Robust non-fragile guaranteed cost control for singular Markovian jump time-delay systems

Abstract

This paper deals with the problem of robust non-fragile guaranteed cost control for a class of uncertain singular Markovian jump systems with time-varying delays. The time-varying and norm-bounded parameter uncertainties exist in both the state matrix and the derivative matrix. A non-fragile proportional-derivative state feedback controller is designed to guarantee the closed-loop singular Markovian jump system is normal and robustly stochastically stable. By proposing an optimization strategy in the framework of linear matrix inequalities, the upper bound of the cost function can be minimized. Finally, a numerical example is given to demonstrate the effectiveness of the results.

Keywords

Singular Markovian jump systems time-varying delays proportional-derivative state feedback controller non-fragile guaranteed cost control linear matrix inequalities

Introduction

Due to the particularity of the structure, singular systems are complicated, though they are widely applied in many fields such as interconnected systems, economical systems, and so on (Dai, 1989). Although corresponding control problems are difficult since considerations must be given to stability, regularity and non-impulsiveness (continuous-time) or causality (discrete-time) simultaneously, many researchers have devoted time to studying control problems of singular systems because of the theoretical and practical values. Lots of relevant literature has been published; (see, e.g. the work by Chu and Ho, 1999; Duan and Patton, 1999; Lin et al., 2005; Lv et al., 2015; Ma and Zhang, 2009; Ren and Zhang, 2012, 2015; Sakthivel et al., 2015; Wang and Zhang, 2012; Wang et al., 2012, 2013; Wu et al., 2012a,b, 2013; Xia et al., 2008; Xu and Lam, 2006; Xu et al., 2002; Yue and Han, 2004; Zhang and Liu, 2011; Zhuang et al., 2016).

The structure of the derivative matrix can affect the performance of a singular system directly. An arbitrarily small perturbation on the derivative matrix may destabilize the system (Xu and Lam, 2006). Therefore, a traditional state or output feedback controller cannot stabilize a singular system with perturbations in the derivative matrix. Up to now, some studies have been done on singular systems with uncertainties in the derivative matrices (see, e.g. the work by Lin et al., 2005; Lv et al., 2015; Ren and Zhang, 2012, 2015; Wang and Zhang, 2012). A proportional-derivative feedback controller was first employed to normalize and stabilize singular systems with norm-bounded uncertainties in the derivative matrices in the work by Lin et al. (2005). Based on the linear matrix inequalities (LMIs) method, Ren and Zhang (2012, 2015) made some progress on robust guaranteed cost control and $H_{\infty}$ control for singular systems with uncertainties in the derivative matrices, which also used proportional-derivative state feedback controllers (PDSFCs). It is worth mentioning that proportional-derivative feedback controllers have been widely used in singular systems, such as the regularization problem (Chu and Ho, 1999), pole assignment (Duan and Patton, 1999) and impulsive mode elimination (Zhang and Liu, 2011).

On the other hand, Markovian jump systems (MJSs), as one kind of stochastic systems, have their own characteristics and applications. Much effort has been done for the analysis and synthesis of MJSs (see, e.g. the work by Jiao et al., 2016; Li et al., 2014; Lv et al., 2015; Ma and Zhang, 2009; Sakthivel et al., 2015; Shen et al., 2014; Wang and Zhang, 2012; Wang et al., 2013; Wei and Zheng, 2014; Wei et al., 2016; Wu et al., 2012a,b, 2013; Xia et al., 2008, 2012; Xu et al., 2007; Zhang and Lam, 2010; Zhang et al., 2013; Zhao et al., 2009; Zhuang et al., 2016). It is named as the singular Markovian jump system (SMJS) when subsystems of the MJS are singular ones. SMJSs with time-varying delays are very complicated due to systems coupled with time-delay phenomenons. For SMJSs with time-varying delays, Zhuang et al. (2016) considered the $H_{\infty}$ deconvolution filtering design problem; Wu et al. (2013) dealt with the problem of passivity analysis; mixed $H_{\infty}$ and passive control was investigated in the work by Sakthivel et al. (2015). The exponential $H_{\infty}$ filtering design problem was solved by Wang et al. (2013) for SMJSs with mixed mode-dependent time-varying delays. For other control problems or the discrete time case, see the literature (Jiang et al., 2015; Ma et al., 2016; Ma and Zhang, 2009; Wu et al., 2012a,b; Xia et al., 2008).

The guaranteed cost control approach can make the upper bound of a given quadratic cost function minimized. So far, a lot of results on guaranteed cost control for different kinds of systems have existed (see, e.g. the work by Lv et al., 2015; Park, 2004; Petersen and McFarlane, 1994; Ren and Zhang, 2012; Wang et al., 2012; Yang et al., 2010; Zhao et al., 2009). Lv et al. (2015) investigated the guaranteed cost control and impulsive control problems for SMJSs with uncertainties in both systems matrices (the derivative matrices included) and transition rate matrices by PDSFCs. An $H_{\infty}$ guaranteed cost state feedback controller was provided to make the SMJS with a time-varying delay regular, impulse free and mean-square exponentially stable by strict LMIs in Wang et al. (2012). However, according to the authors’ knowledge, the guaranteed cost control problems for uncertain SMJSs with time-varying delays have not appeared in the literature when there are uncertainties in the derivative matrices.

In this paper, we investigate the problem of robust non-fragile guaranteed cost control for a class of uncertain SMJSs with time-varying delays via PDSFCs. The main work is stated in the following.

A non-fragile PDSFC is used to ensure that the considered SMJS is normal and robustly stochastically stable. Norm-bounded additive and multiplicative controller gain perturbations are considered, respectively.

The free-weighting-matrix approach in the work by He et al. (2007) is utilized to get the existence condition of a non-fragile PDSFC. We introduce two single integral terms into the constructed stochastic Lyapunov-Krasovskii functional (LKF), and some useful terms are not ignored when measuring the upper bound of the derivative of the constructed stochastic LKF.

The guaranteed cost obtained depends on the initial condition of the SMJS and the choice of guaranteed cost control laws. By selecting an optimal controller, we minimize the upper bound of the guaranteed cost.

The remaining part of the paper is organized as follows. The ‘Problem formulation’ section is the problem formulation which gives the system model, some definitions and useful lemmas. In the ‘Main results’ section, our main results are obtained by LMIs, where the closed-loop SMJS with a time-varying delay becomes normal and robustly stochastically stable and the upper bound of the cost function gets minimized by a robust non-fragile PDSFC. A numerical example is presented in the ‘Numerical example’ section to illustrate the effectiveness of our results and the ‘Conclusions’ section gives the conclusion of the paper.

Problem formulation

Let ${r_{t}, t \geq 0}$ be a continuous-time Markovian process with right continuous trajectories and taking values in a finite set $S = {1, 2, \dots, N}$ with transition probability matrix $Π = {π_{ij}}$ given by

P [r_{t + h} = j ∣ r_{t} = i] = {\begin{matrix} π_{i j} h + o (h), & j \neq i \\ 1 + π_{ii} h + o (h), & j = i \end{matrix}

where $h > 0, lim_{h \to 0} o (h) / h = 0$ and $π_{ij} \geq 0$ for $j \neq i$ , is the transition rate from mode i at time t to mode j at time $t + h$ and $π_{i i} = - \sum_{j = 1, j \neq i}^{N} π_{ij}$ .

Fix a probability space $(Ω, F, P)$ and consider a continuous-time uncertain SMJS with a time-varying delay, which is described by

\begin{matrix} (E (r_{t}) + Δ E (r_{t})) \overset{\cdot}{x} (t) = (A (r_{t}) + Δ A (r_{t})) x (t) + (A_{d} (r_{t}) \\ + Δ A_{d} (r_{t})) x (t - d (t)) + B (r_{t}) u (t) \\ x (t) = ϕ (t), \forall t \in [- τ, 0] \end{matrix}

(1)

where $x (t) \in R^{n}$ is the state vector; $u (t) \in R^{m}$ is the control input; the matrix $E \in R^{n \times n}$ may be singular and it is assumed that $rank (E) = r \leq n$ ; $ϕ (t)$ is a compatible vector valued continuous function; $d (t)$ is a time-varying continuous function that satisfies

0 \leq d (t) \leq τ, \overset{\cdot}{d} (t) \leq μ < \infty

(2)

where $τ$ is the upper bound of the time delay and $μ$ is the time delay variation rate. $A (r_{t}), A_{d} (r_{t}) and B (r_{t})$ are known real constant matrices with appropriate dimensions for each $r_{t} \in S$ . For notational simplicity, in the sequel, for each possible $r_{t} = i \in S$ , a matrix $A (r_{t})$ is denoted as $A_{i}$ .

For any value $r_{t} = i \in S$ , $Δ E_{i}, Δ A_{i} and Δ A_{d i}$ are unknown matrices representing parameter uncertainties, and are assumed to be of the form

[\begin{matrix} Δ E_{i} & Δ A_{i} & Δ A_{d i} \end{matrix}] = M_{1 i} F_{1} (t) [\begin{matrix} N_{1 i} & N_{2 i} & N_{3 i} \end{matrix}]

(3)

where $M_{1 i}, N_{1 i}, N_{2 i} and N_{3 i}$ are known real constant matrices and $F_{1} (t)$ is the uncertain matrix satisfying $F_{1}^{T} (t) F_{1} (t) \leq I$ .

In this paper, we design a PDSFC for the system given by equation (1) of the form

u (t) = K_{ai} x (t) - K_{ei} \overset{\cdot}{x} (t)

(4)

where $K_{ai}$ and $K_{ei}$ are the state feedback gain and the derivative state feedback gain matrices with appropriate dimensions, respectively. Since gain perturbations may arise when the controller given by equation (4) is implemented into the system given by equation (1), the actual controller will be of the following form

u (t) = (K_{ai} + Δ K_{ai} (t)) x (t) - (K_{ei} + Δ K_{ei} (t) \overset{\cdot}{x} (t)

(5)

where $Δ K_{ei} (t) and Δ K_{ai} (t)$ are the controller gain perturbations for each $i \in S$ . In this paper, we consider the following two classes of controller gain perturbations.

$Δ K_{ei} (t), Δ K_{ai} (t)$ are with the norm-bounded additive form

[\begin{matrix} Δ K_{ei} (t) & Δ K_{ai} (t) \end{matrix}] = M_{2 i} F_{2} (t) [\begin{matrix} D_{1 i} & D_{2 i} \end{matrix}],

(6)

where $M_{2 i}, D_{1 i} and D_{2 i}$ are known real constant matrices for each $i \in S$ and $F_{2} (t)$ is the uncertain matrix satisfying $F_{2}^{T} (t) F_{2} (t) \leq I$ .

$Δ K_{ei} (t) and Δ K_{ai} (t)$ are with the norm-bounded multiplicative form

[\begin{matrix} Δ K_{ei} (t) & Δ K_{ai} (t) \end{matrix}] = M_{3 i} F_{3} (t) [\begin{matrix} D_{3 i} K_{ei} & D_{3 i} K_{ai} \end{matrix}]

(7)

where $M_{3 i} and D_{3 i}$ are known real constant matrices for each $i \in S$ and $F_{3} (t)$ is the uncertain matrix satisfying $F_{3}^{T} (t) F_{3} (t) \leq I$ .

Substituting the controller given by equation (5) into the system given by equation (1), we can get the following closed-loop SMJS

\begin{matrix} E_{ci} \overset{\cdot}{x} (t) = A_{ci} x (t) + {\bar{A}}_{d i} x (t - d (t)) \\ x (t) = ϕ (t), \forall t \in [- τ, 0] \end{matrix}

(8)

where

E_{ci} = E_{i} + Δ E_{i} + B_{i} (K_{ei} + Δ K_{ei})

(9)

A_{ci} = A_{i} + Δ A_{i} + B_{i} (K_{ai} + Δ K_{ai})

(10)

{\bar{A}}_{d i} = A_{d i} + Δ A_{d i}

(11)

Motivated by the work by Ren and Zhang (2012), the guaranteed cost function is defined by

J = E {\int_{0}^{\infty} [x^{T} (t) W_{1 i} x (t) + \overset{\cdot}{x} {(t)}^{T} W_{2 i} \overset{\cdot}{x} (t) + u^{T} (t) Z_{i} u (t)] dt}

(12)

where $W_{1 i}, W_{2 i}, Z_{i}$ are given positive definite matrices for each $i \in S$ .

Throughout this paper, we need the following definitions and lemmas.

Definition 1. (Ren and Zhang, 2012; Lv et al., 2015). Consider the uncertain SMJS given by equation (1). If there exist a controller given by equation (5) and a positive scalar $J_{0}$ such that for all admissible uncertainties, the derivative matrix $E_{ci}, \forall i \in S$ , in the system given by equation (8) is invertible, system equation (8) is robustly stochastically stable, and the corresponding value of the cost function given by equation (12) satisfies $J \leq J_{0}$ , then $J_{0}$ is said to be a guaranteed cost, andequation (5) is said to be a robust non-fragile guaranteed cost PDSFC for the system given byequation (1).

Remark 1. As stated in Definition 1, when the derivative matrix $E_{ci}, \forall i \in S$ becomes invertible by PDSFC equation (5), the system given byequation (8) will be a normal MJS. So we introduce the following definition of the stochastic stability for the system given byequation (8).

Definition 2. (Xu et al., 2007). The system given byequation (8) is said to be stochastically stable if for finite $ϕ \in R^{n}$ defined on $[- τ, 0]$ , and $r_{0} \in S$ , the following is satisfied

lim_{t \to \infty} E {\int_{0}^{t} x^{T} (s, ϕ, r_{0}) x (s, ϕ, r_{0}) d s} < \infty

where $x (t, ϕ, r_{0})$ denotes the solution to the system given by equation (8) at time t under the initial conditions $ϕ (t)$ and $r_{0}$ .

Lemma 1. (Petersen, 1987). Given a symmetric matrix Z and matrices X and Y of appropriate dimensions, then

Z + X F (t) Y + (X F (t) Y)^{T} < 0

for all $F (t)$ satisfying $F^{T} (t) F (t) \leq I$ , if and only if there exists a scalar $ϵ > 0$ such that

Z + ϵ X X^{T} + ϵ Y^{T} Y < 0

Lemma 2. (Zhou and Khargonekar, 1988). For matrices X, Y and $J > 0$ with appropriate dimensions, the following inequality holds

XY + Y^{T} X^{T} \leq XJ X^{T} + Y^{T} J^{- 1} Y

Main results

First, we provide the existence condition of a robust non-fragile guaranteed cost PDSFC for the system given by equation (1).

Theorem 1. For prescribed scalars $μ$ and $τ$ , equation (5) is a robust non-fragile guaranteed cost PDSFC for the system given by equation (1), if there exists matrices $P_{i} > 0, Q_{1} > 0, Q_{2} > 0, R > 0, T_{1 i}, T_{2 i}, H_{1 i}, H_{2 i}, H_{3 i}, S_{1 i}, S_{2 i} and S_{3 i}$ such that the following inequality holds for each $i \in S$

Φ_{0 i} = [\begin{matrix} Π_{11 i} & Π_{12 i} & Π_{13 i} & - H_{1 i} & τ S_{1 i} & τ H_{1 i} \\ ★ & Π_{22 i} & Π_{23 i} & - H_{2 i} & τ S_{2 i} & τ H_{2 i} \\ ★ & ★ & Π_{33 i} & - H_{3 i} & τ S_{3 i} & τ H_{3 i} \\ ★ & ★ & ★ & - Q_{2} & 0 & 0 \\ ★ & ★ & ★ & ★ & - τ R & 0 \\ ★ & ★ & ★ & ★ & ★ & - τ R \end{matrix}] < 0

(13)

where

\begin{matrix} Π_{11 i} = \sum_{i = 1}^{N} π_{ij} P_{j} - T_{1 i} A_{ci} - A_{ci}^{T} T_{1 i}^{T} + Q_{1} + Q_{2} + S_{1 i} \\ + S_{1 i}^{T} + W_{1 i} + {\bar{K}}_{ai}^{T} Z_{i} {\bar{K}}_{ai} \\ Π_{12 i} = P_{i} + T_{1 i} E_{ci} - A_{ci}^{T} T_{2 i}^{T} + S_{2 i}^{T} - {\bar{K}}_{ai}^{T} Z_{i} {\bar{K}}_{ei} \\ Π_{13 i} = - T_{1 i} {\bar{A}}_{d i} - S_{1 i} + S_{3 i}^{T} + H_{1 i} \\ Π_{22 i} = T_{2 i} E_{ci} + E_{ci}^{T} T_{2 i}^{T} + τ R + W_{2 i} + {\bar{K}}_{ei}^{T} Z_{i} {\bar{K}}_{ei} \\ Π_{23 i} = - T_{2 i} {\bar{A}}_{d i} - S_{2 i} + H_{2 i} \\ Π_{33 i} = - (1 - μ) Q_{1} - S_{3 i} - S_{3 i}^{T} + H_{3 i} + H_{3 i}^{T} \\ {\bar{K}}_{ai} = K_{ai} + Δ K_{ai} (t) {\bar{K}}_{ei} = K_{ei} + Δ K_{ei} (t) \end{matrix}

(13)

In this case, the upper bound of the cost function given by equation (12) satisfies

J_{0} = x_{0}^{T} P_{r_{0}} x_{0} + \int_{- τ}^{0} x^{T} (s) (Q_{1} + Q_{2}) x (s) d s + \int_{- τ}^{0} \int_{θ}^{0} {\overset{\cdot}{x}}^{T} (s) R \overset{\cdot}{x} (s) d s d θ

Proof From equation (13), we have $Π_{22 i} < 0$ . Then we can get by the positive definiteness of $R, W_{2 i} and Z_{i}$ that

T_{2 i} E_{ci} + E_{ci}^{T} T_{2 i}^{T} < 0

which implies that the derivative matrix $E_{ci}$ is invertible for all admissible uncertainties. Definite a new process ${(x_{t}, r_{t}), t \geq 0}$ by $x_{t} (s) = x (t + s), - 2 τ \leq s \leq 0$ , then ${(x_{t}, r_{t}), t \geq 0}$ is a Markov process with initial state $(ϕ (\cdot), r_{0})$ . Now choose the following stochastic LKF for the system given by equation (1)

\begin{matrix} V (x_{t}, r_{t}, t) = x^{T} (t) P_{i} x (t) + \int_{t - d (t)}^{t} x^{T} (s) Q_{1} x (s) d s \\ + \int_{t - τ}^{t} x^{T} (s) Q_{2} x (s) d s \\ + \int_{- τ}^{0} \int_{t + θ}^{t} {\overset{\cdot}{x}}^{T} (s) R \overset{\cdot}{x} (s) dsd θ \end{matrix}

(14)

For any matrices $T_{1 i}$ , $T_{2 i}$ with appropriate dimensions, we have

2 [x^{T} (t) T_{1 i} + {\overset{\cdot}{x}}^{T} (t) T_{2 i}] [E_{ci} \overset{\cdot}{x} (t) - A_{ci} x (t) - A_{d i} x (t - d (t))] = 0

(15)

From the Leibniz–Newton formula, the following equations hold for any matrices $S_{1 i}$ , $S_{2 i}$ , $S_{3 i}$ , $H_{1 i}$ , $H_{2 i}$ and $H_{3 i}$ with appropriate dimensions

2 [x^{T} (t) S_{1 i} + {\overset{\cdot}{x}}^{T} (t) S_{2 i} + x^{T} (t - d (t)) S_{3 i}] [x (t) - \int_{t - d (t)}^{t} \overset{\cdot}{x} (s) d s - x (t - d (t))] = 0

(16)

\begin{matrix} 2 [x^{T} (t) H_{1 i} + {\overset{\cdot}{x}}^{T} (t) H_{2 i} + x^{T} (t - d (t)) H_{3 i}] [x (t - d (t)) - \int_{t - τ}^{t - d (t)} \overset{\cdot}{x} (s) d s - x (t - τ)] = 0 \end{matrix}

(17)

On the other hand, we can get the following equation

- \int_{t - τ}^{t} {\overset{\cdot}{x}}^{T} (s) R \overset{\cdot}{x} (s) d s = - \int_{t - d (t)}^{t} {\overset{\cdot}{x}}^{T} (s) R \overset{\cdot}{x} (s) d s - \int_{t - τ}^{t - d (t)} {\overset{\cdot}{x}}^{T} (s) R \overset{\cdot}{x} (s) d s .

(18)

Let $L$ be the weak infinitesimal generator of the random process ${x_{t}, r_{t}}$ . For each $i \in S$ , we have

L V (x_{t}, r_{t}, t) = 2 x^{T} (t) P_{i} \overset{\cdot}{x} (t) + \sum_{j = 1}^{N} π_{i j} x^{T} (t) P_{j} x (t) + x^{T} (t) (Q_{1} + Q_{2}) x (t) - (1 - \overset{\cdot}{d} (t)) x^{T} (t - d (t)) Q_{1} x (t - d (t)) - x^{T} (t - τ) Q_{2} x (t - τ) + τ {\overset{\cdot}{x}}^{T} (t) R \overset{\cdot}{x} (t) - \int_{t - τ}^{t} {\overset{\cdot}{x}}^{T} (s) R \overset{\cdot}{x} (s) d s

Then

\begin{array}{l} L V (x_{t}, r_{t}, t) \leq 2 x^{T} (t) P_{i} \overset{\cdot}{x} (t) + \sum_{j = 1}^{N} π_{i j} x^{T} (t) P_{j} x (t) + x^{T} (t) (Q_{1} + Q_{2}) x (t) \\ - (1 - μ) x^{T} (t - d (t)) Q_{1} x (t - d (t)) - x^{T} (t - τ) Q_{2} x (t - τ) \\ + τ {\overset{\cdot}{x}}^{T} (t) R \overset{\cdot}{x} (t) - \int_{t - d (t)}^{t} {\overset{\cdot}{x}}^{T} (s) R \overset{\cdot}{x} (s) d s - \int_{t - τ}^{t - d (t)} {\overset{\cdot}{x}}^{T} (s) R \overset{\cdot}{x} (s) d s \\ + 2 [x^{T} (t) T_{1 i} + {\overset{\cdot}{x}}^{T} (t) T_{2 i}] [E_{c i} \overset{\cdot}{x} (t) - A_{c i} x (t) - A_{d i} x (t - d (t))] \\ + 2 [x^{T} (t) S_{1 i} + {\overset{\cdot}{x}}^{T} (t) S_{2 i} + x^{T} (t - d (t)) S_{3 i}] [x (t) - \int_{t - d (t)}^{t} \overset{\cdot}{x} (s) d s - x (t - d (t))] \\ + 2 [x^{T} (t) H_{1 i} + {\overset{\cdot}{x}}^{T} (t) H_{2 i} + x^{T} (t - d (t)) H_{3 i}] [x (t - d (t)) - \int_{t - τ}^{t - d (t)} \overset{\cdot}{x} (s) d s - x (t - τ)] \\ \leq ξ^{T} (t) (Φ_{1 i} + τ S_{i} R^{- 1} S_{i}^{T} + τ H_{i} R^{- 1} H_{i}^{T}) ξ (t) \\ - \int_{t - d (t)}^{t} (ξ^{T} (t) S_{i} + {\overset{\cdot}{x}}^{T} (s) R) R^{- 1} \\ (S_{i}^{T} ξ (t) + R \overset{\cdot}{x} (s)) d s - \int_{t - τ}^{t - d (t)} (ξ^{T} (t) H_{i} + {\overset{\cdot}{x}}^{T} (s) R) \\ R^{- 1} (H_{i}^{T} ξ (t) + R \overset{\cdot}{x} (s)) d s \\ - (x^{T} (t) W_{1 i} x (t) + \overset{\cdot}{x} {(t)}^{T} W_{2 i} \overset{\cdot}{x} (t) + u^{T} (t) Z_{i} u (t)) \end{array}

(19)

where

\begin{matrix} Φ_{1 i} = [\begin{matrix} Π_{11 i} & Π_{12 i} & Π_{13 i} & - H_{1 i} \\ ★ & Π_{22 i} & Π_{23 i} & - H_{2 i} \\ ★ & ★ & Π_{33 i} & - H_{3 i} \\ ★ & ★ & ★ & - Q_{2} \end{matrix}] \\ ξ^{T} (t) = [\begin{matrix} x^{T} (t) & {\overset{\cdot}{x}}^{T} (t) & x^{T} (t - d (t)) & x^{T} (t - τ) \end{matrix}] \\ S_{i}^{T} = [\begin{matrix} S_{1 i}^{T} & S_{2 i}^{T} & S_{3 i}^{T} & 0 \end{matrix}] H_{i}^{T} = [\begin{matrix} H_{1 i}^{T} & H_{2 i}^{T} & H_{3 i}^{T} & 0 \end{matrix}] \end{matrix}

Since $R > 0$ , $W_{1 i} > 0$ , $W_{2 i} > 0$ and $Z_{i} > 0$ , then the last three terms in equation (19) are less than 0. By equation (13) and the Schur complement, we can get $L V (x_{t}, r_{t}, t) < 0$ , which ensures the stochastic stability of the system given by equation (1) (see Xu et al., 2007). Then from equations (13) and (19), we obtain

L V (x_{t}, r_{t}, t) + x^{T} (t) W_{1 i} x (t) + \overset{\cdot}{x} (t)^{T} W_{2 i} \overset{\cdot}{x} (t) + u^{T} (t) Z_{i} u (t) < 0

(20)

for any $ξ (t) \neq 0$ and all admissible uncertainties. Using Dynkin’s formula, from equation (20), for each $i \in S$ we have

\begin{matrix} J = E {\int_{0}^{\infty} [x^{T} (t) W_{1 i} x (t) + \overset{\cdot}{x} {(t)}^{T} W_{2 i} \overset{\cdot}{x} (t) + u^{T} (t) Z_{i} u (t)] dt} \\ \leq E {\int_{0}^{\infty} L V (x_{t}, r_{t}, t) dt} \leq V (x_{0}, r_{0}, 0) = x_{0}^{T} P_{r_{0}} x_{0} \\ + \int_{- τ}^{0} x^{T} (s) (Q_{1} + Q_{2}) x (s) d s + \int_{- τ}^{0} \int_{θ}^{0} {\overset{\cdot}{x}}^{T} (s) R \overset{\cdot}{x} (s) dsd θ \end{matrix}

This completes proof.

Remark 2. In Theorem 1, the free-weighting-matrix approach by He et al. (2007) is used to estimate the upper bound of the derivative of the LKF. Two single integral terms are introduced to the constructed stochastic LKF and some useful terms of the derivative of the LKF are not ignored. Paying attention to the inequality given by (13), we can not get a LMI-based condition for solving feedback gains of the PDSFC directly. Motivated by Ren and Zhang (2012, 2015), next we will utilize congruence transformations and free-connection weighting matrices to the inequality given by (13), and a new existence condition of the PDSFC will be gotten in the following corollary.

Corollary 1. For prescribed scalars $μ$ and $τ$ , equation (5) is a robust non-fragile guaranteed cost PDSFC for the system given by (1), if there exists matrices $V_{1 i} > 0, L_{1} > 0, L_{2} > 0, \tilde{R} > 0, V_{2 i}, V_{3 i}, {\tilde{H}}_{1 i}, {\tilde{H}}_{2 i}, {\tilde{H}}_{3 i}, {\tilde{S}}_{1 i}, {\tilde{S}}_{2 i} and {\tilde{S}}_{3 i}$ such that the following inequality holds for each $i \in S$

Φ_{2 i} = [\begin{matrix} Ω_{1 i} & Ω_{2 i} & Ω_{4 i} & Ω_{6 i} \\ ★ & Ω_{3 i} & 0 & 0 \\ ★ & ★ & Ω_{5} & 0 \\ ★ & ★ & ★ & Ω_{7 i} \end{matrix}] < 0

(21)

where

\begin{array}{l} Ω_{1 i} = [\begin{matrix} {\bar{Π}}_{11 i} & {\bar{Π}}_{12 i} & 0 & 0 \\ ★ & {\bar{Π}}_{22 i} & - {\bar{A}}_{d i} L_{1} & 0 \\ ★ & ★ & - (1 - μ) L_{1} & 0 \\ ★ & ★ & ★ & - L_{2} \end{matrix}] \\ Ω_{2 i} = [\begin{matrix} V_{1 i} W_{1 i} & V_{2 i} W_{2 i} & V_{1 i} & V_{1 i} & τ V_{2 i} \\ 0 & V_{3 i} W_{2 i} & 0 & 0 & τ V_{3 i} \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{matrix}] \\ Ω_{4 i} = [\begin{matrix} τ {\tilde{S}}_{1 i} & τ {\tilde{H}}_{1 i} & {\tilde{H}}_{1 i} & {\tilde{S}}_{1 i} & 0 & V_{1 i} \\ τ {\tilde{S}}_{2 i} & τ {\tilde{H}}_{2 i} & {\tilde{H}}_{2 i} & {\tilde{S}}_{2 i} & 0 & 0 \\ τ {\tilde{S}}_{3 i} & τ {\tilde{H}}_{3 i} & {\tilde{H}}_{3 i} & {\tilde{S}}_{3 i} & L_{1} & - L_{1} \\ 0 & 0 & 0 & 0 & - L_{2} & 0 \end{matrix}] \\ Ω_{6 i} = [\begin{matrix} \sqrt{π_{i 1}} V_{1 i} & \cdot \cdot \cdot & \sqrt{π_{i (i - 1)}} V_{1 i} & \sqrt{π_{i (i + 1)}} V_{1 i} & \cdot \cdot \cdot & \sqrt{π_{i N}} V_{1 i} \\ 0 & \cdot \cdot \cdot & 0 & 0 & \cdot \cdot \cdot & 0 \\ 0 & \cdot \cdot \cdot & 0 & 0 & \cdot \cdot \cdot & 0 \\ 0 & \cdot \cdot \cdot & 0 & 0 & \cdot \cdot \cdot & 0 \end{matrix}] \\ Ω_{3 i} = - d i a g {- W_{1 i}, - W_{2 i}, - L_{1}, - L_{2}, τ (R - 2 I)} \\ Ω_{5} = - d i a g {- τ R, - τ R, R - 2 I, R - 2 I, - R, - R} \\ Ω_{7 i} = - d i a g {V_{11}, \dots, V_{1 (i - 1)}, V_{1 (i + 1)}, \dots, V_{1 N}} \\ {\bar{Π}}_{11 i} = π_{i i} V_{1 i} + V_{2 i} + V_{2 i}^{T} + (V_{1 i} {\bar{K}}_{a i}^{T} - V_{2 i} {\bar{K}}_{e i}^{T}) Z_{i} ({\bar{K}}_{a i} V_{1 i}^{T} - {\bar{K}}_{e i} V_{2 i}^{T}) \\ {\bar{Π}}_{12 i} = V_{3 i}^{T} - V_{1 i} A_{c i}^{T} + V_{2 i} E_{c i}^{T} + (V_{2 i} {\bar{K}}_{e i}^{T} - V_{1 i} {\bar{K}}_{a i}^{T}) Z_{i} {\bar{K}}_{e i} V_{3 i}^{T} \\ {\bar{Π}}_{22 i} = E_{c i} V_{3 i}^{T} + V_{3 i} E_{c i}^{T} + V_{3 i} {\bar{K}}_{e i}^{T} Z_{i} {\bar{K}}_{e i} V_{3 i}^{T} \end{array}

${\bar{K}}_{ai}$ and ${\bar{K}}_{ei}$ are given in Theorem 1.

In this case, the upper bound of the cost function given by equation (12) satisfies

J_{0} = x_{0}^{T} V_{1 r_{0}}^{- 1} x_{0} + \int_{- τ}^{0} x^{T} (s) (L_{1}^{- 1} + L_{2}^{- 1}) x (s) d s + \int_{- τ}^{0} \int_{θ}^{0} {\overset{\cdot}{x}}^{T} (s) R \overset{\cdot}{x} (s) d s d θ

(22)

Proof From equation (13) and the Schur complement, we can get

\begin{matrix} Φ_{i 0} = [\begin{matrix} {\hat{Π}}_{11 i} & {\hat{Π}}_{12 i} & - T_{1 i} {\bar{A}}_{d i} & 0 \\ ★ & {\hat{Π}}_{22 i} & - T_{2 i} {\bar{A}}_{d i} & 0 \\ ★ & ★ & - (1 - μ) Q_{1} & 0 \\ ★ & ★ & ★ & - Q_{2} \end{matrix}] \\ + diag {(Q_{1} + Q_{2}), 0, 0, 0} + diag {0, τ R, 0, 0} \\ + τ S_{i} R^{- 1} S_{i}^{T} + τ H_{i} R^{- 1} H_{i}^{T} + H_{i} e_{1} + {(H_{i} e_{1})}^{T} \\ + S_{i} e_{2} + {(S_{i} e_{2})}^{T} < 0 \end{matrix}

(23)

where

\begin{matrix} {\hat{Π}}_{11 i} = \sum_{i = 1}^{N} π_{ij} P_{j} - T_{1 i} A_{ci} - A_{ci}^{T} T_{1 i}^{T} + W_{1 i} + {\bar{K}}_{ai}^{T} Z_{i} {\bar{K}}_{ai} \\ {\hat{Π}}_{12 i} = P_{i} + T_{1 i} E_{ci} - A_{ci}^{T} T_{2 i}^{T} - {\bar{K}}_{ai}^{T} Z_{i} {\bar{K}}_{ei} \\ {\hat{Π}}_{22 i} = T_{2 i} E_{ci} + E_{ci}^{T} T_{2 i}^{T} + W_{2 i} + {\bar{K}}_{ei}^{T} Z_{i} {\bar{K}}_{ei} \\ S_{i}^{T} = [\begin{matrix} S_{1 i}^{T} & S_{2 i}^{T} & S_{3 i}^{T} & 0 \end{matrix}] H_{i}^{T} = [\begin{matrix} H_{1 i}^{T} & H_{2 i}^{T} & H_{3 i}^{T} & 0 \end{matrix}] \\ e_{1} = [\begin{matrix} 0 & 0 & I & - I \end{matrix}], e_{2} = [\begin{matrix} I & 0 & - I & 0 \end{matrix}] \end{matrix}

Let

X_{i} = [\begin{matrix} P_{i} & T_{1 i} & 0 & 0 \\ 0 & T_{2 i} & 0 & 0 \\ 0 & 0 & Q_{1} & 0 \\ 0 & 0 & 0 & Q_{2} \end{matrix}], X_{i}^{- 1} = [\begin{matrix} V_{1 i} & V_{2 i} & 0 & 0 \\ 0 & V_{3 i} & 0 & 0 \\ 0 & 0 & L_{1} & 0 \\ 0 & 0 & 0 & L_{2} \end{matrix}]

(24)

and we have $V_{1 i} = P_{i}^{- 1}$ , $V_{2 i} = - P_{i}^{- 1} T_{1 i} T_{2 i}^{- 1}$ , $L_{1} = Q_{1}^{- 1}$ and $L_{2} = Q_{2}^{- 1}$ . Let

X_{i}^{- 1} H_{i} = {\tilde{H}}_{i}, X_{i}^{- 1} S_{i} = {\tilde{S}}_{i}

(25)

where

{\tilde{S}}_{i}^{T} = [\begin{matrix} {\tilde{S}}_{1 i}^{T} & {\tilde{S}}_{2 i}^{T} & {\tilde{S}}_{3 i}^{T} & 0 \end{matrix}], {\tilde{H}}_{i}^{T} = [\begin{matrix} {\tilde{H}}_{1 i}^{T} & {\tilde{H}}_{2 i}^{T} & {\tilde{H}}_{3 i}^{T} & 0 \end{matrix}]

Pre-multiplying and post-multiplying equation (23) by $X_{i}^{- 1}$ and its transpose, respectively, from equations (24) and (25), (23) is equivalent to

\begin{array}{l} Ω_{1 i} - Ω_{6 i} Ω_{7 i}^{- 1} Ω_{6 i}^{T} + [\begin{matrix} V_{1 i} W_{1 i} V_{1 i}^{T} + V_{2 i} W_{2 i} V_{2 i}^{T} & V_{2 i} W_{2 i} V_{3 i}^{T} & 0 & 0 \\ ★ & V_{3 i} W_{2 i} V_{3 i}^{T} & 0 & 0 \\ ★ & ★ & 0 & 0 \\ ★ & ★ & ★ & 0 \end{matrix}] \\ + diag {(V_{1 i} L_{1}^{- 1} V_{1 i}^{T} + V_{1 i} L_{2}^{- 1} V_{1 i}^{T}), 0, 0, 0} + τ [\begin{matrix} V_{2 i} \\ V_{3 i} \\ 0 \\ 0 \end{matrix}] R {[\begin{matrix} V_{2 i} \\ V_{3 i} \\ 0 \\ 0 \end{matrix}]}^{T} + τ {\tilde{S}}_{i} R^{- 1} {\tilde{S}}_{i}^{T} \\ + τ {\tilde{H}}_{i} R^{- 1} {\tilde{H}}_{i}^{T} + {\tilde{H}}_{i} e_{1} X_{i}^{- T} + {({\tilde{H}}_{i} e_{1} X_{i}^{- T})}^{T} \\ + {\tilde{S}}_{i} e_{2} X_{i}^{- T} + {({\tilde{S}}_{i} e_{2} X_{i}^{- T})}^{T} < 0 \end{array}

(26)

From Lemma 2 we can get

\begin{array}{l} {\tilde{H}}_{i} e_{1} X_{i}^{- T} + {({\tilde{H}}_{i} e_{1} X_{i}^{- T})}^{T} \leq {\tilde{H}}_{i} R {\tilde{H}}_{i}^{T} + e_{1} X^{- T} R^{- 1} {(e_{1} X_{i}^{- T})}^{T} \\ = {\tilde{H}}_{i} R {\tilde{H}}_{i}^{T} + [\begin{matrix} 0 \\ 0 \\ L_{1} \\ - L_{2} \end{matrix}] R^{- 1} {[\begin{matrix} 0 \\ 0 \\ L_{1} \\ - L_{2} \end{matrix}]}^{T} \end{array}

(27)

Similarly

{\tilde{S}}_{i} e_{2} X_{i}^{- T} + {({\tilde{S}}_{i} e_{2} X_{i}^{- T})}^{T} \leq {\tilde{S}}_{i} R {\tilde{S}}_{i}^{T} + [\begin{matrix} V_{1 i} \\ 0 \\ - L_{1} \\ 0 \end{matrix}] R^{- 1} {[\begin{matrix} V_{1 i} \\ 0 \\ - L_{1} \\ 0 \end{matrix}]}^{T}

(28)

Because $R > 0$ , we have $- R^{- 1} < R - 2 I$ . Then by the Shur complement, from equations (27) and (28), we can deduce that equations (26) or (13) holds if equations (21) is true. This completes the proof.

In the following, we will get a solvable LMI-based condition for the robust non-fragile guaranteed cost PDSFC for the system given by equation (1) with controller gain perturbations of the norm-bounded additive form given by equation (6) or of norm-bounded multiplicative form given by equation (7), respectively.

Theorem 2. For prescribed scalars $μ$ and $τ$ , equation (5) is a robust non-fragile guaranteed cost PDSFC with controller gain perturbations of the norm-bounded additive form given byequation (6) for the system given byequation (1), if there exists matrices $V_{1 i} > 0, L_{1} > 0, L_{2} > 0, \tilde{R} > 0, V_{2 i}, V_{3 i}, U_{1 i}, U_{2 i}, {\tilde{H}}_{1 i}, {\tilde{H}}_{2 i}, {\tilde{H}}_{3 i}, {\tilde{S}}_{1 i}, {\tilde{S}}_{2 i} and {\tilde{S}}_{3 i}$ , scalars $δ_{1 i} > 0, δ_{2 i} > 0$ such that the following inequality holds for each $i \in S$

Φ_{3 i} = [\begin{matrix} {\bar{Ω}}_{1 i} & Θ_{1 i} & Ω_{2 i} & Ω_{4 i} & Ω_{6 i} \\ ★ & Θ_{2 i} & 0 & 0 & 0 \\ ★ & ★ & Ω_{3 i} & 0 & 0 \\ ★ & ★ & ★ & Ω_{5} & 0 \\ ★ & ★ & ★ & ★ & Ω_{7 i} \end{matrix}] < 0

(29)

where

\begin{array}{l} {\bar{Ω}}_{1 i} = [\begin{matrix} {\tilde{Π}}_{11 i} & {\tilde{Π}}_{12 i} & 0 & 0 \\ ★ & {\tilde{Π}}_{22 i} & - A_{d i} L_{1} & 0 \\ ★ & ★ & - (1 - μ) L_{1} & 0 \\ ★ & ★ & ★ & - L_{2} \end{matrix}] \\ Θ_{1 i} = [\begin{matrix} - U_{1 i} Z_{i} & - V_{1 i} N_{2 i}^{T} + V_{2 i} N_{1 i}^{T} & 0 & - V_{1 i} D_{2 i}^{T} + V_{2 i} D_{1 i}^{T} \\ U_{2 i} Z_{i} & V_{3 i} N_{1 i}^{T} & δ_{2 i} B_{i} M_{2 i} & V_{3 i} D_{1 i}^{T} \\ 0 & - L_{1} N_{3 i}^{T} & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix}] \\ Θ_{2 i} = [\begin{matrix} - Z_{i} & 0 & δ_{2 i} Z_{i} M_{2 i} & 0 \\ ★ & - δ_{1 i} I & 0 & 0 \\ ★ & ★ & - δ_{2 i} I & 0 \\ ★ & ★ & ★ & - δ_{2 i} I \end{matrix}] \\ {\tilde{Π}}_{11 i} = π_{i i} V_{1 i} + V_{2 i} + V_{2 i}^{T} \\ {\tilde{Π}}_{12 i} = V_{3 i}^{T} - V_{1 i} A_{i}^{T} + V_{2 i} E_{i}^{T} - U_{1 i} B_{i}^{T} \\ {\bar{Π}}_{22 i} = E_{i} V_{3 i}^{T} + V_{3 i} E_{i}^{T} + B_{i} U_{2 i}^{T} + U_{2 i} B_{i}^{T} + δ_{1 i} M_{1 i} M_{1 i}^{T} \end{array}

$Ω_{2 i}, Ω_{3 i}, Ω_{4 i}, Ω_{5 i}, Ω_{6 i} and Ω_{7 i}$ are given in equation (21) . In this case, the upper bound of the cost function given byequation (12) satisfies

J_{0} = x_{0}^{T} V_{1 r_{0}}^{- 1} x_{0} + \int_{- τ}^{0} x^{T} (s) (L_{1}^{- 1} + L_{2}^{- 1}) x (s) d s + \int_{- τ}^{0} \int_{θ}^{0} {\overset{\cdot}{x}}^{T} (s) R \overset{\cdot}{x} (s) d s d θ

and the gains of the guaranteed cost PDSFC given by equation (5) are

K_{ai} = (U_{1 i} + V_{2 i} V_{3 i}^{- 1} U_{2 i})^{T} V_{1 i}^{- T}, K_{ei} = U_{2 i}^{T} V_{3 i}^{- 1}

(30)

Proof From equation (30), we can get $U_{1 i} = V_{1 i} K_{ai}^{T} - V_{2 i} K_{ei}^{T}$ and $U_{2 i} = V_{3 i} K_{ei}^{T}$ . By the Shur complement and equations (3), (6) and (21), we have $Ω_{1 i} < 0$ is equivalent to

\begin{array}{l} [\begin{matrix} π_{i i} V_{1 i} + V_{2 i} + V_{2 i}^{T} & V_{3 i}^{T} - V_{1 i} A_{i}^{T} + V_{2 i} E_{i}^{T} - U_{1 i} B_{i}^{T} & 0 & 0 & - U_{1 i} Z_{i} \\ ★ & E_{i} V_{3 i}^{T} + V_{3 i} E_{i}^{T} + B_{i} U_{2 i}^{T} + U_{2 i} B_{i}^{T} & - A_{d i} L_{1} & 0 & U_{2 i} Z_{i} \\ ★ & ★ & - (1 - μ) L_{1} & 0 & 0 \\ ★ & ★ & ★ & - L_{2} & 0 \\ ★ & ★ & ★ & ★ & - Z_{i} \end{matrix}] \\ + [\begin{matrix} 0 \\ M_{1 i} \\ 0 \\ 0 \\ 0 \end{matrix}] F_{1} (t) {[\begin{matrix} - V_{1 i} N_{2 i}^{T} + V_{2 i} N_{1 i}^{T} \\ V_{3 i} N_{1 i}^{T} \\ - L_{1} N_{3 i}^{T} \\ 0 \\ 0 \end{matrix}]}^{T} + [\begin{matrix} - V_{1 i} N_{2 i}^{T} + V_{2 i} N_{1 i}^{T} \\ V_{3 i} N_{1 i}^{T} \\ - L_{1} N_{3 i}^{T} \\ 0 \\ 0 \end{matrix}] F_{1}^{T} (t) {[\begin{matrix} 0 \\ M_{1 i} \\ 0 \\ 0 \\ 0 \end{matrix}]}^{T} \\ + [\begin{matrix} 0 \\ B_{i} M_{2 i} \\ 0 \\ 0 \\ Z_{i} M_{2 i} \end{matrix}] F_{2} (t) {[\begin{matrix} - V_{1 i} D_{2 i}^{T} + V_{2 i} D_{1 i}^{T} \\ V_{3 i} D_{1 i}^{T} \\ 0 \\ 0 \\ 0 \end{matrix}]}^{T} + [\begin{matrix} - V_{1 i} D_{2 i}^{T} + V_{2 i} D_{1 i}^{T} \\ V_{3 i} D_{1 i}^{T} \\ 0 \\ 0 \\ 0 \end{matrix}] F_{2}^{T} (t) {[\begin{matrix} 0 \\ B_{i} M_{2 i} \\ 0 \\ 0 \\ Z_{i} M_{2 i} \end{matrix}]}^{T} < 0 \end{array}

(31)

By Lemma 1 and the Shur complement, from equation (31), it is easy to get that equation (21) holds if there exist scalars $δ_{1 i} > 0, δ_{2 i} > 0$ such that equation (29) holds. This completes the proof.

Theorem 3. For prescribed scalars $μ$ and $τ$ , equation (5) is a robust non-fragile guaranteed cost PDSFC with controller gain perturbations of the norm-bounded multiplicative form given byequation (7) for the system given byequation (1), if there exists matrices $V_{1 i} > 0, L_{1} > 0, L_{2} > 0, \tilde{R} > 0, V_{2 i}, V_{3 i}, U_{1 i}, U_{2 i}, {\tilde{H}}_{1 i}, {\tilde{H}}_{2 i}, {\tilde{H}}_{3 i}, {\tilde{S}}_{1 i}, {\tilde{S}}_{2 i} and {\tilde{S}}_{3 i}$ , scalars $δ_{1 i} > 0, δ_{2 i} > 0$ such that the following inequality holds for each $i \in S$

Φ_{4 i} = [\begin{matrix} {\bar{Ω}}_{1 i} & {\bar{Θ}}_{1 i} & Ω_{2 i} & Ω_{4 i} & Ω_{6 i} \\ ★ & {\bar{Θ}}_{2 i} & 0 & 0 & 0 \\ ★ & ★ & Ω_{3 i} & 0 & 0 \\ ★ & ★ & ★ & Ω_{5} & 0 \\ ★ & ★ & ★ & ★ & Ω_{7 i} \end{matrix}] < 0

(32)

where

\begin{matrix} {\bar{Θ}}_{1 i} = [\begin{matrix} - U_{1 i} Z_{i} & - V_{1 i} N_{2 i}^{T} + V_{2 i} N_{1 i}^{T} & 0 & - U_{1 i} D_{3 i}^{T} \\ U_{2 i} Z_{i} & V_{3 i} N_{1 i}^{T} & δ_{2 i} B_{i} M_{3 i} & U_{2 i} D_{3 i}^{T} \\ 0 & - L_{1} N_{3 i}^{T} & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix}] \\ {\bar{Θ}}_{2 i} = [\begin{matrix} - Z_{i} & 0 & δ_{2 i} Z_{i} M_{3 i} & 0 \\ ★ & - δ_{1 i} I & 0 & 0 \\ ★ & ★ & - δ_{2 i} I & 0 \\ ★ & ★ & ★ & - δ_{2 i} I \end{matrix}] \end{matrix}

${\bar{Ω}}_{1 i}$ is given in equation (29) and $Ω_{2 i}, Ω_{3 i}, Ω_{4 i}, Ω_{5 i}, Ω_{6 i} and Ω_{7 i}$ are given in equation (21). In this case, the upper bound of the cost function given byequation (12) satisfies

J_{0} = x_{0}^{T} V_{1 r_{0}}^{- 1} x_{0} + \int_{- τ}^{0} x^{T} (s) (L_{1}^{- 1} + L_{2}^{- 1}) x (s) d s + \int_{- τ}^{0} \int_{θ}^{0} {\overset{\cdot}{x}}^{T} (s) R \overset{\cdot}{x} (s) dsd θ

and the gains of the guaranteed cost PDSFC given by equation (5) are

K_{ai} = (U_{1 i} + V_{2 i} V_{3 i}^{- 1} U_{2 i})^{T} V_{1 i}^{- T}, K_{ei} = U_{2 i}^{T} V_{3 i}^{- 1}

(33)

Proof By following a similar line as in the proof of Theorem 2, it is easy to get Theorem 3.

Remark 3. By solving LMI equations (29) or (32), we can get the gains of the PDSFC given byequation (5). Noting that the guaranteed cost $J_{0}$ is related to the constructed stochastic LKF given byequation (14). Both the initial condition of the system given byequation (1) and the choice of guaranteed cost control laws can effect the upper bound of $J_{0}$ . By the method of Park (2004) and Yang et al. (2010), we solve an optimization problem to obtain a controller which can minimize the upper bound of $J_{0}$ in the following.

Theorem 4. The upper bound of guaranteed cost $J_{0}$ is minimized with a robust non-fragile guaranteed cost PDSFC given by equation (5) for system (1), if the following optimization problem

\min_{β, Γ_{1}, Γ_{2}} {β + tr (Γ_{1} + Γ_{2})}

(34)

subject to

(a) LMI equations (29) or (32);

(b) $[\begin{matrix} - β & x_{0}^{T} \\ ★ & - V_{r_{0}} \end{matrix}] < 0$

(c) $[\begin{matrix} - Γ_{1} & Λ_{1}^{T} & Λ_{1}^{T} \\ ★ & - L_{1} & 0 \\ ★ & ★ & - L_{2} \end{matrix}] < 0$

(d) $[\begin{matrix} - Γ_{2} & Λ_{2}^{T} \\ ★ & R - 2 I \end{matrix}] < 0$

has a positive solution set ${\bar{β}, {\bar{V}}_{r_{0}}, {\bar{Γ}}_{1}, {\bar{Γ}}_{2}, {\bar{L}}_{1}, {\bar{L}}_{2}, \bar{R}}$ , where

\int_{- τ}^{0} x (s) x^{T} (s) d s = Λ_{1} Λ_{1}^{T}, \int_{- τ}^{0} (s + τ) \overset{\cdot}{x} (s) {\overset{\cdot}{x}}^{T} (s) d s = Λ_{2} Λ_{2}^{T}

(35)

Proof. From Theorems 2 or 3, condition $(a)$ is obvious. By the Schur complement, conditions $(b)$ to $(d)$ are equivalent to $x_{0}^{T} V_{r_{0}}^{- 1} x_{0} < β$ , $Λ_{1}^{T} (L_{1}^{- 1} + L_{2}^{- 1}) Λ_{1} < Γ_{1}$ , $Λ_{2}^{T} R Λ_{2} < Γ_{2}$ . Noting equations (22) and (35), it is easy to obtain that

\begin{matrix} \int_{- τ}^{0} x^{T} (s) (L_{1}^{- 1} + L_{2}^{- 1}) x (s) d s = tr (\int_{- τ}^{0} x^{T} (s) (L_{1}^{- 1} + L_{2}^{- 1}) x (s) d s) \\ = tr ((L_{1}^{- 1} + L_{2}^{- 1}) \int_{- τ}^{0} x (s) x^{T} (s) d s) = tr ((L_{1}^{- 1} + L_{2}^{- 1}) Λ_{1} Λ_{1}^{T}) \\ = tr (Λ_{1}^{T} (L_{1}^{- 1} + L_{2}^{- 1}) Λ_{1}) < tr (Γ_{1}) \end{matrix}

Similarly, we have

\int_{- τ}^{0} \int_{θ}^{0} {\overset{\cdot}{x}}^{T} (s) R \overset{\cdot}{x} (s) dsd θ = \int_{- τ}^{0} (s + τ) {\overset{\cdot}{x}}^{T} (s) R \overset{\cdot}{x} (s) d s = tr (Λ_{2}^{T} R Λ_{2}) < tr (Γ_{2})

Finally, we can get $J_{0} \leq β + tr (Γ_{1} + Γ_{2})$ , which implies the optimization problem given by equation (34) can lead to the minimization of the guaranteed cost for the system given by equation (1). This completes the proof.

Remark 4. The optimization problem given by equation (34) is given in Theorem 4 to make the upper bound of the guaranteed cost minimized, which can be solved by the Matlab YALMIP Toolbox (Löfberg, 2004).

Numerical example

To show the effectiveness of our results, consider the SMJS given by equation (1) with two modes, the system parameters are described as follows

\begin{matrix} E_{1} = [\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{matrix}], E_{2} = [\begin{matrix} 0 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}], A_{1} = [\begin{matrix} 1.5 & 0 & 1 \\ 0.6 & 1 & 2 \\ 3 & 0.5 & 0 \end{matrix}] \\ A_{2} = [\begin{matrix} - 1 & 0.5 & 0 \\ 0.2 & 0.5 & - 0.4 \\ 0 & 0.1 & - 0.6 \end{matrix}], B_{1} = [\begin{matrix} 1 \\ 0.5 \\ 0.6 \end{matrix}], B_{2} = [\begin{matrix} 0.2 \\ 1 \\ 0.5 \end{matrix}] \\ A_{d 1} = [\begin{matrix} 0.2 & 0.05 & 0.15 \\ 0.1 & 0.25 & 0.1 \\ 0.3 & 0.5 & 0.01 \end{matrix}], A_{d 2} = [\begin{matrix} 0.3 & 0.1 & 0.01 \\ 0.5 & 0.15 & 0.25 \\ 0.2 & 0.1 & 0.05 \end{matrix}] \end{matrix}

The norm-bounded uncertainties in equation (3) are

\begin{matrix} M_{11} = {[\begin{matrix} 0.3 & 0.1 & 0.25 \end{matrix}]}^{T} M_{12} = {[\begin{matrix} 0.2 & 0.5 & 0.1 \end{matrix}]}^{T} \\ N_{11} = [\begin{matrix} 0.2 & 0.15 & 0.3 \end{matrix}] N_{21} = [\begin{matrix} 0.6 & 0.5 & 0.2 \end{matrix}] \\ N_{31} = [\begin{matrix} 0.1 & 0.15 & 0.2 \end{matrix}] N_{12} = [\begin{matrix} 0.1 & 0.25 & 0.2 \end{matrix}] \\ N_{22} = [\begin{matrix} 0.4 & 0.35 & 0.1 \end{matrix}] N_{32} = [\begin{matrix} 0.3 & 0.05 & 0.1 \end{matrix}] \end{matrix}

Since $1 \leq rank (E_{i} + Δ E_{i}) \leq 3$ for $i = 1, 2$ , the unforced system is not normal. Let $W_{11} = W_{12} = 0.2 I$ , $W_{21} = W_{22} = 0.1 I$ , $Z_{1} = Z_{2} = 0.2$ and $τ = μ = 0.5$ , and the transition probability matrix is given as $Π = [\begin{matrix} - 2 & 2 \\ 3 & - 3 \end{matrix}]$ .

Case 1. Consider norm-bounded additive controller gain perturbations in equation (6) with

\begin{matrix} M_{21} = 0.4, D_{11} = [\begin{matrix} 0.5 & 0.4 & 0.6 \end{matrix}], D_{21} = [\begin{matrix} 0.5 & 0.2 & 0.35 \end{matrix}] \\ M_{22} = 0.5, D_{12} = [\begin{matrix} 0.2 & 0.1 & 0.4 \end{matrix}], D_{22} = [\begin{matrix} 0.3 & 0.4 & 0.15 \end{matrix}] \end{matrix}

Suppose that the initial condition is $ϕ (t) = {[\begin{matrix} e^{t} & e^{t} & - e^{t} \end{matrix}]}^{T}$ and $r_{0} = 1$ . From equations (22) and (29), we can get that the guaranteed cost $J_{0} = 55.6642$ . On the other hand, by solving the optimization problem given by equation (34), it can be obtained that the guaranteed cost $J_{0} = 29.1802$ . Thus, the optimization problem given by equation (34) is effective. From equation (34), we can get that the gains of the guaranteed cost PDSFC given by equation (5) are

\begin{matrix} K_{a 1} = [\begin{matrix} - 121.6146 & 986.6983 & 969.0306 \end{matrix}], K_{e 1} = [\begin{matrix} - 9.1044 & - 0.1124 & - 142.7256 \end{matrix}] K_{a 2} = [\begin{matrix} - 278.3925 & - 176.6408 & - 97.3110 \end{matrix}], K_{e 2} = [\begin{matrix} 17.1308 & 7.5452 & 5.2834 \end{matrix}] \end{matrix}

Under the above PDSFC, for the closed-loop SMJS given by equation (8) we can obtain $rank (E_{ci}) = 3$ , $i = 1, 2$ , which reveals that the closed-loop system is normal.

In the following, simulation results are given in Figures 1, 2 and 3 to demonstrate the effectiveness of our methods with $F_{1} (t) = \sin (t)$ , $F_{2} (t) = \cos (t)$ and $d (t) = (τ + τ \sin (2 μ t / τ)) / 2$ . Figure 1 is the jump mode $r_{t}$ , Figure 2 is the system state $x (t)$ and the cost function J is shown in Figure 3.

Case 2. Consider the norm-bounded multiplicative controller gain perturbations in equation (7) with

M_{31} = 0.25, D_{31} = 0.4, M_{32} = 0.3, D_{32} = 0.2

For the initial condition $ϕ (t) = {[\begin{matrix} e^{t} & 0 & - e^{t} \end{matrix}]}^{T}$ and $r_{0} = 1$ , similarly, it can be calculated by equations (22) and (32) that the guaranteed cost $J_{0} = 38.9101$ , while the guaranteed cost obtained by the optimization problem given by equation (34) is 23.7391. Therefore, the effectiveness of the optimization problem given by equation (34) is shown again. From equation (34), we can get that the gains of guaranteed cost PDSFC given by equation (5) are

\begin{matrix} K_{a 1} = [\begin{matrix} 69.3195 & 968.7111 & 821.7452 \end{matrix}], \\ K_{e 1} = [\begin{matrix} - 18.0439 & - 9.4922 & - 106.1533 \end{matrix}] \\ K_{a 2} = [\begin{matrix} - 918.0077 & - 646.5378 & - 246.6062 \end{matrix}], \\ K_{e 2} = [\begin{matrix} 36.7319 & 9.6703 & 6.5603 \end{matrix}] \end{matrix}

It is easy to get $rank (E_{ci}) = 3$ , $i = 1, 2$ , for the closed-loop SMJS given by equation (8). So the closed-loop system is also normal. With $F_{1} (t) = \sin (t)$ , $F_{3} (t) = \cos (t)$ and $d (t) = (τ + τ \sin (2 μ t / τ)) / 2$ , simulation results are given in Figures 4, 5 and 6. The jump mode $r_{t}$ and system state $x (t)$ are shown in Figures 4 and 5, respectively; Figure 6 is the cost function J.

Conclusions

In this paper, the problem of robust non-fragile guaranteed cost control for a class of uncertain SMJSs with time-varying delays has been investigated in the framework of LMIs. Since both the state matrix and the derivative matrix contain time-varying norm-bounded parameter uncertainties, a non-fragile PDSFC has been provided to ensure that the closed-loop SMJS is normal and robustly stochastically stable. Besides, the upper bound of the guaranteed cost has been minimized by solving an optimization problem. A numerical example has demonstrated the effectiveness of our results.

Footnotes

Appendix

Acknowledgements

The authors would like to thank the Editor, the Associate Editor and the two anonymous reviewers for their valuable comments and suggestions which have helped to greatly improve the paper.

Funding

This work was supported by the National Science Foundation of China (grant numbers 61374086 and 61403199), the Fundamental Research Funds for the Central Universities of China (grant number 30916015105), the Natural Science Foundation of Jiangsu Province (grant number BK20140770), the Postdoctoral Science Foundation of China (grant number 2016M602112) and the Natural Science Foundation of Shandong Province (grant number ZR2016FQ09 and ZR2016 JL025).

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