Partially mode-dependent $H_{\infty }$ filtering for discrete-time Markovian jump systems with partly unknown transition probabilities

Abstract

This paper is concerned with the partially mode-dependent $H_{\infty }$ filtering problem for discrete-time Markovian jump systems with partly unknown transition probabilities via different techniques, where the unknown elements are estimated. New version of bounded real lemma for discrete-time Markovian jump systems with partly unknown transition probabilities is presented. Based on the obtained criterion and via a stochastic variable satisfying Bernoulli random binary distribution, new $H_{\infty }$ filter with partially mode-dependent characterization is established in terms of linear matrix inequalities (LMIs). Finally, numerical examples are given to show the effectiveness of the proposed design method.

Introduction

Filtering is a class of important approaches to estimate the state information when the plant is disturbed, and many approaches were proposed for filter design. Compared with traditional Kalman filtering method such as in [1], [2], [3], [4], the main advantage of $H_{\infty }$ filtering approach is it does not require the exact knowledge of the statics of the external noise signals and is insensitive to the uncertainties both in exogenous signal statics and in dynamic models. Over the past decades, the $H_{\infty }$ filtering problem has attracted considerable attention, and a number of reports was investigated in literature, e.g., [5], [6], [7], [8], [9], [10], [11], [12], [13], [14].

On the other hand, many practical dynamics, e.g., solar thermal central receivers, robotic manipulator systems, aircraft control systems, economic systems, and so on, experience abrupt changes in their structures, whose parameters are caused by phenomena such as component failures or repairs, changes in subsystem interconnections and sudden environmental changes. This class of systems named Markovian jump systems (MJSs) has the advantage of better representing these practical systems with different structures due to random abrupt changes. Many important issues have been investigated for MJSs, see, e.g., [15], [16], [17], [18], [19], [20], [21]. Different from the aforementioned references with ideal assumption that the transition matrix is known exactly, the stability and stabilization problems for MJSs without or with time delays were investigated in [22], [23], while authors in [24] further considered the mode-dependent $H_{\infty }$ filtering problem for discrete-time MJSs (DMJSs). The main technique developed in [22], [23], [24] only considered the accessible transition probabilities via separating known elements from unknown elements, which needed to solve more LMIs. On the other hand, another more realistic $H_{\infty }$ filter design methods without mode observation for continuous-time MJSs were established in [25], [26], where system mode is totally ignored even if the system mode can be obtained sometimes, while the mode-independent robust stabilization for nonlinear MJSs was considered in [27]. In some practical applications, however, the mode information is transmitted through unreliable networks, it may be lost or observed simultaneously. That means the system mode is neither totally accessible nor inaccessible. In this case, the mode-independent filtering design method will be more conservative. Thus, the aforementioned two observations motivate the current research.

In this paper, the $H_{\infty }$ filtering problem for DMJSs with partly unknown transition probabilities is considered, where the desired filter is also partially mode-dependent. Firstly, we restudy the $H_{\infty }$ problem for DMJSs with partly unknown transition probabilities. In contrast to earlier results obtained in [22], [23], [24], another new technique is developed to deal with the unknown elements in transition matrix, where the unknown elements are estimated and less LMIs are needed. Moreover, a new $H_{\infty }$ filtering model referred to as partially mode-dependent filter is presented. This filtering design method bridges two extreme cases, that is, mode-independent and mode-dependent $H_{\infty }$ filtering design schemes. Compared with mode-independent $H_{\infty }$ filter, it is illustrated in this paper that the partially mode-dependent $H_{\infty }$ filter is advantageous for reducing conservatism. It is also advantageous over the mode-dependent filtering method in terms of reducing the burden of the data transmission. That means we can observe the system mode with some probability instead of measuring them online exactly. Examples are demonstrated to show the advantage and efficiency of the proposed approaches.

Notation: ${\mathbb{R}}^n$ denotes the n-dimensional Euclidean space, and ${\mathbb{R}}^{m\times n}$ is the set of all $m\times n$ real matrices. The superscript “T” denotes matrix transposition. $\parallel ·\parallel$ refers to the Euclidean vector norm or spectral matrix norm. ${\mathbb{Z}}^{+}$ represents the set of positive integers. $\mathcal{E}\{·\}$ is the expectation operator with respect to some probability measure. In symmetric block matrices, we use “*” to denote the parts that can be introduced by symmetry, that is,

$$\left[\begin{array}{cc}\hfill L\hfill & \hfill N\hfill \\ \hfill *\hfill & \hfill R\hfill \end{array}\right]=\left[\begin{array}{cc}\hfill L\hfill & \hfill N\hfill \\ \hfill N^T\hfill & \hfill R\hfill \end{array}\right]\text{.}$$