Delay-dependent non-fragile robust dissipative filtering for uncertain nonlinear stochastic singular time-delay systems with Markovian jump parameters

Abstract

The problem of delay-dependent non-fragile and robust dissipative filtering is investigated for a class of uncertain nonlinear stochastic singular time-delay Itô-type systems with Markovian jump parameters. With the system uncertainty and the filter gain perturbations, the closed-loop filtering error system is robust asymptotically stable and satisfies the dissipation performance. By constructing a Lyapunov-Krasovskii function and applying the Itô differential formula to compute the differential function along the system, a sufficient condition for the existence of a robust and non-fragile dissipative filter is derived in the form of linear matrix inequality (LMI). An illustrative numerical example is provided to demonstrate the effectiveness of the proposed approach.

1 Introduction

As an important kind of hybrid systems, Markovian jump systems have received increasing attention in the past few years due to the fact that they have strong practical relevance in mechanical systems, economics systems with human operators, and other engineering areas [1–4]. Partially mode-dependent $H_{\mathrm{\infty }}$ filter design problem was tackled for discrete-time Markovian jump systems with partly unknown transition probabilities in [5].

Note that stochastic systems with Brownian motions, governed by the Itô differential equations, have attracted many researchers’ attention over the past decades due to the extensive application of stochastic modeling in mechanical systems, economics, and other areas. In [6], the $H_{\mathrm{\infty }}$ filtering problem was solved for uncertain stochastic time-delay systems with sector-bounded nonlinearities. In [7], the joint state filtering and parameter estimation problem was solved for linear stochastic time-delay systems. In [8], the robust $H_{\mathrm{\infty }}$ filtering problem was solved for discrete nonlinear stochastic systems with time-varying delay. In [9], the problem of non-fragile $H_{\mathrm{\infty }}$ filtering was thoroughly studied for uncertain stochastic time-delay systems.

For the stochastic case, the Markovian switching problem has received considerable attention, and a number of traditional approaches have been proposed in the literature. In [10], the delay-range-dependent robust $H_{\mathrm{\infty }}$ filtering problem was neatly solved for uncertain stochastic systems with mode-dependent time delays and Markovian jump parameters. In [11], the non-fragile $H_{\mathrm{\infty }}$ filtering problem for uncertain stochastic time-delay systems with Markovian jump parameters was thoroughly studied. In [12], the robust $H_{\mathrm{\infty }}$ exponential filtering was exploited to uncertain stochastic time-delay systems with Markovian switching and nonlinearities. It is well known that stochastic time-delay systems with Markovian jump parameters play a very important role in digital signal analysis and processing. However, despite its importance, up to now, the delay-dependent non-fragile robust dissipative filtering problem for general uncertain stochastic time-delay systems with Markovian jump parameters has not been fully investigated and the relevant results have been very few.

On the other hand, a great deal of attention has been devoted to the study of singular systems over the past decades. Singular systems are referred to as descriptor systems, implicit systems, generalized state-space systems, or semi-state systems. The singular Markovian jump time-delay systems have also been investigated by many researchers. In [13], the robust exponential stability was studied for uncertain singular Markovian jump time-delay systems. In [14], the delay-dependent $H_{\mathrm{\infty }}$ filtering problem for singular Markovian jump time-delay systems was thoroughly investigated.

The notion of dissipativity plays an important role in systems, circuits, networks, and control theory. The dissipativity contains small gain and passivity as its special cases and is mainly used for stability analysis for nonlinear systems. The main motivation for the study of a general dissipative control problem is that it offers flexibility for gain and phase performance trade off. Moreover, it is effective to deal with robust and nonlinear control. In [15], the passivity control for a kind of T-S fuzzy descriptor system is presented. In [16], the dissipative filtering for discrete fuzzy systems was thoroughly studied. In [17], the dissipative filtering for linear discrete-time systems via LMI was studied. In [18], the robust dissipative filtering for continuous-time polytopic uncertain neutral systems was studied. Unfortunately, to the best of the authors’ knowledge, up to now, the delay-dependent non-fragile robust dissipative filtering for uncertain nonlinear stochastic singular time-delay systems with Markovian jump parameters has rarely been reported.

In this paper, the delay-dependent non-fragile robust dissipative filter design method for uncertain nonlinear stochastic singular time-delay systems with Markovian jump parameters is considered. The system under study involves parameter uncertainties, stochastic disturbances, time-varying delays and inherent sector-like nonlinearities. Note that, among different descriptions of the nonlinearities, the so-called sector nonlinearity [19] has gained much attention for deterministic systems, the control analysis and model reduction problems have been studied; see [20, 21]. By establishing a Lyapunov-Krasovskii function and applying the Itô differential formula, a new delay-dependent bounded real lemma for nonlinear stochastic singular time-delay and Markovian jump systems is derived. Neither model transformation nor bounding technique for cross terms is involved. Based on the obtained bounded real lemma, the existence condition of a robust dissipative filter and a filter design method are presented by LMI approach. Also, it is shown that the proposed filter design method is widely applicable to singular systems and non-singular systems by a numerical example.

2 System description and preliminary lemma

Given a probability space $(\mathrm{\Omega },F,P)$, where Ω is the sample space, F is the σ algebra of subsets of the sample space and P is the probability measure on F. Over this probability space $(\mathrm{\Omega },F,P)$, we consider the following uncertain linear stochastic systems with Markovian jump parameters and mode-dependent time delays:

$$\begin{array}{r}\begin{array}{rl}Edx(t)=& [A(r(t),t)x(t)+A_d(r(t),t)x(t-d(t))+B(r(t),t)f(x(t))\\ +B_d(r(t),t)f_d\left(x(t-d(t))\right)+C(r(t),t)v(t)]dt\\ +[G(r(t),t)x(t)+G_d(r(t),t)x(t-d(t))+H(r(t),t)f(x(t))\\ +H_d(r(t),t)f_d\left(x(t-d(t))\right)+D(r(t),t)v(t)]d\omega (t),\end{array}\\ \begin{array}{rl}dy(t)=& [\overline{A}(r(t),t)x(t)+{\overline{A}}_d(r(t),t)x(t-d(t))+\overline{B}(r(t),t)f(x(t))\\ +{\overline{B}}_d(r(t),t)f_d\left(x(t-d(t))\right)+\overline{C}(r(t),t)v(t)]dt\\ +[\overline{G}(r(t),t)x(t)+{\overline{G}}_d(r(t),t)x(t-d(t))+\overline{H}(r(t),t)f(x(t))\\ +{\overline{H}}_d(r(t),t)f_d\left(x(t-d(t))\right)+\overline{D}(r(t),t)v(t)]d\omega (t),\end{array}\\ \begin{array}{rl}z(t)=& L(r(t),t)x(t)+J(r(t),t)x(t-d(t))+B_z(r(t),t)f(x(t))\\ +H_z(r(t),t)f_d\left(x(t-d(t))\right)+K(r(t),t)v(t),\end{array}\\ \begin{array}{r}x(t)=\phi (t),t\in [-d,0],\end{array}\end{array}$$

(1)

where $x(t)\in R^n$ is the state; $v(t)\in R^p$ is the disturbance input which belongs to $L_2[0,\mathrm{\infty })$; $y(t)\in R^p$ is the measurement; $z(t)\in R^q$ is the signal to be estimated; and $\omega (t)$, independent of the Markov process, is a one-dimensional standard Wiener process. E is a singular square matrix, and $rankE=r<n$. $A(r(t),t)$, $A_d(r(t),t)$, $B(r(t),t)$, $B_d(r(t),t)$, $C(r(t),t)$, $G(r(t),t)$, $G_d(r(t),t)$, $H(r(t),t)$, $H_d(r(t),t)$, $D(r(t),t)$, $\overline{A}(r(t),t)$, ${\overline{A}}_d(r(t),t)$, $\overline{B}(r(t),t)$, ${\overline{B}}_d(r(t),t)$, $\overline{C}(r(t),t)$, $\overline{G}(r(t),t)$, ${\overline{G}}_d(r(t),t)$, $\overline{H}(r(t),t)$, ${\overline{H}}_d(r(t),t)$, $\overline{D}(r(t),t)$, $L(r(t),t)$, $J(r(t),t)$, $B_z(r(t),t)$, $H_z(r(t),t)$, $K(r(t),t)$ are governed by the Markov process $r(t)$, and $K(r(t),t)$ is symmetric. $r(t)$ is 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 $\mathrm{\Pi }={[{\pi }_{ij}]}_{i,j\in S}$ given by

$$P\{r(t+\mathrm{\Delta })=j|r(t)=i\}=\{\begin{array}{cc}{\pi }_{ij}\mathrm{\Delta }+o(\mathrm{\Delta }),\hfill & i\ne j,\hfill \\ 1+{\pi }_{ij}\mathrm{\Delta }+o(\mathrm{\Delta }),\hfill & i=j,\hfill \end{array}$$

where $\mathrm{\Delta }>0$, ${lim}_{\mathrm{\Delta }\to 0}\frac{o(\mathrm{\Delta })}{\mathrm{\Delta }}=0$; ${\pi }_{ij}$ for $i\ne j$ is the transition rate from mode i at time t to mode j and ${\pi }_{ii}=-{\sum }_{i\ne j}{\pi }_{ij}$. $d(t)$ is the time-varying delay when the mode is in $r(t)$ and satisfies

$$0\le d_i(t)\le d_{1i},{\dot{d}}_i(t)\le h_i<1,\mathrm{\forall }r(t)=i,i\in S,$$

(2)

where $d_{1i}$, $h_i$ are known real constant scalars, $d_1=max\{d_{1i},i\in S\}$. $\phi (t)$ denotes a vector-valued initial continuous function defined on the interval $[-d_1,0]$.

When $r(t)=i$, let $A_i$ denote $A(r(t),t)$; so, for each $r(t)=i\in S$,

$$\begin{array}{r}A(r(t),t)=A_i+\mathrm{\Delta }{A}_i,A_d(r(t),t)=A_{di}+\mathrm{\Delta }{A}_{di},B(r(t),t)=B_i+\mathrm{\Delta }{B}_i,\\ B_d(r(t),t)=B_{di}+\mathrm{\Delta }{B}_{di},C(r(t),t)=C_i+\mathrm{\Delta }{C}_i,G(r(t),t)=G_i+\mathrm{\Delta }{G}_i,\\ G_d(r(t),t)=G_{di}+\mathrm{\Delta }{G}_{di},H(r(t),t)=H_i+\mathrm{\Delta }{H}_i,H_d(r(t),t)=H_{di}+\mathrm{\Delta }{H}_{di},\\ D(r(t),t)=D_i+\mathrm{\Delta }{D}_i,\overline{A}(r(t),t)={\overline{A}}_i+\mathrm{\Delta }{\overline{A}}_i,{\overline{A}}_d(r(t),t)={\overline{A}}_{di}+\mathrm{\Delta }{\overline{A}}_{di},\\ \overline{B}(r(t),t)={\overline{B}}_i+\mathrm{\Delta }{\overline{B}}_i,{\overline{B}}_d(r(t),t)={\overline{B}}_{di}+\mathrm{\Delta }{\overline{B}}_{di},\overline{C}(r(t),t)={\overline{C}}_i+\mathrm{\Delta }{\overline{C}}_i,\\ \overline{G}(r(t),t)={\overline{G}}_i+\mathrm{\Delta }{\overline{G}}_i,{\overline{G}}_d(r(t),t)={\overline{G}}_{di}+\mathrm{\Delta }{\overline{G}}_{di},\overline{H}(r(t),t)={\overline{H}}_i+\mathrm{\Delta }{\overline{H}}_i,\\ {\overline{H}}_d(r(t),t)={\overline{H}}_{di}+\mathrm{\Delta }{\overline{H}}_{di},\overline{D}(r(t),t)={\overline{D}}_i+\mathrm{\Delta }{\overline{D}}_i,L(r(t),t)=L_i+\mathrm{\Delta }{L}_i,\\ J(r(t),t)=J_i+\mathrm{\Delta }{J}_i,B_z(r(t),t)=B_{zi}+\mathrm{\Delta }{B}_{zi},H_z(r(t),t)=H_{zi}+\mathrm{\Delta }{H}_{zi},\\ K(r(t),t)=K_i+\mathrm{\Delta }{K}_i,\end{array}$$

(3)

where $A_i$, $A_{di}$, $B_i$, $B_{di}$, $C_i$, $G_i$, $G_{di}$, $H_i$, $H_{di}$, $D_i$, ${\overline{A}}_i$, ${\overline{A}}_{di}$, ${\overline{B}}_i$, ${\overline{B}}_{di}$, ${\overline{C}}_i$, ${\overline{G}}_i$, ${\overline{G}}_{di}$, ${\overline{H}}_i$, ${\overline{H}}_{di}$, ${\overline{D}}_i$, $L_i$, $J_i$, $B_{zi}$, $H_{zi}$, $K_i$ are known real constant matrices describing the nominal system; $\mathrm{\Delta }{A}_i$, $\mathrm{\Delta }{A}_{di}$, $\mathrm{\Delta }{B}_i$, $\mathrm{\Delta }{B}_{di}$, $\mathrm{\Delta }{C}_i$, $\mathrm{\Delta }{G}_i$, $\mathrm{\Delta }{G}_{di}$, $\mathrm{\Delta }{H}_i$, $\mathrm{\Delta }{H}_{di}$, $\mathrm{\Delta }{D}_i$, $\mathrm{\Delta }{\overline{A}}_i$, $\mathrm{\Delta }{\overline{A}}_{di}$, $\mathrm{\Delta }{\overline{B}}_i$, $\mathrm{\Delta }{\overline{B}}_{di}$, $\mathrm{\Delta }{\overline{C}}_i$, $\mathrm{\Delta }{\overline{G}}_i$, $\mathrm{\Delta }{\overline{G}}_{di}$, $\mathrm{\Delta }{\overline{H}}_i$, $\mathrm{\Delta }{\overline{H}}_{di}$, $\mathrm{\Delta }{\overline{D}}_i$, $\mathrm{\Delta }{L}_i$, $\mathrm{\Delta }{J}_i$, $\mathrm{\Delta }{B}_{zi}$, $\mathrm{\Delta }{H}_{zi}$, $\mathrm{\Delta }{K}_i$ are unknown matrices representing time-varying parameter uncertainties, and the admissible uncertainties are assumed to be modeled in the form

$$\left[\begin{array}{ccccc}\mathrm{\Delta }{A}_i& \mathrm{\Delta }{A}_{di}& \mathrm{\Delta }{B}_i& \mathrm{\Delta }{B}_{di}& \mathrm{\Delta }{C}_i\\ \mathrm{\Delta }{G}_i& \mathrm{\Delta }{G}_{di}& \mathrm{\Delta }{H}_i& \mathrm{\Delta }{H}_{di}& \mathrm{\Delta }{D}_i\\ \mathrm{\Delta }{\overline{A}}_i& \mathrm{\Delta }{\overline{A}}_{di}& \mathrm{\Delta }{\overline{B}}_i& \mathrm{\Delta }{\overline{B}}_{di}& \mathrm{\Delta }{\overline{C}}_i\\ \mathrm{\Delta }{\overline{G}}_i& \mathrm{\Delta }{\overline{G}}_{di}& \mathrm{\Delta }{\overline{H}}_i& \mathrm{\Delta }{\overline{H}}_{di}& \mathrm{\Delta }{\overline{D}}_i\\ \mathrm{\Delta }{L}_i& \mathrm{\Delta }{J}_i& \mathrm{\Delta }{B}_{zi}& \mathrm{\Delta }{H}_{zi}& \mathrm{\Delta }{K}_i\end{array}\right]=\left[\begin{array}{c}{M}_{i1}\\ M_{i2}\\ M_{i3}\\ M_{i4}\\ M_{i5}\end{array}\right]{\mathrm{\Delta }}_1\left[\begin{array}{ccccc}{N}_{i1}& N_{i2}& N_{i3}& N_{i4}& N_{i5}\end{array}\right],$$

(4)

where $M_{i1}$, $M_{i2}$, $M_{i3}$, $M_{i4}$, $M_{i5}$, $N_{i1}$, $N_{i2}$, $N_{i3}$, $N_{i4}$, $N_{i5}$ are known real constant matrices and ${\mathrm{\Delta }}_1$ is the uncertain time-varying matrix satisfying ${\mathrm{\Delta }}_1^T{\mathrm{\Delta }}_1\le I$.

The vector-valued nonlinear functions f, $f_d$, are assumed to satisfy the following sector-bounded conditions:

$$\begin{array}{r}{[f(x(t))-{\hat{R}}_{1}x(t)]}^T[f(x(t))-{\hat{R}}_{2}x(t)]\le 0,\mathrm{\forall }x(t)\in R^n,\\ {[f_d\left(x(t-d(t))\right)-{\hat{S}}_{1}x(t-d(t))]}^T[f_d\left(x(t-d(t))\right)-{\hat{S}}_{2}x(t-d(t))]\\ \le 0,\mathrm{\forall }x(t-d(t))\in R^n,\end{array}$$

(5)

where ${\hat{R}}_1,{\hat{R}}_2,{\hat{S}}_1,{\hat{S}}_2\in R^{n\times n}$ are known real constant matrices, and $\hat{R}={\hat{R}}_1-{\hat{R}}_2$, $\hat{S}={\hat{S}}_1-{\hat{S}}_2$ are symmetric positive definite matrices.

Remark 1 When E is a unit matrix, $B_i=0$, $B_{di}=0$, $H_i=0$, $H_{di}=0$, $D_i=0$, ${\overline{B}}_i=0$, ${\overline{B}}_{di}=0$, ${\overline{H}}_i=0$, ${\overline{H}}_{di}=0$, ${\overline{D}}_i=0$, $J_i=0$, $B_{zi}=0$, $H_{zi}=0$, $K_i=0$, $\mathrm{\Delta }{B}_i=0$, $\mathrm{\Delta }{B}_{di}=0$, $\mathrm{\Delta }{G}_i=0$, $\mathrm{\Delta }{G}_{di}=0$, $\mathrm{\Delta }{H}_i=0$, $\mathrm{\Delta }{H}_{di}=0$, $\mathrm{\Delta }{D}_i=0$, $\mathrm{\Delta }{\overline{B}}_i=0$, $\mathrm{\Delta }{\overline{B}}_{di}=0$, $\mathrm{\Delta }{\overline{G}}_i=0$, $\mathrm{\Delta }{\overline{G}}_{di}=0$, $\mathrm{\Delta }{\overline{H}}_i=0$, $\mathrm{\Delta }{\overline{H}}_{di}=0$, $\mathrm{\Delta }{\overline{D}}_i=0$, $\mathrm{\Delta }{L}_i=0$, $\mathrm{\Delta }{J}_i=0$, $\mathrm{\Delta }{B}_{zi}=0$, $\mathrm{\Delta }{H}_{zi}=0$, $\mathrm{\Delta }{K}_i=0$, system (1) was studied in [10]. The system in this paper is a class of stochastic time-delay systems broader than others.

In this paper, we consider a full-order filter with the following form:

(6)

(7)

where $\hat{x}(t)\in R^n$ is the filter state and $\hat{z}(t)\in R^q$ is the estimated vector, $E_{fi}$, $A_{fi}$, $B_{fi}$, $C_{fi}$, $L_{fi}$, $J_{fi}$ are the desired filter matrices to be designed, $E_{fi}$ may be a singular square matrix, $\mathrm{\Delta }{E}_{fi}$, $\mathrm{\Delta }{A}_{fi}$, $\mathrm{\Delta }{B}_{fi}$, $\mathrm{\Delta }{C}_{fi}$, $\mathrm{\Delta }{L}_{fi}$, $\mathrm{\Delta }{J}_{fi}$ are the filter gain variations with the following form:

$$\left[\begin{array}{ccc}\mathrm{\Delta }{E}_{fi}& \mathrm{\Delta }{A}_{fi}& \mathrm{\Delta }{B}_{fi}\\ \mathrm{\Delta }{C}_{fi}& \mathrm{\Delta }{L}_{fi}& \mathrm{\Delta }{J}_{fi}\end{array}\right]=\left[\begin{array}{c}{M}_{i6}\\ M_{i7}\end{array}\right]{\mathrm{\Delta }}_2\left[\begin{array}{ccc}{N}_{i6}& N_{i7}& N_{i8}\end{array}\right],$$

(8)

where $M_{i6}$, $M_{i7}$, $N_{i6}$, $N_{i7}$, $N_{i8}$ are known real constant matrices and ${\mathrm{\Delta }}_2$ is unknown time-varying matrix function satisfying ${\mathrm{\Delta }}_2^T{\mathrm{\Delta }}_2\le I$.

Remark 2 When $E_{fi}$ is a unit matrix, and $C_{fi}=0$, $J_{fi}=0$, $\mathrm{\Delta }{E}_{fi}=0$, $\mathrm{\Delta }{C}_{fi}=0$, $\mathrm{\Delta }{J}_{fi}=0$, the filter (6) was studied in [9]. The objective in this paper improves the function of the filter.

Applying this filter to system (1), we obtain the following filtering error system:

$$\{\begin{array}{c}\begin{array}{rl}Edx(t)=& [Ax(t)+A_{d}x(t-d(t))+Bf(x(t))+B_d{f}_d(x(t-d(t)))+Cv(t)]dt\\ +[Gx(t)+G_{d}x(t-d(t))+Hf(x(t))\\ +H_d{f}_d(x(t-d(t)))+Dv(t)]d\omega (t),\end{array}\hfill \\ \begin{array}{rl}\tilde{E}d\tilde{x}(t)=& [\tilde{A}\tilde{x}(t)+{\tilde{A}}_d\tilde{x}(t-d(t))+\tilde{B}f(x(t))+{\tilde{B}}_d{f}_d(x(t-d(t)))+\tilde{C}v(t)]dt\\ +[\tilde{G}W\tilde{x}(t)+{\tilde{G}}_{d}W\tilde{x}(t-d(t))+\tilde{H}f(x(t))\\ +{\tilde{H}}_d{f}_d(x(t-d(t)))+\tilde{D}v(t)]d\omega (t),\end{array}\hfill \\ e(t)=\tilde{L}\tilde{x}(t)+\tilde{J}\tilde{x}(t-d(t))+{\tilde{B}}_{zi}f(x(t))+{\tilde{H}}_{zi}f(x(t-d(t)))+\tilde{K}v(t),\hfill \end{array}$$

(9)

where

$$\begin{array}{c}\tilde{x}(t)={[x^T(t),{(x(t)-\hat{x}(t))}^T]}^T,e(t)=z(t)-\hat{z}(t),\hfill \\ \tilde{E}={\tilde{E}}_i+\mathrm{\Delta }{\tilde{E}}_i,A=A_i+\mathrm{\Delta }{A}_i,A_d=A_{di}+\mathrm{\Delta }{A}_{di},\hfill \\ B=B_i+\mathrm{\Delta }{B}_i,B_d=B_{di}+\mathrm{\Delta }{B}_{di},C=C_i+\mathrm{\Delta }{C}_i,\hfill \\ G=G_i+\mathrm{\Delta }{G}_i,G_d=G_{di}+\mathrm{\Delta }{G}_{di},H=H_i+\mathrm{\Delta }{H}_i,\hfill \\ H_d=H_{di}+\mathrm{\Delta }{H}_{di},D=D_i+\mathrm{\Delta }{D}_i,\tilde{A}={\tilde{A}}_i+\mathrm{\Delta }{\tilde{A}}_i,\hfill \\ {\tilde{A}}_d={\tilde{A}}_{di}+\mathrm{\Delta }{\tilde{A}}_{di},\tilde{B}={\tilde{B}}_i+\mathrm{\Delta }{\tilde{B}}_i,{\tilde{B}}_d={\tilde{B}}_{di}+\mathrm{\Delta }{\tilde{B}}_{di},\hfill \\ \tilde{C}={\tilde{C}}_i+\mathrm{\Delta }{\tilde{C}}_i,\tilde{G}={\tilde{G}}_i+\mathrm{\Delta }{\tilde{G}}_i,{\tilde{G}}_d={\tilde{G}}_{di}+\mathrm{\Delta }{\tilde{G}}_{di},\hfill \\ \tilde{H}={\tilde{H}}_i+\mathrm{\Delta }{\tilde{H}}_i,{\tilde{H}}_d={\tilde{H}}_{di}+\mathrm{\Delta }{\tilde{H}}_{di},\tilde{D}={\tilde{D}}_i+\mathrm{\Delta }{\tilde{D}}_i,\hfill \\ \tilde{L}={\tilde{L}}_i+\mathrm{\Delta }{\tilde{L}}_i,\tilde{J}={\tilde{J}}_i+\mathrm{\Delta }{\tilde{J}}_i,{\tilde{B}}_{zi}=B_{zi}+\mathrm{\Delta }{B}_{zi},\hfill \\ {\tilde{H}}_{zi}=H_{zi}+\mathrm{\Delta }{H}_{zi},\tilde{K}=K_i+\mathrm{\Delta }{K}_i,\hfill \\ {\tilde{E}}_i=\left[\begin{array}{cc}E& 0\\ 0& E_{fi}\end{array}\right],{\tilde{A}}_i=\left[\begin{array}{cc}{A}_i& 0\\ A_i-A_{fi}-B_{fi}{\overline{A}}_i& A_{fi}\end{array}\right],\hfill \\ {\tilde{A}}_{di}=\left[\begin{array}{cc}{A}_{di}& 0\\ A_{di}-C_{fi}-B_{fi}{\overline{A}}_{di}& C_{fi}\end{array}\right],\hfill \\ {\tilde{B}}_i=\left[\begin{array}{c}{B}_i\\ B_i-B_{fi}{\overline{B}}_i\end{array}\right],{\tilde{B}}_{di}=\left[\begin{array}{c}{B}_{di}\\ B_{di}-B_{fi}{\overline{B}}_{di}\end{array}\right],\hfill \\ {\tilde{C}}_i=\left[\begin{array}{c}{C}_i\\ C_i-B_{fi}{\overline{C}}_i\end{array}\right],{\tilde{G}}_i=\left[\begin{array}{c}{G}_i\\ G_i-B_{fi}{\overline{G}}_i\end{array}\right],\hfill \\ {\tilde{G}}_{di}=\left[\begin{array}{c}{G}_{di}\\ G_{di}-B_{fi}{\overline{G}}_{di}\end{array}\right],{\tilde{H}}_i=\left[\begin{array}{c}{H}_i\\ H_i-B_{fi}{\overline{H}}_i\end{array}\right],\hfill \\ {\tilde{H}}_{di}=\left[\begin{array}{c}{H}_{di}\\ H_{di}-B_{fi}{\overline{H}}_{di}\end{array}\right],{\tilde{D}}_i=\left[\begin{array}{c}{D}_i\\ D_i-B_{fi}{\overline{D}}_i\end{array}\right],\hfill \\ {\tilde{L}}_i=\left[\begin{array}{cc}{L}_i-L_{fi}& L_{fi}\end{array}\right],{\tilde{J}}_i=\left[\begin{array}{cc}{J}_i-J_{fi}& J_{fi}\end{array}\right],W=\left[\begin{array}{cc}I& 0\end{array}\right],\hfill \\ \mathrm{\Delta }{\tilde{E}}_i=\left[\begin{array}{cc}0& 0\\ 0& \mathrm{\Delta }{E}_{fi}\end{array}\right],\mathrm{\Delta }{\tilde{A}}_i=\left[\begin{array}{cc}\mathrm{\Delta }{A}_i& 0\\ \mathrm{\Delta }{A}_i-\mathrm{\Delta }{A}_{fi}-\mathrm{\Delta }{B}_{fi}{\overline{A}}_i-B_{fi}\mathrm{\Delta }{\overline{A}}_i& \mathrm{\Delta }{A}_{fi}\end{array}\right],\hfill \\ \mathrm{\Delta }{\tilde{A}}_{di}=\left[\begin{array}{cc}\mathrm{\Delta }{A}_{di}& 0\\ \mathrm{\Delta }{A}_{di}-\mathrm{\Delta }{C}_{fi}-\mathrm{\Delta }{B}_{fi}{\overline{A}}_{di}-B_{fi}\mathrm{\Delta }{\overline{A}}_{di}& \mathrm{\Delta }{C}_{fi}\end{array}\right],\hfill \\ \mathrm{\Delta }{\tilde{B}}_i=\left[\begin{array}{c}\mathrm{\Delta }{B}_i\\ \mathrm{\Delta }{B}_i-\mathrm{\Delta }{B}_{fi}{\overline{B}}_i-B_{fi}\mathrm{\Delta }{\overline{B}}_i\end{array}\right],\mathrm{\Delta }{\tilde{B}}_{di}=\left[\begin{array}{c}\mathrm{\Delta }{B}_{di}\\ \mathrm{\Delta }{B}_{di}-\mathrm{\Delta }{B}_{fi}{\overline{B}}_{di}-B_{fi}\mathrm{\Delta }{\overline{B}}_{di}\end{array}\right],\hfill \\ \mathrm{\Delta }{\tilde{C}}_i=\left[\begin{array}{c}\mathrm{\Delta }{C}_i\\ \mathrm{\Delta }{C}_i-\mathrm{\Delta }{B}_{fi}{\overline{C}}_i-B_{fi}\mathrm{\Delta }{\overline{C}}_i\end{array}\right],\mathrm{\Delta }{\tilde{G}}_i=\left[\begin{array}{c}\mathrm{\Delta }{G}_i\\ \mathrm{\Delta }{G}_i-\mathrm{\Delta }{B}_{fi}\overline{G}-B_{fi}\mathrm{\Delta }{\overline{G}}_i\end{array}\right],\hfill \\ \mathrm{\Delta }{\tilde{G}}_{di}=\left[\begin{array}{c}\mathrm{\Delta }{G}_{di}\\ \mathrm{\Delta }{G}_{di}-\mathrm{\Delta }{B}_{fi}{\overline{G}}_{di}-B_{fi}\mathrm{\Delta }{\overline{G}}_{di}\end{array}\right],\mathrm{\Delta }{\tilde{H}}_i=\left[\begin{array}{c}\mathrm{\Delta }{H}_i\\ \mathrm{\Delta }{H}_i-\mathrm{\Delta }{B}_{fi}{\overline{H}}_i-B_{fi}\mathrm{\Delta }{\overline{H}}_i\end{array}\right],\hfill \\ \mathrm{\Delta }{\tilde{H}}_{di}=\left[\begin{array}{c}\mathrm{\Delta }{H}_{di}\\ \mathrm{\Delta }{H}_{di}-\mathrm{\Delta }{B}_{fi}{\overline{H}}_{di}-B_{fi}\mathrm{\Delta }{\overline{H}}_{di}\end{array}\right],\mathrm{\Delta }{\tilde{D}}_i=\left[\begin{array}{c}\mathrm{\Delta }{D}_i\\ \mathrm{\Delta }{D}_i-\mathrm{\Delta }{B}_{fi}{\overline{D}}_i-B_{fi}\mathrm{\Delta }{\overline{D}}_i\end{array}\right],\hfill \\ \mathrm{\Delta }{\tilde{L}}_i=\left[\begin{array}{cc}\mathrm{\Delta }{L}_i-\mathrm{\Delta }{L}_{fi}& \mathrm{\Delta }{L}_{fi}\end{array}\right],\mathrm{\Delta }{\tilde{J}}_i=\left[\begin{array}{cc}\mathrm{\Delta }{J}_i-\mathrm{\Delta }{J}_{fi}& \mathrm{\Delta }{J}_{fi}\end{array}\right].\hfill \end{array}$$

Denote ${〈x,Hy〉}_T={\int }_0^T{x}^T(t)Hy(t)dt$, in this article, we focus our attention on the quadratic supply rate

$$E(v,e,T)={〈e,Qe〉}_T+2{〈e,Sv〉}_T+{〈v,Rv〉}_T,$$

where Q, S, R are appropriately dimensioned, and Q, R are symmetric matrices.

Definition 1 [18]

If filtering error system (9) is asymptotically stable and for all $T>0$ and $0\ne v(t)\in L_2[0,\mathrm{\infty })$, there exists $\alpha >0$ such that the following inequality is well defined, then

$$E(v,e,T)>\alpha {〈v,v〉}_T.$$

(10)

Definition 2 Given a set of suited dimension real matrices Q, S, R, where Q, R are symmetric matrices, system (6) is called a robust dissipative filter of uncertain system (1) if there exist $A_{fi}$, $B_{fi}$, $C_{fi}$, $L_{fi}$, $J_{fi}$, $E_{fi}$ such that

• (a) the augmented system (9) with $v(t)=0$ is robust asymptotically stable for all uncertainties;

• (b) the filtering error system is strict robust $(Q,S,R)$ dissipative.

Our aim is to determine parameters $A_{fi}$, $B_{fi}$, $C_{fi}$, $L_{fi}$, $J_{fi}$, $E_{fi}$ such that system (6) is a robust dissipative filter for uncertain system (1).

Lemma 1 [8]

Given a set of suited dimension real matrices Q, H, E, Q is a symmetric matrix, such that

$$Q+HFE+E^T{F}^T{H}^T<0$$

for all F satisfies $F^{T}F\le I$ if and only if there exists $\epsilon >0$ such that

$$Q+\epsilon H{H}^T+{\epsilon }^{-1}{E}^{T}E<0.$$

Lemma 2 (Schur complement)

Given a symmetric matrix $S=\left[\begin{array}{cc}{S}_{11}& S_{12}\\ S_{12}^T& S_{22}\end{array}\right]$, where $S_{11}\in R^{r\times r}$, the following three conditions are equivalent:

• (i) $S<0$;

• (ii) $S_{11}<0$, $S_{22}-S_{12}^T{S}_{11}^{-1}{S}_{12}<0$;

• (iii) $S_{22}<0$, $S_{11}-S_{12}{S}_{22}^{-1}{S}_{12}^T<0$.

3 Main results

Theorem 1 Given matrices Q, S, R, where Q, R are symmetric and Q is negative definite and with all uncertainties, the robust dissipative filtering problem is solved for the uncertain Markovian jump system (9) if there exist symmetric matrices $P_l>0$, $P_i>0$, $U>0$, $U_i>0$, and $\tilde{U}>0$, positive scalars ${\lambda }_1$, ${\lambda }_2$, and any appropriately dimensioned matrices $N_i={[{\overline{N}}_{1i}^T{\overline{N}}_{2i}^T]}^T$, $M_i={[{\overline{M}}_{1i}^T{\overline{M}}_{2i}^T]}^T$, $X=X^T=\left[\begin{array}{cc}{X}_{11}& X_{12}\\ \ast & X_{22}\end{array}\right]\ge 0$, and under zero conditions for nonzero $v(t)\in L_2(0,\mathrm{\infty })$, such that the following LMI holds:

(11)

(12)

(13)

where

$$\begin{array}{c}{\mathrm{\Omega }}_{11i}=\sum _{l=1}^N{\pi }_{il}{E}^T{P}_l+P_{l}A+A^T{P}_l-{\lambda }_1{\breve{R}}_1,{\mathrm{\Omega }}_{16i}=-{\lambda }_1{\breve{R}}_2+P_{l}B,\hfill \\ {\mathrm{\Omega }}_{33i}=\sum _{j=1}^N{\pi }_{ij}{\tilde{E}}^T{P}_j+W^{T}UW+U_i+d_1{W}^T\tilde{U}W+P_i\tilde{A}+{\tilde{A}}^T{P}_i+d_1{X}_{11}+{\overline{N}}_{1i}+{\overline{N}}_{1i}^T,\hfill \\ {\mathrm{\Omega }}_{34i}=P_i{\tilde{A}}_d+d_1{X}_{12}-{\overline{N}}_{1i}+{\overline{N}}_{2i}^T+{\overline{M}}_{1i},{\mathrm{\Omega }}_{38i}=P_i\tilde{C}-{\tilde{L}}^{T}S,{\mathrm{\Omega }}_{313i}=W^T{\tilde{G}}^T{P}_i,\hfill \\ {\mathrm{\Omega }}_{44i}=-(1-h_i)W^{T}UW+d_1{X}_{22}-{\overline{N}}_{2i}-{\overline{N}}_{2i}^T+{\overline{M}}_{2i}^T+{\overline{M}}_{2i},{\mathrm{\Omega }}_{413i}=W^T{\tilde{G}}_d^T{P}_i,\hfill \\ {\mathrm{\Omega }}_{88i}=-S^T\tilde{K}-{\tilde{K}}^{T}S-R,\hfill \\ {\psi }_1=\left[\begin{array}{cccc}{X}_{11}& X_{12}& 0& {\overline{N}}_{1i}\\ \ast & X_{22}& 0& {\overline{N}}_{2i}\\ \ast & \ast & W^T\tilde{U}W& 0\\ \ast & \ast & \ast & 0\end{array}\right],{\psi }_2=\left[\begin{array}{cccc}{X}_{11}& X_{12}& 0& {\overline{M}}_{1i}\\ \ast & X_{22}& 0& {\overline{M}}_{2i}\\ \ast & \ast & W^T\tilde{U}W& 0\\ \ast & \ast & \ast & 0\end{array}\right].\hfill \end{array}$$

Proof For each $r(t)=i\in S$, choose a Lyapunov-Krasovskii functional for system (9) as

$$V(x,\tilde{x},t,i)=V_1(x,\tilde{x},t,i)+V_2(x,\tilde{x},t,i)+V_3(x,\tilde{x},t,i)+V_4(x,\tilde{x},t,i)+V_5(x,\tilde{x},t,i),$$

where

$$\begin{array}{c}{V}_1(x,\tilde{x},t,i)=x^T(t)E^T{P}_{l}x(t),\hfill \\ V_2(x,\tilde{x},t,i)={\tilde{x}}^T(t){\tilde{E}}^T{P}_i\tilde{x}(t),\hfill \\ V_3(x,\tilde{x},t,i)={\int }_{t-d(t)}^t{\tilde{x}}^T(s)W^{T}UW\tilde{x}(s)ds,\hfill \\ V_4(x,\tilde{x},t,i)={\int }_{t-d_1}^t{\tilde{x}}^T(s)U_i\tilde{x}(s)ds,\hfill \\ V_5(x,\tilde{x},t,i)={\int }_{-d_1}^0{\int }_{t+\theta }^t{\tilde{x}}^T(s)W^T\tilde{U}W\tilde{x}(s)dsd\theta ,\hfill \end{array}$$

and $P_l>0$, $P_i>0$, $U>0$, $U_i>0$, $\tilde{U}>0$. $V_3(x,\tilde{x},t,i)$, $V_4(x,\tilde{x},t,i)$, $V_5(x,\tilde{x},t,i)$ are governed by t, so we have $V_{\tilde{x}\tilde{x}}(\tilde{x},t,i)=P_l+P_i$.

When $v(t)\equiv 0$,

$$\begin{array}{c}\begin{array}{rl}L{V}_1(x,\tilde{x},t,i)=& x^T(t)\sum _{l=1}^N{\pi }_{il}{E}^T{P}_{l}x(t)\\ +2x^T(t)P_l[Ax(t)+A_{d}x(t-d(t))+Bf(x(t))+B_d{f}_d\left(x(t-d(t))\right)]\\ +\{{[Gx(t)+G_{d}x(t-d(t))+Hf(x(t))+H_d{f}_d\left(x(t-d(t))\right)]}^T\\ \times P_l[Gx(t)+G_{d}x(t-d(t))+Hf(x(t))+H_d{f}_d\left(x(t-d(t))\right)]\},\end{array}\hfill \\ \begin{array}{rl}L{V}_2(x,\tilde{x},t,i)=& {\tilde{x}}^T(t)\sum _{j=1}^N{\pi }_{ij}{\tilde{E}}^T{P}_j\tilde{x}(t)\\ +2{\tilde{x}}^T(t)P_i[\tilde{A}\tilde{x}(t)+{\tilde{A}}_d\tilde{x}(t-d(t))+\tilde{B}f(x(t))+{\tilde{B}}_d{f}_d\left(x(t-d(t))\right)]\\ +\{{[\tilde{G}W\tilde{x}(t)+{\tilde{G}}_{d}W\tilde{x}(t-d(t))+\tilde{H}f(x(t))+{\tilde{H}}_d{f}_d\left(x(t-d(t))\right)]}^T\\ \times P_i[\tilde{G}W\tilde{x}(t)+{\tilde{G}}_{d}W\tilde{x}(t-d(t))+\tilde{H}f(x(t))+{\tilde{H}}_d{f}_d\left(x(t-d(t))\right)]\},\end{array}\hfill \\ L{V}_3(x,\tilde{x},t,i)={\tilde{x}}^T(t)W^{T}UW\tilde{x}(t)-(1-\dot{d}(t)){\tilde{x}}^T(t-d(t))W^{T}UW\tilde{x}(t-d(t)),\hfill \\ L{V}_4(x,\tilde{x},t,i)={\tilde{x}}^T(t)U_i\tilde{x}(t)-{\tilde{x}}^T(t-d_1)U_i\tilde{x}(t-d_1),\hfill \\ \begin{array}{rl}L{V}_5(x,\tilde{x},t,i)=& d_1{\tilde{x}}^T(t)W^T\tilde{U}W\tilde{x}(t)-{\int }_{t-d(t)}^t{\tilde{x}}^T(s)W^T\tilde{U}W\tilde{x}(s)ds\\ -{\int }_{t-d_1}^{t-d(t)}{\tilde{x}}^T(s)W^T\tilde{U}W\tilde{x}(s)ds.\end{array}\hfill \end{array}$$

Using the Newton-Leibniz formula, for any appropriately dimensioned matrices $N_i$, $M_i$, the following equations are true:

(14)

(15)

where ${\xi }_1(t)={[{\tilde{x}}^T(t){\tilde{x}}^T(t-d(t))]}^T$.

On the other hand, for any semi-positive definite matrix $X=X^T=\left[\begin{array}{cc}{X}_{11}& X_{12}\\ \ast & X_{22}\end{array}\right]\ge 0$, the following equation holds:

$$\begin{array}{rl}0& ={\int }_{t-d_1}^t{\xi }_1^T(t)X{\xi }_1(t)ds-{\int }_{t-d_1}^t{\xi }_1^T(t)X{\xi }_1(t)ds\\ =d_1{\xi }_1^T(t)X{\xi }_1(t)-{\int }_{t-d(t)}^t{\xi }_1^T(t)X{\xi }_1(t)ds-{\int }_{t-d_1}^{t-d(t)}{\xi }_1^T(t)X{\xi }_1(t)ds.\end{array}$$

(16)

In addition, it is clear that the following equation is also true:

$$\begin{array}{rl}{\int }_{t-d_1}^t{\xi }_1^T(t)W^T\tilde{U}W{\xi }_1(t)ds=& {\int }_{t-d(t)}^t{\xi }_1^T(t)W^T\tilde{U}W{\xi }_1(t)ds\\ +{\int }_{t-d_1}^{t-d(t)}{\xi }_1^T(t)W^T\tilde{U}W{\xi }_1(t)ds.\end{array}$$

(17)

Notice that (5) implies

$$\begin{array}{r}{\left[\begin{array}{c}x(t)\\ f(x(t))\end{array}\right]}^T\left[\begin{array}{cc}{\breve{R}}_1& {\breve{R}}_2\\ {\breve{R}}_2^T& I\end{array}\right]\left[\begin{array}{c}x(t)\\ f(x(t))\end{array}\right]\le 0,\\ {\left[\begin{array}{c}x(t-d(t))\\ f_d(x(t-d(t)))\end{array}\right]}^T\left[\begin{array}{cc}{\breve{S}}_1& {\breve{S}}_2\\ {\breve{S}}_2^T& I\end{array}\right]\left[\begin{array}{c}x(t-d(t))\\ f_d(x(t-d(t)))\end{array}\right]\le 0,\end{array}$$

(18)

where ${\breve{R}}_1=({\hat{R}}_1^T{\hat{R}}_2+{\hat{R}}_2^T{\hat{R}}_1)/2$, ${\breve{R}}_2=-({\hat{R}}_1^T+{\hat{R}}_2^T)/2$, ${\breve{S}}_1=({\hat{S}}_1^T{\hat{S}}_2+{\hat{S}}_2^T{\hat{S}}_1)/2$, ${\breve{S}}_2=-({\hat{S}}_1^T+{\hat{S}}_2^T)/2$.

Then, adding the terms on the right of Eqs. (14)-(16) and (18) to $V(x,\tilde{x},t,i)$ yields

$$\begin{array}{c}LV(x,\tilde{x},t,i)\hfill \\ \le LV(x,\tilde{x},t,i)-{\lambda }_1{\left[\begin{array}{c}x(t)\\ f(x(t))\end{array}\right]}^T\left[\begin{array}{cc}{\breve{R}}_1& {\breve{R}}_2\\ {\breve{R}}_2^T& I\end{array}\right]\left[\begin{array}{c}x(t)\\ f(x(t))\end{array}\right]\hfill \\ -{\lambda }_2{\left[\begin{array}{c}x(t-d(t))\\ f_d(x(t-d(t)))\end{array}\right]}^T\left[\begin{array}{cc}{\breve{S}}_1& {\breve{S}}_2\\ {\breve{S}}_2^T& I\end{array}\right]\left[\begin{array}{c}x(t-d(t))\\ f_d(x(t-d(t)))\end{array}\right]\hfill \\ +2{\xi }_1^T(t)N_i[\tilde{x}(t)-\tilde{x}(t-d(t))-{\int }_{t-d(t)}^t\dot{\tilde{x}}(s)ds]\hfill \\ +2{\xi }_1^T(t)M_i[\tilde{x}(t-d(t))-\tilde{x}(t-d_1)-{\int }_{t-d_1}^{t-d(t)}\dot{\tilde{x}}(s)ds]\hfill \\ +d_1{\xi }_1^T(t)X{\xi }_1(t)-{\int }_{t-d(t)}^t{\xi }_1^T(t)X{\xi }_1(t)ds-{\int }_{t-d_1}^{t-d(t)}{\xi }_1^T(t)X{\xi }_1(t)ds\hfill \\ \le {\xi }^T(t)H_i\xi (t),\hfill \end{array}$$

where

(19)

Via Lyapunov-Krasovskii, according to (13), if $H_i<0$ and $LV(x,\tilde{x},t,i)<0$, system (9) is robust asymptotically stable.

When $v(t)\ne 0$,

$$LV(x,\tilde{x},v,t,i)\le \varsigma {(t)}^T(t)H_i^{\mathrm{\prime }}\varsigma (t),$$

where

$$\begin{array}{c}\begin{array}{rl}\varsigma (t)=& [\begin{array}{cccccc}{x}^T(t)& x^T(t-d(t))& {\tilde{x}}^T(t)& {\tilde{x}}^T(t-d(t))& {\tilde{x}}^T(t-d_1)& f(x(t))\end{array}\\ {\begin{array}{cccc}{f}_d(x(t-d(t)))& v^T(t)& {\int }_{t-d(t)}^t{\eta }^T(t,s)ds& {\int }_{t-d_1}^{t-d(t)}{\eta }^T(t,s)ds\end{array}]}^T,\end{array}\hfill \\ H_i^{\mathrm{\prime }}=\left[\begin{array}{ccccccccc}{H}_{11i}& H_{12i}& 0& 0& 0& H_{16i}& P_l{B}_d+G^T{P}_l{H}_d& H_{18i}^{\mathrm{\prime }}& 0\\ \ast & H_{22i}& 0& 0& 0& G_d^T{P}_{l}H& -{\lambda }_2{\breve{S}}_2+G_d^T{P}_l{H}_d& G_d^T{P}_{l}D& 0\\ \ast & \ast & H_{33i}& H_{34i}& -{\overline{M}}_{1i}& H_{36i}& P_i{\tilde{B}}_d+W^T{\tilde{G}}^T{P}_i{\tilde{H}}_d& H_{38i}^{\mathrm{\prime }}& 0\\ \ast & \ast & \ast & H_{44i}& -{\overline{M}}_{2i}& W^T{\tilde{G}}_d^T{P}_i\tilde{H}& W^T{\tilde{G}}_d^T{P}_i{\tilde{H}}_d& H_{48i}^{\mathrm{\prime }}& 0\\ \ast & \ast & \ast & \ast & -U_i& 0& 0& 0& 0\\ \ast & \ast & \ast & \ast & \ast & H_{66i}& H_{67i}& H_{68i}^{\mathrm{\prime }}& 0\\ \ast & \ast & \ast & \ast & \ast & \ast & H_{77i}& H_{78i}^{\mathrm{\prime }}& 0\\ \ast & \ast & \ast & \ast & \ast & \ast & \ast & H_{88i}^{\mathrm{\prime }}& 0\\ \ast & \ast & \ast & \ast & \ast & \ast & \ast & \ast & -\psi \end{array}\right]\hfill \\ H_{18i}^{\mathrm{\prime }}=P_{l}C+G^T{P}_{l}D,H_{38i}^{\mathrm{\prime }}=P_i\tilde{C}+W^T{\tilde{G}}^T{P}_i\tilde{D},\hfill \\ H_{48i}^{\mathrm{\prime }}=W^T{\tilde{G}}_d^T{P}_i\tilde{D},H_{68i}^{\mathrm{\prime }}={\tilde{H}}^T{P}_i\tilde{D}+H^T{P}_{l}D,\hfill \\ H_{78i}^{\mathrm{\prime }}={\tilde{H}}_d^T{P}_i\tilde{D}+H_d^T{P}_{l}D,H_{88i}^{\mathrm{\prime }}={\tilde{D}}^T{P}_i\tilde{D}+D^T{P}_{l}D,\hfill \\ \psi =\left[\begin{array}{c}{\psi }_1\\ {\psi }_2\end{array}\right].\hfill \end{array}$$

Furthermore, we consider the dissipation performance. For the zero initial condition

$$J(T,\alpha )={\int }_0^T[e^T(t)Qe(t)+2e^T(t)Sv(t)+v^T(t)(R-\alpha I)v(t)]dt,$$

it is easy to get $J(T,\alpha )=E(v,e,T)-\alpha {〈v,v〉}_T$.

Consider the augmented system (9) is strict robust $(Q,S,R)$ dissipative if and only if there exists $\alpha >0$, with all $T>0$ and under zero conditions for nonzero $v(t)\in L_2[0,\mathrm{\infty })$, such that $J(T,\alpha )\ge 0$.

$$\begin{array}{r}{\varsigma }^T(t)H_i^{\mathrm{\prime }}\varsigma (t)-J(T,\alpha )\\ ={\int }_0^T[{\xi }^T(t)H_i\xi (t)-e^T(t)Qe(t)-2e^T(t)Sv(t)-v^T(t)(R-\alpha I)v(t)]dt\\ ={\int }_0^T[{\varsigma }^T(t)({\mathrm{\Upsilon }}_i+\alpha diag(0,0,0,0,0,0,0,I,0,0))\varsigma (t)]dt,\end{array}$$

where

$$\begin{array}{c}{\mathrm{\Upsilon }}_i=\left[\begin{array}{ccccccccc}{\mathrm{\Upsilon }}_{11i}& P_l{A}_d+G^T{P}_l{G}_d& 0& 0& 0& {\mathrm{\Upsilon }}_{16i}& {\mathrm{\Upsilon }}_{17i}& {\mathrm{\Upsilon }}_{18i}& 0\\ \ast & -{\lambda }_2{\breve{S}}_1+G_d^T{P}_l{G}_d& 0& 0& 0& G_d^T{P}_{l}H& {\mathrm{\Upsilon }}_{27i}& G_d^T{P}_{l}D& 0\\ \ast & \ast & {\mathrm{\Upsilon }}_{33i}& {\mathrm{\Upsilon }}_{34i}& -{\overline{M}}_{1i}& {\mathrm{\Upsilon }}_{36i}& {\mathrm{\Upsilon }}_{37i}& {\mathrm{\Upsilon }}_{38i}& 0\\ \ast & \ast & \ast & {\mathrm{\Upsilon }}_{44i}& -{\overline{M}}_{2i}& {\mathrm{\Upsilon }}_{46i}& {\mathrm{\Upsilon }}_{47i}& {\mathrm{\Upsilon }}_{48i}& 0\\ \ast & \ast & \ast & \ast & -U_i& 0& 0& 0& 0\\ \ast & \ast & \ast & \ast & \ast & {\mathrm{\Upsilon }}_{66i}& {\mathrm{\Upsilon }}_{67i}& {\mathrm{\Upsilon }}_{68i}& 0\\ \ast & \ast & \ast & \ast & \ast & \ast & {\mathrm{\Upsilon }}_{77i}& {\mathrm{\Upsilon }}_{78i}& 0\\ \ast & \ast & \ast & \ast & \ast & \ast & \ast & {\mathrm{\Upsilon }}_{88i}& 0\\ \ast & \ast & \ast & \ast & \ast & \ast & \ast & \ast & -\psi \end{array}\right],\hfill \\ {\mathrm{\Upsilon }}_{11i}=\sum _{l=1}^N{\pi }_{il}{E}^T{P}_l+P_{l}A+A^T{P}_l+G^T{P}_{l}G-{\lambda }_1{\breve{R}}_1,\hfill \\ {\mathrm{\Upsilon }}_{16i}=-{\lambda }_1{\breve{R}}_2+P_{l}B+G^T{P}_{l}H,\hfill \\ {\mathrm{\Upsilon }}_{17i}=P_l{B}_d+G^T{P}_l{H}_d,{\mathrm{\Upsilon }}_{18i}=P_{l}C+G^T{P}_{l}D,{\mathrm{\Upsilon }}_{27i}=-{\lambda }_2{\breve{S}}_2+G_d^T{P}_l{H}_d,\hfill \\ \begin{array}{rl}{\mathrm{\Upsilon }}_{33i}=& \sum _{j=1}^N{\pi }_{ij}{\tilde{E}}^T{P}_j+W^{T}UW+U_i+d_1{W}^T\tilde{U}W+P_i\tilde{A}\\ +{\tilde{A}}^T{P}_i+W^T{\tilde{G}}^T{P}_i\tilde{G}W+d_1{X}_{11}+{\overline{N}}_{1i}+{\overline{N}}_{1i}^T-{\tilde{L}}^{T}Q\tilde{L},\end{array}\hfill \\ {\mathrm{\Upsilon }}_{34i}=P_i{\tilde{A}}_d+W^T{\tilde{G}}^T{P}_i{\tilde{G}}_{d}W+d_1{X}_{12}-{\overline{N}}_{1i}+{\overline{N}}_{2i}^T+{\overline{M}}_{1i}-{\tilde{L}}^{T}Q\tilde{J},\hfill \\ {\mathrm{\Upsilon }}_{36i}=P_i\tilde{B}+W^T{\tilde{G}}^T{P}_i\tilde{H}-{\tilde{L}}^{T}Q{\tilde{B}}_{zi},{\mathrm{\Upsilon }}_{37i}=P_i{\tilde{B}}_d+W^T{\tilde{G}}^T{P}_i{\tilde{H}}_d-{\tilde{L}}^{T}Q{\tilde{H}}_{zi},\hfill \\ {\mathrm{\Upsilon }}_{38i}=P_i\tilde{C}+W^T{\tilde{G}}^T{P}_i\tilde{D}-{\tilde{L}}^{T}Q\tilde{K}-{\tilde{L}}^{T}S,\hfill \\ \begin{array}{rl}{\mathrm{\Upsilon }}_{44i}=& -(1-h_i)W^{T}UW+W^T{\tilde{G}}_d^T{P}_i{\tilde{G}}_{d}W+d_1{X}_{22}\\ -{\overline{N}}_{2i}-{\overline{N}}_{2i}^T+{\overline{M}}_{2i}^T+{\overline{M}}_{2i}-{\tilde{J}}^{T}Q\tilde{J},\end{array}\hfill \\ {\mathrm{\Upsilon }}_{46i}=W^T{\tilde{G}}_d^T{P}_i\tilde{H}-{\tilde{J}}^{T}Q{\tilde{B}}_{zi},{\mathrm{\Upsilon }}_{47i}=W^T{\tilde{G}}_d^T{P}_i{\tilde{H}}_d-{\tilde{J}}^{T}Q{\tilde{H}}_{zi},\hfill \\ {\mathrm{\Upsilon }}_{48i}=W^T{\tilde{G}}_d^T{P}_i\tilde{D}-{\tilde{J}}^{T}Q\tilde{K}-{\tilde{J}}^{T}S,{\mathrm{\Upsilon }}_{66i}={\tilde{H}}^T{P}_i\tilde{H}-{\lambda }_{1}I+H^T{P}_{l}H-{\tilde{B}}_{zi}^{T}Q{\tilde{B}}_{zi},\hfill \\ {\mathrm{\Upsilon }}_{67i}={\tilde{H}}^T{P}_i{\tilde{H}}_d+H^T{P}_l{H}_d-{\tilde{B}}_{zi}^{T}Q{\tilde{H}}_{zi},\hfill \\ {\mathrm{\Upsilon }}_{68i}={\tilde{H}}^T{P}_i\tilde{D}-{\tilde{B}}_{zi}^{T}Q\tilde{K}-{\tilde{B}}_{zi}^{T}S+H^T{P}_{l}D,\hfill \\ {\mathrm{\Upsilon }}_{77i}={\tilde{H}}_d^T{P}_i{\tilde{H}}_d-{\lambda }_{2}I+H_d^T{P}_l{H}_d-{\tilde{H}}_{zi}^{T}Q{\tilde{H}}_{zi},\hfill \\ {\mathrm{\Upsilon }}_{78i}={\tilde{H}}_d^T{P}_i\tilde{D}-{\tilde{H}}_{zi}^{T}Q\tilde{K}-{\tilde{H}}_{zi}^{T}S+H_d^T{P}_{l}D,\hfill \\ {\mathrm{\Upsilon }}_{88i}={\tilde{D}}^T{P}_i\tilde{D}-{\tilde{K}}^{T}Q\tilde{K}-S^T\tilde{K}-{\tilde{K}}^{T}S-R+D^T{P}_{l}D.\hfill \end{array}$$

When ${\mathrm{\Upsilon }}_i<0$, there must exist $\alpha >0$ small enough such that ${\mathrm{\Upsilon }}_i+\alpha diag(0,0,0,0,0,0,0,I,0,0)<0$. Hence, there exists $\alpha >0$ for all $T>0$ and $0\ne v(t)\in L_2[0,\mathrm{\infty })$, we have $J(T,\alpha )\ge 0$. By the Schur complement formula, (13) is completed. □

Based on the sufficient conditions above, the following criterion can be obtained readily.

Theorem 2 Given matrices Q, S, R, where Q, R are symmetric and Q is negative definite and with all uncertainties, the robust dissipative filtering problem is solved for the uncertain Markovian jump system (9) if there exist positive scalars ${\lambda }_1$, ${\lambda }_2$ and ${\epsilon }_{ki}$ ($k=1,2$), where i is taking values in a finite set $S=\{1,2,\dots ,N\}$, symmetric matrices $P_l>0$, $X_i>0$, $Y_i>0$, $U>0$, $U_i>0$ and $\tilde{U}>0$, matrices $W_i$, $Z_i$, $V_i$, $R_i$, and any appropriately dimensioned matrices $N_i={[{\overline{N}}_{1i}^T{\overline{N}}_{2i}^T]}^T$, $M_i={[{\overline{M}}_{1i}^T{\overline{M}}_{2i}^T]}^T$, $X=X^T=\left[\begin{array}{cc}{X}_{11}& X_{12}\\ \ast & X_{22}\end{array}\right]\ge 0$ such that the following LMI holds:

$$\begin{array}{r}{E}^T{P}_l=P_{l}E\ge 0,\\ E^T{X}_i=X_{i}E\ge 0,\\ R_i^T=R_i\ge 0,\\ \left[\begin{array}{ccc}{\mathrm{\Theta }}_{1i}& {\mathrm{\Theta }}_{2i}& {\mathrm{\Theta }}_{3i}\\ \ast & {\mathrm{\Theta }}_{4i}& 0\\ \ast & \ast & {\mathrm{\Theta }}_{4i}\end{array}\right]<0,\end{array}$$

(20)

where

$$\begin{array}{c}{\mathrm{\Theta }}_{1i}=\left[\begin{array}{cccccccccccccccc}{\mathrm{\Theta }}_{11}& {\mathrm{\Theta }}_{12}& 0& 0& 0& 0& 0& 0& {\mathrm{\Theta }}_{19}& {\mathrm{\Theta }}_{110}& P_l{C}_i& 0& 0& {\mathrm{\Theta }}_{114}& 0& 0\\ \ast & {\mathrm{\Theta }}_{22}& 0& 0& 0& 0& 0& 0& 0& {\mathrm{\Theta }}_{210}& 0& 0& 0& {\mathrm{\Theta }}_{214}& 0& 0\\ \ast & \ast & {\mathrm{\Theta }}_{33}& {\mathrm{\Theta }}_{34}& {\mathrm{\Theta }}_{35}& {\mathrm{\Theta }}_{36}& {\mathrm{\Theta }}_{37}& {\mathrm{\Theta }}_{38}& {\mathrm{\Theta }}_{39}& {\mathrm{\Theta }}_{310}& {\mathrm{\Theta }}_{311}& 0& {\mathrm{\Theta }}_{313}& 0& {\mathrm{\Theta }}_{315}& {\mathrm{\Theta }}_{316}\\ \ast & \ast & \ast & {\mathrm{\Theta }}_{44}& {\mathrm{\Theta }}_{45}& {\mathrm{\Theta }}_{46}& {\mathrm{\Theta }}_{47}& {\mathrm{\Theta }}_{48}& {\mathrm{\Theta }}_{49}& {\mathrm{\Theta }}_{410}& {\mathrm{\Theta }}_{411}& 0& {\mathrm{\Theta }}_{413}& 0& 0& 0\\ \ast & \ast & \ast & \ast & {\mathrm{\Theta }}_{55}& {\mathrm{\Theta }}_{56}& {\mathrm{\Theta }}_{57}& {\mathrm{\Theta }}_{58}& 0& 0& {\mathrm{\Theta }}_{511}& 0& {\mathrm{\Theta }}_{513}& 0& {\mathrm{\Theta }}_{515}& {\mathrm{\Theta }}_{516}\\ \ast & \ast & \ast & \ast & \ast & {\mathrm{\Theta }}_{66}& {\mathrm{\Theta }}_{67}& {\mathrm{\Theta }}_{68}& 0& 0& {\mathrm{\Theta }}_{611}& 0& {\mathrm{\Theta }}_{613}& 0& 0& 0\\ \ast & \ast & \ast & \ast & \ast & \ast & {\mathrm{\Theta }}_{77}& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ \ast & \ast & \ast & \ast & \ast & \ast & \ast & {\mathrm{\Theta }}_{88}& 0& 0& 0& 0& 0& 0& 0& 0\\ \ast & \ast & \ast & \ast & \ast & \ast & \ast & \ast & {\mathrm{\Theta }}_{99}& 0& {\mathrm{\Theta }}_{911}& 0& {\mathrm{\Theta }}_{913}& {\mathrm{\Theta }}_{914}& {\mathrm{\Theta }}_{915}& {\mathrm{\Theta }}_{916}\\ \ast & \ast & \ast & \ast & \ast & \ast & \ast & \ast & \ast & {\eta }_1& {\eta }_2& 0& {\eta }_3& {\eta }_4& {\eta }_5& {\eta }_6\\ \ast & \ast & \ast & \ast & \ast & \ast & \ast & \ast & \ast & \ast & {\eta }_7& 0& {\eta }_8& {\eta }_9& {\eta }_{10}& {\eta }_{11}\\ \ast & \ast & \ast & \ast & \ast & \ast & \ast & \ast & \ast & \ast & \ast & -\psi & 0& 0& 0& 0\\ \ast & \ast & \ast & \ast & \ast & \ast & \ast & \ast & \ast & \ast & \ast & \ast & Q& 0& 0& 0\\ \ast & \ast & \ast & \ast & \ast & \ast & \ast & \ast & \ast & \ast & \ast & \ast & \ast & {\eta }_{12}& 0& 0\\ \ast & \ast & \ast & \ast & \ast & \ast & \ast & \ast & \ast & \ast & \ast & \ast & \ast & \ast & {\eta }_{13}& 0\\ \ast & \ast & \ast & \ast & \ast & \ast & \ast & \ast & \ast & \ast & \ast & \ast & \ast & \ast & \ast & -Y_i\end{array}\right],\hfill \\ {\mathrm{\Theta }}_{2i}=\left[\begin{array}{ccccccc}{P}_l{M}_{i1}& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\\ 0& X_i{M}_{i1}& 0& 0& 0& 0& 0\\ 0& Y_i{M}_{i1}-Z_i{M}_{i3}& -M_{\mathrm{\Sigma }}& -Y_i{M}_{i7}& -Y_i{M}_{i6}& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\\ 0& -S^T{M}_{i5}& 0& 0& 0& S^T{M}_{i7}& 0\\ 0& 0& 0& 0& 0& 0& 0\\ 0& Q{M}_{i5}& 0& 0& 0& -Q{M}_{i7}& 0\\ P_l{M}_{i2}& 0& 0& 0& 0& 0& 0\\ 0& X_i{M}_{i2}& 0& 0& 0& 0& 0\\ 0& Y_i{M}_{i2}-Z_i{M}_{i4}& 0& 0& 0& 0& Y_i{M}_{i6}\end{array}\right],\hfill \\ {\mathrm{\Theta }}_{3i}=\left[\begin{array}{ccccccc}{\epsilon }_{1i}{N}_{i1}^T& 0& 0& 0& 0& 0& 0\\ {\epsilon }_{1i}{N}_{i2}^T& 0& 0& 0& 0& 0& 0\\ 0& {\epsilon }_{1i}{N}_{i1}^T& 0& 0& {\epsilon }_{2i}{N}_{i7}^T+{\epsilon }_{2i}{\overline{A}}_i^T{N}_{i8}^T& {\epsilon }_{2i}{N}_{i7}^T& -{\epsilon }_{2i}{\overline{G}}_i^T{N}_{i8}^T\\ 0& 0& {\epsilon }_{2i}{N}_{\mathrm{\Sigma }}^T& 0& -{\epsilon }_{2i}{N}_{i7}^T& -{\epsilon }_{2i}{N}_{i7}^T& 0\\ 0& {\epsilon }_{1i}{N}_{i2}^T& 0& {\epsilon }_{2i}{N}_{i6}^T& {\epsilon }_{2i}{\overline{A}}_{di}^T{N}_{i8}^T& {\epsilon }_{2i}{N}_{i8}^T& -{\epsilon }_{2i}{\overline{G}}_{di}^T{N}_{i8}^T\\ 0& 0& 0& -{\epsilon }_{2i}{N}_{i6}^T& 0& -{\epsilon }_{2i}{N}_{i8}^T& 0\\ 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\\ {\epsilon }_{1i}{N}_{i3}^T& {\epsilon }_{1i}{N}_{i3}^T& 0& 0& {\epsilon }_{2i}{\overline{B}}_i^T{N}_{i8}^T& 0& -{\epsilon }_{2i}{\overline{H}}_i^T{N}_{i8}^T\\ {\epsilon }_{1i}{N}_{i4}^T& {\epsilon }_{1i}{N}_{i4}^T& 0& 0& {\epsilon }_{2i}{\overline{B}}_{di}^T{N}_{i8}^T& 0& -{\epsilon }_{2i}{\overline{H}}_{di}^T{N}_{i8}^T\\ {\epsilon }_{1i}{N}_{i5}^T& {\epsilon }_{1i}{N}_{i5}^T& 0& 0& {\epsilon }_{2i}{\overline{C}}_i^T{N}_{i8}^T& 0& -{\epsilon }_{2i}{\overline{D}}_i^T{N}_{i8}^T\\ 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\end{array}\right],\hfill \\ {\mathrm{\Theta }}_{4i}=diag\{-{\epsilon }_{1i}I,-{\epsilon }_{1i}I,-{\epsilon }_{2i}I,-{\epsilon }_{2i}I,-{\epsilon }_{2i}I,-{\epsilon }_{2i}I,-{\epsilon }_{2i}I\},\hfill \\ {\mathrm{\Theta }}_{11}=\sum _{l=1}^N{\pi }_{il}{E}^T{P}_l+P_l{A}_i+A_i^T{P}_l-{\lambda }_1{\breve{R}}_1,{\mathrm{\Theta }}_{12}=P_l{A}_{di},{\mathrm{\Theta }}_{19}=-{\lambda }_1{\breve{R}}_2+P_l{B}_i,\hfill \\ {\mathrm{\Theta }}_{110}=P_l{B}_{di},{\mathrm{\Theta }}_{114}=G_i^T{P}_l,{\mathrm{\Theta }}_{22}=-{\lambda }_2{\breve{S}}_1,\hfill \\ {\mathrm{\Theta }}_{210}=-{\lambda }_2{\breve{S}}_2,{\mathrm{\Theta }}_{214}=G_{di}^T{P}_l,\hfill \\ {\mathrm{\Theta }}_{33}=\sum _{j=1}^N{\pi }_{ij}{E}^T{X}_j+U+U_{i1}+d_1\tilde{U}+X_i{A}_i+A_i^T{X}_i+d_1{X}_{1111}+N_{1i11}+N_{1i11}^T,\hfill \\ {\mathrm{\Theta }}_{34}=A_i^T{Y}_i-W_i^T-{\overline{A}}_i^T{Z}_i^T+d_1{X}_{1112}+N_{1i12}+N_{1i21}^T,\hfill \\ {\mathrm{\Theta }}_{35}=X_i{A}_{di}+d_1{X}_{1211}-N_{1i11}+N_{2i11}^T+M_{1i11},{\mathrm{\Theta }}_{36}=d_1{X}_{1212}-N_{1i12}+N_{2i21}^T+M_{1i12},\hfill \\ {\mathrm{\Theta }}_{37}=-M_{1i11},{\mathrm{\Theta }}_{38}=-M_{1i12},{\mathrm{\Theta }}_{39}=X_i{B}_i,\hfill \\ {\mathrm{\Theta }}_{310}=X_i{B}_{di},{\mathrm{\Theta }}_{311}=X_i{C}_i-L_i^{T}S+L_{fi}^{T}S,\hfill \\ {\mathrm{\Theta }}_{313}=L_i^{T}Q-L_{fi}^{T}Q,{\mathrm{\Theta }}_{315}=G_i^T{X}_i,{\mathrm{\Theta }}_{316}=G_i^T{Y}_i-{\overline{G}}_i^T{Z}_i^T,\hfill \\ {\mathrm{\Theta }}_{44}=\sum _{j=1}^N{\pi }_{ij}{R}_j+W_i+W_i^T+U_{i2}+d_1{X}_{1122}+N_{1i22}+N_{1i22}^T,\hfill \\ {\mathrm{\Theta }}_{45}=Y_i{A}_{di}-V_i-Z_i{\overline{A}}_{di}+d_1{X}_{1221}-N_{1i21}+N_{2i12}^T+M_{1i21},\hfill \\ {\mathrm{\Theta }}_{46}=V_i+d_1{X}_{1222}-N_{1i22}+N_{2i22}^T+M_{1i22},{\mathrm{\Theta }}_{47}=-M_{1i21},{\mathrm{\Theta }}_{48}=-M_{1i22},\hfill \\ {\mathrm{\Theta }}_{49}=Y_i{B}_i-Z_i{\overline{B}}_i,{\mathrm{\Theta }}_{410}=Y_i{B}_{di}-Z_i{\overline{B}}_{di},\hfill \\ {\mathrm{\Theta }}_{411}=Y_i{C}_i-Z_i{\overline{C}}_i-L_{fi}^{T}S,{\mathrm{\Theta }}_{413}=L_{fi}^{T}Q,\hfill \\ {\mathrm{\Theta }}_{55}=-(1-h_i)U+d_1{X}_{2211}-N_{2i11}-N_{2i11}^T+M_{2i11}^T+M_{2i11},\hfill \\ {\mathrm{\Theta }}_{56}=d_1{X}_{2212}-N_{2i12}-N_{2i21}^T+M_{2i21}^T+M_{2i12},{\mathrm{\Theta }}_{57}=-M_{2i11},{\mathrm{\Theta }}_{58}=-M_{2i12},\hfill \\ {\mathrm{\Theta }}_{511}=-J_i^{T}S+J_{fi}^{T}S,{\mathrm{\Theta }}_{513}=J_i^{T}Q-J_{fi}^{T}Q,\hfill \\ {\mathrm{\Theta }}_{515}=G_{di}^T{X}_i,{\mathrm{\Theta }}_{516}=G_{di}^T{Y}_i-{\overline{G}}_{di}^T{Z}_i^T,\hfill \\ {\mathrm{\Theta }}_{66}=d_1{X}_{2222}-N_{2i22}-N_{2i22}^T+M_{2i22}^T+M_{2i22},{\mathrm{\Theta }}_{67}=-M_{2i21},{\mathrm{\Theta }}_{68}=-M_{2i22},\hfill \\ {\mathrm{\Theta }}_{611}=-J_{fi}^{T}S,{\mathrm{\Theta }}_{613}=J_{fi}^{T}Q,{\mathrm{\Theta }}_{77}=-U_{i1},\hfill \\ {\mathrm{\Theta }}_{88}=-U_{i2},{\mathrm{\Theta }}_{99}=-{\lambda }_{1}I,{\mathrm{\Theta }}_{911}=-B_{zi}^{T}S,\hfill \\ {\mathrm{\Theta }}_{913}=B_{zi}^{T}Q,{\mathrm{\Theta }}_{914}=H_i^T{P}_l,{\mathrm{\Theta }}_{915}=H_i^T{X}_i,\hfill \\ {\mathrm{\Theta }}_{916}=H_i^T{Y}_i-{\overline{H}}_i^T{Z}_i^T,{\eta }_1=-{\lambda }_{2}I,{\eta }_2=-H_{zi}^{T}S,\hfill \\ {\eta }_3=H_{zi}^{T}Q,{\eta }_4=H_{di}^T{P}_l,{\eta }_5=H_{di}^T{X}_i,\hfill \\ {\eta }_6=H_{di}^T{Y}_i-{\overline{H}}_{di}^T{Z}_i^T,{\eta }_7=-K_i^{T}S-S^T{K}_i-R,\hfill \\ {\eta }_8=K_i^{T}Q,{\eta }_9=D_i^T{P}_l,{\eta }_{10}=D_i^T{X}_i,\hfill \\ {\eta }_{11}=D_i^T{Y}_i-{\overline{D}}_i^T{Z}_i^T,{\eta }_{12}=-P_l,{\eta }_{13}=-X_i,\hfill \\ \begin{array}{rl}{M}_{\mathrm{\Sigma }}=& [\begin{array}{cccc}{\pi }_{i1}{Y}_1{M}_{16}& {\pi }_{i2}{Y}_2{M}_{26}& \cdots & {\pi }_{i,i-1}{Y}_{i-1}{M}_{i-1,6}\end{array}\\ \begin{array}{ccc}{\pi }_{i,i+1}{Y}_{i+1}{M}_{i+1,6}& \cdots & {\pi }_{i,N}{Y}_N{M}_{N,6}\end{array}],\end{array}\hfill \\ N_{\mathrm{\Sigma }}={\left[\begin{array}{cccccc}{N}_{16}^T{N}_{26}^T& \cdots & N_{i-1,6}^T& N_{i+1,6}^T& \cdots & N_{N,6}^T\end{array}\right]}^T,\hfill \\ \psi =\left[\begin{array}{c}{\psi }_1\\ {\psi }_2\end{array}\right],{\psi }_1=\left[\begin{array}{ccccc}{X}_{11}& X_{12}& 0& 0& {\overline{N}}_{1i}\\ \ast & X_{22}& 0& 0& {\overline{N}}_{2i}\\ \ast & \ast & \tilde{U}& 0& 0\\ \ast & \ast & \ast & 0& 0\\ \ast & \ast & \ast & \ast & 0\end{array}\right],\hfill \\ {\psi }_2=\left[\begin{array}{ccccc}{X}_{11}& X_{12}& 0& 0& {\overline{M}}_{1i}\\ \ast & X_{22}& 0& 0& {\overline{M}}_{2i}\\ \ast & \ast & \tilde{U}& 0& 0\\ \ast & \ast & \ast & 0& 0\\ \ast & \ast & \ast & \ast & 0\end{array}\right],\hfill \\ U_i=\left[\begin{array}{c}{U}_{i1}\\ U_{i2}\end{array}\right],X_{11}=\left[\begin{array}{cc}{X}_{1111}& X_{1112}\\ X_{1121}& X_{1122}\end{array}\right],\hfill \\ X_{12}=\left[\begin{array}{cc}{X}_{1211}& X_{1212}\\ X_{1221}& X_{1222}\end{array}\right],X_{22}=\left[\begin{array}{cc}{X}_{2211}& X_{2212}\\ X_{2221}& X_{2222}\end{array}\right],\hfill \\ {\overline{N}}_{1i}=\left[\begin{array}{cc}{N}_{1i11}& N_{1i12}\\ N_{1i21}& N_{1i22}\end{array}\right],{\overline{N}}_{2i}=\left[\begin{array}{cc}{N}_{2i11}& N_{2i12}\\ N_{2i21}& N_{2i22}\end{array}\right],\hfill \\ {\overline{M}}_{1i}=\left[\begin{array}{cc}{M}_{1i11}& M_{1i12}\\ M_{1i21}& M_{1i22}\end{array}\right],{\overline{M}}_{2i}=\left[\begin{array}{cc}{M}_{2i11}& M_{2i12}\\ M_{2i21}& M_{2i22}\end{array}\right].\hfill \end{array}$$

In this case, a suitable robust dissipative filter in the form of (6) has parameters as follows:

$$A_{fi}=Y_i^{-1}{W}_i,B_{fi}=Y_i^{-1}{Z}_i,C_{fi}=Y_i^{-1}{V}_i,L_{fi},J_{fi},E_{fi}=Y_i^{-1}{R}_i.$$

Remark 3 The robust dissipative filter design problem is solved in Theorem 2 for the uncertain stochastic singular time-delay systems with Markovian jump parameters. We derive an LMI-based sufficient condition for the existence of full-order filters that ensure asymptotic stability and strict robust $(Q,S,R)$ dissipativity of the resulting filtering error system and reduce the effect of the disturbance input on the estimated signal to a prescribed level for all admissible uncertainties. The feasibility of the filter design problem can be readily checked by the solvability of an LMI, which is dependent on the upper bound of the time-varying delays. The solvability of such a delay-dependent LMI can be readily checked by resorting to the Matlab LMI Tool box. In the next section, an illustrative example is provided to show the potential of the proposed techniques.

Remark 4 When the finite set $S=\{1\}$, the uncertain linear stochastic system (1) is a generalized uncertain stochastic singular time-delay system without Markovian jump parameters.

4 Numerical value example

Example In this section, a numerical example is presented to demonstrate the usefulness of the developed method on the design of a robust dissipative filter for the uncertain stochastic singular time-delay systems with Markovian jump parameters.

Consider system (1) with the following parameters:

$$\begin{array}{c}{A}_1=\left[\begin{array}{cc}1& 0\\ 1& 0\end{array}\right],A_{d1}=\left[\begin{array}{cc}2& 1\\ 1& 2\end{array}\right],B_1=\left[\begin{array}{cc}3& 2\\ 4& 1\end{array}\right],\hfill \\ B_{d1}=\left[\begin{array}{cc}3& 2\\ 4& 1\end{array}\right],C_1=\left[\begin{array}{cc}1& 3\\ 1& 2\end{array}\right],\hfill \\ G_1=\left[\begin{array}{cc}2& 4\\ 1& 2\end{array}\right],G_{d1}=\left[\begin{array}{cc}1& 2\\ 1& 1\end{array}\right],H_1=\left[\begin{array}{cc}2& 1\\ 1& 2\end{array}\right],\hfill \\ H_{d1}=\left[\begin{array}{cc}2& 1\\ 1& 2\end{array}\right],D_1=\left[\begin{array}{cc}2& 1\\ 1& 2\end{array}\right],\hfill \\ {\overline{A}}_1=\left[\begin{array}{cc}1& 0\\ 1& 0\end{array}\right],{\overline{A}}_{d1}=\left[\begin{array}{cc}2& 1\\ 1& 2\end{array}\right],{\overline{B}}_1=\left[\begin{array}{cc}3& 2\\ 4& 1\end{array}\right],\hfill \\ {\overline{B}}_{d1}=\left[\begin{array}{cc}3& 2\\ 4& 1\end{array}\right],{\overline{C}}_1=\left[\begin{array}{cc}1& 3\\ 1& 2\end{array}\right],\hfill \\ {\overline{G}}_1=\left[\begin{array}{cc}2& 4\\ 1& 2\end{array}\right],{\overline{G}}_{d1}=\left[\begin{array}{cc}1& 2\\ 1& 1\end{array}\right],{\overline{H}}_1=\left[\begin{array}{cc}2& 1\\ 1& 2\end{array}\right],\hfill \\ {\overline{H}}_{d1}=\left[\begin{array}{cc}2& 1\\ 1& 2\end{array}\right],{\overline{D}}_1=\left[\begin{array}{cc}2& 1\\ 1& 2\end{array}\right],\hfill \\ L_1=\left[\begin{array}{cc}1& 0\\ 1& 0\end{array}\right],J_1=\left[\begin{array}{cc}2& 1\\ 1& 2\end{array}\right],B_{z1}=\left[\begin{array}{cc}3& 2\\ 4& 1\end{array}\right],\hfill \\ H_{z1}=\left[\begin{array}{cc}3& 2\\ 4& 1\end{array}\right],K_1=\left[\begin{array}{cc}1& 3\\ 3& 1\end{array}\right],\hfill \\ M_{11}=\left[\begin{array}{c}0.01\\ 0.02\end{array}\right],M_{12}=\left[\begin{array}{c}-0.02\\ 0.02\end{array}\right],M_{13}=\left[\begin{array}{c}-0.02\\ 0.01\end{array}\right],\hfill \\ M_{14}=\left[\begin{array}{c}0.01\\ 0.02\end{array}\right],M_{15}=\left[\begin{array}{c}-0.02\\ 0.02\end{array}\right],\hfill \\ N_{11}=\left[\begin{array}{cc}0.01& 0.02\end{array}\right],N_{12}=\left[\begin{array}{cc}-0.02& 0.02\end{array}\right],N_{13}=\left[\begin{array}{cc}-0.02& 0.01\end{array}\right],\hfill \\ N_{14}=\left[\begin{array}{cc}0.01& 0.02\end{array}\right],N_{15}=\left[\begin{array}{cc}-0.02& 0.02\end{array}\right],\hfill \\ A_2=\left[\begin{array}{cc}1& 0\\ 1& 0\end{array}\right],A_{d2}=\left[\begin{array}{cc}2& 1\\ 1& 2\end{array}\right],B_2=\left[\begin{array}{cc}3& 2\\ 4& 1\end{array}\right],\hfill \\ B_{d2}=\left[\begin{array}{cc}3& 2\\ 4& 1\end{array}\right],C_2=\left[\begin{array}{cc}1& 3\\ 1& 2\end{array}\right],\hfill \\ G_2=\left[\begin{array}{cc}2& 4\\ 1& 2\end{array}\right],G_{d2}=\left[\begin{array}{cc}1& 2\\ 1& 1\end{array}\right],H_2=\left[\begin{array}{cc}2& 1\\ 1& 2\end{array}\right],\hfill \\ H_{d2}=\left[\begin{array}{cc}2& 1\\ 1& 2\end{array}\right],D_2=\left[\begin{array}{cc}2& 1\\ 1& 2\end{array}\right],\hfill \\ {\overline{A}}_2=\left[\begin{array}{cc}1& 0\\ 1& 0\end{array}\right],{\overline{A}}_{d2}=\left[\begin{array}{cc}2& 1\\ 1& 2\end{array}\right],{\overline{B}}_2=\left[\begin{array}{cc}3& 2\\ 4& 1\end{array}\right],\hfill \\ {\overline{B}}_{d2}=\left[\begin{array}{cc}3& 2\\ 4& 1\end{array}\right],{\overline{C}}_2=\left[\begin{array}{cc}1& 3\\ 1& 2\end{array}\right],\hfill \\ {\overline{G}}_2=\left[\begin{array}{cc}2& 4\\ 1& 2\end{array}\right],{\overline{G}}_{d2}=\left[\begin{array}{cc}1& 2\\ 1& 1\end{array}\right],{\overline{H}}_2=\left[\begin{array}{cc}2& 1\\ 1& 2\end{array}\right],\hfill \\ {\overline{H}}_{d2}=\left[\begin{array}{cc}2& 1\\ 1& 2\end{array}\right],{\overline{D}}_2=\left[\begin{array}{cc}2& 1\\ 1& 2\end{array}\right],\hfill \\ L_2=\left[\begin{array}{cc}1& 0\\ 1& 0\end{array}\right],J_2=\left[\begin{array}{cc}2& 1\\ 1& 2\end{array}\right],B_{z2}=\left[\begin{array}{cc}3& 2\\ 4& 1\end{array}\right],\hfill \\ H_{z2}=\left[\begin{array}{cc}3& 2\\ 4& 1\end{array}\right],K_2=\left[\begin{array}{cc}1& 3\\ 3& 1\end{array}\right],\hfill \\ M_{21}=\left[\begin{array}{c}0.01\\ 0.02\end{array}\right],M_{22}=\left[\begin{array}{c}-0.02\\ 0.02\end{array}\right],M_{23}=\left[\begin{array}{c}-0.02\\ 0.01\end{array}\right],\hfill \\ M_{24}=\left[\begin{array}{c}0.01\\ 0.02\end{array}\right],M_{25}=\left[\begin{array}{c}-0.02\\ 0.02\end{array}\right],\hfill \\ N_{21}=\left[\begin{array}{cc}0.01& 0.02\end{array}\right],N_{22}=\left[\begin{array}{cc}-0.02& 0.02\end{array}\right],N_{23}=\left[\begin{array}{cc}-0.02& 0.01\end{array}\right],\hfill \\ N_{24}=\left[\begin{array}{cc}0.01& 0.02\end{array}\right],N_{25}=\left[\begin{array}{cc}-0.02& 0.02\end{array}\right],\hfill \\ {\hat{R}}_1=\left[\begin{array}{cc}1& 0\\ 1& 0\end{array}\right],{\hat{R}}_2=\left[\begin{array}{cc}2& 1\\ 1& 2\end{array}\right],{\hat{S}}_1=\left[\begin{array}{cc}3& 2\\ 4& 1\end{array}\right],{\hat{S}}_2=\left[\begin{array}{cc}3& 2\\ 4& 1\end{array}\right],\hfill \end{array}$$

and the time-varying delay $d_i(t)$ satisfies (2) with $d_1=1$, $h_1=0.2$, $h_2=0.5$. The transition probability matrix is supposed to be

$$M=\left[\begin{array}{cc}-0.5& 0.5\\ 0.3& -0.3\end{array}\right].$$

The filter uncertainties are in the form of (8) with

$$\begin{array}{c}{M}_{16}=\left[\begin{array}{c}0.01\\ 0.02\end{array}\right],M_{17}=\left[\begin{array}{c}-0.02\\ 0.02\end{array}\right],N_{16}=\left[\begin{array}{cc}0.01& 0.02\end{array}\right],\hfill \\ N_{17}=\left[\begin{array}{cc}-0.02& 0.02\end{array}\right],N_{18}=\left[\begin{array}{cc}-0.02& 0.01\end{array}\right],\hfill \\ M_{26}=\left[\begin{array}{c}0.01\\ 0.02\end{array}\right],M_{27}=\left[\begin{array}{c}-0.02\\ 0.02\end{array}\right],N_{26}=\left[\begin{array}{cc}0.01& 0.02\end{array}\right],\hfill \\ N_{27}=\left[\begin{array}{cc}-0.02& 0.02\end{array}\right],N_{28}=\left[\begin{array}{cc}-0.02& 0.01\end{array}\right].\hfill \end{array}$$

Suppose $Q=diag(-1,-1)$, $R=diag(1,1)$, $S=diag(1,1)$. With the above parameters and by using the Matlab LMI Tool box, we solve LMI (20) and obtain

$$\begin{array}{c}{A}_{f1}=\left[\begin{array}{cc}-0.4506& -0.1244\\ -0.1277& -0.5264\end{array}\right],B_{f1}=\left[\begin{array}{cc}1.0000& -0.0000\\ -0.0000& 1.0000\end{array}\right],\hfill \\ C_{f1}=\left[\begin{array}{cc}-0.0818& -0.0242\\ -0.0236& -0.0946\end{array}\right],\hfill \\ L_{f1}=\left[\begin{array}{cc}1.9268& -0.4997\\ -0.2300& 0.0943\end{array}\right],J_{f1}=\left[\begin{array}{cc}1.0139& 0.4922\\ 0.4915& 1.0000\end{array}\right],\hfill \\ E_{f1}=\left[\begin{array}{cc}1.4887& -0.0282\\ -0.1388& 1.4305\end{array}\right],\hfill \\ A_{f2}=\left[\begin{array}{cc}-0.4564& -0.1298\\ -0.1310& -0.5322\end{array}\right],B_{f2}=\left[\begin{array}{cc}1.0000& -0.0000\\ -0.0000& 1.0000\end{array}\right],\hfill \\ C_{f2}=\left[\begin{array}{cc}-0.0825& -0.0245\\ -0.0241& -0.0950\end{array}\right],\hfill \\ L_{f2}=\left[\begin{array}{cc}1.9246& -0.5001\\ -0.2324& 0.0931\end{array}\right],J_{f2}=\left[\begin{array}{cc}1.0153& 0.4942\\ 0.4950& 1.0009\end{array}\right],\hfill \\ E_{f2}=\left[\begin{array}{cc}2.0232& -0.0278\\ -0.1384& 1.9660\end{array}\right].\hfill \end{array}$$

In order to show the advantage of the proposed method, we consider the filtering with $\mathrm{\Delta }{A}_{fi}=0$, $\mathrm{\Delta }{B}_{fi}=0$, $\mathrm{\Delta }{C}_{fi}=0$, $\mathrm{\Delta }{L}_{fi}=0$, $\mathrm{\Delta }{J}_{fi}=0$, $\mathrm{\Delta }{E}_{fi}=0$. And all the other parameters are the same as those above. We can obtain the filter parameters as follows:

$$\begin{array}{c}{A}_{f1}=\left[\begin{array}{cc}-0.4490& -0.1154\\ -0.1189& -0.5210\end{array}\right],B_{f1}=\left[\begin{array}{cc}1.0000& -0.0000\\ -0.0000& 1.0000\end{array}\right],\hfill \\ C_{f1}=\left[\begin{array}{cc}-0.0833& -0.0230\\ -0.0223& -0.0952\end{array}\right],\hfill \\ L_{f1}=\left[\begin{array}{cc}1.9240& -0.4949\\ -0.2145& 0.0921\end{array}\right],J_{f1}=\left[\begin{array}{cc}1.0133& 0.4907\\ 0.4904& 0.9980\end{array}\right],\hfill \\ E_{f1}=\left[\begin{array}{cc}1.4740& -0.0306\\ -0.1412& 1.4153\end{array}\right],\hfill \\ A_{f2}=\left[\begin{array}{cc}-0.4557& -0.1216\\ -0.1227& -0.5275\end{array}\right],B_{f2}=\left[\begin{array}{cc}1.0000& -0.0000\\ -0.0000& 1.0000\end{array}\right],\hfill \\ C_{f2}=\left[\begin{array}{cc}-0.0840& -0.0233\\ -0.0228& -0.0957\end{array}\right],\hfill \\ L_{f2}=\left[\begin{array}{cc}1.9228& -0.4955\\ -0.2194& 0.0912\end{array}\right],J_{f2}=\left[\begin{array}{cc}1.0147& 0.4934\\ 0.4937& 1.0000\end{array}\right],\hfill \\ E_{f2}=\left[\begin{array}{cc}1.4743& -0.0301\\ -0.1408& 1.4158\end{array}\right].\hfill \end{array}$$

When $\mathrm{\Delta }{A}_{fi}=0$, $\mathrm{\Delta }{B}_{fi}=0$, $\mathrm{\Delta }{C}_{fi}=0$, $\mathrm{\Delta }{L}_{fi}=0$, $\mathrm{\Delta }{J}_{fi}=0$, $\mathrm{\Delta }{E}_{fi}=0$, by Theorem 2, we can obtain $d_{1max}$ is $1.0192\times {10}^8$. But in this paper, when $\mathrm{\Delta }{A}_{fi}\ne 0$, $\mathrm{\Delta }{B}_{fi}\ne 0$, $\mathrm{\Delta }{C}_{fi}\ne 0$, $\mathrm{\Delta }{L}_{fi}\ne 0$, $\mathrm{\Delta }{J}_{fi}\ne 0$, $\mathrm{\Delta }{E}_{fi}\ne 0$, we can obtain that ${\hat{d}}_{1max}$ is $1.3732\times {10}^8$. It is easy to get $d_1<{\hat{d}}_1$. Therefore, we can see the advantage of the proposed method.

5 Conclusion

The robust dissipative filtering problem of the stochastic singular systems with Markovian jump parameters influenced by both unknown deterministic and time-varying disturbance inputs and noise is treated in this paper. A numerical example is given to illustrate the usefulness and effectiveness of the proposed design method. The results of this paper can be utilized in several ways, as stand-alone fault-tolerant state estimation techniques, as an essential part of fault-tolerant state feedback controller design, as fault-sensitive filters or as a bank of filters for multiple model fault detection and isolation schemes.