# Finite-Time Stability and Stabilization of Switched Linear Time-Varying Systems with Time-Varying Delay

Authors: Lulu Feng; Ping Zhao
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

This paper deals with the finite-time stability (FTS) of switched linear time-varying (SLTV) systems with time-varying delay. Firstly, based on Lyapunov–Krasovskii functional technique and average dwell time (ADT) approach, a sufficient criterion on FTS for SLTV systems with time-varying delay is obtained. For the SLTV system with delay and control input, based on the criterion, a state feedback controller is designed such that the closed-loop system is finite-time stable (FTS). Finally, an example is employed to illustrate the validity of our results.


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## Body

## 1. Introduction

Switched systems are a special kind of hybrid dynamical systems, which consist of several subsystems and a switching law that orchestrates the switching between the subsystems [1]. It has been widely applied in intelligent traffic control [2], formation flying [3], sensor networks [4, 5], robotics [6, 7], and so on. In practice, time delays widely exist in many switched systems, which may cause some unexpected errors or even lead to crash; therefore, stability analysis for switched systems with time delay is particularly important and has been extensively studied. Most of the existing results focus on Lyapunov asymptotic stability (LAS). As we all know, LAS is defined over infinite-time interval. However, due to large state amplitude in the transient process, some asymptotic stable systems may be useless. Thus, it is more meaningful to study the stability in a fixed time interval. The concept of finite-time stability (FTS) was first introduced in 1960s [8, 9]. A system is said to be finite-time stable (FTS) if its state does not exceed a certain bound during a specified time interval for a given bound on the initial condition. For such stability, there are many practical applications such as networked control systems [10] and network congestion control [11].Up to now, many important results on FTS have been reported [12–15]. For instance, Ren et al. proposed a suitable event-driven communication scheme for the networked switched systems, and finite-time boundedness and input-output FTS are simultaneously considered [16]. Yu et al. investigated FTS of switched positive linear systems with time-varying delays using the ADT method, and a sufficient condition on FTS is provided for the case of time-invariant systems [17].We can see that the majority of the existing literatures related to finite-time stability analysis are mainly on time-invariant systems. However, many systems in practical applications are time-varying. For the active suspension system in auto diving, the vehicle sprung and unsprung masses vary with the loading conditions. To increase the control accuracy, this system should be represented by a time-varying system [18]. Therefore, the stability analysis for time-varying systems is very important and necessary. In the past few decades, the stability theory of linear time-invariant systems has developed considerably [19, 20], while that of linear time-varying (LTV) systems is comparatively slow. This is because the analysis of LTV systems is much more complicated than that of linear time-invariant systems. The state transition matrixes of LTV systems, which are generally needed to ascertain the properties of stability, are impossible to be derived in addition to particular cases [21–23]. So, we must find another way to analyze the stability of SLTV systems with time-varying delays. With the help of Lyapunov functions, a series of differential Lyapunov-inequality-based sufficient conditions is derived for FTS analysis. And, as is well known, general Lyapunov–Krasovskii functionals with more information on time delay are helpful for reducing conservatism [24, 25]. Therefore, in this paper, we analyze FTS of SLTV systems with time-varying delays using the Lyapunov–Krasovskii functional method.In this paper, the main contributions of this paper can be summarized as follows: (1) sufficient conditions on FTS of the switched system will be given under the case of LTV systems; (2) a state feedback controller will be designed for the SLTV system with time-varying delays.The paper is organized as follows. The problem to be considered is formulated in Section2. In Section 3, FTS criterions are derived in terms of linear matrix inequalities (LMIs). In Section 4, a state feedback controller will be designed which can guarantee the FTS of the systems. In Section 5, a numerical example will be given to show the effectiveness of the proposed method. Section 6 concludes the paper.


## 2. Preliminaries

ℝn and ℝn×m denote, respectively, n-dimensional real space and n×m-dimensional real matrix space. λminR denotes the minimal of all eigenvalues of matrix R and λmaxR denotes the maximum of all eigenvalues of matrix R. RT denotes the transpose of matrix R. I denotes identity matrix with an appropriate dimension.Consider the following SLTV systems:(1)x˙t=Aσttxt+Bσttxt−dt,t∈0,Tf,xs=φs,∀s∈−d,0,where x⋅∈ℝn is the system state and Tf is a positive constant which represents the end time of system operation; switching signal σt is a piecewise constant function that maps from 0,Tf into the index set M:=1,2,…,m; Ai,Bii∈M are system matrices with appropriate dimensions; dt is the bounded time delay with 0≤dt≤d and d˙t≤ρ<1 (d and ρ are positive constants). The system state is denoted by xt,σ0,φ with initial mode σ0=σ0 and initial state φs,s∈−d,0. When clear from the context, we write xt instead of xt,σ0,φ.For convenience, we shall introduce the following definitions.Definition 1.
System (1) is said to be FTS with respect to c1,c2,Tf if(2)sup−d≤s≤0φTsφs≤c1⟹xTtxt≤c2,∀t∈0,Tf,where c1 and c2 are given positive constants with c1≤c2.Remark 1.
Definition1 implies that if system (1) is FTS, the state must be within the prescribed bound in the fixed time interval, and constants c1 and c2 are usually assigned according to the actual situation of the specific problem. In engineering applications, there are many systems that work on a finite-time interval, such as the climb process of hypersonic aircraft, and it must reach supersonic speed within the fixed time interval; in the process of spacecraft rendezvous and docking, the two spacecrafts must be combined in space orbit and structurally connected into a whole within the fixed time interval.Definition 2.
LetNσt,T denote the switching number of σt over t,T, ∀T>t≥0. If Nσt,T≤N0+T−t/τD holds for some τD>0 and N0≥0, then τD and N0 are called ADT and chattering bound, respectively.


## 3. Finite-Time Stability Analysis

In this section, using Lyapunov–Krasovkii functional technique and ADT approach, a sufficient condition on FTS of system (1) will be given as follows.Theorem 1.
System (1) is FTS with respect to c1,c2,Tf if, for any i∈M, there are positive definite matrix-valued functions Pit,Qit,Zit∈ℝn×n, positive constant λ, and constant β>1, such that the following inequalities hold:(3)ΨitPitBitBiTtPit−eλdQit1−ρ<0,(4)Pit≤βPjt,Qit≤βQjt,Zit≤βZjt,∀i,j∈M,(5)Q˙it≤0,(6)Z˙it≤0,and the ADT satisfies(7)τD>Tflnβlnc2+lnλp−λTf−lnη1+N0lnβ,where(8)Ψit=AiTtPit+P˙it+PitAit+Qit+dZit−λPit,λp=mini∈MλminPit,η1=α1c1+deλdα2c1+d2eλdα3c1,α1=maxi∈MλmaxPi0,α2=maxi∈MλmaxQi0,α3=maxi∈MλmaxZi0.Proof.
Take the following Lyapunov–Krasovkii functional candidate:(9)Vσtt,xt=Vσt,1t,xt+Vσt,2t,xt+Vσt,3t,xt,where(10)Vσt,1t,xt=xTtPσttxt,Vσt,2t,xt=∫t−dttxTseλt−sQσttxsds,Vσt,3t,xt=∫−d0∫t+θtxTseλt−sZσttxsdsdθ.
Calculating the derivatives ofVσt,1t, Vσt,2t, and Vσt,3t with respect to t along the trajectories of system (1), we can obtained(11)V˙σt,1t,xt=xTtAσtTtPσtt+P˙σtt+PσttAσttxt+xTt−dtBσtTtPσttxt+xσtTtPσttBσttxt−dt,V˙σt,2t,xt=λVσt,2t+xTtQσttxt−eλdtxTt−dtQσttxt−dt1−d˙t+∫t−dtteλt−sxTsQ˙σttxsds,V˙σt,3t,xt=λVσt,3t−∫t−dteλt−sxTsZσttxsds+∫−d0∫t+θteλt−sxTsZ˙σttxsdsdθ+dxTtZσttxt.
Furthermore,(12)V˙σtt,xt−λVσtt,xt=xTtAσtTtPσtt+P˙σtt+PσttAσtt+Qσtt+dZσtt−λPσttxt+xTt−dtBσtTtPσttxt−eλdtQσttxt−dt1−d˙t+xTtPσttBσttxt−dt−∫t−dteλt−sxTsZσttxsds+∫t−dtteλt−sxTsQ˙σttxsds+∫−d0∫t+θteλt−sxTsQ˙σttxsdsdθ≤xTtAσtTtPσtt+P˙σtt+PσttAσtt+Qσtt+dZσtt−λPσttxt+xTt−dtBσtTtPσttxt−eλdQσttxt−dt1−ρ+xTtPσttBσttxt−dt−∫t−dteλt−sxTsZσttxsds+∫t−dtteλt−sxTsQ˙σttxsds+∫−d0∫t+θteλt−sxTsQ˙σttxsdsdθ.
Inequality (12) can be rewritten as follows:(13)V˙σtt,xt−λVσtt,xt≤xtxt−dtTΨσttPσttBσttBσtTtPσtt−eλdQσtt1−ρxtxt−dt−∫t−dtxTseλt−sZσttxsds+∫t−dtxTseλt−sQ˙σttxsds+∫−d0∫t+θtxTseλt−sZ˙σttxsdsdθ.
From conditions (3), (5), and (6), we can obtain(14)V˙σtt,xt−λVσtt,xt<0.
Sinced/dte−λtVit,xt=e−λtV˙it,xt−λVit,xt, inequality (14) can be transformed into(15)ddte−λtVσtt,xt<0.
Lettk stand for the instant of kth switching and tk− denote the instant just before tk, σt=σtk,∀t∈tk,tk+1. Integrating both sides of (15) from tk to t, it follows that(16)Vσtkt,xt≤eλt−tkVσtktk,xtk,t∈tk,tk+1.
From (4) and the continuity of xt, the following inequalities hold:(17)Vσtkt,xt≤βeλt−tkVσtk−tk,xtk,Vσtk−tk,xtk≤eλtk−tk−1Vσtk−1tk−1,xtk−1.
So,(18)Vσtt,xt≤βeλt−tk−1Vσtk−1tk−1,xtk−1.
Iteratively, we can obtain(19)Vσtt,xt≤βNσ0,teλtVσ00,x0.
It is obvious thatVσ00,x0 can be written as follows:(20)Vσ00,x0=xT0Pσ00x0+∫−d00xTseλt−sQσ00xsds+∫−d0∫θ0xTseλt−sZσ00xsdsdθ,(21)≤α1c1+deλdα2c1+d2eλdα3c1,where α1=maxi∈MλmaxPi0, α2=maxi∈MλmaxQi0, and α3=maxi∈MλmaxZi0. By (19) and (21),(22)Vσtt,xt≤βNσ0,teλtα1c1+deλdα2c1+d2eλdα3c1.
According to (9),(23)Vσtt,xt>xTtPσttxt≥λminPσttxTtxt≥λpxTtxt.
Then, assuming that the total switching number ofσt over 0,Tf is N. From (22) and (23),(24)xTtxt<βNσ0,teλtα1c1+deλdα2c1+d2eλdα3c1λp−1=βNσ0,teλtη1λp−1<βNeλTfη1λp−1.
From the definition of ADT,N=Nσ0,Tf≤N0+Tf/τD. By inequality (7), we have(25)Tflnβlnc2+lnλp−λTf−lnη1+N0lnβ<τD.
It can be transformed into(26)N0+TfτD<lnc2+lnλp−λTf−lnη1lnβ,i.e.,(27)N<lnc2+lnλp−λTf−lnη1lnβ.
So,Nlnβ<lnc2λp/eλTfη1, and(28)βNeλTfη1λp−1<c2.
Consequently, from (24), xTtxt<c2 for t∈0,Tf. System (1) is FTS with respect to c1,c2,Tf.


## 4. Finite-Time Stability via State Feedback

In this section, the LMIs conditions, introduced in Theorem1, will be exploited to design a state feedback controller for the SLTV control system with time-varying delay. The resultant closed-loop system could be proved to be FTS. The problem is described concretely as follows.Given the time interval0,Tf, let us consider the SLTV control system:(29)x˙t=Aσttxt+Bσttxt−dt+Gσtut,t∈0,Tf,xs=φs,∀s∈−τ,0,where u⋅ represents the control input. The aim of this section is to find a memoryless, state feedback control law ut=Kσttxt, such that the resultant closed-loop system(30)x˙t=Aσtt+GσttKσttxt+Bσttxt−dt,t∈0,Tf,xs=φs,∀s∈−τ,0,is FTS with respect to c1,c2,Tf.The following theorem states the sufficient conditions for the FTS of the closed-loop system (30) with respect to c1,c2,Tf.Theorem 2.
System (30) is FTS with respect to c1,c2,Tf if there exist piecewise continuously differentiable positive definite matrix-functions Di⋅, Hi⋅, and Ji⋅ and a piecewise continuous matrix-function Li⋅ with appropriate dimensions such that the following LMIs hold, for t∈0,Tf,(31)ΘitBitBiTtΠit<0,(32)βDit≥Djt,Hit≤βHjt,Jit≤βJjt,∀i,j∈M,(33)H˙it≤0,(34)J˙it≤0,and the ADT satisfies(35)τD>Tflnβlnc2+lnγp−λTf−lnδ1+N0lnβ,where(36)Θit=DitAiTt−D˙it+AitDit+Hit+dJit−λDit+GitLit+LiTtGiTt,Πit=−eλdHit1−ρ,μ1=maxi∈MλmaxDi−10,μ2=maxi∈MλmaxDi−10Hi0Di−10,μ3=maxi∈MλmaxDi−10Ji0Di−10,γp=mini∈MλminDi−1t,and the state feedback controller gain Kit=LitDi−1t,∀i∈M.Proof.
To prove the finite time stability of system (30), we only need to prove that system (30) satisfies all the conditions of Theorem 1. Concretely, we only need to prove conditions (3)–(6) are equivalent to (31)–(34), separately.
Now, we prove the equivalence of (3) and (31). For system (30), from (3), we have(37)ΦitPitBitBiTtPit−eλdQit1−ρ<0,where Φit=Aσtt+GσttKσttTPit+P˙it+PitAσtt+GσttKσtt+Qit+dZit−λPit. Let Dit=Pi−1t; we pre- and postmultiply (37) by the block tridiagonal matrix Dit,I, and we obtain(38)ϒitBitBiTtΨit<0,where(39)ϒit=DitAiTt+DitGiTtKiTt−D˙it+AitDit+GitKitDit+DitQitDit+dDitZitDit−λDit,Ψit=−eλdDitQittDit1−ρ.
Then, fromKit=LitDi−1t, Hit=DitQitDit, and Jit=DitZitDit, it can be obtained that Θit=ϒit and Πit=Ψit, and further, (31) and (3) are equivalent. In the same way, equivalence between (4) and (32) is straightforward. Similarly, (5) and (33) are equivalent, (6) and (34) are equivalent. Therefore, from Theorem 1, the closed-loop system (30) is FTS.Remark 2.
Here, it is assumed that there is no delay between the controllers and the subsystems, i.e., the control act on the subsystems without delay.


## 5. Numerical Example

In this section, a numerical example will be given to illustrate the validity of our results.Example 1.
Consider the following SLTV system:(40)x˙t=Aσttxt+Bσttxt−dt+Gσtut,t∈0,Tf,xs=φs,∀s∈−τ,0,where σt∈M=1,2, dt=0.0001t,(41)A1=0.01t0.01t0.1cost0.1,B1=0.01−0.01t0.1sint0.01,G1=14,A2=0.01−0.01t0.1sint0.01,B2=0.01t0.01t0.1cost0.1,G2=41.
Letc1=1,c2=8, and Tf=10 be given. The switching law of system (40) is shown in Figure 1, and it satisfies the ADT parameters τD=1.25 and N0=1. With this switching law, Figure 2 shows the system state of system (40) without control and with initial state −0.10.5T. From Figure 2, it can be seen that the system state is unbounded and is not FTS.
From inequality (7), we let λ=0.1andβ=1.05. Applying Theorem 2, we design a finite time stable state feedback controller ut=Kσttxt. Figure 3 shows the time traces for the two elements of state feedback controller gain Kσtt. It can be seen from Figure 4 that, with the state feedback controller, the system state with initial state −0.10.5T satisfies xTtxt<c2=8 over 0,10.Figure 1
Switching law.Figure 2
State of system (40) with u≡0.Figure 3
Time traces of the elements in state feedback controller gainKit=k1t,k2t.Figure 4
State of the resultant closed-loop system of system (40).Remark 3.
SincePt,Qt,andZt are continuously time varying, LMIs cannot be solved directly. In order to get a computationally tractable problem, we discretize the time interval into equally spaced subintervals. If the length of the subintervals is sufficient small, the continuous functions can be approximated by piecewise functions. Then, the LMIs can be solved. This approximating method can also be found in [26, 27]. Concretely, discretize 0,Tf into equally spaced time instances tii=0,1,2,…,N with t0=0,tN=Tf, and tk−tk−1≜ε=Tf/N. Thus, on the time interval tk−1,tk with small enough ε, the time-varying matrix-valued functions Pt,Qt,andZt can be denoted as Pt=Ptk,Qt=Qtk,andZt=Ztk, for t∈tk−1,tk. Then, P˙t can be denoted as Ptk−Ptk−1/ε with sufficiently small ε. Similarly, Q˙tandZ˙t can be replaced by Qtk−Qtk−1/εandZtk−Ztk−1/ε.


## 6. Conclusions

This paper focuses on the FTS analysis of SLTV systems with time-varying delays. For this kind of systems, based on the Lyapunov–Krasovskii functional technique and ADT method, a sufficient criterion on FTS is obtained. Based on this criterion, using the LMI method, a state feedback controller is designed such that the system is FTS. Finally, a numerical example illustrates the validity of the obtained results. The innovation mainly lies in the studying of FTS for the switched system on the basis of the LTV system, which has become more and more important now. And, a time-varying state feedback controller is designed. Future efforts will be devoted to obtain less conservative criteria for the finite-time stability of linear time-varying system and extend the FTS theory to the neural network context.



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