ A_{2} = [\begin{matrix} 2.2 & - 0.8 \\ 0 & 0.5 \end{matrix}], A_{3} = [\begin{matrix} 0.2 \\ 0.1 \end{matrix}] \\ D = [\begin{matrix} 0 & 0.1 \\ 0.2 & 0.1 \end{matrix}], D_{1} = [\begin{matrix} - 0.1 & 0.2 \\ - 0.1 & - 0.4 \end{matrix}], \\ D_{2} = [\begin{matrix} 0.8 & 1 \\ - 0.6 & 2.5 \end{matrix}], C = [\begin{matrix} 0.2 & 0.1 \end{matrix}] \\ M_{1} = [\begin{matrix} 0.1 \\ 0.2 \end{matrix}], N_{1} = [\begin{matrix} 0.1 & 0 \end{matrix}], N_{2} = [\begin{matrix} 0.2 & 0.1 \end{matrix}], \\ N_{3} = [\begin{matrix} 0 & 0.1 \end{matrix}] N_{4} = [\begin{matrix} 0.2 & 0 \end{matrix}], N_{5} = [\begin{matrix} 0.1 & 0.3 \end{matrix}], \\ N_{6} = [\begin{matrix} 0.1 & 0.3 \end{matrix}] \end{matrix}

For different d_M with d_m = 0, by solving LMI (40), we can get the minimum controller performance γ in Table 1 and Table 2, respectively, in which “—” means that the results are not applicable to the corresponding cases.

Table 1.

Minimum γ for different d_M with d_m = 0 and σ = 1.

d_M	1	2	3	4	10
(Xu et al., 2004)	0.0891	0.1375	0.3644	—	—
Corollary 1	0.0006	0.0009	0.0011	0.0012	0.0019

Table 2.

Minimum γ for different d_M with d_m = 0 and σ = 0.5.

d_M	2	4	6	8	10
(Xu et al., 2004)	0.0866	0.1350	0.3388	—	—
Corollary 1	0.0009	0.0012	0.0015	0.0017	0.0019

Table 1 conclusively shows that our results are less conservative than those in Xu et al. (2004). When σ = 0.5, d_m = 2 and d_M = 4, γ= 0.1, the H_∞ controller gain matrix is given as follows

K = [\begin{matrix} 0.8217 & - 0.1334 \\ - 0.1060 & - 0.0099 \end{matrix}]

With the initial condition x(0) = [1 2]^T, the trajectory of system states and the controlled output z(k) are plotted in Figure 1 and Figure 2, respectively. The disturbance input is given as v(k) = 2e ^−k/2 sin(k/3) and F(k) = sin(k).

Example 2. Consider the system Σ with the following the matrices:

Mode 1

\begin{matrix} A_{1} = [\begin{matrix} - 0.3 & 0 \\ 0 & 0.4 \end{matrix}], A_{11} = [\begin{matrix} 0.1 & 0.2 \\ - 0.3 & 0.1 \end{matrix}], \\ A_{21} = [\begin{matrix} 0.2 & - 0.1 \\ 0.2 & 0.2 \end{matrix}], A_{31} = [\begin{matrix} 0.1 & 0 \\ 0 & - 0.3 \end{matrix}] \\ D_{1} = [\begin{matrix} - 0.2 & 0.1 \\ 0.2 & 0.3 \end{matrix}], D_{11} = [\begin{matrix} 0.2 & 0 \\ 0 & 0.3 \end{matrix}], \\ D_{21} = [\begin{matrix} - 0.4 & 0.3 \\ 0.1 & 0.2 \end{matrix}], D_{31} = [\begin{matrix} 0.1 & 0 \\ 0 & 0.2 \end{matrix}] \\ C_{1} = [\begin{matrix} 0 & 0.1 \\ 0 & 0 \end{matrix}], C_{11} = [\begin{matrix} 0.1 & 0 \\ 0.1 & 0.2 \end{matrix}], \\ C_{21} = [\begin{matrix} - 0.1 & 0.1 \\ 0.2 & - 0.2 \end{matrix}] \end{matrix}

Mode 2

\begin{matrix} A_{2} = [\begin{matrix} 0.3 & 0 \\ 0 & 0.2 \end{matrix}], A_{12} = [\begin{matrix} 0.3 & 0.1 \\ 0.1 & - 0.2 \end{matrix}], \\ A_{22} = [\begin{matrix} 0.15 & - 0.2 \\ 0 & 0.1 \end{matrix}], A_{32} = [\begin{matrix} 0.2 & 0.1 \\ 0 & - 0.3 \end{matrix}] \\ D_{2} = [\begin{matrix} 0.2 & - 0.2 \\ 0 & 0.1 \end{matrix}], D_{12} = [\begin{matrix} - 0.1 & 0.3 \\ 0.2 & 0.1 \end{matrix}], \\ D_{22} = [\begin{matrix} 0.1 & - 0.2 \\ 0 & 0.1 \end{matrix}], D_{32} = [\begin{matrix} 0.2 & 0.1 \\ 0 & - 0.2 \end{matrix}] \\ C_{2} = [\begin{matrix} 0.2 & 0 \\ 0 & 0.3 \end{matrix}], C_{12} = [\begin{matrix} 0.2 & 0.1 \\ - 0.1 & 0.3 \end{matrix}], \\ C_{22} = [\begin{matrix} 0.2 & - 0.2 \\ 0.1 & 0.2 \end{matrix}] \end{matrix}

Mode 3

\begin{matrix} A_{3} = [\begin{matrix} - 0.4 & 0 \\ 0 & - 0.3 \end{matrix}], A_{13} = [\begin{matrix} - 0.2 & 0.3 \\ 0.1 & 0.2 \end{matrix}], \\ A_{23} = [\begin{matrix} - 0.2 & 0 \\ 0 & 0.3 \end{matrix}], A_{33} = [\begin{matrix} 0.1 & 0 \\ 0 & - 0.2 \end{matrix}] \\ D_{3} = [\begin{matrix} 0 & 0.2 \\ 0.2 & 0.3 \end{matrix}], D_{13} = [\begin{matrix} - 0.1 & 0.2 \\ 0.1 & - 0.3 \end{matrix}], \\ D_{23} = [\begin{matrix} 0.3 & 0 \\ 0.2 & - 0.3 \end{matrix}], D_{33} = [\begin{matrix} 0.1 & 0.2 \\ 0 & - 0.1 \end{matrix}] \\ C_{3} = [\begin{matrix} - 0.2 & 0.1 \\ 0.15 & - 0.1 \end{matrix}], C_{13} = [\begin{matrix} 0.1 & 0.15 \\ 0 & - 0.1 \end{matrix}], \\ C_{23} = [\begin{matrix} - 0.1 & 0.2 \\ 0.2 & 0.1 \end{matrix}] \end{matrix}

with

\begin{matrix} M_{11} = 0.1 I, M_{12} = - 0.15 I, M_{13} = - 0.1 I, M_{21} = - 0.1 I, \\ M_{22} = 0.1 I, M_{23} = - 0.15 I \\ N_{11} = 0.1 I, N_{12} = - 0.1 I, N_{13} = - 0.2 I, N_{21} = 0.2 I, \\ N_{22} = 0.1 I, N_{23} = - 0.1 I \\ N_{31} = - 0.1 I, N_{32} = 0.2 I, N_{33} = - 0.15 I, N_{41} = - 0.1 I, \\ N_{42} = 0.15 I, N_{43} = 0.1 I \\ N_{51} = 0.2 I, N_{52} = - 0.2 I, N_{53} = - 0.1 I, N_{61} = 0.15 I, \\ N_{62} = 0.2 I, N_{63} = - 0.1 I \\ N_{71} = 0, N_{72} = 0.1 I, N_{73} = - 0.1 I \end{matrix}

The transition probabilities matrix is given as follows:

Π = [\begin{matrix} 0.1 & 0.5 & 0.4 \\ 0.2 & 0.3 & 0.5 \\ 0.1 & 0.2 & 0.7 \end{matrix}]

The purpose of this example is to illustrate the validity of non-fragile H_∞ controller designed in Theorem 4. For different d_M with d_m = 0 and σ = 1, the lower bound of noise attenuation level γ are given in Table 3.

Table 3.

Lower bound of γ for different d_M with d_m = 0 and σ = 1.

d_M	2	4	6	8	10
Theorem 4	0.5390	0.7622	0.9335	1.0779	1.2501

For given d_m = 2, d_M = 4 and σ = 1, by Theorem 4, we get the minimum allowance value is γ = 0.6601, and the non-fragile H_∞ controller gain matrices of (35) is given as follows

K_{1} = [\begin{matrix} 1.7302 & 4.1196 \\ 0.1140 & 0.5488 \end{matrix}], K_{2} = [\begin{matrix} - 0.0887 & 6.8734 \\ - 0.2612 & 3.4475 \end{matrix}], K_{3} = [\begin{matrix} - 0.9798 & 0.0880 \\ 0.1004 & 3.1776 \end{matrix}]

With the initial conditions x(0) = [3 1]^T, the trajectory of system states, the controlled output z(k) and one of the possible realisations of the Markovian jumping modes are plotted in Figure 3, Figure 4 and Figure 5, respectively. The disturbance input is given as $v (k) = {[\begin{matrix} \frac{1}{4^{k}} & \frac{1}{2^{k}} \end{matrix}]}^{T}$ , and F_i (k) = sin(k),F_1i(k) = sin(k),i = 1,2,3.

It is clearly observed from the simulation results above that all the expected objectives are well achieved.

Conclusion

The robust stochastic stabilisation and H_∞ control problem for uncertain discrete-time stochastic systems with Markovian jumping parameters and time-varying delay is investigated in this paper. The parameter uncertainties are assumed to be norm-bounded. On the basis of the Lyapunov-Krasovskii functional, delay-segment-dependent sufficient conditions for the solvability of the proposed problems are formulated in terms of LMIs. Finally, numerical examples have been provided to demonstrate the effectiveness and usefulness of the proposed design method. Our future work would be the extension of the present results to more complex discrete-time stochastic Markov jump systems, such as discrete-time network-based stochastic Markov jump systems (Lu et al., 2013) and discrete-time stochastic Markov jump with randomly occurring uncertainties and sensor failures (Shen et al., 2014).

Footnotes

Funding

This work was supported by the Natural Science Foundation of Jiangsu Province (No. BK20130239) and the Research Fund for the Doctoral Program of Higher Education of China (No. 20130094120015).

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Robust non-fragile stabilisation and H ∞ control for uncertain discrete-time stochastic systems with Markovian jump parameters and time-varying delay

Abstract

Keywords

Introduction

Problem description

Robust non-fragile stabilization

Robust non-fragile H∞ control

Numerical examples

Conclusion

Footnotes

Funding

References

Robust non-fragile H_∞ control