Adaptive sliding mode control for persistent dwell-time switched nonlinear systems with matched/mismatched uncertainties and its application☆
Introduction
For many practical systems, the phenomena of nonlinear uncertainties have always been inevitable, and control problems for these nonlinear systems with uncertainties have been at the frontier of research in recent decades. To achieve the control objectives, various control approaches have emerged, such as control [1], sliding mode control (SMC) [2], etc. For related literature, please refer to Dong et al. [3], 4], Wang et al. [5], Dong et al. [6], Cheng et al. [7], Wang et al. [8]. However, most of the existing control methods have the limitation that the uncertainty is assumed to have a known upper bound [9], [10], [11]. Nevertheless, obtaining the accurate upper bound of the uncertainty in practice is hard to be implemented, which causes these control methods powerless at this point. Fortunately, because of the ability to handle unknown parameters, the adaptive control method has attracted academic attention, and many excellent achievements have emerged. For example, considering uncertain nonlinear systems, the authors in Li and Liu [12] studied the adaptive output feedback control problem under an event-triggered mechanism. In [13], the authors investigated the adaptive tracking control problem for a constrained stochastic nonlinear system in a finite time sense, where the parameter uncertainty and input saturation were also considered. For more related literature, please refer to Fang et al. [14], Yang et al. [15], Tao et al. [16] and the references therein.
It is worth mentioning that the majority of the aforementioned literature considers non-switched systems, which contradicts some phenomena that actual systems may be affected by various external factors and may exhibit switching characteristics [17], [18], [19], [20], [21], [22]. Moreover, the study of switched systems is of practical relevance considering that switched controllers can easily handle more complex systems compared to single-mode controllers [23]. As far as switched systems are concerned, different switching mechanisms have tremendous impacts on the system performance, and unsuitable switching mechanisms can even render the system unstable [24]. Currently, two types of switching mechanisms are dominant, i.e., stochastic switching containing Markov process and deterministic switching containing dwell-time (DT) [25], [26], [27], [28] and average dwell-time (ADT) [29] switchings. For deterministic switching, DT and ADT are extensively employed, but both switching rules suffer from the problem of restricted flexibility, and their number of switchings in a finite amount of time is somewhat limited. Fortunately, it has been pointed out that persistent dwell-time (PDT) switching is a more general deterministic switching, which can include DT and ADT switchings as its special cases by regulating the corresponding parameters [30], [31]. In essence, PDT switching consists of multiple stages, where each stage consists of a -portion containing fast frequency switching and a -portion containing slow frequency switching. Therefore, PDT switching can be used to describe some systems with both fast and slow switching frequencies. To mention only a few, in literature [32], under a round-robin protocol, the authors studied the state estimation problem of PDT switched coupled networks. For singularly perturbed systems, the authors in Wang et al. [33] introduced the PDT switching signal and investigated the dissipative control problem.
In another promising area, SMC has received much attention from both engineering and academic communities due to its excellent properties such as strong robustness and fast convergence, and related achievements are fruitful [34], [35], [36], [37], [38]. However, most of the existing literature assumes that the nonlinearity has known bounds, which is not general for practical applications. For example, based on the event-triggered mechanism, the authors in Fan and Wang [34] investigated the SMC problem of a multi-input Takagi–Sugeno (T-S) fuzzy system whose uncertainty term was assumed to satisfy with and being known scalars. In [35], the authors studied the issue of SMC for network control systems while packet dropouts were considered, with external perturbations having known bounds by default. In [36], the authors addressed the SMC problem for uncertain systems where mismatched disturbances and communication constraints were also considered, and the nonlinear function was assumed to have a known upper bound . In view of this, the adaptive SMC approach has gradually emerged due to its ability of estimating unknown parameters [39]. In [40], the authors employed the adaptive SMC method to solve the stability problem of Markov jump nonlinear systems, where the case of actuator failure was considered. In [41], the authors studied the adaptive SMC issue for T-S fuzzy systems with mismatched uncertainties and external perturbations. Nonetheless, the associated adaptive SMC problem has not been investigated when involving the PDT switching mechanism, which motivates the research in this paper.
Inspired by the above discussions, this paper determines to study the adaptive SMC problem of continuous-time PDT switched nonlinear systems while considering matched and mismatched uncertainties in the system modeling. The main contributions are exemplified as follows:
- (i)
A more general PDT switching strategy is introduced to describe the switching of system parameters, and both matched and mismatched uncertainties are considered, where the upper bound of the matched uncertainty is unknown.
- (ii)
The matched uncertainty-independent and low-dimensional sliding mode dynamics (LDSMD) is obtained based on a linear sliding surface and a reduced-order method. Then, sufficient conditions are derived by the Lyapunov function and PDT switching analysis techniques guaranteeing that LDSMD is globally uniformly exponentially stable (GUES).
- (iii)
A novel switched adaptive SMC law is constructed, which not only ensures that the system state trajectories can reach the sliding surface but also estimates the unknown upper bound parameters of the matched uncertainty.
The rest of this paper is composed of the following parts. Section 2 is the preliminaries, including the plant description and some other necessary concepts. Section 3 presents the main results of the stability and reachability analysis. In Section 4, two examples are given to perform the validity verification. Section 5 summarizes the full paper.
The notations used in this paper are standard. : the rank of matrix . The others notations can be found in Zhang et al. [42].
Section snippets
Preliminaries
In this section, PDT switching rule, plant description, sliding surface design, and some necessary concepts are given.
Main results
This section is divided into two parts. In the first part, sufficient conditions about LDSMD Eq. (6) being GUES are derived. In the second part, an adaptive switched SMC law is constructed to ensure the reachability of sliding surface Eq. (4).
Simulation examples
For validation of the feasibility of the method employed in this paper, a numerical example and a practical example involving an RLC series circuit are provided. The related parameters for generating a mode sequence that conforms to the PDT switching mechanism are as follows:which are used in two examples, and the sequence diagram is shown in Fig. 2. Example 1 Consider the continuous-time PDT switched system Eq. (1) with , whose parameters are selected as
Conclusion
The problem of adaptive sliding mode control of persistent dwell-time switched nonlinear systems with matched and mismatched uncertainties has been investigated in this paper. Moreover, considering that the upper bound of the matched uncertainty is unknown, it has been estimated by the adaptive control method. Then, the reduced-order sliding mode dynamics has been derived based on the constructed linear sliding surface, followed by sufficient conditions according to the Lyapunov function and
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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This work was supported by the National Natural Science Foundation of China under Grants 61703004, 62173001, the Major Natural Science Foundation of Higher Education Institutions of Anhui Province under grant KJ2020ZD28, the Natural Science Foundation for Excellent Young Scholars of Anhui Province 2108085Y21, the Youth Fundation of Anhui University of Technology under Grant QZ201911.