Finite-time dissipative filter design for discrete-time stochastic interval systems with time-varying delays

The finite-time dissipative filtering problem for a kind of discrete-time stochastic interval system with time-varying delays whose parameters are taken in a convex hull is investigated in this paper. Taking a representative subsystem from a stochastic convex hull system, based on convex analysis and matrix theory, a new interval matrix method is proposed to study the finite-time dissipative filter problem, which can deduct the conservativeness. Then, the finite-time dissipative filter is designed by employing a complex Lyapunov-Krasovskii functional together with the improved Wirtinger inequality technique. Correspondingly, some novel sufficient conditions are obtained to ensure the filtering error system with time-varying delays robustly stochastically finite-time bounded and the dissipative index is satisfied. Next, the desired filter gains are achieved in terms of linear matrix inequalities. Finally, the effectiveness of the designed filter is demonstrated by a numerical example with simulations.


Introduction
Strictly speaking, time delays are ubiquitous in control systems. 1,2 As an effective method to characterize systems, time delays have catched many researchers' attention. The research on system with time delays originated in the 1950s, there have existed lots of research results up to now. [3][4][5] Furthermore, discretetime systems, especially discrete-time systems with time-varying delays, are expanded rapidly. [6][7][8] In recent years, some novel skills for the stability analysis of discrete-time systems with time-varying delays have been raised. For example, Zhang et al. 9 have proposed the free-weighting-matrix method. A novel stability condition for discrete-time systems with interval timevarying delays was investigated by employing improved Wirtinger inequality method. 10 The concept of dissipative was first proposed by Willemns. 11 When the supply is greater than the storage for a system, the system is called dissipative system. Dissipative systems are thought to be open systems far from thermodynamic equilibrium state, which explain many previously unexplained phenomena in nature. For linear and nonlinear systems, dissipative theory plays a unique role in the aspect of designing feedback controllers. It is not only a powerful tool for researching economic systems and complex systems but also extensively used in solving the robust control problems of various systems. [12][13][14] Finite-time stability was first proposed, 15 since then lots of researchers begun to study the finite-time control. In 2012, a novel global finite-time stability feedback controller was proposed. 16 By combining the adaptive fuzzy control method with backstepping technique, an adaptive fuzzy finite-time backstepping control method was given. 17 As a matter of fact, the system reaches a dissipative state in a finite-time interval is more practical. The finite-time dissipative problem has faster convergence speed and better robustness. Thus, the finite-time dissipative problem has become increasingly valuable. [18][19][20][21] In practical engineering problems, we have to optimize the estimation of stochastic variables based on the observation process, which is called as the filtering problem. In the past few decades, Wiener filter, Kalman filter, and other filtering methods have been proposed. It is worth noting that Kalman filter is a crucial research object. On one hand, it is always one of the most commonly used ways for state estimation because of its small computation and recursive realtime processing. On the other hand, it can be used in radio, computers, almost all videos or communication equipment and even in many engineering applications like radars and computer vision. [22][23][24] Considering this, many results have been investigated. [25][26][27][28] For example, the time-varying reliable H ' filter design problem for semi-Markov jump systems over a lossy network was studied. 25 The problem of event-triggered filtering for a class of nonlinear cyber-physical systems with deception attacks was considered. 26 As we known, there always exist uncertainties due to errors in systems modeling and changes in operating conditions. One way of describing uncertainty of the system is parameter uncertainty, so the system with interval parameter and stochastic interval parameter has attracted great attention of researchers. [29][30][31] For example, the finite-time stabilization problem of memristor-based inertial neural networks with time-varying delays via interval matrix method has been studied. 29 Inspired by the above analysis, this paper is mainly devoted to research the finite-time dissipative filtering problem for a kind of discrete-time stochastic interval system, which is different from the existing literature. 32,12 The main contributions of this paper are highlighted as follows. First, interval parameters are embedded into the discrete-time stochastic interval system, and the timevarying delays are taken into account in exploring its finite-time dissipative filtering problem. Second, improved Wirtinger inequality is further proposed as a novel summation inequality to reduce the conservatism of finite-time dissipative filtering. Third, there is few results on finite-time dissipative filter design for discrete-time systems by the interval matrix method.
The remainder of this paper is organized as follows. In section 2, the discrete-time stochastic convex hull system with time-varying delays is proposed and transformed to the discrete-time stochastic interval system with time-varying delays. Then, a linear filter is constructed to obtain the filter error stochastic system. Next, the relevant assumptions, definitions, and lemmas are given. Section 3 is the main section, which mainly analyzes the problem of finite-time dissipative filtering, and gives some novel sufficient conditions. Finally, the validity of the above method and the accuracy of the conclusion are illustrated by numerical simulation. A summary is provided in the last section.
Notations: R s stands for s-dimensional Euclidean space; I and 0 denotes the identity matrix and zero matrix with appropriate dimensions; Ã denotes the symmetric elements of symmetric matrix; P . 0 means that P is a positive definite and symmetric matrix; Q À1 denotes the inverse of matrix Q; E stands for mathematical expectation.

Problem formulation and preliminaries
Consider the following discrete-time stochastic convex hull system: where x(k) 2 R n , y(k) 2 R m , and z(k) 2 R q are the state vector, measured output, and control output respectively; v(k) 2 R p is the disturbance input, which belongs to L 2 ½0, ', where L 2 ½0, ' denotes the space of nonanticipatory square-summable stochastic process with respect to (F k ) k . 0. v(k) is a scalar Brownian motion on a complete probability space (O, F, P) with E(v(k)) = 0, E(v 2 (k)) = 1: d(k) is the time-varying delay satisfying 04d 1 4d(k)4d 2 4', where d 1 and d 2 are given nonzero constants. And G i1 , G j1 (i, j = 0, 1), C 0 and C d are given matrix with appropriate dimensions.
Let co½A 0 , A 0 =f(a ij (k)) 2 R n3n ja ij 4a ij (k)4 a ij ,i,j = 1, 2, ÁÁÁ, ng, where the matrix A 0 =(a ij ) n3n and A 0 =( a ij ) n3n are assumed to be known. By applying the theories of literature, 33 H j (i, j=0,d, p,q) in stochastic convex hull system (1), such that Obviously, we have : Similar to the existing literature, 34 by employing interval matrix method, one follows that where F(k) is an unknown time-varying matrix with F T (k)F(k)6I. Remark 1. Noted that the interval matrix method not only overcomes the influence of countless subsystems switching casually in the system (1), but also greatly reduces computational complexity and conservativeness, which makes the results different from the existing ones. [35][36][37][38][39] Denote Obviously, the uncertain parameters (3) can be rewritten as Furthermore, we design the linear filter for (2) as follow: wherex(k) 2 R n ,ẑ(k) 2 R q are the estimates of x(k) and z(k), respectively; A K , B K , and C K are the filter gain matrix to be calculated. Denote j(k) = x T (k)x T (k) Â Ã T and the output error e(k) = z(k) Àẑ(k). The filter error system can be described as: Meanwhile, the admissible conditions with the uncertainties are formulated as: Assumption 1. The disturbance input vector v(k) is assumed to be time-varying and satisfies the following constraint where K . 0 is a given constant.
Definition 1. 40 Given positive constants c 1 , c 2 with c 2 . c 1 , K and a symmetric matrix L, the filter error system (5) is defined to be robustly stochastically finitetime bounded with respect to (c 1 , Definition 2. 35 The filter error system (5) is robustly stochastically finite-time dissipative with respect to (c 1 , c 2 , K, L, d), for given a constant b . 0, matrix S . 0, R . 0 and Q, such that (i) the filter error system (5) is robustly stochastically finite-time bounded with respect to (c 1 , c 2 , K, L, d); (ii) under the zero initial state, the following condition holds Lemma 1. 10 For a symmetric matrix Z . 0 and a variable j 2 ½a, b, where b À a>1, the following inequality holds Remark 2. In this paper, the inequality in Lemma 1 is called the improved Wirtinger inequality. We know that the Wirtinger inequality is applied to the stability analysis of continuous-time systems. However, the improved Wirtinger inequality, which combines with the efficient representation of the improved reciprocal convex combination inequality, devotes to the stability analysis of discrete-time systems. 10 Lemma 2. 41 (Schur complement). Given three matrix Lemma 3. 42 Define that S, D, E are real matrix and F(k) is a matrix function satisfying F T (k)F(k)6I. There exists e . 0, it holds that:

Main results
In this section, based on the complex Lyapunov-Krasovskii functional, improved Wirtinger inequality, some complicated mathematical skills, and linear matrix inequality technique, some novel sufficient conditions are presented to guarantee the robustly stochastically finite-time dissipative for the filter error system (5).

<
: Under conditions (8) and (9), combining (20) and (21) together, it can be concluded that Therefore, from Definition 1, we conclude that the filter error system (5) is robustly stochastically finite-time bounded. Next, sufficient conditions of the robustly stochastically finite-time dissipative for the filter error system (5) are given. Firstly, take into account the performance index as By using the steps of (18), we derive E DVðkÞ À ðg À 1ÞVðkÞ À JðkÞ f g From Schur complement technique, the linear matrix inequality (7) are obtained from (23). Therefore, if (7) What's more, VðkÞ>0ðk ¼ 1; 2; Á Á Á ; KÞ, we have which implies Therefore, by Definition 2, it shows that the filter error system (5) is robustly stochastically finite-time dissipative. The proof is completed.

Remark 3.
As we all know, the stability of the system can be easily determined by choosing a reasonable Lyapunov functional in the stability analysis of the system. In this paper, the Lyapunov functional (11) containing triple summation terms is considered. It is evident that the considered Lyapunov functional increases the computational complexity. Meanwhile, the improved Wirtinger inequality in Lemma 1 is applied in calculating the forward difference and inequality scaling in Theorem 1. In fact, the considered Lyapunov functional in this paper can take fully the advantage of the improved Wirtinger inequality and reduces the conservatism caused by the optimal setting of linear matrix inequality with increasing the upper bound of time delays in control theory together with the improved Wirtinger inequality in the filter design, as discussed in Zhang et al., 4 Chen and Sun. 43 Now, the design of the finite-time dissipative filter gain is presented in the following theorem.

Numerical simulation
A numerical example with simulation results is provided in this section to ensure the effectiveness of the designed filter. Consider the systems (2)  To illustrate the effectiveness of the designed filter, the simulation is carried out. Furthermore, the simulation results are presented in Figures 1 to 6, in which Figures  1 and 2 are the state trajectories x i (k) and their filterŝ x i (k) (i = 1, 2) with different initial conditions. The tracking error state estimate of z(k) Àẑ(k) is shown in Figure 3. It is evident that the error state estimate trends to 0, that is to say, the designed filter is observed to be effective.    Figure 4 gives the trajectory of E j T ðkÞLjðkÞ È É . In Figure 4, when k 2 (0, 80), it means in finite-time, the trajectory of E j T ðkÞLjðkÞ È É is not beyond the provided upper bound c 2 = 30, which illustrates the the filter error system (5) is robustly stochastically finite-time bounded and dissipative. The exogenous disturbance input v(k) is chosen as v T (k) = e À0:06k sin(0:6k) 0 Â Ã , it is clearly that E P 80 k¼0 v T ðkÞvðkÞ n o = 4:1247 \ 4:3, which is verified in Figure 5.
To validate the dissipative performance in the proposed finite-time interval, the discrete-time dissipative performance index can be defined as b(k) = P K k = 0 ½e T (k)Qe(k) + 2e T (k)Sv(k) + v T (k)Rv(k) P K k = 0 v T (k)v(k) : From Figure 6, we can conclude that b(k) is always higher than b Ã = 1:5, which implies (6) holds. It can be seen from the simulation results that the designed filter can effectively satisfy desired tracking performance.

Conclusion
The finite-time dissipative filtering problem for a kind of discrete-time stochastic interval system with timevarying delays whose parameters are taken in a convex hull has been addressed in this paper. With the help of the complex Lyapunov functional and improved Wirtinger inequality technique, the finite-time dissipative filter has also been given. Some new sufficient conditions are obtained to ensure the filter error system (5) robustly stochastically finite-time bounded and dissipative. Finally, a numerical example with simulations has demonstrated the effectiveness of the desired filter.
In the future, it is well worth to further investigate the event-triggered finite-time dissipative filtering problem for a class of Markov switching systems via interval matrix methods. We are also interested in exploring the fault detection filter design problem and the finitetime filtering problem for networked control systems via interval matrix method. Furthermore, the integration of our research with practical applications could be considered.

Declaration of conflicting interests
The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.