Stability of switched systems with admissible switching signals


 In this paper, stability of switched systems is investigated for a class of switching signals which meet some admissibility conditions. Firstly, the admissible edge-dependent divergence time is defined in terms of admissible transition edges and it will vary with the compensation bounds. Then the admissible edge-dependent bounded maximum average dwell time (BMADT) is imposed on switching signals. As a result, a sufficient condition is obtained for globally uniformly exponential stability of switched nonlinear systems with such switching signals. Secondly, by setting the equal compensation bounds for the same reaching subsystems, the mode-dependent divergence time is defined, and then the mode-dependent BMADT is proposed. A stability condition under the mode-dependent BMADT is established. These stability results are then applied to switched linear systems. The numerical example is presented to show that the proposed techniques are less restrictive and more flexible in application, compared with the BMADT.


Introduction
Switched systems have attracted great attention, due to their widespread applications in various fields [1][2][3][4]. Switched systems may be classified to three types: all subsystems are stable, all subsystems are unstable, and some subsystems are unstable [5][6][7][8][9][10][11]. Note that stability of subsystems does not guarantee stability of a switched system and a switching signal governing selection of subsystems is crucial for stability of the system. A switching signal is classified by causes determining whether to switch: state of subsystems or time. When a switched system with all subsystems being unstable is stabilized under a state-dependent switching sequence, it is common to find a stable convex combination [12][13][14]. However, designing a state-dependent switching signal requires the full state information.
In contrast, time-dependent switching signals are easier to be designed [15][16][17][18]. When there are stable and unstable subsystems, slow switching and fast switching are used. The representative techniques include the dwell time (DT) [19], the average dwell time (ADT) [20], the mode-dependent ADT [21] and the admissible edge-dependent ADT [22,23]. The main idea behind them is to activate stable subsystems for sufficiently long to suppress the state divergence made by unstable subsystems. However, the stability conditions involve priori knowledge of stability of subsystems, and they fail on switched systems with all subsystems being unstable.
For switched systems with all subsystems being unstable, Mao et al. [11] proposed the divergence time and the bounded maximum average dwell time (BMADT). The time spans which are less than lower compensation bound or more than upper compensation bound have divergence time, then the BMADT is put forward based on the divergence time. Under the BMADT switching, the state divergence caused by switches with divergence time are compensated for by those without divergence time. It means that there is no need to know stability of subsystems in advance and the BMADT can be applied to any switched system regardless of whether or not its subsystems are stable or not. However, the BMADT switching ignores the mapping from time to subsystems, resulting in conservativeness. Relationships between switching time instants and activated subsystems are involved [24] in admissible transition edges, which also reveals the change of activated subsystems before and after a switching time instant. It has been shown that use of admissible transition edges is effective in generalizing the stability criteria of switched systems [22,23].
In view of the above observations, we think that admissible transition edges can relax the restrictions of the BMADT switching for better stability conditions. Firstly, compensation bounds are allowed to be local for different admissible transition edges. Then the admissible edge-dependent divergence time is defined, by which the admissible edge-dependent BMADT is established. Then, a stability condition is proposed via the admissible edge-dependent BMADT, where the suppression of divergent state vary in different admissible transition edges, contributing to fewer restrictions and more application flexibility than that in [11]. In addition, the case that ignores the differences of starting subsystems is considered, then compensation bounds are only consistent with reaching subsystems. Hence, the admissible edge-dependent divergence time is turned into the mode-dependent divergence time. Thus, the modedependent BMADT is proposed, which is utilized to derive a stability criterion of switched nonlinear systems. Based on these results, stability criteria of switched linear systems are established to better illustrate the effectiveness of proposed admissible switching signals.
The rest of the paper is organized as follows. Section 2 introduces descriptions of switched systems and the definition of the BMADT. Section 3 contains definitions of the admissible edge-dependent BMADT and modedependent BMADT, and stability conditions for nonlinear and linear switched systems. Section 4 presents a numerical example. Conclusions are given in Section 5.

Problem Formulation and Preliminaries
Consider the switched nonlinear system described bẏ where x(t) ∈ R n is the system state, f : R n → R n , is a locally Lipschitz nonlinear function, σ(t) : [t 0 , ∞) → M = {1, 2, ..., m}, is the switching signal, m is the number of subsystems, switching time instants are denoted by t s , s = 1, 2, ..., ∆ s = t s −t s−1 is the dwell time of s-th switching. Assume that σ is continuous from the right everywhere, the system (1) has the solution for certain σ and the initial condition x 0 , which is denoted by x(t), and the origin is an equilibrium of the system (1), i.e., f σ (0) = 0. The stability of system (1) is studied in this paper.

Definition 1[21]
The equilibrium x = 0 of system (1) is globally uniformly exponentially stable (GUES) under certain switching signal σ and the initial condition The stability of system (1) depends on the switching signal σ(t). Choose t and t with t > t > 0. Definé and T (t, t, t) is called [11] the divergence time over the interval [t 0 , t]. Let N (t) be the number of switchings in [t 0 , t]. Choose τ t,t > 0. If the switching signal satisfy τ t,t is called [11] the bounded maximum average dwell time (BMADT).
Choose t, t and τ t,t , then the switching signal σ(t) is restricted by (2)-(5). The first switching time instant satisfies t 1 ∈ (0, 2(τ t,t + t)]. Let It is seen that t, t and τ t,t are assigned globally over the entire time horizon, ignoring the information of the mapping: σ : Thus, admissible transition edges are introduced in [24], which look into how switches happen among sub- Then (i, j) denotes the admissible transition edge whenever the switching from the i-th subsystem to the j-th subsystem is admissible, where i and j represent the starting and reaching subsystem, respectively.

Stability Analysis
Consider a given switching sequence. Choose t i,j and and Let N i,j (t) be the number of switchings, T i,j (t) be the running time, and ∆ 1,i,j be the first time interval of (i, j) in [t 0 , t]. Choose τ i,j t i,j ,ti,j > 0. If the switching signal satisfy To see the difference of the admissible edge-dependent BMADT from BMADT, consider Example 1 again. (3,2). Then the switching signal σ(t) is restricted by (6)-(9). The first switching time instant satisfies t 1 ∈ (0, 2(τ 1,2 In order to study stability of the system (1), Lyapunov-like functions are applied [24]. Suppose that there exist Lyapunov-like functions V j (x(t)), j = 1, 2, ..., m, positive constant α j > 0, j = 1, 2, ..., m, positive con- and two functions, κ 1 and κ 2 in class κ ∞ , such that Let denote the number of switchings whose corresponding dwell time is less or not less than t i,j of (i, j) in [t 0 , t], and τ i,j t i,j ,ti,j satisfies Theorem 1 If there are Lyapunov-like functions V j (x(t)), j = 1, 2, ..., m, which satisfy (10)- (12), then the system (1) is GEUS under the switching signal satisfying (8), (9) and (13), for any admissible edge-dependent Proof: Let t p , t q ∈ {t s }, be the switching time instant whose corresponding dwell time is not less than t σ(tp−1),σ(tp) or less than t σ(tq−1),σ(tq) , respectively. Then, for any t satisfying t 0 < ... < t l ≤ t < t l+1 , l = N (t). It is seen from (11) and (12) that Let i, j be the values of σ(t) on [t s−1 , t s ) and [t s , t s+1 ), respectively. Then (15) becomes ) .
Finally, it follows from (10) that ∥x 0 ∥, and thus the system (1) is GUES. The proof is completed. From (12), it is seen that if ∆ s < t i,j , i, j ∈ M, i ̸ = j, µ i,j is applied, which means energy functions increase at switching instants. When ∆ s ≥ t i,j , λ i,j is utilized, which is effective in dissipating upward energy functions. Apparently, the involvement of admissible transition edges contributes to more application flexibility than that in [11] as the compensations of energy functions vary in different admissible transition edges.
In order to better demonstrate the effectiveness of the proposed admissible edge-dependent BMADT, we simplify the system (1) to a switched linear system: where A σ(t) , σ(t) ∈ M, are known real constant matrices with appropriate dimensions. The linear matrix inequalities (LMIs) are required to obtain the stability criteria of the system (16). Suppose there exist matrices P j > 0, j = 1, 2, ..., m, positive constant α j > 0, j = 1, 2, ..., m, positive constants 0 < λ i,j < 1, and µ i,j > 1, i, j ∈ M, i ̸ = j, such that and Corollary 1 If there are positive matrices P j , j = 1, 2, ..., m, which satisfy (17)- (19), the system (16) is GEUS under the switching signal satisfying (8), (9) and (20), for any admissible edge-dependent BMADT, Proof: Suppose that there are P j , j = 1, 2, ..., m, satisfying (17)- (19). Choose the Lyapunov-like functions as Then, (11) and (12) are obtained and It follows from a similar argument to the proof of Theorem 1 that ∥x 0 ∥, and thus the system (16) is GUES. The proof is completed. If the differences of starting subsystems are ignored: and T (t, t j , t j ) is called the mode-dependent divergence time over the interval [t 0 , t]. t j , t j are named lower and upper compensation bounds of subsystem j. Let N j (t) be the number of switchings, T j (t) be the running time, and ∆ 1,j be the first time interval of subsystem j in [t 0 , t].
Choose τ j t j ,tj > 0. If the switching signal satisfy τ j t j ,tj is called the mode-dependent BMADT. Compared with the admissible edge-dependent BM-ADT, the mode-dependent BMADT omits the differences of starting subsystems. τ j t j ,tj , t j and t j are only relevant to reaching subsystems.
The compensations of energy functions differ in diverse subsystems. However, the differences of starting subsystems are omitted, µ j and λ j only vary with reaching subsystems. It results in less application flexibility than Theorem 1.
Remark 1 Note that (19) and (33) are true only under the condition t i,j ≤ s ≤ t i,j and t j ≤ s ≤ t j , respectively, which cannot be solved by the traditional LMI toolbox. Whereas, setting sample points in the time interval is able to solve it.

Numerical Example
Consider the system (16). Let σ(t) . The switching signal is constructed with Corollary 1 to stabilize the system. Choose α j , j = 1, 2, 3  The switching signal and the divergence time switching signal divergence time Fig. 1 The divergence time and the switching signal under admissible edge-dependent BMADT switching The switching system with the initial state T is simulated, and the state trajectories are depicted in Fig 2. The state converges to zero as t → ∞, which indicates that the admissible edge-dependent BMADT switching signal can stabilize the system (16).
The mode-dependent BMADT switching signal is constructed with Corollary 3 to stabilize the system (16). Choose α j , j = 1, 2, 3 The feasible solutions are obtained by Corollary 3. It is observed that the choose of parameters is much less than that in Corollary 1. Simultaneously, there are t j = t i,j , t j = t i,j , λ j = λ i,j , µ j = µ i,j , i, j ∈ M, i ̸ = j. A mode-dependent BMADT switching signal is generated under t j , t j , λ j and µ j . The divergence time and the switching signal are shown in Fig 3. The state trajectories with initial state x 0 = [3, −5], are shown in Fig 4. It is clear that the mode-dependent BMADT switching signal can stabilize the switched system.

Conclusion
Stability of switched systems with admissible switching signals is studied in this paper. Stability results under the admissible edge-dependent BMADT switching are general in that compensation bounds and admissible edge-dependent BMADT are not confined to be global. In addition, there is a large range of choosing parameters in stabilizing switched systems. With those parameters, the compensations of divergent state are different for diverse admissible transition edges. Besides, stability criteria under the mode-dependent BMADT are more restrictive than those under the admissible edgedependent BMADT, because the differences of starting subsystems are omitted.