The new stability criteria of discrete-time switched systems with an improved mode dependent average dwell time approach
Introduction
Loosely speaking, a switched system is one of hybrid systems in nature, and its compositions are some subsystems and a switching signal. It can model many dynamic systems, and frequently occurs in some fields, for example in biological systems, computer communication network, chemical processes, power electronics [1], neural network [2], etc. Stability analysis of switched systems has received a lot of attention over the past decades, and it has gotten some interesting results, see the literatures [3], [4], [5]. As we know, most of the existing results are based and focused on the continuous-time case, although switched systems with discrete-time subsystems appear frequently in the real world, for example, a piecewise affine system, networks with TCP, and flight systems [6]. In the last decade, discrete-time switched systems (DTSSs) have more drawn considerable interests and attentions in system and control communities [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23].
So far, most stability criteria for DTSSs, just like continuous-time switched systems, fall into two categories, i.e., the stability with the switching signal being free and the stability with the signal being constrained. A common Lyapunov function is a basic method for the former, while the method called multiple Lyapunov functions (MLFs) may obtain some excellent results with the latter. Since slow switching, one of constrained switching, for DTSSs is frequently encountered in practice, the stability problem under slow switching becomes a hot spot. There are three main methods for slow switching, namely dwell time (DT) [8], [9], average dwell time (ADT) [10], [11], [12], [13] and MDADT [14], [15], [16], [17], [18], [19], [20]. These results are conservative to some extent. Most literatures consider the ADT scheme as a special case of the MDADT scheme, but the two are quite different. It should be noticed that the MDADT scheme does not present a more general result than the ADT. Different from ADT, MDADT is dependent on each subsystem, which has two sides: advantages and disadvantages [24]. A new stability scheme compatible with the two is worth further investigation. As far as the authors know, few literatures have been done on this issue.
Motivated by the above literatures, we investigate a more general stability scheme for DSTSs, which is compatible with ADT and MDADT. There are two reasons for us to consider DTSSs with two subsystems. One is that the structure is simple to study. The other is that the results obtained can be easily extended to DTSSs with multiple subsystems under certain conditions. The contribution of this paper lies in two aspects. On the one hand, a more general MDADT scheme is given, which is compatible with the existing schemes. On the other hand, the new stability conclusions with less conservatism have been obtained for DSTSs. In addition, comparing with those exciting results, the advantages and compatibility of the results in the paper are presented. It should be pointed out that, although the new scheme in this paper is based on the premise that each subsystem is stable and time-delay-free, it can easily be extended directly to DSTSs with unstable subsystems and time-delay by some simple modifications.
The brief paper is organized as follows. In Section 2, some preliminaries and problem formulation are given. Section 3 obtains some stability and stabilization criteria of DSTSs. Based on three examples, comparisons between the existing results and ours are presented in Section 4. Finally, we conclude the article in Section 5.
In this paper, denotes the set of real numbers, denotes the set of integers, denotes the set of positive integers. represents the set of n dimensional real column vectors, and stands for the space of n × n matrices with real entries. For ‖x‖ is Euclidean vector norm of x. In addition, ‖A‖ denotes the spectral norm of A, where . C1 denotes the space of continuously differentiable functions. Class κ∞ stands for a unbounded function if it is continuous, strictly increasing, and The notation ∈ ( ∉ ) means “in” (“not in”). ∀ represents “for all”. In the set X with finite elements, max {X} (min {X}) means the maximum (minimum) value of X. In addition, the notation ⇒(⇔) means “imply” (“be equivalent to”) and lim denotes the limit inferior. If A, B is two given sets, then A ∩ B stands for the intersection of A and B. For P > 0 (P ≥ 0) denotes the matrix P being positive definite (positive semidefinite).
Section snippets
Preliminaries and problem formulation
Consider DSTSs where is the system state, denotes the initial state, k0 ≥ 0 stands for the initial time, switching signal is a piecewise constant function.
If Eq. (1) is linear, it will be degenerated into the following form where are two constant matrices of the subsystems. Remark 2.1 There are several reasons for studying a DTSS with only two subsystems. First, a DSTS can model many practical
Main results
The section presents main results of the paper. Although many results about the stability problem for DSTSs have been obtained, they only present some sufficient conditions with some conservatism. The aim of the paper is to get some less conservative results.
Consider the switched system (1). Let ki be the ith switched times of switched signal σ, and supposeNo loss of generality, suppose the first subsystem is activated on
Numerical examples
In the section, three examples are considered to demonstrate the effectiveness of the proposed scheme. Example 4.1 Consider DSTSs (2) and corresponding system matrices described byIt is obvious that both the subsystems are stable. Some comparisons between the existing results and ours are presented in Table 1. From Table 1, the following facts are valid. (1). If the switching signal has MDADT then we have
Conclusion
This paper mainly investigates the new stability criteria for DSTSs. Since those existing results can be regarded as some special forms of ours under certain conditions, the results of the paper have less conservatism and a larger feasible region in practice. It is worth mentioning that the results obtained may easily be extended to other cases, such as discrete-time switched positive systems, DSTSs with delay, and DSTSs with unstable subsystems, etc. It is one of our future research directions
Acknowledgments
This work is supported partially by the Scientifc and Technological Innovation Programs of Higher Education Institutions in Shanxi (2017149).
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