Global State-Feedback Control for Switched Nonlinear Time-Delay Systems via Dynamic Gains

In this paper, the problem of global state-feedback control is investigated for a class of switched nonlinear time-delay systems. In order to obtain a less-conservative common dynamic gain update law across subsystems, we construct different dynamic gain update laws for individual subsystems. Based on multiple Lyapunov function approach and adding one power integrator technique, the delay-independent controllers for all subsystems and a proper switching law are designed to guarantee that the states of the switched nonlinear time-delay systems can be globally asymptotically to the origin; meanwhile, all the signals of the closed-loop system are bounded. Finally, an example is provided to demonstrate the effectiveness of the proposed method.


Introduction
A switched system is a branch of hybrid system. It consists of multiple subsystems that are either continuous-time or discrete-time ones and a switching law that defines specific subsystems active at instants of time. Switched nonlinear systems are commonly used in practice, such as robot control systems, networked control systems, and aircraft control systems [1][2][3][4][5]. In recent years, the design of switching strategies as well as stability and stabilization issues are critical in control theory and engineering and have achieved considerable results ( [6][7][8][9][10][11][12][13][14][15][16][17] and the references therein). To address these issues, many methods have been presented, such as the common Lyapunov function, the single Lyapunov function method, the average dwell time scheme, and the multiple Lyapunov function method and switched Lyapunov functions. In particular, multiple Lyapunov function method along with the selection of appropriate switching signals provides an effective way for stability analysis and stabilization of switched nonlinear systems [18].
As is well known, time-delays are experienced from time to time in practical control systems, such as ecological systems and industrial procedures, which may cause system instability (see [19,20] and the references therein). For nonswitched nonlinear systems, high-order nonlinear systems without time-delays have been thoroughly investigated in [21]. In the case of growth constraints, the problem of output feedback control was considered in [22,23] for highorder time-delay systems.
Motivated by [23][24][25][26], we construct a memoryless statefeedback controller and abolish the upper bound of the timedelay. For each subsystem, a dynamic gain is introduced in the procedure of the recursive design. With the dynamic gains introduced, some other negative terms are found in the derivative of the Lyapunov function, which may offset stronger system nonlinearities. Although prior knowledge of the upper bound of the time-delay is not required, the system can be regulated to its origin, while all the closedloop signals are bounded. e main contributions of this paper are summarized as follows: (i) without any growth condition on the time-delay systems nonlinearities, delayindependent, nonsmooth but C 0 state-feedback controllers with dynamic gains are developed, which regulate the states of the time-delay system, while keeping the boundedness of the closed-loop system. (ii) e issue of global stabilization for switched nonlinear time-delay systems is studied for the first time by combining multiple Lyapunov function, adding one power integrator method and dynamic gains technique.

Notations.
roughout this paper, R n denotes the n-dimensional real space; R + refers to the set of all nonnegative real numbers; C i denotes the set of all functions with continuous ith partial derivatives; and l i � (l 1 , . . . , l i ) T .

Preliminaries and Problem Statement
In this paper, we consider the problem of the global statefeedback control for switched nonlinear time-delay system described by is a piecewise switching signal taking its values in a finite set M � 1, . . . , m { }, m being the number of subsystems. For i � 1, . . . , n, k ∈ M, p ik are positive odd integers, u k ∈ R is the control input of the kth subsystem. f ik : R 2i ⟶ R and g ik : R n ⟶ R, are C 1 functions with f ik (0, 0) � 0 and g ik (0) � 0. ζ(s) ∈ R n is a continuous function, and takes its values in [− d, 0]. Moreover, it is assumed that the state of system (1) will not jump at the moment of switching, i.e., the trajectory x(t) is a continuous one.
To address the state-feedback control problem of the switched nonlinear time-delay systems (1), the following useful lemmas are given which will be used throughout the paper.
Lemma 1 (see [26]). For positive real numbers p and q, there exists a positive real-valued function 9(x, y) such that Lemma 2 (see [21]). For x, y ∈ R and an integer m ≥ 1, the following inequality holds |x + y| m ≤ 2 m− 1 x m + y m , If m is a positive odd, then Lemma 3 (see [25]). For a continuous function f(x, y), In this paper, we firstly consider the switched nonlinear time-delay systems described by which is a special case of system (1) with g iσ(t) (x) � 0. en, we will design a set of controllers for all subsystems and a switching law to implement global regulation of system (1).
Since f ik (·) is C 1 and f ik (0, 0) � 0, it can be obtained from Lemma 3 that there exist smooth functions φ ikj (x i ) and

Controller Design
Based on the idea of multiple Lyapunov function and adding one power integrator technique, the controller design procedure is summarized as follows.
In the following, we will show that (30) is also held at step i. With this regard, consider the following Lyapunov-Krasovskii functional as where l i ≥ 1 is a dynamic gain to be determined later and Following the same line, we can obtain that W ik (l i− 2 , x i ) is C 1 , and where λ ik > 0 is a constant.

Choose the Lyapunov-Krasovskii functional
(37) e application of (33)-(37) gives rise to Similar to Step 2, we design the gain update law as where By (39), it is easily concluded that Utilizing (32) and (40), it can be verified that 6 Mathematical Problems in Engineering Substituting (39) and (41) into (38) yields en, we choose the virtual controller where α ik (·), β ik (·) are positive C ∞ functions.
Step n. During the inductive argument, there exists a dynamic state-feedback controller such that

Stability Analysis
For simplicity, let V k � V nk and V k � V nk . Now, we are ready to present the main result of this paper. (1), there are the following dynamic state-feedback controllers as _ L � η k (L, x), L ∈ R n− 1 ,

Conclusion
is paper discussed the global state-feedback control problem for a class of switched nonlinear time-delay systems.
is issue can be solved by using explicitly-constructed coordinate transformations and dynamic gains. e designed controllers can be guaranteed that all signals of the system are bounded; meanwhile, the system state globally converges to the origin. In the future, we will further investigate the problem of adaptive output feedback control for switched nonlinear time-delay systems under weaker conditions. In addition, inspired by [29,30], we will further discuss the problem of output feedback control for a class of more general switched nonlinear systems with unknown growth rates.

Data Availability
In our paper, we only use MATLAB for simulation. erefore, we can only provide simulation programming  which can be obtained from the corresponding author upon request.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this paper.