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

Authors: Hui Ye; Bin Jiang; Hao Yang; Gui-Hua Zhao
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

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.


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## Body

## 1. 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 [1–5]. 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 ([6–17] 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 high-order time-delay systems.Motivated by [23–26], we construct a memoryless state-feedback controller and abolish the upper bound of the time-delay. 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 closed-loop signals are bounded. The main contributions of this paper are summarized as follows: (i) without any growth condition on the time-delay systems nonlinearities, delay-independent, nonsmooth but C0 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) The 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.
Throughout this paper,ℝn denotes the n-dimensional real space; ℝ+ refers to the set of all nonnegative real numbers; Ci denotes the set of all functions with continuous ith partial derivatives; and l¯i=l1,…,liT.


## 2. Preliminaries and Problem Statement

In this paper, we consider the problem of the global state-feedback control for switched nonlinear time-delay system described by(1)x˙it=xi+1piσtt+fiσtx¯it,x¯it−d+giσtxt,i=1,…,n−1,x˙nt=uσtpnσtt+fnσtxt,xt−d,xs=ζs,s∈−d,0,where x=x1,…,xnT∈ℝn is the system state. For i=1,…,n, x¯i=x1,…,xiT and x¯it−d=x1t−d,…,xit−dT. d≥0 is an unknown system time-delay. σt 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, pik are positive odd integers, uk∈ℝ is the control input of the kth subsystem. fik:ℝ2i⟶ℝ and gik:ℝn⟶ℝ, are C1 functions with fik0,0=0 and gik0=0. ζs∈ℝn 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 xt 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 numbersp and q, there exists a positive real-valued function ϱx,y such that(2)xpyq≤pp+qϱx,yxp+q+qp+qϱx,y−p/qyp+q.Lemma 2 (see [21]).
Forx,y∈ℝ and an integer m≥1, the following inequality holds(3)x+ym≤2m−1xm+ym,x+y1/m≤x1/m+y1/m≤2m−1/mx+y1/m.
Ifm is a positive odd, then(4)x−ym≤2m−1xm−ym.Lemma 3 (see [25]).
For a continuous functionfx,y, there are C∞ functions ax≥0, by≥0, cx≥1 and dx≥1, such that fx,y≤ax+by, fx,y≤cxdy.In this paper, we firstly consider the switched nonlinear time-delay systems described by(5)x˙it=xi+1piσtt+fiσtx¯it,x¯it−d,i=1,…,n−1,x˙nt=uσtpnσtt+fnσtxt,xt−d,xs=ζs,s∈−d,0,which is a special case of system (1) with giσtx=0. Then, we will design a set of controllers for all subsystems and a switching law to implement global regulation of system (1).Sincefik⋅ is C1 and fik0,0=0, it can be obtained from Lemma 3 that there exist smooth functions φikjx¯i and ψikjx¯it−d, i=1,…,n,k∈M,j=1,…,i, such that(6)fik⋅≤∑j=1iφikjx¯jxj+∑j=1iψikjx¯jt−dxjt−d.


## 3. Controller Design

Based on the idea of multiple Lyapunov function and adding one power integrator technique, the controller design procedure is summarized as follows.Step 1.
Letξ1k=x1 and choose the Lyapunov function V˜1kx1,l1=1/21+1/l1ξ1k2 with l1t≥1 being a dynamic gain to be determined in the next step. By simple calculation, the time derivative of V˜1k yields(7)V˜˙1kx1,l1=1+1l1ξ1kx2p1k+f1k−l˙12l12ξ1k2≤1+1l1ξ1kx2k∗p1k+2ξ1kξ2k+2ξ1kf1k−l˙12l12ξ1k2,where x2k∗p1k is the first virtual controller to be design later and ξ2k=x2p1k−x2k∗p1k.
It can be obtained from (6) that there exist C∞ functions φ1k1x1 and ψ1k1x1t−d such that(8)f1k≤φ1k1x1x1+ψ1k1x1t−dx1t−d.
Consequently, one has(9)2ξ1kf1k≤2ξ1k2φ1k1x1+ξ1k2+ξ1k2t−dψ1k12x1t−d.
Consider the Lyapunov–Krasovskii functional(10)V1k=V˜1k+∫t−dtξ1k2μψ1k12xμdμ,with (7) and (9) in mind, the time derivative of V1k becomes(11)V˙1k≤1+1l1ξ1kx2k∗p1k+ξ2k2−l˙12l12ξ1k2+ξ1k22+2φ1k1x1+ψ1k12x1.
Design the virtual controller as(12)x2k∗p1k=−ξ1kn−12+3+2φ1k1x1+ψ1k12x1+α1kx1≜−ξ1kβ1kx1,where α1kx1,β1kx1∈C∞. This together with l1t≥1 leads to(13)V˙1k≤−1+α1kx1ξ1k2−n−12ξ1k2+ξ2k2−l˙12l12ξ1k2.Step 2.
Take the Lyapunov–Krasovskii functional into consideration(14)V˜2k=V1k+1l11+1l2W2k+12l1l2ξ1k2,where W2k=∫x2k∗x2μp1k−x2k∗p1k2−1/p1kdμ and l2t≥1 is a dynamic gain to be determined later. Then, one has(15)∂W2k∂x2=ξ2k2−1/p1k,∂W2k∂x1=1p1k−2∂x2k∗p1k∂x1∫x2k∗x2μp1k−x2k∗p1k1−1/p1kdμ,λ2kx2−x2k∗2p1k≤W2kx1,x2≤2p1k−1ξ2k2,where λ2k>0 is a constant for k∈M.
Due toljt≥1,j=1,2, combining (13)–(15), one has(16)V˜˙2k≤−1+α1kx1ξ1k2−n−12ξ1k2+ξ2k2−l˙12l12ξ1k2+1l11+1l2ξ2k2−1/p1kx3k∗p2k+x3p2k−x3k∗p2k+2l1ξ2k2−1/p1kf2k+∂W2k∂x1x˙1+1l1l2ξ1kx˙1−l˙1l12W2k−l˙1l2+l1l˙2l12l22W2k+ξ1k22,where x3k∗p2k is the second virtual controller.
From (6), (12), (15), and Lemmas 1 and 2, one has(17)2l1ξ2k2−1/p1kf2k≤ξ2k2Ψ2k,1x¯2+1l1ξ1k2t−dΨ2k,2∗x1t−d+1l1ξ1k2Ψ2k,2x1+ξ2k2t−dΨ2k,1∗x¯2t−d,(18)2l1∂W2k∂x1x˙1+1l1l2ξ1kx˙1≤ξ2k2Φ2k,1x¯2+1l1ξ1k2Φ2k,2x1+1l1ξ1k2t−dΦ2k∗x1t−d,where Ψ2k,j, Ψ2k,j∗, Φ2k,j, j=1,2, and Φ2k∗ are nonnegative C∞ functions for ∀k∈M.
Constructing the Lyapunov–Krasovskii functional(19)V2k=V˜2k+∫t−dtξ2k2μΨ2k,1∗x¯2μdμ+∫t−dt1l1μξ1k2μΨ2k,2∗x1μ+Φ2k∗x1μdμ,which together with (16) and (17) yields(20)V˙2k≤−1+α1kx1ξ1k2−n−12ξ1k2−l˙12l12ξ1k2+1l1ξ1k2Ψ2k,2x1+Ψ2k,2∗x1+Φ2k,2x1+Φ2k∗x1+1l1t−1l1t−dξ1k2t−d×Ψ2k,2∗x1t−d+Φ2k∗x1t−d+1l11+1l2ξ2k2−1/p1kx3k∗p2k+x3p2k−x3k∗p2k+ξ2k23+Ψ2k,1x¯2+Ψ2k,1∗x¯2+Φ2k,1x¯2−l˙1l12W2k−l˙1l2+l1l˙2l12l22W2k+ξ1k22.
To guarantee the boundedness ofl1, we design the following gain update law as(21)l˙1=max−l12+2l1η1kx1,0,l10=1,where η1kx1=Ψ2k,2x1+Ψ2k,2∗x1+Φ2k,2x1+Φ2k∗x1, k∈M.
From (21), it can be concluded that(22)0<l˙1≤2l1η1kx1,l˙1≥−l12+2l1η1kx1,l1t≥l1t−d≥1,which leads to(23)l˙12l12ξ1k2≤12ξ1k2−1l1ξ1k2η1kx1,1l1t−1l1t−d≤0,−l˙1l12W2k−l˙1l2+l1l˙2l12l22W2k+ξ1k22≤−l˙2l1l22W2k+ξ1k22.
Substituting (21) and (23) into (20), it yields(24)V˙2k≤−1+α1kx1ξ1k2−n−22ξ1k2+1l11+1l2ξ2k2−1/p1kx3k∗p2k+ξ2k23+Ψ2k,1x¯2+Ψ2k,1∗x¯2+Φ2k,1x¯2−l˙2l1l22W2k+ξ1k22+1l11+1l2ξ2k2−1/p1kx3p2k−x3k∗p2k.
We design the virtual controller(25)x3k∗p2k=−l1ξ2k1/p1kn+3+α2kx¯2+Ψ2k,1x¯2+Ψ2k,1∗x¯2+Φ2k,1x¯2≜−l1ξ2kβ2kx¯21/p1k.
By Lemmas 1 and 2, one has(26)2ξ2k2−1/p1kx3p2k−x2∗p2k≤ξ2k2+b2ξ23,where ξ2=x3p1kp2k−x2∗p1kp2k and b2>0 is a constant.
Combining (24)–(26), one has(27)V˙2k≤−∑j=121+αjkx¯jξjk2−n−2ξ1k22+ξ2k2+b3kξ3k2−l˙2l1l22W2k+ξ1k22.
Inductive steps: assume at stepi−1, there is a Lyapunov–Krasovskii functional Vi−1,k, k∈M, a chain of dynamic gains lj≥1, j=1,…,i−1, depicted by(28)l˙1=max−l12+2l1η1kx1,0,l10=1,l˙j=max−δjklj2+2ljηjkl¯j−1,x¯j,0,lj0=1,j=2,…,i−2where δjk=1/2p1k⋯pj−1,k−1, ηjk⋅, j=1,…,i−2, k∈M are positive C∞ functions and a series of virtual controllers x1k∗,…,xik∗ given by(29)x1k∗=0,ξ1k=x1−x1k∗,x2k∗p1k=−ξ1kβ1kx1,ξ2k=x2p1k−x2k∗p1k,⋮xik∗p1k⋯pi−1,k=−l1⋯li−2p1k⋯pi−2,kξi−1,kβi−1,kl¯i−3,x¯i−1,ξik=xip1k⋯pi−1,k−xik∗p1k⋯pi−1,k,with βjk⋅, j=1,…,i−1, k∈M being positive C∞ functions such that(30)V˙i−1,k≤−∑j=1i−11+αjkx¯jξjk2+bikξik2−n−i+1ξ1k22+∑j=2i−1ξjk2−l˙i−1l1⋯li−2li−12W2kx¯2+⋯+Wi−1,kl¯i−3,x¯i−1+ξ1k22,where αjk⋅ j=1,…,i−1 are C∞ functions and bik>0.
In the following, we will show that (30) is also held at step i. With this regard, consider the following Lyapunov–Krasovskii functional as(31)V˜ik=Vi−1,k+1l1⋯li−11+1liWikl¯i−1,x¯i+1l1⋯li∑j=2i−1Wjkl¯j−2,x¯j+12ξ1k2,where li≥1 is a dynamic gain to be determined later and Wik=∫xik∗xiμp1k⋯pi−1,k−xik∗p1k⋯pi−1,k2−1/p1k⋯pi−1,kdμ.
Following the same line, we can obtain thatWikl¯i−2,x¯i is C1, and(32)∂Wik∂xi=ξik2−1/p1k⋯pi−1,k,∂Wik∂xj=1p1k⋯pi−1,k−2∂xik∗p1k⋯pi−1,k∂xj×∫xik∗xiμp1k⋯pi−1,k−xik∗p1k⋯pi−1,k1−1/p1k⋯pi−1,kdμ,∂Wik∂lj=1p1k⋯pi−1,k−2∂xik∗p1k⋯pi−1,k∂lj×∫xik∗xiμp1k⋯pi−1,k−xik∗p1k⋯pi−1,k1−1/p1k⋯pi−1,kdμ,λikxi−xik∗2p1k…pi−1,k≤Wikl¯i−2,x¯i≤2p1k⋯pi−1,k−1ξik2,where λik>0 is a constant.
Due tolj≥1, and (30)–(32), one has(33)V˜˙ik≤∑j=1i−11+αjkx¯jξjk2−n−i+1ξ1k22+∑j=2i−1ξjk2+bikξik2−l˙i−1l1⋯li−2li−12∑j=2i−1Wjk+ξ1k22+1+lil1⋯liξik2−1/p1k⋯pi−1,kxi+1,k∗pik+xi+1pik−xi+1,k∗pik+2l1⋯li−1ξik2−1/p1k⋯pi−1,kfik+∑j=1i−1∂Wik∂xjx˙j+∑j=1i−2∂Wik∂ljl˙j+1l1⋯li∑j=2i−1∑p=1j∂Wjk∂xpx˙p+∑p=1j−2∂Wjk∂lpl˙p+ξ1kx˙1−∑p=1i−1l1⋯l˙p⋯li−1l12⋯li−12Wik−∑p=1il1⋯l˙p⋯lil12⋯li2∑j=1iWjk+ξ1k22.
Similarly, we can obtain the following inequalities(34)2l1⋯li−1ξik2−1/p1k⋯pi−1,kfik≤ξik2Ψik,1l¯i−2,x¯i+ξik2t−d×Ψik,1∗l¯i−2t−d,x¯it−d+1l1⋯li−1ξ1k2+∑j=2i−1xj−xjk∗2p1k⋯pj−1,k×Ψik,2l¯i−2,x¯i−1+1l1⋯li−1ξ1k2t−d+∑j=2i−1xjt−d−xjk∗t−d2p1k⋯pj−1,k×Ψik,2∗l¯i−2t−d,x¯i−1t−d,(35)2l1⋯li−1∑j=1i−1∂Wik∂xjx˙j+∑j=1i−2∂Wik∂ljl˙j≤ξik2Φik,1l¯i−2,x¯i+ξik2t−d×Φik,1∗l¯i−2t−d,x¯it−d+1l1⋯li−1ξ1k2+∑j=2i−1xj−xjk∗2p1k⋯pj−1,k×Φik,2l¯i−2,x¯i−1+1l1⋯li−1ξ1k2t−d+∑j=2i−1xjt−d−xjk∗t−d2p1k⋯pj−1,k×Φik,2∗l¯i−2t−d,x¯i−1t−d,(36)1l1⋯li∑j=2i−1∑p=1j∂Wjk∂xpx˙p+∑p=1j−2∂Wjk∂lpl˙p+ξ1kx˙1≤Ωikl¯i−2,x¯i−1l1⋯li−1ξ1k2+∑j=2i−1xj−xjk∗2p1k⋯pj−1,k+ξik2+1l1⋯li−1Ωik∗l¯i−2t−d,x¯i−1t−d×ξ1k2t−d+∑j=2i−1xjt−d−xjk∗t−d2p1k⋯pj−1,k,where Ψik,j, Ψik,j∗, Φik,j, Φik,j∗, Ωik, and Ωik∗, j=1,2, k∈M, are nonnegative C∞ functions.
Choose the Lyapunov–Krasovskii functional(37)Vik=V˜ik+∫t−dtξik2μΨik,1∗l¯i−2μ,x¯iμ+Φik,1∗l¯i−2μs,x¯iμdμ+∫t−dt1l1μ⋯li−1μ×ξ1k2μ+∑j=2i−1xjμ−xjk∗μ2p1k⋯pj−1,k×Ψik,2∗l¯i−2μ,x¯i−1μ+Φik,2∗l¯i−2μ,x¯i−1μ+Ωik∗l¯i−2μ,x¯i−1μdμ.
The application of (33)–(37) gives rise to(38)V˙ik≤∑j=1i−11+αjkx¯jξjk2+bikξik2−n−i+1ξ1k22+∑j=2i−1ξjk2−l˙i−1l1⋯li−2li−12∑j=2i−1Wjk+ξ1k22+1l1⋯li−1ξ1k2+∑j=2i−1xj−xjk∗2p1k⋯pj−1,k×Ψik,2l¯i−2,x¯i−1+Ψik,2∗l¯i−2,x¯i−1+Φik,2l¯i−2,x¯i−1+Φik,2∗l¯i−2,x¯i−1+Ωikl¯i−2,x¯i−1+Ωik∗l¯i−2,x¯i−1+1l1⋯li−1−1l1t−d⋯li−1t−d×ξ1k2t−d+∑j=2i−1xjt−d−xjk∗t−d2p1k⋯pj−1,k×Ψik,2∗l¯i−2t−d,x¯i−1t−d+Φik,2∗l¯i−2t−d,x¯i−1t−d+Ωik∗l¯i−2t−d,x¯i−1t−d+1+lil1⋯liξik2−1/p1k⋯pi−1,kxi+1,k∗pik+xi+1pik−xi+1,k∗pik+ξik21+Ψik,1l¯i−2,x¯i+Ψik,1∗l¯i−2,x¯i+Φik,1l¯i−2,x¯i+Φik,1∗l¯i−2,x¯i−∑p=1i−1l1⋯l˙p⋯li−1l12⋯li−12Wik−∑p=1il1⋯l˙p⋯lil12⋯li2×∑j=1iWjk+ξ1k22.
Similar to Step 2, we design the gain update law as(39)l˙i−1=max−δi−1,kli−12+2li−1ηi−1,kl¯i−2,x¯i−1,0,where li−10=1, δi−1,k=1/2p1k⋯pi−2,k−1 and ηi−1,kl¯i−2,x¯i−1=ci−1,kΨik,2+Ψik,2∗+Φik,2+Φik,2∗+Ωik+Ωik∗ with ci−1,k=max1,1/2λ2k,…,1/2λi−1,k, k∈M.
By (39), it is easily concluded that(40)0≤l˙i−1≤2li−1ηi−1,k⋅,l˙i−1≥−δi−1,kli−12+2li−1ηi−1,k⋅,li−1t≥li−1t−d≥1.
Utilizing (32) and (40), it can be verified that(41)−l˙i−1l1⋯li−2li−12∑j=2i−1Wjk+ξ1k22≤ξ1k22+∑j=2i−1ξjk2−ηi−1,k2ci−1,kl1⋯li−1×ξ1k2+∑j=2i−1xj−xjk∗2p1k⋯pj−1,k,1l1t⋯li−1t−1l1t−d⋯li−1t−d≤0−∑p=1i−1l1⋯l˙p⋯li−1l12⋯li−12Wik−∑p=1il1⋯l˙p⋯lil12⋯li2×∑j=1iWjk+ξ1k22≤−l˙il1⋯li−1li2∑j=1iWjk+ξ1k22.
Substituting (39) and (41) into (38) yields(42)V˙ik≤−∑j=1i−11+αjkx¯jξjk2−n−iξ1k22+∑j=2i−1ξjk2+1+lil1⋯liξik2−1/p1k⋯pi−1,kxi+1,k∗pik+xi+1pik−xi+1,k∗pik+ξik21+bik+Ψik,1+Ψik,1∗+Φik,1+Φik,1∗−l˙il1⋯li−1li2∑j=1iWjk+ξ1k22.
Then, we choose the virtual controller(43)xi+1∗pik=−l1⋯li−1ξik1/p1k⋯pi−1,kn−i+3+bik+αikx¯i+Ψik,1+Ψik,1∗+Φik,1+Φik,1∗≜−l1⋯li−1ξikβikl¯i−2,x¯i1/p1k⋯pi−1,k,where αik⋅,βik⋅ are positive C∞ functions.
Bylj≥1, j=1,…,i−1, and Lemmas 1-2, one has(44)2l1⋯li−1ξik2−1/p1k⋯pi−1,kxi+1pik−xi+1,k∗pik≤ξik2+bi+1,kξi+1,k2,where ξi+1,k=xi+1p1k⋯pik−xi+1,k∗p1k⋯pik and bi+1,k>0 is a constant.
Substituting (43) and (44) into (42) leads to(45)V˙ik≤∑j=1i−1+αjkx¯jξjk2−n−iξ1k22+∑j=2i−1ξjk2+bi+1,kξi+1,k2−l˙il1⋯li−1l12∑j=2iWjk+ξ1k22.
This completes the inductive proof.Stepn.
During the inductive argument, there exists a dynamic state-feedback controller(46)uk=xn+1,k∗=−l1⋯ln−11/pnkξnk1/p1k⋯pnk3+bnk+αnkx+Ψnk,1+Ψnk,1∗+Φnk,1+Φnk,1∗1/pnk≜−l1⋯ln−11/pnkξnkβnkl¯n−2,x1/p1k⋯pnk,such that(47)V˙nk≤−∑j=1n1+αjkx¯jξjk2.


## 4. Stability Analysis

For simplicity, letV˜k=V˜nk and Vk=Vnk. Now, we are ready to present the main result of this paper.Theorem 1.
For switched nonlinear time-delay system (1), there are the following dynamic state-feedback controllers as(48)L˙=ηkL,x,L∈ℝn−1,uk=βkL,x,k∈M,where ηk:ℝn−1×ℝn⟶ℝn−1 and βk:ℝn−1×ℝn⟶ℝ are continuous functions, and positive C∞ functions αikx¯i and βikl¯i−2,x¯i, i=1,…,n, k∈M, and continuous functions γkpx>0, k,p∈M such that(49)∂V˜k∂xGkx−∑j=1nαjkx¯jξjk2+∑p=1mγkpxVp−Vk<0,∀x≠0,where ∂V˜k/∂x=∂V˜k/∂x1,…,∂V˜k/∂xn and Gkx=g1kx,…,gn−1,kx,0T, and an appropriate switching law (52) to enable the system state converge to origin while keeping the boundedness of the closed-loop system.Proof.
For system (1), we choose the Lyapunov–Krasovskii functional Vk and design the controllers as(50)uk=xn+1,k∗=−l1⋯ln−11/pnkξnk1/p1k⋯pnk3+bnk+αnkx+Ψnk,1+Ψnk,1∗+Φnk,1+Φnk,1∗+ϒkx1/pnk≜−l1⋯ln−11/pnkξnkβnkl¯n−2,x1/p1k⋯pnk,with ϒkx being positive C∞ function, such that(51)V˙k≤∂V˜k∂xGkx−∑j=1n1+αjkx¯jξjk2−ξnk2ϒx.
Next, the switching law is chosen as(52)σt=argmink∈MVk.
Combining (49), (51), and (52), one has(53)V˙k≤−∑j=1nξjk2<0,∀x≠0.
On the contrary, usingσt=argmink∈MVk, one has(54)Vσtj−1tj=Vσtjtjwhere tj is the jth switching instant. This together with (53) yields for ∀t∈tj,tj+1(55)Vσtjt≤Vσtjtj=Vσtj−1tj≤Vσtj−1tj−1=Vσtj−2tj−1≤⋯≤Vσ00.
The use of (31) and (32) can estimate the lower bound of Vk. It can be easily verified that Vk≥1/2cnkx12+1/l1x2−x2k∗2p1k+⋯+1/l1⋯ln−1xn−xnk∗2p1k⋯pn−1,k, which together with (55), implies the boundedness of x1, xi−xik∗2pik⋯pi−1,k/l1⋯li−1, i=2,…,n, k∈M.
By (21), it can be easily found out that the dynamic gain l1t follows a monotone and nondecreasing pattern. Then, l1t is shown to be bounded by a contradiction argument. Suppose limt⟶+∞l1t=+∞. By continuity, it is clear that η1kx1 is bounded. Hence, there exists a constant T>0, such that −l12+2l1η1kx1≤0 on T,+∞. This together with (21) implies l˙1t=0 on T,+∞, which apparently goes against limt⟶+∞l1t=+∞. Thus, l1t is bounded. This conversely indicates that x2−x2k∗ is bounded, so is x2 in light of (29). In a recursive manner, it is relatively easy to verify that lit and xi are bounded based on (28) and (29).
Finally, by (53), it is easily seen that Vkt is monotonically nonincreasing and bounded below by zero, and hence limt⟶+∞Vkt exists and is finite. Thus, from (54) we can obtain(56)∫0+∞ξik2tdt≤−∫0∞V˙σtsds=−∫0t1V˙σ0sds+⋯+∫tj−1tjV˙σtj−1sds+∫tj+∞V˙σtjsds=Vσ00−Vσ0t1+⋯+Vσtj−1tj−1−Vσtj−1tj+Vσtjtj−limt⟶+∞Vσtjt=Vσ00−limt⟶+∞Vσtjt≤Vσ00≤+∞,∀t≥0.
From the boundedness ofl1,…,ln−1 and x1,…,xn, it can be deduced that ξ˙ik is bounded. Thus, by Barbalat’s lemma, it can be concluded that limt⟶+∞ξikt=0, which together with (4) leads to limt⟶+∞xt=0.Remark 1.
Condition (49) is weaker than the assumption in [27, 28]. That is, when gik=0, i=1,…,n, d=0 and Vl=Vk, l,k∈M, (1) is naturally satisfied. This is described in Theorem 1 in [2]. In addition, the flexible choice of αikx¯i and βikl¯i−2,x¯i, i=1,…,n, k∈M, increases the flexibility of (49).Remark 2.
For switched nonlinear time-delay systems (1), the stability analysis based on a common Lyapunov function method breaks down, and we shift to employ multiple Lyapunov functions method to effectively solve global state-feedback control problem. In addition, the construction of coordinate changes and the dynamic gain update law of choice plays an important role in the design of delay-independent controllers of all subsystems.Remark 3.
For nonlinear switched systems without time-delay, the triangular form in [27, 28] is a special case of system (1) when gikx=0, pik=1 and d=0. For nonlinear switched systems without time-delay, the system in [2] is a special case of nontriangular structure when d=0. For nonswitched time-delay nonlinear systems, the system in [25] is restricted to a triangular structure when gikx=0, M=1.Remark 4.
When the constant time-delayd becomes the time-varying time-delay dt, Theorem 1 still holds under the assumption: For each i=1,…,n, d:ℝ+⟶ℝ satisfies 0≤dt≤d¯ and d˙t≤τ<1, where d¯>0 is an unknown constant and τ>0 is a constant. Moreover, Theorem 1 can also be extended to the more general switched nonlinear systems with multiple time-delays of the form x˙i=hiσtx¯ixi+1piσt+fiσtx¯i,x1t−d1t,…,xit−dit+giσtx, i=1,…,n−1,x˙n=hnσtxuσtpnσt+fnσtx,x1t−d1t,…,xt−dit,xs=ζs,s∈−d,0, under the assumptions: (i) hiσt, i=1,…,n, are C1 functions and have the same sign. (ii) For each i=1,…,n, di:ℝ+⟶ℝ satisfies 0≤dit≤d¯ and d˙it≤τ<1, where d¯>0 is an unknown constant and τ>0 is a constant.


## 5. An Illustrative Example

Consider the switched nonlinear time-delay system:(57)x˙1=x2p1σt+f1σtx1,x1t−d+g1σtx1,x2,x˙2=uσtp2σt+f2σtx,xt−d,where x=x1,x2T, σ:0,∞⟶1,2, p11=1, p21=5, f11=x1sinx1t−d, g11=x1x2+3x2, f21=x12x2t−dsinx1, p12=3, p22=1, f12=x1, g12=2x1x2+5x2−x12x2, f22=1/2x25t−d.Clearly, none of the subsystems of (57), even if d=0, is stabilizable by any smooth feedback because their linearization at the origin follows a mode that is uncontrollable and relevant to a positive eigenvalue. For this reason, it can be inferred that when d>0, there is a need to control the delay-involved switched system (57) using the nonsmooth feedback. By Theorem 1, a delay-independent dynamic state-feedback controller can be constructed of the following form as l˙1=max−l12+2l1η1kx1,0,l10=1, uk=−l11/p2kξ2kβ2kx1/p1kp2k,k∈1,2, with η1kx1 and β2k being smooth functions to be determined. For simulation, let ξ1k=x1, ξ2k=x2p1k+ξ1kβ1kx1, and(58)Vk=121+1l1ξ1k2+∫t−dt−ξ1kβ1kx11/p1kμp1k+ξ1kβ1kx12−1/p1kdμ+∫t−dtξ2k2μΨ2k,1∗x¯2μdμ+∫t−dt1l1μξ1k2μΨ2k,2∗x1μ+ϒ2k∗x1μdμ.The switching law is chosen asσt=argmink∈MVk. It is clear that the choice of η11x1=9x12+30, η12x1=9x14/3+60, β1k=9/2, β21=1/l1x12+35, β22=1/l1x2d3+9/2x14/3+64+6x23+9/2x14/3, Ψ21,1∗=1/l1x12, Ψ22,1∗=1/l1x2d3+9/2x14/3, Ψ2k,2∗=0, ϒ2k∗=0.5 renders(59)V˙k≤−∑j=1nξjk2<0,∀x≠0.As proved in Theorem 1, the signals of the closed-loop system can be bounded in the manner ofx1⟶l1⟶x2 and limt⟶+∞xt=0. Figures 1–4 illustrate the reliability of the proposed control approach when d=sint. Figures 5–8 illustrate the reliability of the proposed control approach when d=0.1. Through comparison, it can be found that whether the delay is time-varying or constant does not affect the stability of the system, but will affect the stability time.Figure 1
Trajectories ofx1 and x2 with d=sint.Figure 2
Trajectories ofV1 and V2 with d=sint.Figure 3
System switching signalσt with d=sint.Figure 4
Dynamic gainl1 with d=sint.Figure 5
Trajectories ofx1 and x2 with d=0.1.Figure 6
Trajectories ofV1 and V2 with d=0.1.Figure 7
System switching signalσt with d=0.1.Figure 8
Dynamic gainl1 with d=0.1.


## 6. Conclusion

This paper discussed the global state-feedback control problem for a class of switched nonlinear time-delay systems. This issue can be solved by using explicitly-constructed coordinate transformations and dynamic gains. The 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.



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