A new result on observer-based sliding mode control design for a class of uncertain It stochastic delay systems
Introduction
It has been widely witnessed that control of stochastic systems has increasingly received much attention as one of the most practically meaningful systems in both academic research and application fields [1]. It is worthwhile pointing out that, in recent years, the study of stochastic delay systems (SDS) has inspired a new wave of research under a very crucial factor that time-delay may frequently occur during the whole operation process, such as chemical processes, networked control systems, etc. Accordingly, a great deal of work has been devoted to the stability and stabilization of It SDS [2], [3]. Meanwhile, the associated control design problems have been explored for the systems in parallel with the development of system control theory, e.g., H∞ control and filtering [4], [5], [6].
Due to its various attractive features such as quick response, good transient performance, particularly, the invariance against matched uncertainties, and wide applications to various complex systems [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], sliding mode control (SMC) [18], [19] has been well known as an effective robust control strategy for uncertain and incompletely modelled systems. It is noted that a growing interest has been devoted to the extension of SMC to accommodate the SDS. Also, it is a fact that uncertain nonlinearity may occur through the system control channels (i.e., the matched uncertainty in SMC theory [18]), due to the variation of the control components and the structural parameters as well as the existence of the inevitably external disturbance, and this will also affect the systems performance directly or indirectly, and even leads to instability. Some representative results regarding the SMC of SDS include SMC of uncertain SDS [14], SMC for uncertain SDS with H∞ performance index [15] to deal with a limitation (i.e., there exists a matrix G with appropriate dimension satisfying for all t ≥ 0); further, robust SMC of uncertain SDS has been considered where such restrictive condition to the most existing results is removed in [16].
It should be noted that most existing results for the SDS are obtained upon the premise that the system states are accessible, despite the efficacy of SMC. In many cases, consider that the state variables may not be totally acquired or even knotty to be estimated via output measurement, the observer-based SMC, also called sliding mode observer (SMO) strategy [20], has been developed and excellently implemented in various cases [21], [22], [23], [24], [25], [26], [27], [28], [29]. In particular, by using the SMO approach, a class of Markovian jump systems against actuator faults with quantized measurements and unknown actuator faults was concerned in [21]. In view of a new observer-based SMC design, a class of nonlinear delay systems was investigated in [25], and recently in [26] robust H∞ control for uncertain singular time-delay systems was studied via a novel SMO synthesis. It is noted that designs of the new sliding surface and/or new-form observers were developed in those works, which may not be extended to stochastic control systems for certain technical reasons directly. To the best of the authors’ knowledge, vast majority of the existing routes for the SMO are that: state observer is designed to generate the original state with assistance of the control input and/or its compensator to restrain the uncertainties and nonlinearity of the system and make the closed-loop systems operate stably. In detail, the design principle leads to that the estimation error system does not contain the control input in general, and then the closed-loop systems can maintain the desirable characteristics on the predesigned sliding surface through the observer and its error system when the associated sliding mode controller is employed. As a result, the achievements using this SMO-idea have been widely applied for SDS [28], [29]. By following along the lines of [25], [26], a new SMO-based scheme is presented in this paper, which may be a worthy addition of the SMO approach for the SDS. The key novelty covers the following:
(a) A particular state observer is designed without any control terms compared with the existing results on SDS.
(b) A novel integral sliding surface design is established on the basis of the new observer and the outputs in such a way that specific sliding mode dynamics (SMDs) of the closed-loop systems is reconstructed.
(c) A sufficient criterion for expected performance of the underlying SMDs is proposed with an easy-to-test LMI framework.
(d) A novel associated reaching motion controller is then synthesized to adaptively ensure the sliding mode phase so as to accommodate the desirable effects of the control strategy.
As such, the proposed scheme is feasible for analysing the stability of the unmeasured system state through the original system itself and its error system. In the other words, an improved procedure is created from the fact that if the stability of the original system and its error system can be ensured, then the observer state can also tend to be stable as it is. All these features distinguish the present scheme from the existing literatures on SDS.
The rest of the paper is organized as follows. Section 2 describes the research problems and preliminaries. Section 3 presents the main results of the new scheme. Section 4 selects a bench test example to demonstrate the results with computational experiments. Section 5 draws brief summary of the study and potential research expansion.
Notations: Throughout the paper and unless specified, let be a completed probability space with a natural filtration where Ω is a sample space, is the σ-algebra of subset of the sample space, and is the probability measure on . is the expectation operator with respect to the probability measure . If A is a vector or matrix, its transpose is denoted by AT, and the symmetric elements of the matrix is denoted by “ * ”. X > Y means that the matrix is positive definite. sym{X} is denoted as sym . If M is a matrix, its operator norm is denoted by λmax(M) and λmin(M) represent its maximum and minimum eigenvalues, respectively. Tr{ · } denotes the trace of a matrix. diag{ · } represents a block-diagonal matrix. stands for the space of square integral vector functions over [0, ∞). Let d > 0 and denote the family of all continuous -valued functions on . Let be the family of all -measurable bounded -valued random variables, and be the family of all -valued -adapted process such that a.s. Let be the family of processes in such that .
Section snippets
System description and preliminaries
Consider the following uncertain It-type stochastic delay systems (SDS) [15], [16] described by
where is the state vector, is the control input, is the system output, and v(t) ∈ Rq represents exogenous disturbance which belongs to . d > 0 is the time-delay, and ω(t) is a standard scalar Brownian motion defined on a completed probability
Main results
This section presents the main results of the SMO enhanced adaptive control of SDS, which includes step by step details of the major analytical development. And the novelty of the developed scheme with comparison of relevant literatures is presented in Remarks 1–3, respectively.
Illustrative example
In this section, a specific bench test example is provided to further demonstrate the performance of the developed scheme in terms of computational experiments.
Example 1
Consider the mathematical model of a water-quality dynamic systems [15], [16] subject to environmental noises and external disturbance with the form in Eq. (1), where x1(t) and x2(t) stand for the concentrations of two main types of pollutant sources, namely algae and ammonia products, respectively; u(t)
Conclusions
The problems of SMO design for uncertain SDS with unmeasured states, nonlinearity and external disturbance have been studied in this paper. The key features of the scheme lie in the design of a particular state observer, integral-type sliding surface and the associated adaptive SMC law for the SDS. By Lemma 1, the easy-to-check LMIs condition has been established to ensure the mean-square exponential stability of the SMDs enforced on the sliding surface. If a non-affine model, e.g., nonlinear
Acknowledgments
The authors would like to thank the editors and anonymous referees for their constructive comments and suggestions. This work was partially supported by the National Natural Science Foundation of China (nos. 61273188, 61374079, 61473097, 61603204, 41306002), the Natural Science Foundation of Shandong Province under grant ZR2017MF055, ZR2016FP03 and the Qingdao Application Basic Research Project under grant 16-5-1-22-jch.
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