Quantized asynchronous extended dissipative observer-based sliding mode control for Markovian jump TS fuzzy systems

Abstract

This article is dedicated to the issue of asynchronous adaptive observer-based sliding mode control for a class of nonlinear stochastic switching systems with Markovian switching. The system under examination is subject to matched uncertainties, external disturbances, and quantized outputs and is described by a TS fuzzy stochastic switching model with a Markovian process. A quantized sliding mode observer is designed, as are two modes-dependent fuzzy switching surfaces for the error and estimated systems, based on a mode dependent logarithmic quantizer. The Lyapunov approach is employed to establish sufficient conditions for sliding mode dynamics to be robust mean square stable with extended dissipativity. Moreover, with the decoupling matrix procedure, a new linear matrix inequality-based criterion is investigated to synthesize the controller and observer gains. The adaptive control technique is used to synthesize asynchronous sliding mode controllers for error and SMO systems, respectively, so as to ensure that the pre-designed sliding surfaces can be reached, and the closed-loop system can perform robustly despite uncertainties and signal quantization error.Finally, simulation results on a one-link arm robot system are provided to show potential applications as well as validate the effectiveness of the proposed scheme.

Introduction

Dynamical systems have often been subjected to random changes in their structures caused by component failures, changes in subsystem interconnections, or environmental changes. The Markovian jump systems (MJSs) are generally adopted to appropriately characterize and model such a class of systems. MJSs have attracted a great deal of works in the past years, see [1], [2], [3], [4], [5], [6], [7], [8]. Nevertheless, although many achievements have been obtained in this research work, the existing results are mainly about linear MJSs, which imposes a considerable limitation in real applications since most practical systems are highly nonlinear. As an alternative way to overcome this limitation, the TS fuzzy-model-based approaches bridge the gap of analysis methods between linear and highly nonlinear MJSs. In this regard, the analysis and synthesis of nonlinear systems have attracted the attention of many researchers in the control field. Due to the strong nonlinear nature of real industrial systems, some nonlinear dynamic systems can be approximated by the overall TS fuzzy model for stability analysis and controller design [9], [10], [11], [12], [13], [14]. Therefore, the application of the TS fuzzy model greatly expands the research field of nonlinear control theory. As a result, rich literature related to controller design, filtering design, and stability analysis on TS MJSs has been published [15], [16], [17].

Quite differently, sliding mode control (SMC) has been considerably employed as a robust control technique for complex engineering systems. Accordingly, SMC drives system trajectories onto a pre-specified sliding surface and forces them to move along it. In addition to its fast response, SMC has the following principal features and advantages: (1) good transient performance; (2) invariance against matched uncertainties; and (3) robustness. In this light, SMC approaches have received a great deal of attention for various classes of systems [18], [19], [20], [21], [22]. Due to the finite sampling rate, an SMC law conceived for continuous-time systems might not be suitable for discrete-time systems as well. Therefore, analyses of discrete-time sliding mode control (DSMC) have contributed to emphasising a new area of research, and some representative works on DSMC have been published in [23], [24], [25].

We can highlight that the results associated with the SMC approaches mentioned above are developed supposing that all the state variables of the system are available for measurement. However, for any reason, the full system states are typically unavailable and an observer should be introduced to estimate the unknown variables. As an attractive approach, the sliding mode observer (SMO) has been recently involved to cope with the state estimation concerns for dirent classes of systems. Accordingly, the SMO has attracted substantial research attention with some outstanding results, which can be found in [26], [27], [28]. To cite a few, the SMO was adopted to synthesize a controller for singular semi-Markovian jumping systems with delay in [29]. Also, the adaptive SMO-based control for non-linear MJSs was studied in [30]. The control problem using SMO for Itô stochastic delay frameworks were analyzed in [31].

As of quite recently, the extended dissipative performance [32] has been introduced to cover both $H_{\infty }$ and $L_2-L_{\infty }$ as well as passivity and $(Q,S,R)-$dissipativity performances in a unified framework. In general, the extended dissipative performance can be reduced to the four above performances by simply choosing different parameter matrices. The generality of extended dissipative performance has led to a large number of related outcomes [33], [34], [35]. It should be pointed out that in engineering systems that employ digital channels for signal communication, signal quantization becomes crucial, especially when the bandwidth and energy are limited. Nevertheless, it may impact the system’s performance when signals are quantized. Thus, it is not surprising that researchers have recently examined the problems of control and filtering using various quantization approaches [4], [36]. For instance, in [37] the author studied quantized non-stationary filtering for networked Markov switching repeated scalar non-linear systems. The paper in [38], discussed the quantification $H_{\infty }$control problem for non-linear stochastic network systems accompanied by probabilistic data missing. In ref [39], the quantized sampled-data control design problem for TS fuzzy networked control system with stochastic cyber-attacks was investigated. A crucial remark to highlight in this study is that the use of networked control systems NCS may lead to transmission delays, stochastic perturbation, and data dropouts which may influence the synchronisation between the system and controller modes in MJSs. To satisfactorily deal with the issue of asynchronous events, many studies have been accomplished based on the hidden Markov model (HMM) [4], [40]. Yet, despite all of this, the asynchronous quantized control problem remains unexplored.

Based on the above bibliographical review, which reveals to us that little progress has been made towards the asynchronous quantized observer-based sliding mode control for nonlinear stochastic switching systems, the following main contributions of this study are outlined:

• Compared with the works in [41], [42], [43], where the observer-based SMC problem for discrete-time nonlinear systems with quantization is considered, the current work is related to the asynchronous sliding mode control issue employing a sliding mode observer with output quantization and asynchronous modes. Moreover, this work investigates the extended dissipativity performance index instead of the $H_{\infty }$index suggested in [41], [42].

• To ensure sliding mode dynamics are stochastically mean square stable and exhibit extended dissipativity [44], sufficient conditions will be derived. For designing controller and observer gains within the LMI framework, we will use a decoupling matrix approach.

• The existing SMC strategies for discrete-time fuzzy systems [45], [46], are based on the assumption that all outputs are available and that the bounds of matched nonlinearities are known. In this study these assumptions are not required, and by implementing the adaptive method, asynchronous SMC laws can be synthesized for systems exhibiting unknown nonlinearities and quantization errors.

The paper is organized as follows. Section 2, describes the research problem and provides some important preliminaries. Section 3 is concerned with the main results that illustrate the design procedure. Section 4 highlights the simulation results via a flexible joint moving by a DC motor to illuminate the significance of the theoretical funding. Section 5 concludes the paper and paths the way for future research.

Table 1 displays the notations and acronyms to be used in this paper.