Conditions for the Stability of Switched Systems Containing Unstable Subsystems

This brief studies the stability of switched systems in which all the subsystems may be unstable. In addition, some of the switching behaviors of the systems are destabilizing. By using the piecewise Lyapunov function method and taking a tradeoff between the increasing scale and the decreasing scale of the Lyapunov function at switching times, the maximum dwell time for admissible switching signals is obtained and the extended stability results for switched systems in a nonlinear setting are first derived. Then, based on the discretized Lyapunov function method, the switching stabilization problem for linear context is solved. By contrasting with the contributions available in the literature, we do not require that all the switching behaviors of the switching system under consideration are stabilizing. More specifically, even if all the subsystems governing the continuous dynamics are not stable and some of the switching behaviors are destabilizing, the stability of the switched system can still be retained. A numerical example is given to illustrate the validity of the proposed results.

stability and the stabilization of switched linear systems. The theoretical results and applications for the stability analysis and stabilization of switched systems have been systematized in [14], to which we refer the reader for research purpose.
On the other hand, the stabilization issue of switched systems with unstable subsystems deservers deep investigation due to its theoretical significance and practical application. In fact, even if a switched system contains unstable subsystems, the stability of it can still be guaranteed as long as the switching signal is properly designed. The paper [15] first studied the topic for switched linear systems with some unstable subsystems by using the average dwell time approach. It was shown that if the average dwell time is chosen sufficiently large and the total activation time of unstable subsystems is relatively small compared with that of stable ones, the stability of the switched systems can be guaranteed. Since then a lot of contributions have been devoted to switched systems subject to unstable subsystems. For instance, asynchronous networked control systems can be modeled by switched systems with both stable and unstable subsystems [16]. To sum up, the main stabilization idea is to keep the stability of the whole switched system through the trade-off between the stable subsystems and the unstable ones [17].
However, when all the subsystems are unstable, the stabilization problem is much more challenging since the trade-off idea obviously fails. To solve this issue, the state-dependent switching strategies such as the min-projection strategy and the largest region function strategy have been proposed. Until recently, the dwell time switching strategy was resorted to discuss the stability of switched linear systems with all unstable subsystems under time-dependent switching. In a word, the idea of these methods is to find a switching signal that renders the switched system stable. Switching stabilization has a wide range of applications in fields like hovercraft [18] and electrochromic films [19] and has been employed in various studies. For example, the input-to-state stability of switched delay systems with all unstable subsystems were retained, provided that the switching signal satisfies a dwell-time upper bound condition in [20]. Work [21] extended the results in [20] into an enlarged class of hybrid delayed systems. In [22], stabilization of switched linear systems composed fully of unstable subsystems under dwell time switching was investigated. It is worth pointing out that all the switching behaviors are required to be stabilizing in the above literatures. In fact, even if some of the switching behaviors are destabilizing, one may exploit the good characteristic of the switching behaviors to make the switched system asymptotically stable. These remarks motivate the present work.
The main objective of this paper is to propose the stabilization conditions of switched systems. The extended stability results for switched systems in a nonlinear setting are first derived by using the piecewise Lyapunov function method and taking a tradeoff between the increasing scale and the decreasing scale of the Lyapunov function at switching times. Then, based on discretized Lyapunov method, the switching stabilization problem for linear context is solved. By contrasting with the results available in the literatures, the contribution of this paper lies in two aspects. First, by further allowing the Lyapunov function to increase during the running time of the active subsystem, no subsystem is required to be stable. Second, it is shown that, even if all the subsystems governing the continuous dynamics are not stable and some of the switching behaviors of the switched systems are destabilizing, the residual stabilizing switching behaviors can successfully stabilize the whole switched system, provided that there are no overly long time intervals between switchings and the destabilizing switchings do not occur too frequently.
The rest of this paper is divided into four sections. Section 2 introduces some preliminaries while Section 3 states our main results regarding the stabilization conditions of switched linear systems. A numerical example is presented in Section 4 and Section 5 concludes the paper.
Notations. S = {1, 2, .., s}. N denotes the set of nonnegative integer numbers and N + = N/{0}. A function γ : R + → R + is said to be of class K if it is continuous, zero at the origin, and strictly increasing. It is of class K ∞ if it is of class K and unbounded. A function β : R + × R + → R + is of class KL if β (·,t) is of class K for each t ≥ 0 and β (s, ·) is decreasing to zero for each s ≥ 0.

II. PRELIMINARIES
Consider the switched nonlinear systeṁ where x(t) ∈ R n x is the state; x(t 0 ) = x 0 is the initial state; σ (t) : [0, ∞) → S is a switching signal. It has the switching sequence {( 0 ,t 0 ), ..., ( k ,t k ), ..., | k ∈ S , k ∈ N}, which means that the k th subsystem is active when t ∈ [t k ,t k+1 ). For ∀i ∈ S , f i : R n x → R n x are smooth functions with f i (0) = 0, and the origin is not a stable equilibrium for any modes i ∈ S . We assume that no jump occurs in the state at a switching time and that only finitely many switchings can occur in any finite interval. For the sake of clarity, we recall the definition of asymptotic stability for switched systems. Definition 1. [10] System (1) with switching signal σ (t) is said to be asymptotically stable if there exists function β ∈ KL, such that for any initial condition x 0 ∈ R n x , the solution of system (1) exists over R + and satisfies In this section, we aim to establish sufficient conditions that guarantee the stability of system (1) when all subsystems of (1) are unstable. In [20], [21], [22], this issue has been studied by requiring that all of the switching behaviors of the switching signal admit stabilizing property to compensate the increment of the multiple Lyapunov functions. In this paper, it is shown that even if some of the switching behaviors play the destabilizing property, the stability of the whole switched system can still be retained. We assume that after m stabilizing switching, the system should undergo a destabilizing switching. Likewise, after n destabilizing switching, the system should undergo a stabilizing switching. For this purpose, we define a new class of switching time sequences. First, we let p l = m+n, m, n ∈ N + , and define the sets To investigate the stability property of system (1), we choose the following piecewise Lyapunov functions . Now, we are ready to establish the stability result for system (1).
Theorem 1. Consider switched nonlinear system (1) with switching signal σ (t). If there exists a piecewise continuous Lyapunov function defined in (2) such that (i) for each i ∈ S , there exist functions α j ∈ K ∞ ( j = 1, 2) and a constant η > 0 such that, for all x ∈ R n x and t ≥ t 0 ≥ 0, (ii) there are two constants 0 < ρ < 1, µ ≥ 1 such that for all Then, switched nonlinear system (1) with switching signal σ (t) is asymptotically stable.
Proof. See the Appendix. Corollary 1. Consider switched system (1) with switching signal σ (t). If there exists a piecewise continuous Lyapunov function defined in (2) such that (i)' for each i ∈ S , there exist functions α j ∈ K ∞ ( j = 1, 2) and a constant η > 0 such that, for all x ∈ R n x and t ≥ t 0 ≥ 0, Then, switched system (1) with switching signal σ (t) is asymptotically stable.
Remark 1. The results in [23] apply to cases where some subsystems of the switched systems are not necessarily stable under the influence of asynchronous switching or input delay. The method in the literature fail to deal with the stability of system (1) in the paper because all the subsystems are allowed to be unstable.

Remark 2.
In some previous results for switched nonlinear systems with all unstable subsystems [20], [21], [22], the switching behaviors are required to decrease the value of the piecewise continuous Lyapunov functions at switching times. However, this viewpoint is too strict since the destabilizing property of some of switching behaviors is not considered. In Theorem 1, even if all the subsystems are unstable and some of the switching behaviors are destabilizing, the residual stabilizing switching behaviors can successfully stabilize the switched system.
Proof. See the Appendix.

IV. NUMERICAL EXAMPLE
In this section, we present one example to illustrate the effectiveness of the proposed method.
Example 1. Most electronic systems, mechanical systems, and aeroengine control systems can be modeled as multiintegrator systems [24]. For simplicity, we consider a given double integrator system as the example and first design a suitable feedback controller for it. Considering the scenario as following, the controller may have several backup controllers, but all of controllers are partly failure or totally failure. The double integrator system will be unstable with any controller. However, we can design a switching law to stabilize the double integrator system.
Consider the double integrator system with the state matrix with Obviously, we can check that all of matrixes are unstable. Choose a switching sequence that satisfies t n+1 −t n = 0.12s, n ∈ N. we see that τ min = τ max = 0.12s. For, L = 8.333, ρ = 0.76, µ = 1.3, and η = 1.9, the state trajectory and the evolution of the Lyapunov function under the switching signal σ are shown in Figs. 1 and 2, respectively.

V. CONCLUSION
We have studied the stability of switched systems in which all the subsystems may be unstable and some of the switching behaviors are destabilizing. By using the piecewise Lyapunov function method and taking a tradeoff between the increasing and the decreasing scales of the Lyapunov function at switching times, the extended stability results for switched systems in a nonlinear setting have been derived. Then, based on the discretized Lyapunov function method, the switching stabilization problem for linear context has been solved.
Proof of Theorem 2. We construct the multiple linear copositive Lyapunov function is positive definite matrix and P i (t), i ∈ S − is time-scheduled positive definite matrix. In order to render the stability condition numerically computable, we divide the interval N kp = [t l k +p ,t l k +p + τ min ), p ∈ Ξ 1 into L segments N q kp = [t l k +p + θ q ,t l k +p + θ q+1 ), q ∈ S of equal length d = τ min /L, and then θ q = qd = qτ min /L, q ∈ S. The continuous matrix function P i (t),t ∈ [t l k +p ,t l k +p+1 ) is chosen to be linear within each segment N q kp , q ∈ S. Let P q i = P i (t l k +p + θ q ), then since the matrix function P i (t), i ∈ S − is piecewise linear in [t l k +p ,t l k +p + τ min ), it can be expressed in terms of the values at segmentation points using a linear interpolation formula, i.e., for 0 ≤ α ≤ 1, q ∈ S, and t ∈ N q kp , where α = (t − t l k +p − θ q )/d. Then, the continuous matrix function P i (t), i ∈ S is completely determined by P q i , q ∈ S {L}, i ∈ S − in the minimal dwell time interval [t l k +p ,t l k +p + τ min ). In the interval [t l k +p + τ min ,t l k +p+1 ), the matrix function P i (t), i ∈ S − is chosen as P i (t) = P L i , i ∈ S . To sum up, the discretized matrix function P i (t), i ∈ S − is expressed as and the corresponding discretized Lyapunov function for the ,t ∈ [t l k +p + τ min ,t l k +p+1 ). One can prove easily that (3) in condition (i) of Theorem 1 is satisfied.
Therefore, considering iv), the stability of system (6) governed by any switching law σ (t) can be established by Theorem 1.