Asynchronously switched control of a class of slowly switched linear systems

Abstract

The stabilization problem for a class of switched linear systems with average dwell time (ADT) switching is reinvestigated in this paper. State-feedback controllers are designed, which takes the more practical case, asynchronous switching, into account, where the so-called “asynchronous switching” indicates that the switchings between the controllers and the system modes are in the presence of a time delay. By combining the asynchronous switching, an improved stabilization approach is given, and existence conditions of the controllers associated with the corresponding ADT switching are formulated in terms of a set of linear matrix inequalities. A numerical example is given to show the validity and potential of the obtained theoretical results.

Introduction

Switched systems belong to a special class of hybrid control systems, which provide a unified framework for mathematical modeling of many practical systems with both continuous dynamics and discrete elements. In recent years, much attention has been devoted to the study of switched systems in both linear and nonlinear contexts [1], [2], [3], [4], [5], [6], [7] due to its many applications in a broad range of areas such as neural networks [8], [9], power electronics [10], and flight control systems [11], to list a few. The systems consist of a family of subsystems, together with a switching signal which specifies the active subsystem at every instant of time [12], [13], [14], [15].

In practice, the switching events in switched systems could be classified into autonomous (uncontrolled) or controlled ones [14], [16], [17], [18], [19], [20]. As pointed out in [21], [22], [23], average dwell time (ADT) switching is a class of controlled switching signals, which can be designed by two parameters, i.e., the increasing coefficient of the Lyapunov function at switching instants and the decay rate of the Lyapunov function during the running time of subsystems. It has been also well known that the ADT scheme characterizes a larger class of stable switching signals than the dwell time (DT) scheme [24], and its extreme case is actually the arbitrary switching. Thus, it is of theoretical and practical significance to investigate control problems for switched systems under ADT switching, although a number of stability results related with ADT switching have been developed [25].

In recent years, another basic problem for switched systems under ADT switching, concerning state-feedback stabilization, has also been widely studied, see for example [26], [27], [28] and the references therein. However, these results are mainly obtained in an ideal scenario that the switchings between the controllers and the system modes are synchronous, which is quite unrealistic. In practice, it takes time to apply the matched controller by detecting the switchings of the controlled plants [29], i.e., the available information of system mode is delayed (see the switching signal channel shown in Fig. 1), which causally introduces the asynchronous switchings between system modes and controller candidates [30]. Thus, it is necessary to consider asynchronous switching for efficient control design. Given the physical presence of asynchronous switching, it is noted that some researchers have focused on the asynchronously switched control for switched systems subjected to both DT switching and ADT switching [25], [31], [32], respectively.

In the presence of asynchronous switching, the switching of controllers to be designed is not consistent with the switching of the controlled plants, therefore, it is somewhat difficult to design the ADT switching by those standard techniques that have been successfully used for general switched systems. To tackle this problem, most authors in the literature utilize a class of Lyapunov-like functions [31], [32] whose energy can rise with a bounded rate for each active mode. Then, while the subsystem is controlled by unmatched controllers, the closed-loop system is allowed to be unstable, i.e., the energy of the engaged Lyapunov-like function rises (but bounded) during this time interval as shown in Fig. 2. In light of the extended Lyapunov-like function, the switching of the closed-loop system can be treated as synchronous switching to the system mode, but the system may be unstable during the unmatched time. However, it is worth mentioning that the results obtained by this method leave room to be improved in two ways. Firstly, the system performance cannot be guaranteed to deteriorate to an unacceptable level during the asynchronous switching time since the closed-loop system is unstable at this moment. Secondly, the minimal ADT should be designed sufficiently large to compensate the energy increment produced during the unmatched time, which is probably still not anticipated.

These naturally give rise to new questions: can the instability of the closed-loop systems be avoided, during the asynchronous switching time? If yes, how to design the controllers and the corresponding ADT switching to ensure the stability for a given switched system?

In this paper, the asynchronously switched control problems for a class of slowly switched linear systems are reinvestigated in both continuous-time and discrete-time contexts. The main contributions of this paper lie in that the asynchronous switchings between system modes and the state-feedback controllers are combined, by which a new design method for the controllers associated with the corresponding ADT switching is given. Benefiting from this new method, the resulting closed-loop systems are designed to be stable during both the matched time and unmatched time. The remainder of the paper is organized as follows. Section 2 reviews necessary definitions and lemmas on the stability analysis of switched systems. In Section 3, the controllers and the corresponding ADT switching are designed, upon which the stabilization conditions are given. A numerical example is presented in Section 4 to demonstrate the feasibility and effectiveness of the proposed techniques.

Notations. In this paper, the notation used is standard. ${\mathbb{R}}^n$ denotes the $n$-dimensional Euclidean space; the notation $\Vert \cdot \Vert$ refers to the Euclidean vector norm. In addition, $\rho \left(A\right)$ stand for the eigenvalues of $A$, $\varrho \left(A\right)$ refer to the real parts of the eigenvalues of $A$, and $ð\left(A\right)$ are the singular values of $A$.