Stabilization of Discrete-Time Planar Switched Linear Systems with Impulse

Abstract

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 transition matrix, 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.

1. Introduction

Recent years have witnessed a rapid progress for switched systems, for example, see monographs [1–3] 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 [11–14]. A multiple Lyapunov functional method was used to study the stability of switched systems with delays in [15].

For systems that switch among a finite set of controllable linear systems, the stabilization problem of continuous-time switched systems with arbitrary switching frequency was studied in [16, 17] by developing an improved estimation on transition matrices. Very recently, the results in [16, 17] were further extended to switched systems with impulses and perturbations [18]. So far, the stability and stabilization problems for switched systems were studied in [19–31], to name a few.

In this paper, motivated by the work in [16–18], 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 discrete-time 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.

2. Preliminaries

Consider the following planar discrete-time linear system: 𝑥(𝑘+1)=𝐺𝑥(𝑘)+𝐻𝑢(𝑘),𝑘≥0, (1) where 𝑥∈𝑅² is the state, 𝑢∈𝑅^𝑚 is the controlled input, and 𝐺 and 𝐻 are matrices of appropriate dimensions.

Under the following linear feedback law: 𝑢(𝑘)=𝐾𝑥(𝑘), (2) the solution of the system (1) takes the form 𝑥(𝑘)=(𝐺+𝐻𝐾)^𝑘𝑥(0), (3) where (𝐺+𝐻𝐾)^𝑘 is called the transition matrix.

When the system (1) is controllable, we first establish an estimation on the transition matrix (𝐺+𝐻𝐾)^𝑘.

Lemma 1. Let 𝐺∈𝑅^2×2 and 𝐻∈𝑅^2×𝑚 be constant matrices such that the pair (𝐺,𝐻) is controllable. Then, for any 0<𝜆<1, there exists a matrix 𝐾∈𝑅^𝑚×2 such that (𝐺+𝐻𝐾)^𝑘≤𝑁𝜆^𝑘−1,𝑘≥0, (4) where 𝑁>0 is a constant, which is independent of 𝜆 and can be estimated precisely in terms of 𝐺 and 𝐻.

Proof. First, we consider the case of single input, that is, 𝑚=1. Noting that (𝐺,𝐻) is controllable, we can choose a feedback matrix 𝐾∈𝑅^1×2 such that eigenvalues 𝜆_𝑖 of 𝐺+𝐻𝐾 satisfy 0<𝜆_𝑖<1,𝜆_𝑖≠𝜆_𝑗,𝑖≠𝑗,𝑖,𝑗=1,2. (5)
In particular, for any 0<𝜆<1, we can choose 𝜆_𝑖=𝜆𝑖,𝑖=1,2. (6)
Set ×𝑎𝐹=𝐺𝐻𝐻10₁1, (7) where 𝑎_𝑖, 𝑖=1,2, are determined by det(𝑆𝐼−𝐺)=𝑠²+𝑎₁𝑠+𝑎₀. (8)
Let 𝐺_𝑐=𝐹⁻¹𝐺𝐹,𝐻_𝑐=𝐹⁻¹𝐻,𝐾_𝑐=𝐾𝐹. (9)
We have that 𝜆_𝑖, 𝑖=1,2, are also eigenvalues of (𝐺_𝑐+𝐻_𝑐𝐾_𝑐), and (𝐺_𝑐,𝐻_𝑐) is in controller canonical form. Let 𝜆𝑃=11₁𝜆₂. (10)
It is not difficult to see that 𝑃⁻¹𝐺_𝑐+𝐻_𝑐𝐾_𝑐𝑃=𝐽=diag𝜆₁,𝜆₂, (11)
that is, 𝑃⁻¹𝐹⁻¹(𝐺+𝐻𝐾)𝐹𝑃=𝐽. (12)
It implies that 𝐺+𝐻𝐾=𝐹𝑃𝐽𝑃⁻¹𝐹⁻¹. (13)
Consequently, (𝐺+𝐻𝐾)^𝑘=𝐹𝑃𝐽^𝑘𝑃⁻¹𝐹⁻¹≤‖𝐹‖𝐹⁻¹‖𝑃‖𝑃⁻¹𝐽^𝑘. (14)
Let 𝑃=[𝑝_𝑖𝑗] and 𝑃⁻¹=[𝑒_𝑖𝑗]. First, we have ‖𝑃‖≤2max_{1≤𝑖,𝑗≤2}𝑝_𝑖𝑗=2. (15)
Second, noting that 𝑃⁻¹=adj𝑃/det𝑃, we get from (6) that 𝑃⁻¹=2−1𝜆2−2𝜆. (16)
So, 𝑃⁻¹≤4𝜆. (17)
Since 𝐽^𝑘≤𝜆^𝑘₁=𝜆^𝑘, (18) substituting (15), (17), and (18) into (14) yields that (𝐺+𝐻𝐾)^𝑘≤‖𝐹‖𝐹⁻¹4×2×𝜆×𝜆^𝑘=𝑁𝜆^𝑘−1, (19) where 𝑁=8‖𝐹‖‖𝐹⁻¹‖, which is independent of 𝜆. Therefore, we have that Lemma 1 holds for the single input case.
For the multiple-input case, one sees that for any 𝑐∈𝑅^𝑚 such that 𝐻𝑐≠0, there exists 𝐾₀∈𝑅^𝑚×2 such that (𝐺+𝐻𝐾₀,𝐻𝑐) is itself controllable. Hence, the conclusion of the single-input case that has been proved above is applicable to the controllable pair (𝐺+𝐻𝐾₀,𝐻𝑐). Therefore, for any 0<𝜆<1, there exists 𝑘₁∈𝑅^1×2 such that 𝐺+𝐻𝐾₀+𝐻𝑐𝑘₁^𝑘≤𝑁⋅𝜆^𝑘−1. (20)
The proof of Lemma 1 is completed.

When (𝐺,𝐻) is uncontrollable, for any given feedback 𝐾, there always exist a positive-definite symmetric matrix 𝑃 and an appropriate constant 𝜇>0 such that (𝐺+𝐻𝐾)^𝑇𝑃(𝐺+𝐻𝐾)−𝑃<𝜇𝑃, (21) which can be solved by using the GEVP solver in the LMI Toolbox of MATLAB [32].

Define the following Lyapunov function: 𝑉(𝑥(𝑘))=𝑥^𝑇(𝑘)𝑃𝑥(𝑘),𝑘≥0. (22)

It is easy to see that 𝜆_min(𝑃)‖𝑥‖²≤𝑉(𝑥)≤𝜆_max(𝑃)‖𝑥‖², (23) where 𝜆_min(𝑃) and 𝜆_max(𝑃) denote the smallest and the largest eigenvalue of the positive definite symmetric matrix 𝑃.

Lemma 2. For system (1), if the pair (𝐺,𝐻) is uncontrollable and (21) holds, then for any given feedback matrix 𝐾, there exists a constant 𝜇>0 such that ‖𝑥(𝑘)‖<1+𝜇𝜆_max(𝑃)𝜆_min(𝑃)^𝑘‖𝑥(0)‖,𝑘≥0. (24)

Proof. Let the Lyapunov function be defined by (22). Along the solution of system (1), we have Δ𝑉(𝑥(𝑘))=𝑥^𝑇(𝑘+1)𝑃𝑥(𝑘+1)−𝑥^𝑇(𝑘)𝑃𝑥(𝑘)=𝑥^𝑇(𝑘)(𝐺+𝐻𝐾)^𝑇𝑃(𝐺+𝐻𝐾)−𝑃𝑥(𝑘). (25)
By (21) and (25), we obtain Δ𝑉(𝑥(𝑘))−𝜇𝑉(𝑥(𝑘))=𝑥^𝑇(𝑘)(𝐺+𝐻𝐾)^𝑇𝑃(𝐺+𝐻𝐾)−𝑃−𝜇𝑃𝑥(𝑘)<0, (26) which implies that 𝑉(𝑥(𝑘+1))<1+𝜇𝑉(𝑥(𝑘)). (27)
By (23) and (27), we have 𝜆_min(𝑃)‖𝑥(𝑘+1)‖²<1+𝜇𝜆_max(𝑃)‖𝑥(𝑘)‖². (28)
Thus, ‖𝑥(𝑘+1)‖<1+𝜇𝜆_max(𝑃)𝜆_min(𝑃)‖𝑥(𝑘)‖. (29)
By induction, we have ‖𝑥(𝑘)‖<1+𝜇𝜆_max(𝑃)𝜆_min(𝑃)^𝑘‖𝑥(0)‖,𝑘≥0. (30)
This completes the proof of Lemma 2.

3. Stabilization of Discrete-Time Planar Switched Impulsive Systems

Now, we study the stabilization of the following discrete-time switched linear system: 𝑥(𝑘+1)=𝐺_𝜎(𝑘)𝑥(𝑘)+𝐻_𝜎(𝑘)𝑢(𝑘),𝑘≠𝑘_𝑗,𝑥𝑘⁺_𝑗=𝐹𝜎𝑘_𝑗,𝜎𝑘_𝑗+1𝑥𝑘_𝑗, (31) where 𝑥∈𝑅² is the state, 𝑢∈𝑅^𝑚 is the input, and 𝜎(𝑘)∈Λ={1,2,…,𝑁} is a switching signal for some positive integer 𝑁>1, which is a piecewise constant function. When 𝜎(𝑘)=𝑖,𝑖∈Λ, system (31) switches to the 𝑖th subsystem. Moreover, 𝐹(⋅,⋅) is a constant matrix, representing the impulse effect on the system at the switching time. Moreover,  0=𝑘₀<𝑘₁<𝑘₂<⋯ denote the discontinuous points (or switching points) of 𝜎(𝑘), and denote 𝜎(𝑘_𝑗−1)=𝑝_𝑗,(𝑗=1,2,…). 𝐺_𝑖 and 𝐻_𝑖, 𝑖∈Λ, are system matrices of appropriate dimensions.

Throughout this paper, we assume that(H1)max_{1≤𝑖,𝑗≤𝑁}‖𝐹(𝑖,𝑗)‖≤𝛿, where 𝛿>0 is a positive constant.

Under the linear feedback law 𝑢(𝑘)=𝐾_𝜎(𝑘)𝑥(𝑘) for 𝑘≠𝑘_𝑗, 𝑗=1,2…, system (31) reduces to the following closed-loop system: 𝑥(𝑘+1)=𝐺_𝜎(𝑘)+𝐻_𝜎(𝑘)𝐾_𝜎(𝑘)𝑥(𝑘),𝑘≠𝑘_𝑗. (32)

Denote the frequency of the switching signal by 𝑓=limsup_𝑘→∞𝑁_𝜎(0,𝑘)𝑘, (33) where 𝑁_𝜎(0,𝑘) is the number of activated subsystems on [0,𝑘]. If (𝐺_𝑖,𝐻_𝑖) is controllable for 𝑖∈Λ, we have the following result.

Theorem 3. Assume that (H1) and (H2) hold and (𝐺_𝑖,𝐻_𝑖) is controllable for 𝑖∈Λ. Then, there exist a set of feedback matrices {𝐾_𝑖}_𝑖∈Λ such that the closed-loop system (32) is asymptotically stable for any switching signal 𝜎 with a frequency 𝑓<1.

Proof. For any 𝑘>0, assume that 𝑘∈(𝑘_𝑖,𝑘_𝑖+1] for some positive integer 𝑖. Note that 𝑓<1. By the definition of 𝑓, we can choose a constant 0<𝛼<1 such that 𝑖+1≤𝑘𝛼 for sufficiently large 𝑘. Without loss of generality, we assume that 𝑖+1≤𝑘𝛼 for 𝑘≥0. Set 𝛽=1−𝛼. It is easy to see that 𝛽>0. Let 𝑐=(𝛿𝑁)^𝛼/𝛽 and 𝜆  be sufficiently small such that 𝜆𝑐<1. For such a choice of 𝜆, by Lemma 1, there exist a set of feedback matrices {𝐾_𝑖}_𝑖∈Λ such that for any 𝑘>𝑘≥0, 𝐺_𝑖+𝐻_𝑖𝐾_𝑖^𝑘−𝑘≤𝑁𝜆^𝑘−𝑘₋₁, (34) where 𝑁=max_𝑖∈Λ𝑁_𝑖. For any 𝑙∈[𝑘_𝑗−1,𝑘_𝑗),(1≤𝑗≤𝑖+1), we have 𝜎(𝑙)=𝜎𝑘_𝑗−1=𝑝_𝑗,𝑥(𝑙+1)=𝐺_𝑝𝑗+𝐻_𝑝𝑗𝐾_𝑝𝑗𝑥(𝑙),𝑥𝑘⁺_𝑗=𝐹𝑝_𝑗,𝑝_𝑗+1𝑥𝑘_𝑗,𝑥𝑘_𝑗=𝐺_𝑝𝑗+𝐻_𝑝𝑗𝐾_𝑝𝑗^{𝑘𝑗−𝑘𝑗−1}𝑥𝑘⁺_𝑗−1. (35)
Since 𝑘_𝑖≤𝑘<𝑘_𝑖+1, that is, 𝑘∈[𝑘_𝑖,𝑘_𝑖+1), we obtain 𝑥(𝑘)=𝐺_𝑝𝑖+1+𝐻_𝑝𝑖+1𝐾_𝑝𝑖+1^{𝑘−𝑘𝑖}𝑥𝑘⁺_𝑖=𝐺_𝑝𝑖+1+𝐻_𝑝𝑖+1𝐾_𝑝𝑖+1^{𝑘−𝑘𝑖}𝐹𝑝_𝑖,𝑝_𝑖+1×𝐺_𝑝𝑖+𝐻_𝑝𝑖𝐾_𝑝𝑖^{𝑘𝑖−𝑘𝑖−1}𝑥𝑘⁺_𝑖−1=𝐺_𝑝𝑖+1+𝐻_𝑝𝑖+1𝐾_𝑝𝑖+1^{𝑘−𝑘𝑖}𝐹𝑝_𝑖,𝑝_𝑖+1×𝐺_𝑝𝑖+𝐻_𝑝𝑖𝐾_𝑝𝑖^{𝑘𝑖−𝑘𝑖−1}×𝐹𝑝_𝑖−1,𝑝_𝑖𝐺_𝑝𝑖−1+𝐻_𝑝𝑖−1𝐾_𝑝𝑖−1^{𝑘𝑖−1−𝑘𝑖−2}𝑥𝑘⁺_𝑖−2=𝐺_𝑝𝑖+1+𝐻_𝑝𝑖+1𝐾_𝑝𝑖+1^{𝑘−𝑘𝑖𝑖}_𝑗=1𝐹𝑝_𝑗,𝑝_𝑗+1×𝐺_𝑝𝑗+𝐻_𝑝𝑗𝐾_𝑝𝑗^{𝑘𝑗−𝑘𝑗−1}𝑥(0). (36)
By the analysis and (H1), we obtain ‖𝑥(𝑘)‖≤𝐺_𝑝𝑖+1+𝐻_𝑝𝑖+1𝐾_𝑝𝑖+1^{𝑘−𝑘𝑖}×^𝑖_𝑗=1𝐹𝑝_𝑗,𝑝_𝑗+1×𝐺_𝑝𝑗+𝐻_𝑝𝑗𝐾_𝑝𝑗^{𝑘𝑗−𝑘𝑗−1}×‖𝑥(0)‖≤𝛿^𝑖𝑁^𝑖+1𝜆^{𝑘−𝑖−1}‖𝑥(0)‖≤𝛿^𝑖+1𝑁^𝑖+1𝜆^−(𝑖+1)𝜆^𝑘‖𝑥(0≤)‖𝛿𝑁𝜆^−1𝑘𝛼𝜆^𝑘‖𝑥(0=())‖𝛿𝑁^𝛼𝑘𝜆^𝛽𝑘‖𝑥(0=())‖𝜆𝑐^𝛽𝑘‖𝑥(0)‖. (37)
Noting that 𝜆𝑐<1 and 𝛽>0, we have that system (31) is stabilizable under arbitrary switching signal with a frequency 𝑓<1. This completes the proof of Theorem 3.

Next, we consider the case when there exist both controllable subsystems and uncontrollable subsystems for system (31). For the sake of convenience, we suppose that(H2)(𝐺₁,𝐻₁),(𝐺₂,𝐻₂),…,(𝐺_𝑝,𝐻_𝑝) are uncontrollable subsystems and (𝐺_𝑝+1,𝐻_𝑝+1),(𝐺_𝑝+2,𝐻_𝑝+2),…,(𝐺_𝐼,𝐻_𝐼) are controllable subsystems, where 1≤𝑝<𝐼.

Denote the switching frequency of those controllable subsystems by 𝑓=limsup_𝑘→∞𝑁_𝜎(0,𝑘)𝑘, (38) where 𝑁_𝜎(0,𝑘) is the number of activated controllable subsystems on [0,𝑘]. Denote the total activation time for those controllable subsystems on [0,𝑘] by 𝑇(0,𝑘). In this paper, we assume that there exists a constant 𝛾>0 such that 𝑇(0,𝑘)≥𝛾𝑘,𝑘≥0. (39)

Similar to the above analysis, for any given feedback matrices 𝐾_𝑖(1≤𝑖≤𝑝), there exist positive definite symmetric matrices 𝑃_𝑖 and positive constants 𝜇_𝑖 such that 𝐺_𝑖+𝐻_𝑖𝐾_𝑖^𝑇𝑃_𝑖𝐺_𝑖+𝐻_𝑖𝐾_𝑖−𝑃_𝑖<𝜇_𝑖𝑃_𝑖,1≤𝑖≤𝑝. (40)

Set 𝜆=max_{1≤𝑖≤𝑝}1+𝜇_𝑖𝜆_max𝑃_𝑖𝜆_min𝑃_𝑖>1. (41)

Theorem 4. Assume that (H2), (39), and (40) hold. Then, there exist a set of feedback matrices {𝐾_𝑖}_𝑖∈Λ such that the closed-loop system (32) is asymptotically stable for any switching signal 𝜎 with a frequency 𝑓<𝛾.

Proof. For any 𝑘>0, assume that 𝑘∈[𝑘_𝑖,𝑘_𝑖+1) for some positive integer 𝑖. Denote the number of activated controllable subsystems in [0,𝑘] by 𝑖₁. Since 𝑓<𝛾. By the definition of 𝑓, there exists a constant 0<𝛼<𝛾 such that 𝑖₁≤𝑘𝛼 for 𝑘≥0 without loss of generality. Set 𝛽=𝛾−𝛼. It is easy to see that 𝛽>0 since 0<𝛼<𝛾. Let the constant 𝑐̃ satisfies 𝛿𝑁𝛼𝜆=𝑐𝛽̃, (42) where 𝜆 is determined by (41) under any given feedback matrices {𝐾_𝑖}_{1≤𝑖≤𝑝}.
For any 𝑘≥0, assume that 𝜎(𝑘_𝑗−1)=𝑝_𝑗, 1≤𝑗≤𝑖+1 and 𝑘_𝑖≤𝑘<𝑘_𝑖+1. By (35), we have 𝑥(𝑘)=𝐺_𝑝𝑖+1+𝐻_𝑝𝑖+1𝐾_𝑝𝑖+1^{𝑘−𝑘𝑖}𝑥𝑘⁺_𝑖=𝐺_𝑝𝑖+1+𝐻_𝑝𝑖+1𝐾_𝑝𝑖+1^{𝑘−𝑘𝑖}𝐹𝑝_𝑖,𝑝_𝑖+1𝑥𝑘_𝑖. (43)
If 𝑝_𝑗∈{𝑝+1,𝑝+2,…,𝑁}, by (34) and (H1), we have ‖𝑥(𝑘)‖≤𝐺_𝑝𝑖+1+𝐻_𝑝𝑖+1𝐾_𝑝𝑖+1^{𝑘−𝑘𝑖}⋅𝐹𝑝_𝑖,𝑝_𝑖+1⋅𝑥𝑘_𝑖≤𝛿𝑁𝜆^{𝑘−𝑘𝑖−1}𝑥𝑘_𝑖. (44)
If 𝑝_𝑗∈{1,2,…,𝑝}, by Lemma 2, (41), and (H1), we have ‖𝑥(𝑘)‖≤1+𝜇𝜆_max(𝑃)𝜆_min(𝑃)^{𝑘−𝑘𝑖}𝛿𝑥𝑘_𝑖≤𝛿𝜆^{𝑘−𝑘𝑖}𝑥𝑘_𝑖. (45)
Based on the same analysis, there exists a feedback matrix 𝐾_𝑝𝑚−1 for each 𝑚∈{1,2,…,𝑗} such that 𝑥𝑘_𝑚≤𝛿𝑁𝜆^{𝑘𝑚−𝑘𝑚−1}𝑥𝑘_𝑚−1,𝜎𝑘_𝑚−1∈,𝛿𝜆𝑝+1,𝑝+2,…,𝑁_{𝑘𝑚−𝑘𝑚−1}𝑥𝑘_𝑚−1,𝜎𝑘_𝑚−1∈1,2,…,𝑝. (46)
We now choose 𝜆 sufficiently small such that 𝜆𝑐̃<1. Under the feedback law 𝑢(𝑘)=𝐾_𝑖𝑚𝑥(𝑡) for 𝑚=0,1,…,𝑗, we get from (39) and (44)–(46) by induction that ‖𝑥(𝑘)‖≤𝛿^𝑘𝑁𝜆^−1𝑖1𝜆^𝑇(0,𝑘)𝜆^𝑘‖𝑥(0)‖≤𝛿^𝑘𝑁𝜆⁻¹𝛼^𝑘𝜆^𝛾𝑘𝜆^𝑘‖𝑥(0=𝛼𝜆)‖𝛿𝑁^𝑘𝜆𝛽_𝑘‖𝑥(0=(̃)𝛽)‖𝑐𝜆^𝑘‖𝑥(0)‖. (47)
Noting that 𝑐𝜆̃<1 and 𝛽>0, we have that system (32) is stabilizable under arbitrary switching signal with a frequency 𝑓<𝛾. This completes the proof of Theorem 4.

4. Examples

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

Example 1. Consider the switched systems (31), with 𝜎(𝑘)=1,2,3, and 𝐺₁=3105,𝐻₁=14,𝐺₂=7410−1,𝐻₂=31,𝐺₃=8351,𝐻₃=51. (48)

It is not difficult to verify that (𝐺_𝑖,𝐻_𝑖) is controllable for 𝑖=1,2,3.

By Lemma 1, we get 𝑁=max_𝑖=1,2,3𝑁_𝑖=161. We choose 𝜆=1/2, then the closed-loop poles of (32) are 1/2 and 1/4. We have the feedback matrices 𝐾₁=,𝐾3.43750.9531₂=,𝐾1.18751.6875₃=1.53020.5991. (49)

Let 𝛿=2 and the switching frequency 𝑓<1. Based on the proof of Theorem 3, we can choose that 𝛼=1/20, then 𝛽=1−𝛼=19()20>0,𝑐=𝛿𝑁^𝛼/𝛽=1.3552. (50)

We can get 𝜆𝑐=0.6776<1. By Theorem 3, the system (32) is asymptotically stable with 𝑓<1.

Example 2. Consider the switched systems (31), with 𝜎(𝑘)=1,2,3, and 𝐺₁=01−10,𝐻₁=10,𝐺₂=1−420,𝐻₂=21,𝐺₃=1001,𝐻₃=10. (51)
It is not difficult to verify that (𝐺_𝑖,𝐻_𝑖) is controllable for 𝑖=1,2, and (𝐺₃,𝐻₃) is uncontrollable.
By Lemma 1, we get 𝑁=max_𝑖=1,2𝑁_𝑖=21. Choose 𝜆=1/10, we have the feedback matrices 𝐾₁=,𝐾−0.15000.9950₂=1.0545−1.2590. (52)
Let 𝛿=3 and 𝛾=3. The switching frequency 𝑓<3, 𝛼=1/10. For given 𝐾₃=[21]. Based on the (40), we can choose 𝜇₃=12 and 𝑃₃=³¹₁₁, then 𝜆=25.0889. (53)
By Theorem 4, we have 𝛽=29̃10>0,𝑐=4.92, (54) then 𝜆𝑐̃=0.492<1. (55)
By Theorem 4, the system (32) is asymptotically stable with 𝑓<3.

5. 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. When there exists at least one uncontrollable subsystem, by using a Lyapunov functional approach, we show that the stabilizability of the switched impulsive system can be retained for the appropriate switching frequency of those controllable subsystems.

Acknowledgments

The authors thank the reviewers for their helpful and valuable comments on this paper. This work was supported by the Natural Science Foundation of Shandong Province under Grant nos. JQ201119 and ZR2010AL002 and the National Natural Science Foundation of China under Grant no. 61174217.