# Global Robust Stabilization Control for Nonlinear Time-Delay Systems with Dead-Zone Input and Complex Dynamics

Authors: Lingrong Xue; Zhen-Guo Liu; Junjun Chen; Fujing Xu
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

This article investigates the robust stabilization problem for nonlinear time-delay systems with dead-zone input and complex dynamics. By flexibly using the inequality technique, the backstepping control method, and skillfully introducing a new Lyapunov–Krasovskii functional, we obtain a stable controller without using unmeasurable signals in the dynamic subsystem. The control system is guaranteed to be stable finally. Two simulation examples are given to verify the control strategy.


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## Body

## 1. Introduction

Many practical models in engineering are nonlinear systems such as the flexible-joint robot [1], the wheeled inverted pendulum [2], and the autonomous underwater vehicle [3]. In the past few years, scholars have focused on studying nonlinear systems, such as [4–11]. For numerous nonlinear systems, the time-delay phenomenon which may lead to system instability often exists and is inevitable [12]. Nonlinear systems sometimes involve complex dynamics, in which the information of the states is not available. For more results on nonlinear systems with complex dynamics, we refer the reader to [13]. Besides the time delay, nonlinear conditions, and complex dynamics, specific control inputs such as the dead-zone input [14, 15] can also have a significant impact on the system. Considering the above facts, the nonlinear time-delay systems with dead-zone input and complex dynamics are investigated in this paper.In recent years, some complicated linear systems have been studied. For example, Xu and Zhang [16] considered the stochastic large population system and presented a novel strategy for linear-quadratic games. However, different from linear systems, control problems of nonlinear systems are often difficult since they have more complicated dynamics. Specially, Guo [17] studied nonlinear chaotic systems and raised a physically implementable controller to solve the projective synchronization problem. In engineering practice, lots of nonlinear systems can be approximated by using linear systems at the origin; thus, the theory of linear systems can be applied. However, some systems may not be linearized at the origin or can be linearized but have uncontrollable Jacobian linearization [18, 19]. So, it is necessary to study nonlinear control design methods for those systems. Besides, time-delay problems cannot be ignored for the system control design, since ignoring it may make the system unstable. Many scholars have studied the associated control design for systems with time delay (for example, see the adaptive control problem [20], the stabilization problem [21], and the tracking problem [22]).In recent years, the control design for systems with complex dynamics has been one of the interesting topics. Particularly, with the choice of a state observer, the adaptive control problem of systems containing complex dynamics was solved in [23]. By utilizing a new Lyapunov function, the adaptive tracking problem was studied in [24] for systems with input saturation and complex dynamics. On the other hand, time delay may bring negative effects to the stability of systems. Therefore, scholars have studied control problems for nonlinear systems with time delay. For systems involving time delay and complex dynamics, in [25], by the neural network method and the Lyapunov–Krasovskii functional, the tracking control design was studied. In [26], a modified strategy of adding a power integrator was applied for stochastic delayed nonlinear systems with complex dynamics. Subsequently, this technique was further applied to systems with uncertainty in [27].In practice, dead zone may exist in the actuator or the control input of the system. There have been some related reports mainly discussing the neural method and the fuzzy method. Specially, the neural control method [28] and the adaptive fuzzy control method [29] were applied to solve adaptive control problems of nonlinear systems. Recently, the Lyapunov–Krasovskii functional control approach has been used to study the nonlinear tracking control of systems containing dead-zone input and implies that the approach is important for systems with time delay, see [30]. However, this method is not extended to solve the robust stabilization problem for systems involving time delay, dead-zone input, and complex dynamics. Also, few studies in the literature considered the robust control problem for the system.The difficulty and the contribution of this paper are as follows:(i)
Considering that the system of this paper involves complicated dynamics, time delay, external disturbances, and input dead zone, the robust stabilization control problem of this paper is more challenging. The existing methods are difficult to solve the problem of this paper. Particularly, the homogeneous domination approach [5] has not proposed a strategy to solve the input dead-zone problem, and the tuning functions-based robust control method [7] has not give a solution to the complicated dynamics. The neural control method [28] and the fuzzy method [29] are difficult to give appropriate bounds for the nonlinear time-delay terms. The continuous control methods in [31, 32] did not provide a strategy to deal with the external disturbances. This article will consider more complicated nonlinear systems and present a control strategy to solve the control problem.(ii)
A new robust stabilization strategy is raised. By recursively selecting a Lyapunov–Krasovskii functional and by presenting a modified backstepping control technique, a stable controller without utilizing the unmeasurable states is successfully constructed.


## 2. Problem Formulation and Preliminaries

Consider the nonlinear system as follows:(1)η˙=Aη+ϕ0x1,x1t−τ,x˙i=ditxi+1+ϕiη,ηt−τ,x¯i,x¯it−τ,i=1,…,n−1,x˙n=dntuv+ϕnη,ηt−τ,x,xt−τ,where η=η1,…,ηm⊤∈Rm is the unmeasurable dynamics and x=x1,…,xn⊤∈ℛn is the state vector which is measurable. x¯i=x1,…,xi⊤. ηt−τ=η1t−τ,…,ηmt−τ⊤ and x¯it−τ=x1t−τ,…,xit−τ⊤. τ is the constant time delay. A∈Rm×m is the Hurwitz matrix that satisfies A⊤P+PA≤−Q, where P and Q are positive definite symmetric matrices. The control coefficients djt∈ℛ,j=1,…,n satisfy d≤djt≤d¯ with d≥0 and d¯≥0. ϕk∈ℛ,k=0,1,…,n, are continuous functions. The dead-zone input uv is(2)uv=mrtv−brt,v≥brt,0,−blt<v<brt,mltv+blt,v≤−blt,where mrt>0,mlt>0,brt>0, and blt>0 the are time-varying functions.To facilitate the problem simply, letϖ=q/p≥0, where q and p are the even integer and the odd integer, respectively. Therefore, define the constants ε1=1 and εj=εj−1+ϖ,j=2,…,n. Next, we need the following assumptions.Assumption 1.
The nonlinear terms satisfy the following:(3)ϕ0x1,x1t−τ≤C0x1+x1t−τ,ϕiη,ηt−τ,x¯i,x¯it−τ≤Cηεi+1+ηt−τεi+1+∑j=1ixjεi+1/εj+∑j=1ixjt−τεi+1/εj+M0t,i=1,…,n,where C0>0 and C>0 are constants and M0t is a bounded disturbance.Assumption 2.
There are constantsm¯r>0,m¯l>0,b¯r>0, and b¯l>0 satisfying(4)m¯r≤mrt,m¯l≤mlt,brt≤b¯r,andblt≤b¯l.
Next, Lemma1 is provided for control design.Lemma 1. (see [30]).
For givenm>0andn>0 and functions ax,y>0andcx,y>0, there holds(5)ax,yxmyn≤cx,yxm+n+nm+nmm+ncx,ym/nax,ym+n/nym+n.


## 3. Main Results

Theorem 1.
For system (1), suppose that Assumptions 1-2 hold. Then, under the following transformation(6)ζ1=x1,ζi=xi−αi−1,αi=−giζiεi+1/εi,i=2,…,n,there exists a robust controller:(7)v=αnm¯r+b¯r,αn>0,0,αn=0,αnm¯l−b¯l,αn<0.where αn=−gnζnεn+1/εn. The controller ensures that the considered system is globally stable.Proof.
We divide the proof into several parts.


### 3.1. Part I: Robust Control Design

We construct the controller by using the modified backstepping technique.Step 1: definingσ=εn and choosing U0=1/ση⊤Pησ, we have(8)U˙0=η⊤Pησ−1η⊤PAη+ϕ0+Aη+ϕ0⊤Pη=η⊤Pησ−1η⊤PA+A⊤Pη+2η⊤Pϕ0.Noting thatλpη2≤η⊤Pη≤λPη2 and η⊤PA+A⊤Pη≤−λqη2, where λp is the minimal eigenvalue of P and λP is the maximal eigenvalues of P. λq is the minimal eigenvalue of Q. Then, introduce ζ1=x1 and define λ=minλp,λq. From Assumption 1 and Lemma 1, we get(9)U˙0≤−λση2σ+2η⊤Pησ−1η⊤Pϕ0=−λση2σ+Δ,(10)Δ=2η⊤Pησ−1η⊤Pϕ0≤2λPσ−1η2σ−2⋅η⋅P⋅ϕ0≤2λPσ−1Pη2σ−1C0ζ1+ζ1t−τ≤1n+2λση2σ+γ01C0,λPζ12σε1+γ02C0,λPe−μτζ12σε1t−τ,whereγ01C0,λP and γ02C0,λP are the positive constants depended on C0 and λP. Substituting (10) into (9), it yields(11)U˙0≤−n+1n+2λση2σ+γ01C0,λPζ12σ/ε1+γ02C0,λPe−μτζ12σ/ε1t−τ.By Lemma1 and Assumption 1, there are constants γ11>0 and M1>0 such that(12)x12σ−ε2/ε1ϕ1≤Cx12σ−ε2/ε1ηε2+ηt−τε2+x1ε2/ε1+x1t−τε2/ε1+M0x12σ−ε2/ε1≤γ11ζ12σ/ε1+1n+2λση2σ+12n+2e−μτλσηt−τ2σ+e−μτζ12σ/ε1t−τ+M02σ/ε2M1.Introduce the transformationζ2=x2−α1, and choose V1=U0+U1+W1+T1, where U1=n/2n+2λσ∫t−τteμs−tηs2σds., W1=ε1/2σ−ϖζ12σ−ϖ/ε1, and T1=n+γ02∫t−τteμs−tζ12σ/ε1sds. μ>0 is a constant. Then, it follows from (11) and (12) that(13)V˙1=U˙0+ζ12σ−ε2/ε1d1x2+ϕ1+n+γ02ζ12σ/ε1−n+γ02e−μτζ12σ/ε1t−τ+n2n+2λση2σ−n2n+2e−μτλσηt−τ2σ−μU1+T1≤−n2n+2λση2σ+d1ζ2ζ12σ−ε2/ε1+d1α1ζ12σ−ε2/ε1+n+γ01+γ02+γ11ζ12σ/ε1+M02σ/ε2M1−n−1e−μτζ12σ/ε1t−τ−n−12n+2e−μτλσηt−τ2σ−μU1+T1.Selecting the virtual controlα1=−1/d2n+γ01+γ02+γ11ζ1ε2/ε1def=−g1ζ1ε2/ε1, and substituting it into (13), it yields that(14)V˙1≤−n2n+2λση2σ−nζ12σ/ε1+d1ζ2ζ12σ−ε2/ε1+M02σ/ε2M1−μU1+T1−n−1e−μτζ12σ/ε1t−τ−n−12n+2e−μτλσηt−τ2σ.Stepk (k=2,…,n−1): for step k−1, suppose that there exist transformations (6) and a Lyapunov–Krasovskii functional Vk−1 such that there exist the following inequality:(15)V˙k−1≤−n−k+22n+2λση2σ−n−k+2∑j=1k−1ζj2σ/εj+dk−1ζkζk−12σ−εk/εk−1+∑j=1k−1M02σ/εj+1Mj−μU1+∑j=1k−1Tj−n−k+1e−μτ∑j=1k−1ζj2σ/εjt−τ−n−k+12n+2e−μτλσηt−τ2σ.Next, we prove it still holds fori=k. Choosing(16)Vk=Vk−1+Wk+Tk,Wk=εk2σ−ϖζk2σ−ϖ/εk,Tk=n−i+1∫t−τteμs−tζk2σ/εksds.introducingζk+1=xk+1−αk, and taking the derivative of Vk, it can be deduced that(17)V˙k≤−n−k+22n+2λση2σ−n−k+2∑j=1k−1ζj2σ/εj+dk−1ζkζk−12σ−εk/εk−1+∑j=1k−1M02σ/εj+1Mj−μU1+∑j=1kTj−n−k+1e−μτ∑j=1kζj2σ/εjt−τ−n−k+12n+2e−μτλσηt−τ2σ+n−k+1ζk2σ/εk+ζk2σ−εk+1/εkϕk−∑j=1k−1∂αk−1∂xjdjxj+1+ϕj+dkζk+1ζk2σ−εk+1/εk+dkαkζk2σ−εk+1/εk.On the basis ofdk−1≤d¯ and Lemma 1, there is a constant γ11>0 such that(18)dk−1ζkζk−12σ−εk/εk−1≤13ζk−12σ/εk−1+γk1ζk2σ/εk.Similarly to the proof of (12), one obtains(19)ζk2σ−εk+1/εkϕk≤Cζk2σ−εk+1/εkηεk+1+ηt−τεk+1+∑j=1kζj−gj−1ζj−1εj/εj−1εk+1/εj+∑j=1kζjt−τ−gj−1ζj−1εj/εj−1t−τεk+1/εj+M0ζk2σ−εk+1/εk≤γk2ζk2σ/εk+14n+2λση2σ+14n+2e−μτλσηt−τ2σ+12e−μτ∑j=1kζj2σ/εjt−τ+M02σ/εk+1Mk1+13∑j=1k−1ζj2σ/εj,whereγk2>0 and Mk1>0 are constants. From Assumption 1, we have(20)djxj+1+ϕj≤dζj+1+gjζjϵj+1/ϵj+Cηϵj+1+ηt−τϵj+1+∑i=1jζi−gi−1ζi−1ϵi/ϵi−1ϵj+1/ϵi+∑i=1jζit−τ−gi−1ζi−1ϵi/ϵi−1t−τϵj+1/ϵi+M0.Utilizing (20) and Lemma 1 and noting that ∂αk−1/∂xj=gjgj+1⋯gk−1εk/εjζjϖ/εjζj+1ϖ/εj+1⋯ζk−1ϖ/εk−1, it follows that(21)∂αk−1∂xjdjxj+1+ϕj≤dζj+1+gjζjεj+1/εj+Cηεj+1+ηt−τεj+1+∑i=1jζi−gi−1ζi−1εi/εi−1εj+1/εi+∑i=1jζit−τ−gi−1ζi−1εi/εi−1t−τεj+1/εi+M0gjgj+1⋯gk−1εkεjζjϖ/εjζj+1ϖ/εj+1⋯ζk−1ϖ/εk−1≤Cj1∑i=1jζiεj+1/εi+Cj2∑i=1jζit−τεj+1/εi+Cj3ηεk+1+Cj4ηt−τεk+1+M0gjgj+1⋯gk−1εkεjζjϖ/εjζj+1ϖ/εj+1⋯ζk−1ϖ/εk−1,whereCjk>0,k=1,…,4 are constants. It can be deduced from (21) and Lemma 1 that(22)−ζk2σ−εk+1/εk∑j=1k−1∂αk−1∂xjdjxj+1+ϕj≤γk3ζk2σ/εk+14n+2λση2σ+14n+2e−μτλσηt−τ2σ+12e−μτ∑j=1kζj2σ/εjt−τ+M02σ/εk+1Mk2+13∑j=1k−1ζj2σ/εj,whereγk3>0 and Mk2>0 are constants. Choosing Mk such that 1/Mk=1/Mk1+1/Mk2 and using (19) and (22), it yields that(23)ζk2σ−εk+1/εkϕk−∑j=1k−1∂αk−1∂xjdjxj+1+ϕj≤γk2+γk3ζk2σ/εk+12n+2λση2σ+12n+2e−μτλσηt−τ2σ+e−μτ∑j=1kζj2σ/εjt−τ+M02σ/εk+1Mk+23∑j=1k−1ζj2σ/εj.Now, we construct the virtual controlαk=−1/d2n−2k+2+γk1+γk2+γk3ζkεk+1/εkdef=−gkζkεk+1/εk. With the help of (17), (18), and (23), it is deduced that(24)V˙k≤−n−k+12n+2λση2σ−n−k+1∑j=1kζj2σ/εj+dkζk+1ζk2σ−εk+1/εk−μU1+∑j=1kTj−n−ke−μτ∑j=1kζj2σ/εjt−τ−n−k2n+2e−μτλσηt−τ2σ+∑j=1kM02σ/εj+1Mj.This completes the control design for stepk=2,…,n−1.Stepn: in this step, we choose Vn=Vn−1+Wn+Tn,Wn=εn/2σ−ϖζn2σ−ϖ/εn and Tn=∫t−τteμs−tζn2σ/εnsds, and using (24), some simple computations lead to(25)V˙n≤−22n+2λση2σ−2∑j=1n−1ζj2σ/εj+dn−1ζnζn−12σ−εn/εn−1+∑j=1n−1M02σ/εj+1Mj−μU1+∑j=1nTj−e−μτ∑j=1nζj2σ/εjt−τ−12n+2e−μτλσηt−τ2σ+ζn2σ/εn+ζn2σ−εn+1/εnϕn−∑j=1n−1∂αn−1∂xjdjxj+1+ϕj+dnuvζn2σ−εn+1/εn.Similar to (18) and (23), by Lemma 1, we obtain that(26)dn−1ζnζn−12σ−εn/εn−1≤13ζn−12σ/εn−1+γn1ζn2σ/εn,(27)ζn2σ−εn+1/εnϕn−∑j=1n−1∂αn−1∂xjdjxj+1+ϕj≤γn2+γn3ζn2σ/εn+12n+2λση2σ+12n+2e−μτλσηt−τ2σ+e−μτ∑j=1nζj2σ/εjt−τ+M02σ/εn+1Mn+23∑j=1n−1ζj2σ/εj,where the constants γn1≥0,γn2≥0,γn3≥0, and Mn≥0. Now, we choose the virtual control αn=−1/d2+γn1+γn2+γn3ζnεn+1/εndef=−gnζnεn+1/εn, which, and (25)–(27), give that(28)V˙n≤−12n+2λση2σ−∑j=1nζj2σ/εj−μU1+∑j=1nTj+∑j=1nM02σ/εj+1Mj+dnuv−αnζn2σ−εn+1/εn.Finally, we choose the control inputv as (7), Then, by using (2), it follows that(29)uv−αn=mrαnm¯r+b¯r−br−αn,αn>0,0,αn=0,mlαnm¯l+b¯l−bl−αn,αn<0,which further renders that dnuv−αnζn2σ−εn+1/εn≤0. Then, using (28), we get(30)V˙n≤−12n+2λση2σ−∑j=1nζj2σ/εj−μU1+∑j=1nTj+∑j=1nM02σ/εj+1Mj.


### 3.2. Part II: Stability Analysis

SinceU0=1/ση⊤Pησ and λpη2≤η⊤Pη≤λPη2, we have η2σ≥σ/λPσU0, which leads to(31)−12n+2λση2σ≤−σ2n+2λPσλσU0.DefiningΔj=2σ−ϖ/2σδjϖ/2σϖ/2σ−ϖεj/2σ−ϖ2σ/2σ−ϖ and Δm=min1≤j≤n1/Δj, it follows from Lemma 1 that Wj=εj/2σ−ϖζj2σ−ϖ/εj≤1/δj+Δjζj2σ/εj, which renders that(32)−∑j=1nζj2σ/εj≤−Δm∑j=1nWj+∑j=1n1δjΔj.Substituting (31) and (32) into (28), it yields that(33)V˙n≤−σ2n+2λPσλσU0−Δm∑j=1nWj+∑j=1n1δjΔj−μU1+∑j=1nTj+∑j=1nM02σ/εj+1Mj≤−ρ1Vn+ρ2,where ρ1=minσ/2n+2λPσλσ,Δm,μ and ρ2=∑j=1n1/δjΔj+∑j=1nM02σ/εj+1/Mj. By the definition of U0,U1, Wi,andTi,i=1,…,n, it follows that(34)U0+U1=1ση⊤Pησ+n2n+2λσ∫t−τteμs−tηs2σds≤λPσση2σ+nτλσ2n+2sup−τ≤k≤0eμkηk+t2σ≤λPσσ+nτλσ2n+2sup−τ≤k≤0ηk+t2σ,∑i=1nWi=∑i=1nεi2σ−ϖζi2σ−ϖ/εi≤εn2σ−ϖ∑i=1nsup−τ≤k≤0ζi2σ−ϖ/εik+t,∑i=1nTi=γ02∫t−τteμs−tζ12σ/ε1sds+∑i=1nn−i+1∫t−τteμs−tζi2σ/εisds≤n+γ02τ∑i=1nsup−τ≤k≤0ζi2σ/εik+t.which lead to(35)Vn=U0+U1+∑i=1nWi+∑i=1nTi≤φ1sup−τ≤k≤0Ξk+t,where the function φ1∈K∞ and Ξt=ηt⊤,ζt⊤⊤. On the other hand, there exists a function φ2∈K∞ such that(36)Vn=U0+U1+∑i=1nWi+∑i=1nTi≥λpσση2σ+∑i=1nεi2σ−ϖζi2σ−ϖ/εi≥φ2Ξt.In view of (28), (35), and (36), and using Lemma 3.2.4 in [33], we obtain(37)V˙n≤Vn0−ρ2ρ1e−ρ1t+ρ2ρ1,which indicates that η and ζ are bounded. From ζ1=x1,ζi=xi−αi−1,α1=−g1ζ1ε2/ε1,andαi=−giζiεi+1/εi,i=2,…,n, the states of η,x1,…,xn are bounded, which indicates the system composed of (1) and (7) is globally stable.


## 3.1. Part I: Robust Control Design

We construct the controller by using the modified backstepping technique.Step 1: definingσ=εn and choosing U0=1/ση⊤Pησ, we have(8)U˙0=η⊤Pησ−1η⊤PAη+ϕ0+Aη+ϕ0⊤Pη=η⊤Pησ−1η⊤PA+A⊤Pη+2η⊤Pϕ0.Noting thatλpη2≤η⊤Pη≤λPη2 and η⊤PA+A⊤Pη≤−λqη2, where λp is the minimal eigenvalue of P and λP is the maximal eigenvalues of P. λq is the minimal eigenvalue of Q. Then, introduce ζ1=x1 and define λ=minλp,λq. From Assumption 1 and Lemma 1, we get(9)U˙0≤−λση2σ+2η⊤Pησ−1η⊤Pϕ0=−λση2σ+Δ,(10)Δ=2η⊤Pησ−1η⊤Pϕ0≤2λPσ−1η2σ−2⋅η⋅P⋅ϕ0≤2λPσ−1Pη2σ−1C0ζ1+ζ1t−τ≤1n+2λση2σ+γ01C0,λPζ12σε1+γ02C0,λPe−μτζ12σε1t−τ,whereγ01C0,λP and γ02C0,λP are the positive constants depended on C0 and λP. Substituting (10) into (9), it yields(11)U˙0≤−n+1n+2λση2σ+γ01C0,λPζ12σ/ε1+γ02C0,λPe−μτζ12σ/ε1t−τ.By Lemma1 and Assumption 1, there are constants γ11>0 and M1>0 such that(12)x12σ−ε2/ε1ϕ1≤Cx12σ−ε2/ε1ηε2+ηt−τε2+x1ε2/ε1+x1t−τε2/ε1+M0x12σ−ε2/ε1≤γ11ζ12σ/ε1+1n+2λση2σ+12n+2e−μτλσηt−τ2σ+e−μτζ12σ/ε1t−τ+M02σ/ε2M1.Introduce the transformationζ2=x2−α1, and choose V1=U0+U1+W1+T1, where U1=n/2n+2λσ∫t−τteμs−tηs2σds., W1=ε1/2σ−ϖζ12σ−ϖ/ε1, and T1=n+γ02∫t−τteμs−tζ12σ/ε1sds. μ>0 is a constant. Then, it follows from (11) and (12) that(13)V˙1=U˙0+ζ12σ−ε2/ε1d1x2+ϕ1+n+γ02ζ12σ/ε1−n+γ02e−μτζ12σ/ε1t−τ+n2n+2λση2σ−n2n+2e−μτλσηt−τ2σ−μU1+T1≤−n2n+2λση2σ+d1ζ2ζ12σ−ε2/ε1+d1α1ζ12σ−ε2/ε1+n+γ01+γ02+γ11ζ12σ/ε1+M02σ/ε2M1−n−1e−μτζ12σ/ε1t−τ−n−12n+2e−μτλσηt−τ2σ−μU1+T1.Selecting the virtual controlα1=−1/d2n+γ01+γ02+γ11ζ1ε2/ε1def=−g1ζ1ε2/ε1, and substituting it into (13), it yields that(14)V˙1≤−n2n+2λση2σ−nζ12σ/ε1+d1ζ2ζ12σ−ε2/ε1+M02σ/ε2M1−μU1+T1−n−1e−μτζ12σ/ε1t−τ−n−12n+2e−μτλσηt−τ2σ.Stepk (k=2,…,n−1): for step k−1, suppose that there exist transformations (6) and a Lyapunov–Krasovskii functional Vk−1 such that there exist the following inequality:(15)V˙k−1≤−n−k+22n+2λση2σ−n−k+2∑j=1k−1ζj2σ/εj+dk−1ζkζk−12σ−εk/εk−1+∑j=1k−1M02σ/εj+1Mj−μU1+∑j=1k−1Tj−n−k+1e−μτ∑j=1k−1ζj2σ/εjt−τ−n−k+12n+2e−μτλσηt−τ2σ.Next, we prove it still holds fori=k. Choosing(16)Vk=Vk−1+Wk+Tk,Wk=εk2σ−ϖζk2σ−ϖ/εk,Tk=n−i+1∫t−τteμs−tζk2σ/εksds.introducingζk+1=xk+1−αk, and taking the derivative of Vk, it can be deduced that(17)V˙k≤−n−k+22n+2λση2σ−n−k+2∑j=1k−1ζj2σ/εj+dk−1ζkζk−12σ−εk/εk−1+∑j=1k−1M02σ/εj+1Mj−μU1+∑j=1kTj−n−k+1e−μτ∑j=1kζj2σ/εjt−τ−n−k+12n+2e−μτλσηt−τ2σ+n−k+1ζk2σ/εk+ζk2σ−εk+1/εkϕk−∑j=1k−1∂αk−1∂xjdjxj+1+ϕj+dkζk+1ζk2σ−εk+1/εk+dkαkζk2σ−εk+1/εk.On the basis ofdk−1≤d¯ and Lemma 1, there is a constant γ11>0 such that(18)dk−1ζkζk−12σ−εk/εk−1≤13ζk−12σ/εk−1+γk1ζk2σ/εk.Similarly to the proof of (12), one obtains(19)ζk2σ−εk+1/εkϕk≤Cζk2σ−εk+1/εkηεk+1+ηt−τεk+1+∑j=1kζj−gj−1ζj−1εj/εj−1εk+1/εj+∑j=1kζjt−τ−gj−1ζj−1εj/εj−1t−τεk+1/εj+M0ζk2σ−εk+1/εk≤γk2ζk2σ/εk+14n+2λση2σ+14n+2e−μτλσηt−τ2σ+12e−μτ∑j=1kζj2σ/εjt−τ+M02σ/εk+1Mk1+13∑j=1k−1ζj2σ/εj,whereγk2>0 and Mk1>0 are constants. From Assumption 1, we have(20)djxj+1+ϕj≤dζj+1+gjζjϵj+1/ϵj+Cηϵj+1+ηt−τϵj+1+∑i=1jζi−gi−1ζi−1ϵi/ϵi−1ϵj+1/ϵi+∑i=1jζit−τ−gi−1ζi−1ϵi/ϵi−1t−τϵj+1/ϵi+M0.Utilizing (20) and Lemma 1 and noting that ∂αk−1/∂xj=gjgj+1⋯gk−1εk/εjζjϖ/εjζj+1ϖ/εj+1⋯ζk−1ϖ/εk−1, it follows that(21)∂αk−1∂xjdjxj+1+ϕj≤dζj+1+gjζjεj+1/εj+Cηεj+1+ηt−τεj+1+∑i=1jζi−gi−1ζi−1εi/εi−1εj+1/εi+∑i=1jζit−τ−gi−1ζi−1εi/εi−1t−τεj+1/εi+M0gjgj+1⋯gk−1εkεjζjϖ/εjζj+1ϖ/εj+1⋯ζk−1ϖ/εk−1≤Cj1∑i=1jζiεj+1/εi+Cj2∑i=1jζit−τεj+1/εi+Cj3ηεk+1+Cj4ηt−τεk+1+M0gjgj+1⋯gk−1εkεjζjϖ/εjζj+1ϖ/εj+1⋯ζk−1ϖ/εk−1,whereCjk>0,k=1,…,4 are constants. It can be deduced from (21) and Lemma 1 that(22)−ζk2σ−εk+1/εk∑j=1k−1∂αk−1∂xjdjxj+1+ϕj≤γk3ζk2σ/εk+14n+2λση2σ+14n+2e−μτλσηt−τ2σ+12e−μτ∑j=1kζj2σ/εjt−τ+M02σ/εk+1Mk2+13∑j=1k−1ζj2σ/εj,whereγk3>0 and Mk2>0 are constants. Choosing Mk such that 1/Mk=1/Mk1+1/Mk2 and using (19) and (22), it yields that(23)ζk2σ−εk+1/εkϕk−∑j=1k−1∂αk−1∂xjdjxj+1+ϕj≤γk2+γk3ζk2σ/εk+12n+2λση2σ+12n+2e−μτλσηt−τ2σ+e−μτ∑j=1kζj2σ/εjt−τ+M02σ/εk+1Mk+23∑j=1k−1ζj2σ/εj.Now, we construct the virtual controlαk=−1/d2n−2k+2+γk1+γk2+γk3ζkεk+1/εkdef=−gkζkεk+1/εk. With the help of (17), (18), and (23), it is deduced that(24)V˙k≤−n−k+12n+2λση2σ−n−k+1∑j=1kζj2σ/εj+dkζk+1ζk2σ−εk+1/εk−μU1+∑j=1kTj−n−ke−μτ∑j=1kζj2σ/εjt−τ−n−k2n+2e−μτλσηt−τ2σ+∑j=1kM02σ/εj+1Mj.This completes the control design for stepk=2,…,n−1.Stepn: in this step, we choose Vn=Vn−1+Wn+Tn,Wn=εn/2σ−ϖζn2σ−ϖ/εn and Tn=∫t−τteμs−tζn2σ/εnsds, and using (24), some simple computations lead to(25)V˙n≤−22n+2λση2σ−2∑j=1n−1ζj2σ/εj+dn−1ζnζn−12σ−εn/εn−1+∑j=1n−1M02σ/εj+1Mj−μU1+∑j=1nTj−e−μτ∑j=1nζj2σ/εjt−τ−12n+2e−μτλσηt−τ2σ+ζn2σ/εn+ζn2σ−εn+1/εnϕn−∑j=1n−1∂αn−1∂xjdjxj+1+ϕj+dnuvζn2σ−εn+1/εn.Similar to (18) and (23), by Lemma 1, we obtain that(26)dn−1ζnζn−12σ−εn/εn−1≤13ζn−12σ/εn−1+γn1ζn2σ/εn,(27)ζn2σ−εn+1/εnϕn−∑j=1n−1∂αn−1∂xjdjxj+1+ϕj≤γn2+γn3ζn2σ/εn+12n+2λση2σ+12n+2e−μτλσηt−τ2σ+e−μτ∑j=1nζj2σ/εjt−τ+M02σ/εn+1Mn+23∑j=1n−1ζj2σ/εj,where the constants γn1≥0,γn2≥0,γn3≥0, and Mn≥0. Now, we choose the virtual control αn=−1/d2+γn1+γn2+γn3ζnεn+1/εndef=−gnζnεn+1/εn, which, and (25)–(27), give that(28)V˙n≤−12n+2λση2σ−∑j=1nζj2σ/εj−μU1+∑j=1nTj+∑j=1nM02σ/εj+1Mj+dnuv−αnζn2σ−εn+1/εn.Finally, we choose the control inputv as (7), Then, by using (2), it follows that(29)uv−αn=mrαnm¯r+b¯r−br−αn,αn>0,0,αn=0,mlαnm¯l+b¯l−bl−αn,αn<0,which further renders that dnuv−αnζn2σ−εn+1/εn≤0. Then, using (28), we get(30)V˙n≤−12n+2λση2σ−∑j=1nζj2σ/εj−μU1+∑j=1nTj+∑j=1nM02σ/εj+1Mj.


## 3.2. Part II: Stability Analysis

SinceU0=1/ση⊤Pησ and λpη2≤η⊤Pη≤λPη2, we have η2σ≥σ/λPσU0, which leads to(31)−12n+2λση2σ≤−σ2n+2λPσλσU0.DefiningΔj=2σ−ϖ/2σδjϖ/2σϖ/2σ−ϖεj/2σ−ϖ2σ/2σ−ϖ and Δm=min1≤j≤n1/Δj, it follows from Lemma 1 that Wj=εj/2σ−ϖζj2σ−ϖ/εj≤1/δj+Δjζj2σ/εj, which renders that(32)−∑j=1nζj2σ/εj≤−Δm∑j=1nWj+∑j=1n1δjΔj.Substituting (31) and (32) into (28), it yields that(33)V˙n≤−σ2n+2λPσλσU0−Δm∑j=1nWj+∑j=1n1δjΔj−μU1+∑j=1nTj+∑j=1nM02σ/εj+1Mj≤−ρ1Vn+ρ2,where ρ1=minσ/2n+2λPσλσ,Δm,μ and ρ2=∑j=1n1/δjΔj+∑j=1nM02σ/εj+1/Mj. By the definition of U0,U1, Wi,andTi,i=1,…,n, it follows that(34)U0+U1=1ση⊤Pησ+n2n+2λσ∫t−τteμs−tηs2σds≤λPσση2σ+nτλσ2n+2sup−τ≤k≤0eμkηk+t2σ≤λPσσ+nτλσ2n+2sup−τ≤k≤0ηk+t2σ,∑i=1nWi=∑i=1nεi2σ−ϖζi2σ−ϖ/εi≤εn2σ−ϖ∑i=1nsup−τ≤k≤0ζi2σ−ϖ/εik+t,∑i=1nTi=γ02∫t−τteμs−tζ12σ/ε1sds+∑i=1nn−i+1∫t−τteμs−tζi2σ/εisds≤n+γ02τ∑i=1nsup−τ≤k≤0ζi2σ/εik+t.which lead to(35)Vn=U0+U1+∑i=1nWi+∑i=1nTi≤φ1sup−τ≤k≤0Ξk+t,where the function φ1∈K∞ and Ξt=ηt⊤,ζt⊤⊤. On the other hand, there exists a function φ2∈K∞ such that(36)Vn=U0+U1+∑i=1nWi+∑i=1nTi≥λpσση2σ+∑i=1nεi2σ−ϖζi2σ−ϖ/εi≥φ2Ξt.In view of (28), (35), and (36), and using Lemma 3.2.4 in [33], we obtain(37)V˙n≤Vn0−ρ2ρ1e−ρ1t+ρ2ρ1,which indicates that η and ζ are bounded. From ζ1=x1,ζi=xi−αi−1,α1=−g1ζ1ε2/ε1,andαi=−giζiεi+1/εi,i=2,…,n, the states of η,x1,…,xn are bounded, which indicates the system composed of (1) and (7) is globally stable.


## 4. Simulation Example

Example 1.
Consider the following nonlinear time-delay system:(38)η˙=−η+0.5x1,x˙1=x2+x15/3t−0.2,x˙2=uv+x27/5+0.3sint,where η is the unmeasurable state, x1andx2 are the measurable states, uv is the dead-zone input given in (2) with br=0.4+0.1sint,bl=0.4−0.1sint, and v is the input of the system. We see that Assumptions 1-2 are satisfied for system (32) with ϖ=2/3,ε1=1,ε2=5/3,ε3=7/3, and b¯r=b¯l=0.5,andm¯r=m¯l=mr=ml=1. Applying the above control method, we choose α2=−15x2+9x15/37/5. Then, the actual controller is constructed as(39)v=α2+b¯r,α2>0,0,α2=0,α2−b¯l,α2<0.
In the simulation, chooseη0=2,x10=0.5,andx20=−2. Figures 1–4 give the trajectories of η,x1,andx2 and the control input v. All signals in systems (38) and (39) are bounded. Hence, the validity of the presented control method is verified.Figure 1
: The trajectory ofη.Figure 2
: The trajectory ofx1.Figure 3
: The trajectory ofx2.Figure 4
: The trajectory ofv.Example 2.
Consider the following system:(40)η˙=−2η+11+η2ηx1,x˙1=x2+x11/3x12/3t−0.5,x˙2=x3−5sinx1−x2+sinx1t−0.5,x˙3=uv+10x2−5x3,where η is the unmeasurable state, x1,x2,x3 are the system states, v is the control input, and uv is the dead-zone input defined in (2) with br=0.5+sintandbl=0.5−sint. It can be deduced that x1η/1+η2≤x1, x11/3x12/3t−0.5≤x1+2/3x1t−0.5, −5sinx1−x2+sinx1t−0.5≤5x1+x2+x1t−0.5, and 10x2−5x3≤10x2+5x3. In this example, ϖ=0,ε1=ε2=ε3=ε4=1, and b¯r=b¯l=1.5,​andm¯r=m¯l==1. Thus, all conditions of Assumptions 1-2 are satisfied. Using the control design method in Section 3, we can design the actual controller as follows:(41)v=α3+b¯r,α3>0,0,α3=0,α3−b¯l,α3<0.where α3=−15x3+10x2+6x1.
In the simulation, the initial conditions are selected asη0=1,x10=0.5,x20=−0.5,andx30=0. Figures 5–9 show the trajectories of η,x1,x2,x3, and the control input v. It can be seen that all signals in systems (34) and (35) are bounded. Hence, the presented control method is effective.Figure 5
: The trajectory ofη.Figure 6
: The trajectory ofx1.Figure 7
The trajectory ofx2.Figure 8
The trajectory ofx3.Figure 9
The trajectory ofv.


## 5. Conclusions

The robust stabilization for nonlinear systems with dead-zone input and time delay has been studied. Because the system involves the dead-zone input, time-delay, disturbance, and unmeasurable states, the stabilization control in this work is more challenging. A robust stable controller has been designed via the Lyapunov–Krasovskii functional and the backstepping technique. Another interesting problem is as follows: When the considered system includes uncertainty parameters, and only the system output is measurable, how can we design the adaptive controller via the output feedback control method?



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