Observer-based robust control for singular switched fractional order systems subject to actuator saturation

https://doi.org/10.1016/j.amc.2021.126538Get rights and content

Highlights

  • The robust stabilization criteria for singular switched FOS subject to actuator saturation under the designed switching law is proposed.

  • An observer to stabilize the closed-loop singular switched FOS is designed and the observer and controller gains are obtained.

  • The stability region by solving an optimization problem in terms of LMIs is estimated.

  • The presented methods are illustrated by the practical and numerical simulations.

Abstract

The present paper proposes the robust control for singular switched fractional order systems (FOS). The main objective of this paper is to design an observer to stabilize the closed-loop singular switched FOS with actuator saturation under the designed switching law. By using generalized singular value decomposition (SVD) and linear matrix inequalities (LMIs), the observer and controller gains are obtained. Then, based on the proposed stabilization conditions, the stability region by solving an optimization problem is estimated. Finally, the presented methods are illustrated by the practical and numerical simulations.

Introduction

FOS and integer order systems are basically synchronous in theory, which has a history of more than 300 years [1], [2]. Currently, FOS have made great progress in both theoretical research and engineering application [3], [4]. Considering its application in biology, mathematics and materials science, FOS have attracted increasing attention to a growing number of scholars. The study of system stability is the fundamental issue for control theory, including fractional order control theory. A basic theory for the stability of FOS is first proposed in Matignon [5]. On this basis, many scholars further propose numerous methods to solve the issue of the stability with FOS [6], [7], [8]. Among them, it is an effective method to determine the stability of FOS by using LMI technology [9], [10]. The existing LMI criteria are divided into two forms: the fractional order interval is within 0<α<1 and 1<α<2 [11], [12]. For the unstable system, we need to design a reasonable controller to make the system stable. Many ways of the controller design have been considered, among which the successful approaches are iterative LMI algorithm [13] and the method involving matrix SVD [14], [15]. The method involving SVD is more convenient than the algorithms.

Switched system is a kind of hybrid dynamic system which chooses continuous variable partial model according to discrete event mechanism, and is an important system model. Switching systems exist in many practical systems, such as radio communications, chemical process, computer disk drives, machine control systems [16], [17], [18]. Stability is an important characteristic of switched systems. There are many valid methods to study the stability and control problems of switched systems, which mainly include three directions: LMI approach; Lyapunov function; average dwell time approach [19], [20], [21], [22], [23], [24]. In addition, when the derivative of the differential equation of switched systems will become switched FOS [25], [26]. Due to the unique properties of FOS, switched FOS tends to more accurately describe the model in reality. With this, the stability study of switching FOS becomes more complex and challenging [27], [28]. Since the stability of switched FOS does not own the Lyapunov function as the counterpart in switched systems. Many results choose the second best method and obtain the stability condition which is like switched systems. The stability domain of these results is the same as that of integer order systems, which is conservative [29], [30], [31], [32]. The results of this manuscript solve this problem to a certains extent. Moreover, when a switched FOS contain at least one singular subsystem, then it will be called a switched singular FOS [33], [34]. However, to the author’s knowledge, there has been relatively few research to study the admissibility of singular switched FOS. Compared with switched FOS, because of the regularity, impulse of the singular systems, the analysis of the admissibility of singular switched FOS becomes more complex and challenging. In addition, actuator saturation often occurs in singular switched FOS and is often ignored [34], [35], [36]. If the controller designed parameters are beyond the working range of the actual systems, the performance of the closed-loop systems will be degraded [37], [38], [39], [40]. Neverness, in reality,such as circuit systems, neural network systems, robot systems are suitable for this kind of systems, which also actively promotes the present work. Therefore, it is necessary to discuss the issue of the stabilization of the switched FOS subject to actuator saturation.

Based on the above mentioned observations, the issue of observer based robust control of singular switched FOS subject actuator saturation has not been reported yet. This paper presents the following contributions:

  • (i)

    The robust stabilization criteria for singular switched FOS with order 0<α<1 subject to actuator saturation under the designed switching law is proposed, which avoids the skew symmetric matrix variable in Ref. [12].

  • (ii)

    By using generalized SVD and LMIs technology, an observer to stabilize the closed-loop singular switched FOS is designed and the observer and controller gains are obtained. And it is more effective to be calculated.

  • (iii)

    According to the obtained stabilization conditions, we estimate the stability region in terms of LMIs. Estimating the stability region of FOSs is somewhat complicated due to the special nature of fractional order. Some the published articles use algorithms to estimate the stability domain [39], which can lead to some difficulties in computational problems. This paper uses mathematical methods to reduce the calculated volume by transforming the problem into an optimization problem.

Notations: XT represents the transpose of the matrix X. The symbols sym(X) indicates the formula of X+XT. |x|=maxs|xs| for xRn, where xs is the sth row of x. a=sin(π2ν), b=cos(π2ν).

Section snippets

Problem formulation and preliminaries

Consider a singular switched uncertain FOS subject to actuator saturation as follows,EDνx(t)=(Aσ(t)+A)x(t)+Bσ(t)sat(uσ(t)(t)),y(t)=Cx(t),where 0<ν<1 is the fractional order. x(t)Rn and y(t)Rq are the system state and the output vector, separately. σ:[0,+)N={1,2,,m} is a switching signal, and m is the number of subsystems. σ=iN. AiRn×n,BiRn×m,CRq×n are known matrices. A is time-variant matrix with uncertaintiesA=MFN,where M,N are known constant matrices, and F is unknown matrices

Main results

In this section, by using the approach of generalized matrix SVD, sufficient conditions for the stabilization of the switched observer is derived in terms of LMIs.

Theorem 1

If there exist matrices P,Wi,Zi,Yi and real positive scalar ρ1 such that (8) and (11) and the following matrix inequalities hold[Σi=1nαisym(AiP+BiΛkWi+BiΛkZi)Σi=1nαiYiCMPMPΣi=1nαiCTYiΣi=1nαisym(AiP+YiC)+ρ1NNT00PTMT0ρ1I0PTMT00ρ1I]<0,[sym(P)εIzisTzis1]0,s=1,2,,m,where Zi=[Zi000],Hi=[H˜i000],Fi=WiP1,H˜i=ZiP1,Li=YiUS1P111S11U1,

Numerical examples

In this section, a practical electrical circuit example and a numerical example are proposed. In Example 1, a electrical circuit example is given to verify the effectiveness of the stabilization criterion of system (25). In Example 2, a numerical example is obtained to proof the availability of the stabilization criterion of system (6).

By solving Theorem 4.6 in Ref. [42], the best value of t>0 is obtained. The invalid output of MATLAB for the result of Theorem 4.6 in Ref. [42] is shown as

Conclusion

The novel LMI-based determination conditions for singular switched FOS subject to actuator saturation with order 0<α<1 have been completely proposed. By using the method involving generalized matrix SVD, the observer-based robust control law is designed to stabilize the closed-loop systems. Then, a method to estimate the stability region in terms of LMIs is provide by solving optimization problem. In the end, to illustrate that the result proposed available, practical and numerical examples are

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