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

+is 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.


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. us, it is more meaningful to study the stability in a fixed time interval. e concept of finite-time stability (FTS) was first introduced in 1960s [8,9]. A system is said to be finitetime 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][13][14][15]. For instance, Ren et al. proposed a suitable event-driven communication scheme for the networked switched systems, and finite-time boundedness and inputoutput 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 timeinvariant 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]. erefore, 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.
is is because the analysis of LTV systems is much more complicated than that of linear time-invariant systems. e 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][22][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]. erefore, 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. e paper is organized as follows. e problem to be considered is formulated in Section 2. 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.

Preliminaries
R n and R n×m denote, respectively, n-dimensional real space and n × m-dimensional real matrix space. λ min (R) denotes the minimal of all eigenvalues of matrix R and λ max (R) denotes the maximum of all eigenvalues of matrix R. R T denotes the transpose of matrix R. I denotes identity matrix with an appropriate dimension.
Consider the following SLTV systems: where x(·) ∈ R n is the system state and T f is a positive constant which represents the end time of system operation; switching signal σ(t) is a piecewise constant function that maps from [0, T f ] into the index set M: � 1, 2, . . . , m { }; A i , B i (i ∈ M) are system matrices with appropriate dimensions; d(t) is the bounded time delay with 0 ≤ d(t) ≤ d and _ d(t) ≤ ρ < 1 (d and ρ are positive constants). e system state is denoted by For convenience, we shall introduce the following definitions.
where c 1 and c 2 are given positive constants with c 1 ≤ c 2 . (1) is FTS, the state must be within the prescribed bound in the fixed time interval, and constants c 1 and c 2 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.

Remark 1. Definition 1 implies that if system
holds for some τ D > 0 and N 0 ≥ 0, then τ D and N 0 are called ADT and chattering bound, respectively.

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.
positive constant λ, and constant β > 1, such that the following inequalities hold: and the ADT satisfies where Proof. Take the following Lyapunov-Krasovkii functional candidate: where Calculating the derivatives of V σ(t),1 (t), V σ(t),2 (t), and V σ(t),3 (t) with respect to t along the trajectories of system (1), we can obtained Furthermore, Mathematical Problems in Engineering Inequality (12) can be rewritten as follows: From conditions (3), (5), and (6), we can obtain Since Let t k stand for the instant of kth switching and t k − denote the instant just before t k , σ(t) � σ(t k ), ∀t ∈ [t k , t k+1 ). Integrating both sides of (15) from t k to t, it follows that From (4) and the continuity of x(t), the following inequalities hold: So, Iteratively, we can obtain It is obvious that V σ 0 (0, x 0 ) can be written as follows: where α 1 � max i∈M λ max (P i (0)) , α 2 � max i∈M λ max (Q i (0))}, and α 3 � max i∈M λ max (Z i (0)) . By (19) and (21), According to (9), en, assuming that the total switching number of σ(t) over [0, T f ] is N. From (22) and (23), From the definition of ADT, N � N σ (0, T f ) ≤ N 0 + T f /τ D . By inequality (7), we have It can be transformed into i.e., So, N ln β < ln(c 2 λ p /e λT f η 1 ), and Consequently, from (24),

Finite-Time Stability via State Feedback
In this section, the LMIs conditions, introduced in eorem 1, will be exploited to design a state feedback controller for the SLTV control system with time-varying delay. e resultant closed-loop system could be proved to be FTS. e problem is described concretely as follows. 4 Mathematical Problems in Engineering Given the time interval [0, T f ], let us consider the SLTV control system: where u(·) represents the control input. e aim of this section is to find a memoryless, state feedback control law is FTS with respect to (c 1 , c 2 , T f ). e following theorem states the sufficient conditions for the FTS of the closed-loop system (30) with respect to (c 1 , c 2 , T f ).

Theorem 2. System (30) is FTS with respect to
, and J i (·) and a piecewise continuous matrix-function L i (·) with appropriate dimensions such that the following LMIs hold, for t ∈ [0, T f ], and the ADT satisfies where and the state feedback controller gain Proof. To prove the finite time stability of system (30), we only need to prove that system (30) satisfies all the conditions of eorem 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 ; we pre-and postmultiply (37) by the block tridiagonal matrix (D i (t), I), and we obtain where , it can be obtained that Θ i (t) � Υ i (t) and Π i (t) � Ψ i (t), 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. erefore, from eorem 1, the closed-loop system (30) is FTS.

Mathematical Problems in Engineering
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.

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: Let c 1 � 1, c 2 � 8, and T f � 10 be given. e switching law of system (40) is shown in Figure 1, and it satisfies the ADT parameters τ D � 1.25 and N 0 � 1. With this switching law, Figure 2 shows the system state of system (40) without control and with initial state (−0.1 0.5) T . From Figure 2, it can be seen that the system state is unbounded and is not FTS.
From inequality (7), we let λ � 0.1 and β � 1.05. Applying eorem 2, we design a finite time stable state feedback controller u(t) � K σ(t) (t)x(t). Figure 3 shows the time traces for the two elements of state feedback controller gain K σ(t) (t). It can be seen from Figure 4 that, with the state feedback controller, the system state with initial state (−0.1 0.5) T satisfies x T (t)x(t) < c 2 � 8 over [0, 10].

Remark 3.
Since P(t), Q(t), and Z(t) 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. en, the LMIs can be solved. is approximating method can also be found in [26,27]. Concretely, discretize [0, T f ] into equally spaced time instances t i (i � 0, 1, 2, . . . , N) with t 0 � 0, t N � T f , and t k − t k−1 ≜ ε � (T f /N). us, on the time interval (t k−1 , t k ] with small enough ε, the time-varying matrix-valued functions P(t), Q(t), and Z(t) can be denoted as P(t) � P(t k ), Q(t) � Q(t k ), and Z(t) � Z(t k ), for t ∈ (t k−1 , t k ]. en, _ P(t) can be denoted as (P(t k ) − P(t k−1 )/ε) with sufficiently small ε. Similarly, _ Q(t) and _ Z(t) can be replaced by (Q(t k ) − Q(t k−1 )/ε) and (Z(t k ) − Z(t k−1 )/ε).

Conclusions
is 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. e 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 timevarying 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.

Data Availability
No data were used to support this study. e authors only used MATLAB for simulation. erefore, simulation programming can be obtained from the corresponding author upon request.

Conflicts of Interest
e authors declare that they have no conflicts of interest.  Mathematical Problems in Engineering 7