Event-Triggered Finite-Time H_∞ Filtering for Discrete-Time Nonlinear Stochastic Systems

Aiqing Zhang, Yunyuan Dong

School of Artificial Intelligence, Jianghan University, Wuhan, China.

DOI: 10.4236/jamp.2023.111002                

Abstract

This paper addresses the problem of event-triggered finite-time H_∞ filter design for a class of discrete-time nonlinear stochastic systems with exogenous disturbances. The stochastic Lyapunov-Krasoviskii functional method is adopted to design a filter such that the filtering error system is stochastic finite-time stable (SFTS) and preserves a prescribed performance level according to the pre-defined event-triggered criteria. Based on stochastic differential equations theory, some sufficient conditions for the existence of H_∞ filter are obtained for the suggested system by employing linear matrix inequality technique. Finally, the desired H_∞ filter gain matrices can be expressed in an explicit form.

Keywords

Event-Triggered Scheme, Discrete-Time Nonlinear Stochastic Systems, Stochastic Finite-Time Stable, Linear Matrix Inequalities (LMIS)

1. Introduction

During the past few decades, there has been a rapidly growing interest in nonlinear stochastic systems. Based on the fundamental stochastic stability theory [1] and Lyapunov-Krasovskii functional [2], some results can be found in the literature [3] - [15]. Specifically, problems of stochastic stabilization and destabilization were studied for nonlinear differential equations by noise and impulsive stochastic nonlinear systems respectively in [4] [6] [10] [14]. References [5] [7] [11] [12] [15] investigated state-feedback and output feedback stabilization problems for stochastic nonlinear systems and stochastic delay nonlinear systems. Fault detection filter and full-order H_∞ filter were provided for nonlinear stochastic systems and nonlinear switched stochastic systems in terms of second-order nonlinear Hamilton-Jacobi inequalities and T-S fuzzy framework respectively in references [3] [8]. Dissipativity and tracking control problems were presented for nonlinear stochastic dynamical systems in references [9] [13].

On the other hand, increasing effort has been paid to the study of event-triggered control (ETC) of nonlinear stochastic systems due to their significance in science and engineering applications. Many important results have been presented for event-triggered control of nonlinear stochastic systems in references [16] - [24]. Dynamic event-triggered control, dynamic self-triggered control and event-triggered stability were investigated for a class of nonlinear stochastic systems by introducing an additional internal dynamic variable in [16] [17]. Based on event-triggered predictive control (ETPC) scheme, a novel discrete-time feedback law was designed for the stabilization of continuous-time stochastic systems with output delay in [24]. The input-to-state practically exponential mean-square stability of stochastic nonlinear delay systems with exogenous disturbances was provided and a framework of event-triggered stabilization was received for the stochastic systems without applying the well-known Lyapunov theorem respectively in [18] and [21]. Periodic event-generators and continuous event-enerators were studied in both static and dynamic cases in [19]. Reference [20] addressed the dynamic event-based fault detection problem of nonlinear stochastic systems influenced by random nonlinearity, data transmission delays and packet dropout. Based on fuzzy technique, the problem of event-triggered optimized control for uncertain nonlinear Itô-type stochastic systems with time-delay was addressed in [22]. The modified unscented Kalman filter was proposed for stochastic nonlinear system with Markov packet dropout in [23].

Although the problem of event-triggered control for nonlinear stochastic systems has been investigated, there has little literature on filtering problem for discrete-time nonlinear stochastic systems. With above inspirations, we aim to propose an event-triggered finite-time filtering scheme for discrete-time nonlinear stochastic systems with exogenous disturbance. We present the definition of SFTS into a class of discrete-time nonlinear stochastic systems. By employing the event-triggered strategy, we construct a detection filter such that the resulting filter error augmented system is SFTS. Sufficient conditions for SFTS of the filter error system is established by constructing the Lyapunov-Krasovskii functional candidate combined with LMIs. The desired event-triggered finite-time filter can be constructed by solving a set of LMIs.

This paper is organized as the following. First, some preliminaries and the problem formulation are introduced in Section 2. In Section 3, in terms of event-triggered technique, a sufficient condition for SFTS of the filter error system is established and a method for designing the corresponding filter is presented. Finally, some conclusions are drawn in Section 4.

Notation: Throughout this paper, the notations used are quite standard. We use $R^n$ to denote the n-dimensional Euclidean space. $R>0$ denotes a symmetric positive definite matrix. The symbol * in a matrix denotes a term that is defined by symmetry of the matrix. I and 0 denote the identity and zero matrices with appropriate dimensions. ${\lambda }_{\max }\left(R\right)$ and ${\lambda }_{\min }\left(R\right)$ denote the maximum and the minimum of the eigenvalues of a real symmetric matrix R. The superscript T denotes the transpose for vectors or matrices. $\Xi \left(P\right)$ is the mathematical expectation of P. Matrices, if not explicitly stated, are with compatible dimensions.

2. Problem Formulation and Preliminaries

We shall consider the following discrete-time nonlinear stochastic system:

$\{\begin{array}{l}x\left(k+1\right)=Ax\left(k\right)+f\left(k,x\left(k\right)\right)+D_{1}v\left(k\right)+g\left(k,x\left(k\right)\right)\varpi \left(k\right)\\ y\left(k\right)=Cx\left(k\right)+D_{2}v\left(k\right)\\ z\left(k\right)=Lx\left(k\right),x\left(0\right)=x_0\in R^n\end{array}$ (1)

where $x\left(k\right)\in R^n$, $y\left(k\right)\in R^m$, $v\left(k\right)\in R^p$, $z\left(k\right)\in R^q$, are state vector, measurement output, external disturbance, and controlled output respectively, $\varpi \left(k\right)$ is a one-dimensional zero-mean process which satisfies

$\Xi \left[\omega \left(k\right)\right]=0,\Xi \left[\omega \left(i\right)\omega \left(j\right)\right]=0,i\ne j,\Xi \left[{\omega }^2\left(k\right)\right]=\delta$ (2)

where $\Xi$ is the expected value. Here $\delta >0$ is a known scalar. The matrices $A,C,D_1{D}_2,L$ are constant matrices with appropriate dimensions.

Assumption 1 The nonlinear functions $f\left(k,x\left(k\right)\right)$ and $g\left(k,x\left(k\right)\right)$ satisfy the following quadratic inequalities:

${\left|f\left(k,x\left(k\right)\right)-f\left(k,\tilde{x}\left(k\right)\right)\right|}^2\le {\epsilon }_1^2{\left|x\left(k\right)-\tilde{x}\left(k\right)\right|}^2$

${\left|g\left(k,x\left(k\right)\right)-g\left(k,\tilde{x}\left(k\right)\right)\right|}^2\le {\epsilon }_2^2{\left|x\left(k\right)-\tilde{x}\left(k\right)\right|}^2$

for all $x\left(k\right),\tilde{x}\left(k\right)\in R^n$, where ${\epsilon }_1,{\epsilon }_2>0$ are constants related to the function $f\left(k,x\left(k\right)\right)$, $g\left(k,x\left(k\right)\right)$.

Assume that ${\left\{t_k\right\}}_{k\in N}$ denotes the triggered instants and there is no time-delay in sampler and actuator, $t_0<t_1<t_2<\cdots <t_k<t_{k+1}$, $t_k\le k<t_{k+1}$. $x\left(t_k\right)$ is the current sampled system state, $t_{k+1}$ is the next sampled instant, which can be determined by the event-trigger, and $x\left(t_0\right)=x_0$ is chosen as the initial sampled state.

In this paper, the event-triggering schemes are described by

$t_{k+1}=\inf \left\{k\ge t_k|e_{y\left(k\right)}^{\text{T}}Q{e}_{y\left(k\right)}-\eta y^{\text{T}}\left(k\right)Qy\left(k\right)>0\right\}$, (3)

where $e_{y\left(k\right)}=y\left(k\right)-y\left(t_k\right)$, $\eta$ is a constant and $Q={\Pi }^{\text{T}}\Pi$ is a symmetric and positive definite matrix with appropriate dimension to be determined.

We now consider the following filter:

$\{\begin{array}{l}\hat{x}\left(k+1\right)=A_f\hat{x}\left(k\right)+B_{f}y\left(t_k\right)\\ \hat{z}\left(k\right)=L_f\hat{x}\left(k\right)\end{array}$ (4)

where $\hat{x}\left(k\right)\in R^n$ is the filter state, and matrices $A_f,B_f,L_f$ are filter parameters with compatible dimensions to be determined.

Define $\bar{x}\left(k\right)=\left[\begin{array}{cc}{x}^{\text{T}}\left(k\right)& {\hat{x}}^{\text{T}}\left(k\right)\end{array}\right]$, $\bar{z}\left(k\right)=z\left(k\right)-\hat{z}\left(k\right)$. Then the filtering error system is

$\{\begin{array}{l}\bar{x}\left(k+1\right)=\bar{A}\bar{x}\left(k\right)+\bar{F}\left(k,x\left(k\right)\right)+\bar{D}\bar{v}\left(k\right)-{\bar{B}}_{f}K{\bar{e}}_{y\left(k\right)}+\bar{G}\left(k,x\left(k\right)\right)\varpi \left(k\right)\\ \bar{z}\left(k\right)=\bar{L}\bar{x}\left(k\right)\end{array}$ (5)

where

$\bar{A}=\left[\begin{array}{cc}A& 0\\ B_{f}C& A_f\end{array}\right]$, $\bar{F}\left(k,x\left(k\right)\right)=\left[\begin{array}{c}f\left(k,x\left(k\right)\right)\\ 0\end{array}\right]$, $\bar{G}\left(k,x\left(k\right)\right)=\left[\begin{array}{c}g\left(k,x\left(k\right)\right)\\ 0\end{array}\right]$,

$\bar{D}=\left[\begin{array}{cc}{D}_1& 0\\ B_f{D}_2& 0\end{array}\right]$, ${\bar{B}}_f=\left[\begin{array}{c}0\\ B_f\end{array}\right]$, $\bar{L}=\left[\begin{array}{cc}L& -L_f\end{array}\right]$, ${\bar{e}}_{y\left(k\right)}=\left[\begin{array}{c}{e}_{y\left(k\right)}\\ 0\end{array}\right]$,

$\bar{v}\left(k\right)=\left[\begin{array}{c}v\left(k\right)\\ 0\end{array}\right]$, $K=\left[\begin{array}{cc}I& 0\end{array}\right]$.

Before providing the main results, we summarize several needed definitions and lemmas from the literature.

Definition 2.1 The filtering error system (5) with event-triggered scheme (3) and $v\left(k\right)=0$ is said to be stochastic finite-time stable (SFTS) with respect to $\left(c_1,c_2,P,N\right)$, where $P>0,0<c_1<c_2$, if the following relation holds:

$\Xi \left[x^{\text{T}}\left(0\right)Px\left(0\right)\right]<c_1\Rightarrow \Xi \left[x^{\text{T}}\left(k\right)Px\left(k\right)\right]<c_2$ for all $k\in 1,2,\cdots ,N$.

Definition 2.2 For $\gamma >0$, suppose the event-triggered residual system in (5) is stochastic finite-time stable (SFTS) with respect to $\left(c_1,c_2,P,N\right)$, then system (5) is said to have a weighted H_∞ attenuation level γ for all nonzero $v\left(k\right)\in l_2\left[0,\infty \right)$, if the following inequality holds:

$\Xi \left\{{\displaystyle {\sum }_{k=k_0}^{\infty }{\bar{z}}^{\text{T}}\left(k\right)\bar{z}\left(k\right)}\right\}<{\gamma }^2{\displaystyle {\sum }_{k=k_0}^{\infty }{\bar{v}}^{\text{T}}\left(k\right)\bar{v}\left(k\right)}$ (6)

Lemma 2.1 ( [25]). Let $\Omega \in R^{n\times n}$ be a symmetric matrix, and let $x\in R^n$, then the following inequality holds

${\lambda }_{\min }\left(\Omega \right)x^{\text{T}}x\le x^{\text{T}}\Omega x\le {\lambda }_{\max }\left(\Omega \right)x^{\text{T}}x$. (7)

Lemma 2.2 (Schur complement [26] [27]) Given a symmetric matrix $\varphi =\left[\begin{array}{cc}{\varphi }_{11}& {\varphi }_{12}\\ {\varphi }_{21}& {\varphi }_{22}\end{array}\right]$, the following three conditions are equivalent to each other:

1) $\varphi <0$ ;

2) ${\varphi }_{11}<0$, and ${\varphi }_{22}-{\varphi }_{12}^{\text{T}}{\varphi }_{11}^{-1}{\varphi }_{12}<0$ ;

3) ${\varphi }_{22}<0$, and ${\varphi }_{11}-{\varphi }_{12}{\varphi }_{22}^{-1}{\varphi }_{12}^{\text{T}}<0$.

3. Main Results

In this section, we focus on stochastic finite-time stable (SFTS) with respect to $\left(c_1,c_2,P,N\right)$ of the event-triggered residual system in (5), and propose sufficient conditions of system performance analysis.

Theorem 3.1 For given constants $\gamma ,\eta ,\delta >0$ and $\mu >1$, suppose that there exist symmetric positive definite matrices $R=P^{\frac{1}{2}}\Omega P^{\frac{1}{2}}$ such that the following LMIs hold

$\Theta <0$ (8)

where

$\Theta =\left[\begin{array}{cccccc}{\Theta }_{11}& {\Theta }_{12}& {\Theta }_{13}& {\Theta }_{14}& {\Theta }_{15}& {\Theta }_{16}\\ *& {\Theta }_{22}& {\Theta }_{23}& 0& 0& 0\\ *& *& -{\gamma }^{2}I& 0& 0& 0\\ *& *& *& -{\eta }^{\frac{1}{2}}I& 0& 0\\ *& *& *& *& -I& 0\\ *& *& *& *& *& -{\Omega }^{-1}\end{array}\right]$

$\frac{{\mu }^{N-k_0}{\lambda }_{\max }\left(\Omega \right)c_1}{{\lambda }_{\min }\left(\Omega \right)}\le c_2$,

${\Theta }_{11}={\bar{A}}^{\text{T}}R\bar{A}+{\bar{A}}^{\text{T}}R{\Psi }_1+{\Psi }_1^{\text{T}}R\bar{A}+{\Psi }_1^{\text{T}}R{\Psi }_1+\delta {\Psi }_2^{\text{T}}R{\Psi }_2-\mu R$,

${\Theta }_{12}=-{\bar{A}}^{\text{T}}R{\bar{B}}_{f}K-{\Psi }_1^{\text{T}}R{\bar{B}}_{f}K$, ${\Theta }_{13}={\bar{A}}^{\text{T}}R\bar{D}+{\Psi }_1^{\text{T}}R\bar{D}$, ${\Psi }_i=\left[\begin{array}{cc}{\epsilon }_i& 0\\ 0& 0\end{array}\right],i=1,2$,

${\Theta }_{14}={\left[\begin{array}{ccc}\bar{\Pi }\bar{C}& 0& \bar{\Pi }{\bar{D}}_2\end{array}\right]}^{\text{T}}$, ${\Theta }_{15}={\left[\begin{array}{ccc}\bar{L}& 0& 0\end{array}\right]}^{\text{T}}$, ${\Theta }_{15}={\left[\begin{array}{ccc}0& 0& P^{\frac{1}{2}}\bar{D}\end{array}\right]}^{\text{T}}$,

${\Theta }_{22}=K^{\text{T}}{\bar{B}}_f^{\text{T}}R{\bar{B}}_{f}K$, ${\Theta }_{23}=-K^{\text{T}}{\bar{B}}_f^{\text{T}}R\bar{D}$, $\bar{Q}=\left[\begin{array}{cc}Q& 0\\ 0& 0\end{array}\right]={\bar{\Pi }}^{\text{T}}\bar{\Pi }$,

${\bar{Q}}_1={\bar{C}}^{\text{T}}{\bar{\Pi }}^{\text{T}}\bar{\Pi }\bar{C}$, ${\bar{Q}}_2={\bar{C}}^{\text{T}}{\bar{\Pi }}^{\text{T}}\bar{\Pi }{\bar{D}}_2$, ${\bar{Q}}_3={\bar{D}}_2^{\text{T}}{\bar{\Pi }}^{\text{T}}\bar{\Pi }{\bar{D}}_2$,

$\bar{\Pi }=\left[\begin{array}{cc}\Pi & 0\end{array}\right]$, $\bar{C}=\left[\begin{array}{cc}C& 0\end{array}\right]$, ${\bar{D}}_2=\left[\begin{array}{cc}{D}_2& 0\end{array}\right]$.

Proof: Consider the following Lyapunov function candidate for system (5):

$V\left(\bar{x}\left(k\right)\right)={\bar{x}}^{\text{T}}\left(k\right)R\bar{x}\left(k\right)$ (9)

Then, based on assumption 1, (8) and Schur complement, it follows that

$\begin{array}{l}\Gamma \left(k\right)\triangleq \Xi \left[V\left(\bar{x}\left(k+1\right)\right)-\mu V\left(\bar{x}\left(k\right)\right)\right]\\ ={\left[\bar{A}\bar{x}\left(k\right)+\bar{F}\left(k,x\left(k\right)\right)-{\bar{B}}_{f}K{\bar{e}}_{y\left(k\right)}+\bar{G}\left(k,x\left(k\right)\right)\varpi \left(k\right)\right]}^{\text{T}}R[\bar{A}\bar{x}\left(k\right)\\ +\bar{F}\left(k,x\left(k\right)\right)-{\bar{B}}_{f}K{\bar{e}}_{y\left(k\right)}+\bar{G}\left(k,x\left(k\right)\right)\varpi \left(k\right)]-\mu {\bar{x}}^{\text{T}}\left(k\right)R\bar{x}\left(k\right)\\ =\left[\begin{array}{cc}{\bar{x}}^{\text{T}}\left(k\right)& {\bar{e}}_{y\left(k\right)}^{\text{T}}\end{array}\right]\left[\begin{array}{cc}{\Gamma }_{11}& {\Gamma }_{12}\\ *& {\Gamma }_{22}\end{array}\right]\left[\begin{array}{c}\bar{x}\left(k\right)\\ {\bar{e}}_{y\left(k\right)}\end{array}\right]<0\end{array}$ (10)

where ${\Gamma }_{11}={\Theta }_{11}$, ${\Gamma }_{12}={\Theta }_{12}$, ${\Gamma }_{22}={\Theta }_{22}$.

Then for $\forall k\in \left[t_k,t_{k+1}\right)$, we have

$\Xi \left[V\left(\bar{x}\left(k\right)\right)\right]\le \text{Ξ}\left[\mu V\left(\bar{x}\left(k-1\right)\right)\right]$. (11)

Proceeding in an iterative fashion, we obtain the following inequality:

$\Xi \left[V\left(\bar{x}\left(k\right)\right)\right]<{\mu }^{k-k_0}\Xi \left[V\left(\bar{x}\left(k_0\right)\right)\right]={\mu }^{k-k_0}\Xi \left[V\left(\bar{x}\left(0\right)\right)\right]\le {\mu }^{N-k_0}{\lambda }_{\max }\left(\Omega \right)c_1$ (12)

On the other hand, it can be derived from (9) and lemma 2.1 that

$\Xi \left[V\left(\bar{x}\left(k\right)\right)\right]\ge {\lambda }_{\min }\left(\Omega \right)\Xi \left[{\bar{x}}^{\text{T}}\left(k\right)P\bar{x}\left(k\right)\right]$. (13)

Thus we have that

$\Xi \left[{\bar{x}}^{\text{T}}\left(k\right)P\bar{x}\left(k\right)\right]\le \frac{{\mu }^{N-k_0}{\lambda }_{\max }\left(\Omega \right)c_1}{{\lambda }_{\min }\left(\Omega \right)}\le c_2$. (14)

According to Definition 2.1, the filter error systems (5) with $v\left(k\right)=0$ is SFTS.

Next, we prove the event-based residual system in (5) satisfies H_∞ performance value.

In view of event condition (3) (8), together with Lemma 2.2, the following inequality can be deduced:

$\begin{array}{c}\mathcal{T}\left(k\right)\triangleq \Xi [V\left(\bar{x}\left(k+1\right)\right)-\mu V\left(\bar{x}\left(k\right)\right)+\eta y^{\text{T}}\left(k\right)Qy\left(k\right)-e_{y\left(k\right)}^{\text{T}}Q{e}_{y\left(k\right)}\\ +{\bar{z}}^{\text{T}}\left(k\right)\bar{z}\left(k\right)-{\gamma }^2{\bar{v}}^{\text{T}}\left(k\right)\bar{v}\left(k\right)]\\ ={\left[\bar{A}\bar{x}\left(k\right)+\bar{F}\left(k,x\left(k\right)\right)+\bar{D}\bar{v}\left(k\right)-{\bar{B}}_{f}K{\bar{e}}_{y\left(k\right)}+\bar{G}\left(k,x\left(k\right)\right)\varpi \left(k\right)\right]}^{\text{T}}\\ \times R[\bar{A}\bar{x}\left(k\right)+\bar{F}\left(k,x\left(k\right)\right)+\bar{D}\bar{v}\left(k\right)-{\bar{B}}_{f}K{\bar{e}}_{y\left(k\right)}+\bar{G}\left(k,x\left(k\right)\right)\varpi \left(k\right)]\end{array}$

$\begin{array}{c}-\mu {\bar{x}}^{\text{T}}\left(k\right)R\bar{x}\left(k\right)+\eta {\left[Cx\left(k\right)+D_{2}v\left(k\right)\right]}^{\text{T}}R\left[Cx\left(k\right)+D_{2}v\left(k\right)\right]\\ -e_{y\left(k\right)}^{\text{T}}Q{e}_{y\left(k\right)}+{\bar{z}}^{\text{T}}\left(k\right)\bar{z}\left(k\right)-{\gamma }^2{\bar{v}}^{\text{T}}\left(k\right)\bar{v}\left(k\right)\\ =\left[\begin{array}{ccc}{\bar{x}}^{\text{T}}\left(k\right)& {\bar{e}}_{y(k)}^{\text{T}}& {\bar{v}}^{\text{T}}\left(k\right)\end{array}\right]\left[\begin{array}{ccc}{T}_{11}& T_{12}& T_{13}\\ *& T_{22}& T_{23}\\ *& *& T_{33}\end{array}\right]\left[\begin{array}{c}\bar{x}\left(k\right)\\ {\bar{e}}_{y\left(k\right)}\\ \bar{v}\left(k\right)\end{array}\right]<0\end{array}$ (15)

where

$T_{11}={\bar{A}}^{\text{T}}R\bar{A}+{\bar{A}}^{\text{T}}R{\Psi }_1+{\Psi }_1^{\text{T}}R\bar{A}+{\Psi }_1^{\text{T}}R{\Psi }_1+\delta {\Psi }_2^{\text{T}}R{\Psi }_2-\mu R+\eta {\bar{Q}}_1+{\bar{L}}^{\text{T}}\bar{L}$,

$T_{12}=-{\bar{A}}^{\text{T}}R{\bar{B}}_{f}K-{\Psi }_1^{\text{T}}R{\bar{B}}_{f}K$, $T_{13}={\bar{A}}^{\text{T}}R\bar{D}+{\Psi }_1^{\text{T}}R\bar{D}+\eta {\bar{Q}}_2$,

$T_{22}=K^{\text{T}}{\bar{B}}_f^{\text{T}}R{\bar{B}}_{f}K-\bar{Q}$, $T_{23}={\Theta }_{23}$, $T_{33}={\bar{D}}^{\text{T}}R\bar{D}+\eta {\bar{Q}}_3-{\gamma }^{2}I$,

$\bar{Q}=\left[\begin{array}{cc}Q& 0\\ 0& 0\end{array}\right]$, ${\bar{Q}}_1=\left[\begin{array}{cc}{C}^{\text{T}}QC& 0\\ 0& 0\end{array}\right]$, ${\bar{Q}}_2=\left[\begin{array}{cc}{C}^{\text{T}}Q{D}_2& 0\\ 0& 0\end{array}\right]$, ${\bar{Q}}_3=\left[\begin{array}{cc}{D}_2^{\text{T}}Q{D}_2& 0\\ 0& 0\end{array}\right]$.

Then we can conclude that (6) holds. It can be concluded that the event-triggered residual system in (5) possesses a prescribed H_∞ performance index proposed in Definition 2.2. Thus the proof is completed.

The following theorem will set forth our filter design method for the system (1).

Theorem 3.2 For given constants $\gamma ,\eta ,\delta >0$ and $\mu >1$, the filtering error system (5) with the event-triggering strategy (3) is SFTS with respect to $\left(c_1,c_2,P,N\right)$ and the error signal satisfies (6), if there exist positive definite matrix

$R=\left[\begin{array}{cc}{R}_{11}& R_{12}\\ R_{12}^T& R_{22}\end{array}\right]$ and matrices $G_1,G_2,G_3$ with appropriate dimensions satisfying:

$\tilde{\Theta }<0$, (16)

where

$\tilde{\Theta }=\left[\begin{array}{cccccc}{\tilde{\Theta }}_{11}& {\tilde{\Theta }}_{12}& {\tilde{\Theta }}_{13}& {\tilde{\Theta }}_{14}& {\tilde{\Theta }}_{15}& {\tilde{\Theta }}_{16}\\ *& {\tilde{\Theta }}_{22}& {\tilde{\Theta }}_{23}& 0& 0& 0\\ *& *& -{\gamma }^{2}I& 0& 0& 0\\ *& *& *& -{\eta }^{\frac{1}{2}}I& 0& 0\\ *& *& *& *& -I& 0\\ *& *& *& *& *& -P^{\frac{1}{2}}{R}^{-1}{P}^{\frac{1}{2}}\end{array}\right]$, (17)

${\tilde{\Theta }}_{11}={\tilde{A}}^{\text{T}}R\tilde{A}+{\tilde{A}}^{\text{T}}R{\Psi }_1+{\Psi }_1^{\text{T}}R\tilde{A}+{\Psi }_1^{\text{T}}R{\Psi }_1+\delta {\Psi }_2^{\text{T}}R{\Psi }_2-\mu R$,

${\tilde{\Theta }}_{12}=-{\tilde{A}}^{\text{T}}R{\tilde{B}}_{f}K-{\Psi }_1^{\text{T}}R{\tilde{B}}_{f}K$, ${\tilde{\Theta }}_{13}={\tilde{A}}^{\text{T}}R\bar{D}+{\Psi }_1^{\text{T}}R\bar{D}$, ${\tilde{\Theta }}_{14}={\Theta }_{14}$,

${\tilde{\Theta }}_{15}=\left[\begin{array}{ccc}\tilde{L}& 0& 0\end{array}\right]$, ${\tilde{\Theta }}_{16}={\Theta }_{16}$, ${\tilde{\Theta }}_{22}={\tilde{B}}_f^{\text{T}}R{\tilde{B}}_{f}K$,

${\tilde{\Theta }}_{23}=-K^{\text{T}}{\tilde{B}}_f^{\text{T}}R\bar{D}\tilde{A}=\left[\begin{array}{cc}A& 0\\ R_{22}^{-1}{G}_{2}C& R_{12}^{-1}{G}_1\end{array}\right]$,

${\tilde{B}}_f=\left[\begin{array}{c}0\\ R_{22}^{-1}{G}_2\end{array}\right]$, $\tilde{L}=\left[\begin{array}{cc}L& -G_3\end{array}\right]$.

Moreover, the suitable filter parameters $A_f,B_f,L_f$ in system (4) can be given by

$A_f=R_{12}^{-1}{G}_1,B_f=R_{22}^{-1}{G}_2,L_f=G_3$. (18)

Proof By Theorem 3.1, let $A_f=R_{12}^{-1}{G}_1,B_f=R_{22}^{-1}{G}_2,L_f=G_3$, then the condition (8) is equivalent to (16).

4. Conclusion

In this paper, we have introduced the concept of SFTS into a class of discrete-time nonlinear stochastic systems with exogenous disturbances. We have addressed the event-triggered finite-time filter designing problem. A sufficient condition is provided to guarantee the SFTS of the filter error system. For the presented event-triggering schemes, the criteria for the event-based filter residual systems with a prescribed performance level γ were established by adopting Lyapunov-Krasovski function method. Sufficient conditions for H_∞ performance analysis and corresponding filter designing technique have been provided in a given finite-time interval in terms of LMIs technique, respectively.

Conflicts of Interest

The authors declare no conflicts of interest regarding the publication of this paper.

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