Quantized $H_{\infty }$ filter design for networked control systems with random nonlinearity and sensor saturation

Abstract

This paper investigates the quantized $H_{\infty }$ filtering problem for networked control systems (NCSs) with multiple influencing factors such as randomly occurring nonlinearities and sensor saturation. The dynamic quantization strategy is employed, and stochastic variable is considered satisfying the Bernoulli distribution. The quantized $H_{\infty }$ filter is designed to ensure the exponentially mean-square stable of the system and to guarantee a prescribed $H_{\infty }$ disturbance attenuation level. Finally, a simulation example is given to demonstrate the effectiveness of the filter design method.

Introduction

For a long time, the filtering problem has become the hot point of research in the control science and signal processing fields [1], [2], [3], [4], [5], [6]. It has played a significant role in the engineering applications such as autonomous navigation, vehicle control, aerospace and power electronics. Generally speaking, filtering theory aims to use the measured output of the system and estimate the signals of the internal control system which cannot be measured directly. In the 1960s, Kalman filter has been proposed [2]. However, the requirement of Kalman filtering theory is that all the statistical characteristics of noise must be known. As a result, the application and popularity of Kalman filtering theory are severely limited. In the 1980s,a new filtering theory, $H_{\infty }$ filtering approach, was presented by Elsayed and Grimble in [3]. Different from Kalman filtering theory, $H_{\infty }$ filtering theory only demands that the energy of noise signal is bounded. Therefore, many rich achievements have been displayed on the $H_{\infty }$ filtering problem like [4], [5], [6].

On the other hand, with the development of network technology, the network has been applied in many fields due to its advantages such as low process cost, reliability, sharing of information resources, flexibility and extensible easily. Nevertheless, compared to the conventional control system, quantization, packet-loss and time-delay become some new research problems in the NCSs field like literature [7], [8], [9], [10], [11], [12], [13]. For quantized research, as early as 1998, [7] began to study the problem of adaptive filtering using quantized output measurements and a description of the quantizer was included in the overall input–output model. In [10], the filter is designed with the dynamic quantization strategy consideration and the needed quantizer range is minimized. Lu et al. [11] address the problem of a reset state observer (RSO)-based control (RSOC) for linear systems using quantized measurements. On the basis of [11], [12] adds a Bernoulli processing to reset the value of the observer with a random way.

Note that the above-mentioned quantized filter design methods are based on the assumption that the system is linear. However, in all the practical engineering application, it is hardly possible to realize this assumption. The fact is that the uncertain factors such as the nonlinearity or saturation will often occur and with randomness. So the system performance will be destroyed. Hence, these uncertain factors must be taken into account in the controller or filter design for NCSs [14], [15]. At present, there are a lot of research results about nonlinearities and saturation [16], [17], [18], [19], [20], [21], [22], [23], [24], [25]. Literature [16] has researched the filtering problem for a class of uncertain Markov jump nonlinear systems. Zhang et al. [21] focus on the filter design for non-uniformly sampled nonlinear systems which can be approximated by Takagi–Sugeno fuzzy systems. Dong et al. [25] further researches the robust $H_{\infty }$ filtering issue for Markovian jump systems (MJSs) when random nonlinearities and sensor saturation happen simultaneously. However, for the quantized filter design problem, to the best of the authors׳ knowledge, there has been no work dealing with the problem with the random nonlinearity and saturation consideration simultaneously, which motivates the study of the paper.

The main task of this paper is to investigate the quantized $H_{\infty }$ filtering problem for NCSs with quantization, randomly occurring nonlinearities and sensor saturation consideration simultaneously. The dynamic quantization strategy is employed and the stochastic nonlinearities are considered satisfying the Bernoulli distribution. Moreover, sufficient conditions based on linear matrix inequalities (LMIs) are given to prove the existence of the quantized $H_{\infty }$ filter which can make the system satisfy exponentially mean-square stable and guarantee a defined $H_{\infty }$ disturbance attenuation level. Finally, the effectiveness of the proposed method is illustrated by a simulation example.

The organization of this paper is as follows. Section 2 introduces the problem of $H_{\infty }$ filtering and some preliminaries. Section 3 proposes the method for $H_{\infty }$ performance analysis and $H_{\infty }$ filter design. In Section 4, an example is presented to illustrate the effectiveness of the proposed method. Finally, Section 5 gives some concluding remarks.

Notation: In this paper, $z\in R^k$ is a vector, the 2-norm of z is defined as $\left|z\right|=\sqrt{\left(z^{T}z\right)}$. The notation $Q>0(\ge 0)$ means that Q is real symmetric and positive definite (semidefinite). For a matrix $\Pi \in R^{m\times n},{\lambda }_{\max }\left(\Pi \right),{\lambda }_{\min }\left(\Pi \right)$ and $\parallel \Pi \parallel$ can be defined as the maximum eigenvalue, the minimum eigenvalue and the largest singular value of matrix Π, respectively. The symbol ⁎ within a matrix represents the symmetric part. $\mathit{Prob}\left\{o\right\}$ stands for the occurrence probability of the event “o”. $E\left\{{\sigma }_k\right\}$ is the expectation of the stochastic variable ${\sigma }_k$.