Zonotope-based interval estimation for discrete-time Markovian jump systems with complex transition probabilities and quantization

Abstract

This paper investigates the interval estimation for a class of discrete-time Markovian jump systems with generally uncertain transition probabilities and unknown-but-bounded disturbance based on the zonotopic approach. First, considering the limited communication bandwidth of the networked system, quantization strategy is exploited to quantify the output estimation error. Then, we study the stochastic stability and ${\mathcal{H}}_{\infty }$ performance of the estimation error system with generally uncertain transition probabilities. Based on the obtained results, new criteria are provided to design the mode-dependent quantized observers. Moreover, with the zonotopic estimation approach, an iterative procedure is raised to construct time-varying zonotopes, based on which the state interval estimation is gained for Markovian jump systems. Finally, a numerical example is proposed to verify the correctness and effectiveness of the proposed method.

Introduction

As a kind of hybrid systems, Markovian jump systems (MJSs) have prominent advantages in modeling some practical systems such as networked control systems, communication systems, power systems, economic systems, etc. Therefore, researches on MJSs, such as stability analysis and controller synthesis, have attracted more and more attention in the past few decades [1], [2], [3], [4], [5], [6], [7]. Many existing results on MJSs are based on the assumption that all transition probabilities (TPs) are completely known. However, many practical control systems cannot guarantee such an ideal condition, which limits the application of theoretical methods proposed before. Hence, it is necessary and worthwhile to discuss the MJSs with partially unknown TPs, and related works regarding this topic have been presented in recent years. To mention a few, Zhang and Boukas [8] put forward the concept of partially unknown TPs, and the ${\mathcal{H}}_{\infty }$ control for discrete-time MJSs with partially TPs was addressed in Zhang and Boukas [9]. In [10], the authors focused on disturbance attenuation and rejection of MJSs with incomplete TPs by constructing an attenuation and rejection controller based on the designed disturbance observer. Furthermore, it is hard to obtain accurate TPs since some values vary between their upper and lower bounds. The concept of general uncertain TPs was proposed in Shen and Yang [11] which combines bounded uncertain TPs [12] and completely unkonwn TPs, and a series of studies have been carried out [13], [14], [15], [16], [17], [18]. For instance, Jiang et al. [13] studied the stability and stabilization of a class of singular semi-Markovian jump systems and some sufficient conditions were given in the form of linear matrix inequalities (LMIs). A reduced-order observer was proposed for fault detection taking the time-varying generally uncertain transition rates into account [14].

In the last decades, state estimation has elicited widespread interest [19], [20], [21], [22] since it can be used for fault diagnosis, optimal control, robust controller design and so on. Particularly, as an important way to estimate the system states, the interval observer approach has drawn considerable attention, and there are numerous meaningful results in recent years. In [23], the authors investigated the interval observer for Markovian jump systems with and without time delays, respectively. In [24], interval observers were designed for linear impulsive systems with dwell time constraints, and a robust state feedback controller was designed based on the designed interval observers. The disturbance interval observer was proposed so that the anti-disturbance control integrating interval observer and sliding mode control was constructed [25]. Under the common assumption that the unmeasurable variables, disturbances and noise were unknown but bounded, an interval observer was designed by Garbouj and Dinh [26], and the ${\mathcal{H}}_{\infty }$ approach was used to improve the accuracy of the interval observer. However, interval observer approach requires that the estimation error systems are cooperative and stable by designing two sub-observers, which is not easy to implement. Although coordinate transformation can solve this problem, transformation matrix does not necessarily exist for some systems. Different from interval observer method, zonotopic set-membership estimation approach can obtain more accurate estimation results since the operations on zonotopes can effectively eliminate wrapping effects in the iterative algorithm, and relax the limitation that the system matrix of the error dynamic system must be cooperative. Recently, there is a rapidly increasing interest regarding on the zonotopic state estimation [27], [28], [29], [30]. In [27], Tang et al. pointed out that the zonotopic estimation approach can make an effective trade-off between estimation accuracy and computational burden. Fan et al. [28] applied the derived zonotopic state estimation results to cyber-physical systems influenced by stealthy deception attacks. In [29], the fault detection and fault isolation were solved for T-S fuzzy systems via zonotope-based interval estimation method. In [30], Huang et al. established a zonotopic estimation criterion for switched systems with ${\mathcal{H}}_{\infty }$ performance, and the effectiveness of the proposed method was verified by a boost converter experiment. However, to the best of authors’ knowledge, related topics on zonotope-based interval estimation for discrete-time MJSs have not been fully considered which arouses our research motivation.

Inspired by the above discussions, this paper aims to propose an interval estimation method based on zonotopic approach for a class of discrete-time MJSs with generally uncertain TPs and quantized measurements. Since TPs often cannot be accurately acquired in actual situation, we consider generally uncertain TPs, including completely unknown cases and uncertain but bounded ones. Using zonotopic approach loosens the limitation of the dynamic error system and reduces the conservatism of the results. First, sufficient conditions of the stochastical stability and ${\mathcal{H}}_{\infty }$ performance are established for the estimation error system. Then, a theorem is presented to deal with the quantized observer design. Furthermore, by utilizing the zonotopic approach, state bounding zonotopes are derived at each time, and the interval estimations are determined for MJSs by means of the deduced ${\mathcal{H}}_{\infty }$ observers and time-varying zonotopes. Finally, the merits and effectiveness of the theoretical results are corroborated by a numerical example.

The rest of this paper is organized as follows. Section 2 gives a problem statement and some useful lemmas. The main results of this paper are introduced in Section 3, including the sufficient conditions for the existence of the ${\mathcal{H}}_{\infty }$ observer and the zonotope-based interval estimation method. Section 4, an example is used to illustrate the effectiveness of the proposed method. Section 5 summarizes the work of this paper.

Notation: Throughout this paper, ${\mathbb{R}}^n$ and ${\mathbb{R}}^{n\times m}$ are the $n$-dimensional Euclidean space and the set of all $n\times m$ real matrices, respectively. For a scalar $a$, $\left|a\right|$ denotes its absolute value. For vector $x$, $y\in {\mathbb{R}}^n$, $\parallel x\parallel$ represents its Euclidean norm, and $x\le y$ indicates $x$ is less than or equal to $y$ elementwise. $⌀$ represents the empty set. $P>0(<0)$ indicates $P$ is symmetric positive (negative) definite matrix. $I_n$ denotes the $n$-dimensional identity matrix and $diag\{A_1,A_2,\dots ,A_n\}$ means a block-diagonal matrix composed of $A_1,A_2,\dots ,A_n$. $(\Omega ,\mathcal{F},\mathcal{P})$ ia a probability space. The notation $*$ refers to symmetric term of a symmetric matrix. “$T$” and “$-1$” stand for the transposition and the inverse of a matrix, respectively. ${\mathcal{L}}_2[0,\infty )$ indicates square-summable discrete vector space in$[0,\infty )$. $E\{·\}$ is the expectation operator. A zonotope $\mathcal{Z}=〈p,H〉$ with center $p\in {\mathbb{R}}^n$ and generator matrix $H\in {\mathbb{R}}^{n\times s}$ is a polytopic set described as the linear image of the $s$-dimensional unit interval $B^s$: $\mathcal{Z}=p\oplus H{B}^s=\{z:p+Hx,x\in B^s,p\in {\mathbb{R}}^n\}$. $\oplus$ stands for Minkowski sum, and $\odot$ is the linear image operator. Given the zonotope $\mathcal{Z}=〈p,H〉$, where $p\in {\mathbb{R}}^n$, $H\in {\mathbb{R}}^{n\times s}$, the followings hold:

$$〈p_1,H_1〉\oplus 〈p_2,H_2〉=〈p_2+p_2,\left[H_1{H}_2\right]〉,$$

$$L\odot 〈p,H〉=〈Lp,LH〉,$$

$$〈p,H〉\in 〈p,\overline{H}〉,$$

where $\overline{H}$ is a diagonal matrix whose diagonal elements are ${\overline{H}}_{i,i}={\sum }_{j=1}^s\left|H_{i,j}\right|$.