Stabilization of Discrete-Time Planar Switched Linear Systems with Impulse

We study the stabilization problem of discrete-time planar switched linear systems with impulse. When all subsystems are controllable, based on an explicit estimation on the state transitionmatrix, we establish a sufficient condition such that the switched impulsive system is stabilizable under arbitrary switching signal with given switching frequency. When there exists at least one uncontrollable subsystem, a sufficient condition is also given to guarantee the stabilization of the switched impulsive system under appropriate switching signal.


Introduction
Recent years have witnessed a rapid progress for switched systems, for example, see monographs [1][2][3] and survey papers [4,5].As usual, a switched system means a type of hybrid dynamic system that consists of a family of continuous-time (discrete-time) subsystems and a switching signal, which determines the switching between subsystems.It is well known that switched systems have a deep background in engineering such as computer disk system [6], robotics [7], power systems [8], air traffic management [9], and automated vehicles [10].
During the last three decades, there is an increasing interest on the stability analysis for switched systems.For stability issues; one important problem is to find conditions that guarantee asymptotic stability of the switched system for arbitrary switching signal.Such a problem is usually studied by using a common Lyapunov functional approach, especially by using a common quadratic Lyapunov functional approach [11][12][13][14].A multiple Lyapunov functional method was used to study the stability of switched systems with delays in [15].
In this paper, motivated by the work in [16][17][18], we study the stabilization problem of discrete-time planar switched linear systems under impulse and arbitrary switching signal with given switching frequency.When all subsystems are controllable, we obtain a discrete analogue of the main result in [17].We also consider the case when there exist both controllable subsystems and uncontrollable subsystems.Before giving our main results, we first establish an estimation on the transition matrix for each controllable subsystem, which plays a key role in the stabilization problem of the switched system.For the uncontrollable subsystems, an estimation on the solution is given by using the Lyapunov functional approach.Then, we show that the discretetime switched impulsive system is also stabilizable under appropriate switching signal when there exist uncontrollable subsystems.
This paper is organized as follows.In Section 2, some preliminaries are formulated.The main results of this paper are given in Section 3. Two examples are worked out in Section 4 to illustrate the main results.Section 5 concludes the paper.

Preliminaries
Consider the following planar discrete-time linear system: where  ∈  2 is the state,  ∈   is the controlled input, and  and  are matrices of appropriate dimensions.Under the following linear feedback law: the solution of the system (1) takes the form where ( + )  is called the transition matrix.
When the system (1) is controllable, we first establish an estimation on the transition matrix ( + )  .Lemma 1.Let  ∈  2×2 and  ∈  2× be constant matrices such that the pair (, ) is controllable.Then, for any 0 <  < 1, there exists a matrix  ∈  ×2 such that      ( + )       ≤  −1 ,  ≥ 0, where  > 0 is a constant, which is independent of  and can be estimated precisely in terms of  and .
Proof.First, we consider the case of single input, that is,  = 1.Noting that (, ) is controllable, we can choose a feedback matrix  ∈  1×2 such that eigenvalues   of  +  satisfy In particular, for any 0 <  < 1, we can choose Set where   ,  = 1, 2, are determined by Let We have that   ,  = 1, 2, are also eigenvalues of (  +     ), and (  ,   ) is in controller canonical form.Let It is not difficult to see that that is, It implies that Consequently, Second, noting that  −1 = adj / det , we get from ( 6) that So, substituting ( 15), (17), and ( 18) into ( 14) yields that where  = 8‖‖‖ −1 ‖, which is independent of .Therefore, we have that Lemma 1 holds for the single input case.
For the multiple-input case, one sees that for any  ∈   such that  ̸ = 0, there exists  0 ∈  ×2 such that ( +  0 , ) is itself controllable.Hence, the conclusion of the single-input case that has been proved above is applicable to the controllable pair ( +  0 , ).Therefore, for any 0 <  < 1, there exists  1 ∈  1×2 such that The proof of Lemma 1 is completed.
When (, ) is uncontrollable, for any given feedback , there always exist a positive-definite symmetric matrix  and an appropriate constant  > 0 such that which can be solved by using the GEVP solver in the LMI Toolbox of MATLAB [32].Define the following Lyapunov function: It is easy to see that where  min () and  max () denote the smallest and the largest eigenvalue of the positive definite symmetric matrix .
Lemma 2. For system (1), if the pair (, ) is uncontrollable and (21) holds, then for any given feedback matrix , there exists a constant  > 0 such that Proof.Let the Lyapunov function be defined by (22).Along the solution of system (1), we have By ( 21) and ( 25), we obtain which implies that By ( 23) and ( 27), we have Thus, By induction, we have This completes the proof of Lemma 2.
Under the linear feedback law () =  () () for  ̸ =   ,  = 1, 2 . .., system (31) reduces to the following closed-loop system: Denote the frequency of the switching signal by By the analysis and (H1), we obtain Noting that  < 1 and  > 0, we have that system (31) is stabilizable under arbitrary switching signal with a frequency  < 1.This completes the proof of Theorem 3.
Denote the switching frequency of those controllable subsystems by where Ñ (0, ) is the number of activated controllable subsystems on [0, ].Denote the total activation time for those controllable subsystems on [0, ] by (0, ).In this paper, we assume that there exists a constant  > 0 such that Similar to the above analysis, for any given feedback matrices   (1 ≤  ≤ ), there exist positive definite symmetric matrices   and positive constants   such that Theorem 4. Assume that (H2), (39), and (40) hold.Then, there exist a set of feedback matrices {  } ∈Λ such that the closed-loop system (32) is asymptotically stable for any switching signal  with a frequency f < .

Examples
In order to illustrate the theoretical result, we consider two examples.

Conclusion
In this paper, the stabilization problem of discrete-time planar switched linear systems with impulse is investigated.When all the subsystems are controllable, we first establish an estimation on the transition matrix for each controllable subsystem, which is a discrete analogue of the corresponding result in [17].By using such an estimation, we prove that the discrete-time switched impulsive system is stabilizable under arbitrary switching signal with a given switching frequency.