Semi-time-dependent stabilization for a class of continuous-time impulsive switched linear systems

In this paper, the stability and stabilization problems of a class of impulsive switched linear systems with mode-dependent persistent dwell-time (MPDT) switching are investigated in continuous-time domain. The proposed switching law is more general, which not only covers the widely probed dwell-time (DT) and average dwell-time (ADT) as special cases but also extends the persistent dwell time (PDT) in the literature. By invoking novel modal partition time-dependent multiple Lyapunov-like functions, the stability criteria for the systems under MPDT switching in nonlinear settings are established as a first attempt, upon which the desired mode-dependent and semi-time-dependent (STD) stabilizing controllers for continuous-time impulsive switched linear systems are designed. It is demonstrated that the control performance is improved as both the length of admissible MPDT and the number of time partition segment are added. Finally, a numerical example is provided to illustrate the potential of the obtained theoretical results.


Introduction
Hybrid systems, which are a class of dynamic systems containing both continuous and discrete dynamics, have received a considerable attention in the past decades, see, for instance, Hasan et al. (2021), Liu et al. (2020), Albea and Seuret (2021) and Li et al. (2018). As two important categories of hybrid systems, impulsive systems and switched systems have received particular attention in the control community, see Luo et al. (2021), Xu et al. (2021), He et al. (2022), Liu and Liang (2013), Zhao et al. (2022), Liang and Xia (2020), Li, Ahn, Guo, et al. (2020) and , respectively. However, in the real world, dynamic systems with impulsive effects and switching have arisen in various disciplines of science and engineering, such as automotive industry, mechanical systems, air traffic control and networked control. Such systems that usually switch of subsystems and abruptly change of states at the switching instants, form a more comprehensive model, i.e. impulsive switched systems (Jiao et al., 2020;Li et al., 2018;Wang et al., 2022). But due to the existence of impulses, switching events will cause oscillations and even instability. Thus it is important to study the stability and stabilization problems of impulsive switched system.
On the other hand, up until now most of works focus on researching switched systems with and without impulsive jumps under dwell-time (DT) or average dwell-time CONTACT Jitai Liang liangjitai@163.com (ADT) switching on account of its theoretical and practical importance (Gao et al., 2021;Wang et al., 2020;Xi & Liu, 2020), to list a few. However, switched systems under persistent dwell-time (PDT) switching signal including DT and ADT switching signal as special cases have been more attractive subject of research since they are suitable to model the behaviours of many practical systems (Shi et al., 2019;Zhao et al., 2022), such as those are suffered both abrupt and intermittent faults (Verhaegen et al., 2010). Meanwhile, it is worth noting that only few results on switched systems without impulsive jumps in the framework of PDT switching signal have been obtained (Chen et al., 2022;Tong et al., 2021;Zhao et al., 2022). Hence, taking impulsive effect into account, the issues of stability analysis and synthesis on impulsive switched systems under PDT switching deserve further investigation since the states of systems abruptly change at the switching instants. Additionally, when applying the Lyapunov function approach to the impulsive switched system under PDT switching in continuous-time domain, the selected Lyapunov function should be capable of capturing the hybrid structure characteristics of the considered systems and the properties of switching signals. Nonetheless, it is necessary to point out that, in most of the works concerning continuous-time impulsive switched system, the resulting stability analysis and synthesis criteria of systems performed by employing time-invariant Lyapunov functions may be conservative. Furthermore, there are only few results on continuous-time switched systems with DT switching in terms of time-vary Lyapunov functions (Chang et al., 2021;Long, 2018;Tang & Li, 2019), let alone the conclusion under more general PDT switching.
Motivated by the aforementioned observations, this paper aims to solve a set of mode-dependent and semi-time-dependent (MDSTD, for short) stabilizing controllers and find the admissible mode-dependent persistent dwell-time (MPDT) switching signal for continuoustime impulsive switched linear systems in terms of novel modal partition time-dependent multiple Lyapunov-like functions method. Moreover, it should be stressed that the larger time partition segment is chosen, the less conservative result can be obtained. The rest of this paper is organized as follows. In Section 2, the considered systems are formulated and basic definitions are introduced. In Section 3, the stability criteria for the continuous-time impulsive switched systems under MPDT switching in nonlinear settings are first provided to design the desired MDSTD stabilizing controllers for continuous-time impulsive switched linear systems such that the resulting closed-loop systems are globally uniformly asymptotically stable. An example is presented to illustrate the efficiency of the proposed method in Section 4. Finally, this paper is concluded in Section 5.
Notations: The notation used throughout this paper is fairly standard. The subscripts 'T' and '−1' stand for matrix transposition and inverse, R n denotes the ndimensional Euclidean space and Z + represents the set of non-negative integers, Z ≥s 1 denotes the sets {k ∈ Z + | k ≥ s 1 } and Z [s 1 ,s 2 ] stands for the sets {k ∈ Z + | s 2 ≥ k ≥ s 1 }, respectively. The notation P > 0 (P ≥ 0) means that P is symmetric and positive (semi-positive) definite. I and 0 represent, respectively, identity matrix and zero matrix. In symmetric block matrices or complex matrix expressions, we use the symbol ' * ' as an ellipsis for the terms that are introduced by symmetry and diag{· · · } stands for a block-diagonal matrix. · is used to refer to the Euclidean vector norm or the spectral norm for matrices. λ max (·) and λ min (·) denote the maximum and minimum eigenvalues, respectively. The operator a gives the nearest integer greater than or equal to a. Let PC PC([t 0 , +∞), R n ) = {x : [t 0 , +∞) → R n , x(t) be continuous everywhere except a finite number of pointst at which each t ≥ 0, β(·, t) is a class K function and for each s ≥ 0, β(s, ·) is nonincreasing and satisfies lim t→∞ β(s, t) = 0.

Preliminaries and problem formulation
Consider a class of impulsive switched linear systems in the form of: . The switching sequences t 0 , t 1 , t 2 , . . . , t s , . . . are unknown a priori, but are known instantly; when t ∈ [t s , t s+1 ), we say the σ (t s )th subsystem is active with running time to be t s+1 − t s .
In this paper, we are concerned with a class of modedependent persistent dwell-time (MPDT) switching signals, for which we recall the following definitions.
Then, combining Definition 2.1 and the definition on PDT switching in Hespanha (2004), the concept of MPDT is given as follows.
Definition 2.2 (Hespanha, 2004): Consider switching instants t 0 , t 1 , . . . , t s , . . .with t 0 = 0. A positive constant τ i is said to be the mode-dependent persistent dwell-time (MPDT), if there exists an infinite number of disjoint intervals of length no smaller than τ i on which σ is constant, and consecutive intervals with this property are separated by no more than T, where T is called the period of persistence.
Remark 2.1: Definition 2.2 means that, within the period of persistence T, the switching among the subsystems can be arbitrary and the running time of any activated subsystem will be all less than τ i . For notation simplicity, in the MPDT switching, we call the interval of length no smaller than τ i as the τ iportion, and the period of persistence as T-portion. Also, we regard the combination of a τ i -portion and a T-portion as a stage of switching. Let T (p) , p ∈ Z + be the actual running time of the T-portion at the pth stage. It holds that

Definition 2.3:
The equilibrium x = 0 of system (1) is globally uniformly asymptotically stable (GUAS) under certain switching signals σ if for initial condition x 0 , there exists a class of KL function κ such that the solution of the system satisfies Therefore, the purpose of this paper is to design a control law u(t) = F(x(t)) and determines a set of switching signals with admissible MPDT such that the resulting closed-loop system is GUAS.

Remark 2.2:
If the operator F(·) in control law is not only mode dependent (Liang & Xia, 2020) but also time dependent (Xiang & Xiao, 2014), which is the least conservative for controller design. However, it is not realistic to infinitely compute and storage such an operator in practice. Therefore, we will be dedicated to conquer the problem.

Main results
To present our main results, a stability criterion for the impulsive switched systems with MPDT switching in nonlinear settings is first established by invoking timedependent multiple Lyapunov-like functions.
Remark 3.1: It should be noted that since the running time of each activated subsystem during the period of persistence is unknown a priori, the worst case of usinĝ Q(T) times of switching during the period of persistence in the derivations of (8) is allowed.
It is straightforward to see from Lemma 3.1 that the adopted multiple Lyapunov-like functions are not only mode dependent, but also time dependent, which are the least conservative for controller design. However, in practice, it is not tractable to infinitely compute and storage of such multiple Lyapunov-like functions. To overcome this problem, in what follows, we shall restrict our attention to novel modal partition time-dependent (MPT) multiple Lyapunov-like functions (MPTMLFs) where V i (d t , x(t)) is supposed to be mode dependent and time vary. d t is a scheduler for the activated subsystem and can be simply computed according to the following rules: (i) in the τ i -portion, where L r arg sup r {t s p +r | t s p +r ≤ t, Such a setting will generate a simplified version of Lemma 3.1 as the following.

Lemma 3.2:
Consider the continuous-time impulsive switched nonlinear system (2) and let λ i > 0, μ i > 0, i ∈ S be given constants. If for a prescribed period of persistence T, there exist class K ∞ functions κ 1 , κ 2 , such that ∀i, j ∈ S, (10)

Remark 3.2:
It is worth mentioning that based on (10) and (11),Q(T) depends on period of persistence T, minimum persistent dwell-time τ i , ∀i ∈ S and maximum seg- For later development of the controller design for system (1), we construct novel MPT quadratic multiple Lyapunov-like functions (MPTQMLFs) where the piecewise continuous matrix function P i (d t ), t ∈ [t s p , t s p+1 ) is chosen to be linear within each segment R t sp ,n , R t sp+Lr ,k , and in the interval [t s p + τ i , t s p +1 ) matrix function P i (d t ) is fixed as a constant matrix P i (t s p + τ i ) P i,H i . Then since the matrix function P i (d t ), i ∈ S is piecewise linear in the interval R t sp ,n and R t sp+Lr ,k , it can be expressed by means of the values at dividing points using a linear interpolation formula. When t ∈ R t sp ,n , n = 0, 1, . . . H i − 1, let σ (t s p ) = i, P i,n P i (t s p + θ n ), When t ∈ R t sp+Lr ,k , k = 0, 1, . . . H j − 2, let σ (t s p +L r ) = j, Hence the piecewise continuous matrix function P i (d t ), i ∈ S is described as The matrix function P σ (t) (d t ) of the above form (22) (20) and (21) is linear interpolation formula. It is convenient for us to verify the stability of system. (iii) The resulting controller via such MPTQMLFs will be model dependent and semi-time-dependent (MDSTD, for short), intuitively, a less conservative result can be achieved. Now, we are ready to propose the following results for the impulsive switched linear systems (1) with MPDT switching on the basis of the above discussions. (1) with u(t) ≡ 0 and let λ i > 0, μ i > 0, i ∈ S be given constants. If for a prescribed period of persistence T, there exist matrices P i,n 0, n = 0, 1, . . . H i , i ∈ S such that for all n = 0, 1, . . . H i − 1, and i ∈ S, the following linear matrix inequalities (LMIs) are satisfied:

Proof: Choose MPTQMLFs
When t ∈ R t sp ,n , n = 0, 1, . . . H i − 1, one haṡ and (24), one obtainṡ On the other hand, when t ∈ [t s p + τ i , t s p +1 ), V i (d t , x(t)) = x T (t)P i,H i x(t), by (25), it holds thaṫ , which implies (13)-(15) is satisfied. In addition, assuming that the system switches from subsystem i to j at time t s p +r when r ∈ Z [2,Q(t sp+1 ,t s p+1 )+1] , condition (27) implies Likewise, it follows from (26) that These imply that (16) and (17) can be satisfied. Therefore, the GUAS of systems (1) governed by switching law (18) can be established on the basis of Lemma 3.2.

Remark 3.4:
It is worth noting that the larger H i is chosen, the denser the division of interval R t sp ,n and R t sp+Lr ,k come about and, intuitively, a less conservative result can be obtained. Now, we focus on studying controller of the form Based on the above theoretical results, we are in a position to give the existence conditions of a set of stabilizing controllers for system (1) under MPDT switching.
Then the close-loop system with controller (28) is GUAS for any switching signal under MPDT satisfying (18), where the controller gains are given as Proof: Theorem 3.3 implies that if the following conditions are satisfied and system (1) is GUAS for any switching signal with MPDT satisfying (18).
to be piecewise linear as P i (d t ) given in (22). Pre-and Postmultiplying both sides of the LMIs in (29) with diag{P i,n , P i,n+1 }, the LMIs in (31) with diag{P i,H i , P i,H i }, the LMIs in (32) with diag{P i,H i , P j,0 }, the LMIs in (33) with diag{P i,m , P j,0 }, the LMIs in (30) with diag{P i,n+1 , P i,n+1 }, and using Schur complement and the matrix inequality, for i ∈ S, n = 0, 1, . . .
Remark 3.5: The main advantage of time-scheduled controller gain K(t) is not only mode dependent but also semi-time-dependent. However, K(t) doesn't vary piecewise linearly in time, since Similarly, we can obtain the following conclusion for the system (1) with PDT switching. Here we omit its proof to avoid the repetition.

Criteria for controller design
Theorem 2 Corollary 2 Switching signal τ * 1 = 0.3, τ * 2 = 0.3 Infeasible that there exist matrices U i,n 0, Y i,n , i ∈ S, n = 0, 1, . . . H such that ∀n = 0, 1, . . . , H − 1, ∀i, j ∈ S, Remark 3.6: Sufficient conditions are derived and formulated in terms of LMIs for given λ i , μ i , H i , i ∈ S in Theorem 3.4, that would reduce the conservativeness to find feasible controllers under MPDT switching for the system than using λ, μ, H corresponding to the scheme of PDT switching.
We also present as below the time-independent controller that can be obtained by the corresponding 'd tindependent' Lyapunov function V i (x(t)), ∀i ∈ S, reduced from (19). The proof can be obtained in a similar vein to the one for Theorem 3.4 and omitted here.
Corollary 3.6: Consider the system (1) and let λ i > 0, μ i > 0, i ∈ S be given constants. For a period of persistence T, suppose that there exist matrices U i 0, Y i , such that ∀i, j ∈ S, then there exists a set of stabilizing controllers such that system (1) is GUAS for any switching signal with MPDT satisfying (6). Moreover, if (47) and (48) have a solution, the controller gains can be given by Remark 3.7: The MPTQMLFs (19) play a crucial role in deriving Theorems 3.3 and 3.4. We note that P i (d t ) and U i (d t ) are piecewise linear in time t, intuitively, which lead to less conservative than time-independent one.

Numerical example
In this section, a numerical example is provided to show the advantage and effectiveness of the obtained theoretical results. Example Consider the impulsive switched systems (1) composed of two subsystems as A 1 = −1.9 3.3 2.6 3.8 , A 2 = 3 5.9 3.8 1.4 , E 1 = 1.7 0.5 1.2 −1.9 , E 2 = 1.6 1 0.8 1.1 , Our purpose here is to design a set of modedependent and semi-time-dependent (MDSTD, for short) stabilizing controllers and find the admissible switching signals with MPDT such that the resulting closed-loop system is stable.
To illustrate the advantages of the proposed MDSTD stabilization, we shall present the design results of switching signals for the sake of comparison. By different strategies and setting the relevant parameters appropriately, the computation results for the system are listed in Table 1.
It is easy to see from Table 1 that, as stated in Remark 3.7, the MLF approach in Corollary 3.6 is infeasible for the systems. It is indicated that the MPTQMLFs method is less conservative than the MLF method.
Generate one possible switching sequences with MPDT property as shown in Figure 1, i.e. T = 2, τ 1 = 1.2, τ 2 = 1.5. The control objective is to reach the equilibrium state by STD and MDSTD stabilizing controllers, respectively, as fast as possible and with minimum overshoot. In Figure 2, the responses of the closed-loop switched system are given. For comparison purposes, four strategies are used. The results plotted in this figure illustrate the efficiency of such strategies. One can also obtain the corresponding state responses of the closedloop system under different parameters H i (or H) as shown in Figure 3. It is shown that (a) (or (b)) is not only fluctuated more greatly than (c) (or (d)) but also converges to the equilibrium state more slowly. Thus, as mentioned in Remark 3.4, we can claim that less conservative results can be obtained as the parameter H i (or H) increases.

Conclusion
The stability and stabilization problems for continuoustime impulsive switched systems with MPDT switching are studied via novel MPTMLFs method. The proposed MPDT switching is less restrict than the traditional PDT switching in practice. The stability results for the continuous-time impulsive switched systems with MPDT are derived in both nonlinear and linear contexts. Moreover, the minimal MPDT for admissible switching signals and the corresponding MDSTD state feedback controller are designed for the systems. The effectiveness of the proposed approach has been demonstrated through a numerical example.

Disclosure statement
No potential conflict of interest was reported by the author.