Inner-approximating domains of attraction for discrete-time switched systems via Multi-step multiple Lyapunov-like functions

Highlights

• An iterative method to estimate domain of attraction for discrete-time switched systems.

• The method is based on iteratively computing the Multi-step multiple Lyapunov-like functions.

• For removing the redundant constraints, we use the homotopy continuation method to precheck the non-emptiness of the Multi-step state subspaces.

• The iterative framework is implemented via S-procedure and sum of squares programming.

• Our method can to some extent get better inner estimations.

Abstract

In this paper, we propose an iterative approach for estimating the domains of attraction for a class of discrete-time switched systems, where the state space is divided into several disjoint regions and each region is described by polynomial inequalities. At first, we introduce the basic concepts of Multi-step state subsequence, Multi-step state subspace, Multi-step basin of attraction and Multi-step multiple Lyapunov-like function. Secondly, beginning with an initial inner estimation, a theoretical framework is proposed for estimating the domain of attraction by iteratively calculating the Multi-step multiple Lyapunov-like functions. Thirdly, notice that the Multi-step state subspaces may be empty sets such that the corresponding constraints in the theoretical framework are redundant, we propose a numerical approach based on the homotopy continuation method to pre-check the non-emptiness of the Multi-step state subspaces, and then under-approximatively realize the framework by using S-procedure and sum of squares programming. At last, we implement our iterative approach and apply it to three discrete-time switched system examples with comparisons to existing methods in the literatures. These computation and comparison results show the advantages of our method.

Introduction

Switched systems, which consist of several continuous-time or discrete-time subsystems and a rule that orchestrates the switching sequences between these subsystems, are a special class of hybrid dynamical systems [1], [2], [3], [4], [5]. In recent decades, researches on switched systems have been paid considerable attention because switched systems can be applied in many realistic environments which exhibit switching features, such as physical systems, networked control systems, biology systems, multi-agent systems, chemical engineering, chaos synchronization, aircraft control, air traffic control, manufacturing systems [6], [7], [8], [9], [10], [11], [12], [13] and so on.

Among numerous interesting researches on the switched systems, the stability analysis is an essential issue, since it is significant not only in pure theoretical analysis [14], [15], [16] but also in actual applications [7], [17], [18]. And for stability analysis, the investigation on the estimation of the domain of attraction of an asymptotically stable equilibrium point is a critical problem, since estimating the domain of attraction plays an important role in actual control systems analysis and design, e.g., we can take advantage of an estimation of the domain of attraction of a chaotic attractor generated by a switched affine system to design a finite time observer [19]. It also has practical applications such as investigating the resilience of ecological systems [20], designing an optimal strategy for cancer treatment [21], [22], and designing controllers for singularly perturbed switched systems subject to actuator saturation [23], etc.

In general, the actual domain of attraction of an asymptotic stable equilibrium point can be neither analytically represented nor exactly obtained by numerical computation. In recent ten years, for continuous-time dynamical systems and continuous-time switched systems, numerous approximation methods have been proposed in the literatures for estimating the domain of attraction. For example, for the continuous-time dynamical systems, there are methods such as invariance principle method [24], [25], rational Lyapunov functions based method [26], differential inclusions based method [27], Hausdorff distance method [28], forward reachable set based method [29], trajectory reversing method [30], Chebyshev polynomial approximation method [31], interval arithmetic based method [32], [33], sum-of-squares optimization method [34], [35], sampling-based method [36], compositions of Lyapunov functions based method [37] and parameter-dependent Lyapunov-like functions based method [38], etc; for the continuous-time switched systems, there are methods such as common Lyapunov functions based method [39], common Lyapunov-like functions based method [40], multiple Lyapunov-like functions based method [41], etc. However, there are relatively few researches on the domains of attraction for discrete-time dynamical systems and discrete-time switched systems. For the discrete-time dynamical systems, there are methods such as occupation measure based method [42], reverse trajectory based method [43], recursive algebraic representation based method [44], etc; for the discrete-time switched systems, there are methods such as interval analysis method [45], common Lyapunov functions based method [46], [47], and common Lyapunov-like functions based method [48], etc.

Although the above methods have shown their efficiency for inner-estimating the domains of attraction for the discrete-time switched systems, but utilizing the common Lyapunov functions or common Lyapunov-like functions to estimate the domain of attraction seems too conservative in practice such that finding inner estimation more precisely to the exact solution is still a challenge. Therefore, inspired by the concept of Lyapunov-like functions introduced in [49], [50], which can be used to calculate inner estimations of the domains of attraction as large as possible, we, in this paper present a sum of squares programming based iterative approach to estimate the domains of attraction of a class of discrete-time switched systems by iteratively calculating the Multi-step multiple Lyapunov-like functions [51], where the state space is divided into several disjoint regions and each region is described by polynomial inequalities. Note that the notion of multi-step evolutions has been previously proposed for nonlinear digital control systems in [52] to confront some of the sampling processes disadvantages; further, it has been proposed for incrementally stable switched systems in [53] to construct efficient multirate symbolic models while reducing the computational complexity.

Firstly, we introduce the concepts of Multi-step state subsequence, Multi-step state subspace, Multi-step basin of attraction and Multi-step multiple Lyapunov-like function, generalizing the corresponding classical 1-step concepts. Secondly, based on an initial inner estimation, a theoretical iterative framework (i.e., Algorithm 1) is proposed for estimating the domain of attraction by iteratively calculating the Multi-step multiple Lyapunov-like functions. Note that the use of Multi-step multiple Lyapunov-like functions can to some extent get better inner estimations of the domain of attraction than the classical 1-step Lyapunov functions based method. Especially, a higher-order truncation based approach is proposed for obtaining an initial inner estimation of the domain of attraction. However, it can be seen from Example 1 that the Multi-step state subspaces may be empty sets such that the corresponding constraints in the theoretical framework will be always true. The occurrence of these redundant constraints will increase not only the conservativeness of our method but also the computational complexity. Thus, for effectively implementing our theoretical iterative framework, we need to remove these unnecessary constraints, that is we need to check the non-emptiness of the Multi-step state subspaces, which is equivalent to checking the existence of solutions to a set of inequalities. Many symbolic computation methods can be used to solve this problem. But, these methods in general have high computational complexity. Therefore, we thirdly present a numerical method named homotopy continuation method [54] to efficiently solve the set of inequalities, arriving at the verification of the non-emptiness of the Multi-step state subspaces. Then, based on the non-empty Multi-step state subspaces, we under-approximately implement the theoretical framework via S-procedure [55] and sum of squares programming, associated with a coordinate-wise iteration idea [56], [57] (i.e., Algorithm 3). In the end, we use the toolboxes SOSTOOLS [58] and SeDuMi [59] to implement our method and apply it to three examples with comparisons to existing methods in the literatures. These computation and comparison results show the advantages of our method.

Note that the Multi-step basin of attraction is different from the basin of attraction for continuous-time switched systems [41], since the Multi-step basin of attraction need not to be an invariant set (see Fig. 1). Thus, Lemma 1 is also different from the Lemma 1 in [41]. Moreover, an initial inner estimation of the domain of attraction can also be obtained by using the methods introduced in the literatures such as [46] instead of our truncation based approach.

The main contributions of this paper lie in:

• Based on the Multi-step basin of attraction, we present a theoretical framework (i.e., Algorithm 1) to iteratively compute the Multi-step basin of attractions by Multi-step multiple Lyapunov-like functions, arriving at inner estimations of the domains of attraction for discrete-time switched systems.

• For some cases, the Multi-step state subspaces may be empty sets such that the corresponding constraints in the theoretical framework will be always true. Therefore, for removing these redundant constraints, we propose the homotopy continuation method for pre-checking the non-emptiness of the Multi-step state subspaces.

• The theoretical framework is under-approximatively implemented via S-procedure and coordinate-wise iteration idea, arriving at a linear sum of squares programming based iterative algorithm for estimating the domain of attraction. Comparisons with the method in [46] show that our method can to some extent obtain better results.

The structure of the paper is organized as follows. In Section 2, we introduce some preliminaries; In Section 3, based on an initial inner estimation, we present a theoretical framework to obtain larger inner estimations of the domains of attraction by iteratively calculating Multi-step multiple Lyapunov-like functions; In Section 4.1, we introduce the homotopy continuation method for pro-checking the non-emptiness of the Multi-step state subspaces, and then we realize this theoretical framework via S-procedure and sum of squares programming in Section 4.2; In Section 5, our approach is implemented and tested on three discrete-time switched system examples with comparisons. We conclude our paper in Section 6.