# Disturbance Attenuation via Output Feedback for Nonlinear Time-Delay Systems with Input Matching Uncertainty

Authors: Chao Guo; Kemei Zhang
Journal: Mathematical Problems in Engineering
 (2018)

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## Abstract

This paper studies the problem of output feedback disturbance attenuation for a class of uncertain nonlinear systems with input matching uncertainty and unknown multiple time-varying delays, whose nonlinearities are bounded by unmeasured states multiplying unknown polynomial-of-output growth rate. By skillfully combining extended state observer, dynamic gain technique, and Lyapunov-Krasovskii theorem, a delay-independent output feedback controller can be developed with only one dynamic gain to guarantee the boundedness of closed-loop system states and the achievement of global disturbance attenuation in theL2-gain sense.


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## Body

## 1. Introduction

The problem of disturbance attenuation via output feedback for nonlinear systems is a relatively meaningful problem in control theory and applications. Compared with the stabilization control and tracking control, fewer results on output feedback disturbance attenuation design have been obtained until now, such as [1–5] and the references therein. It is worth mentioning that, for nonlinear systems with known polynomial-of-output growth rate, the problem of output feedback disturbance attenuation was studied in [5].In this paper, we consider output feedback disturbance attenuation problem of uncertain nonlinear systems as follows:(1)x˙it=xi+1t+fiθ∗t,xt,xt-τit+φiθ∗t,xt⊤ωt,i=1,⋯,n-1,x˙nt=ut+ν+fnθ∗t,xt,xt-τnt+φnθ∗t,xt⊤ωt,yt=x1t,where x=[x1,⋯,xn]⊤∈Rn, u∈R, and y∈R are the system state, control input, and output, respectively. ν∈R is an unknown constant, representing the input matching uncertainty, and let xn+1=ν for notational convenience. θ∗(t):R+→Rm is a vector of continuous time-varying parameters belonging to an unknown bounded set. τi(t):R+→R+,i=1,⋯,n, represent the bounded time-varying delays satisfying τ˙i(t)≤κ<1 for an unknown positive constant κ, and the initial condition is x(t0)=Ψ(t0),t0∈[-τ-,0] with τ-=maxi=1,⋯,n{τi(t),t≥0} and Ψ being a continuous function vector defined on [-τ-,0]. ω(t):R+→Rs is disturbance satisfying ω(t)∈L2[0,+∞). For i=1,⋯,n, functions fi(·) and function vectors φi(·) are continuous in the first argument and locally Lipschitz with respect to the rest of variables.Assumption 1.
Fori=1,⋯,n, there is an unknown positive constant θ and a known positive integer p≥1 such that (2)fi·≤θ1+ytp∑j=1ixjt+θ1+yt-τitp∑j=1ixjt-τit,φi·≤θ1+ytp.During the past decade years, the problem of global output feedback control for uncertain nonlinear or nonlinear time-delay systems with unknown growth rate has been extensively studied with the aid of the dynamic gain technique, and a series of interesting results have been obtained; see [6–11] and the references therein. Specifically, for nonlinear time-delay systems with unknown polynomial-of-output growth rate, [10] achieved the global output feedback stabilization based on only one dynamic gain.However, these results do not consider the input matching uncertainty. In many practical control systems, since input matching uncertainty often causes instability or serious deterioration in the performance of systems, output feedback control of nonlinear systems with input matching uncertainty is an attractive topic in recent years; see [12–17] and the references therein. Reference [12] achieved global output feedback regulation of nonlinear systems with zero dynamics and input matching uncertainty, whose nonlinearities are bounded by unmeasured states multiplying known function of output growth rate. References [13, 14] investigated the problem of global adaptive output feedback stabilization of nonlinear systems with input matching uncertainty, whose uncertain nonlinearities only depend on system output. For a class of uncertain time-varying nonlinear systems with input matching uncertainty, whose nonlinearities are strictly restricted, [15] achieved global output feedback stabilization based on two dynamic gains. Lately, a compact design scheme for nonlinear systems with unknown polynomial-of-output growth rate and input matching uncertainty was proposed in [16] based on only one dynamic gain. Reference [17] studied the output tracking problem for a class of stochastic nonlinear systems with unstable modes.As far as we know, the problem of output feedback disturbance attenuation of uncertain nonlinear systems with input matching uncertainty and unknown multiple time-varying delays, whose nonlinearities are bounded by unmeasured states multiplying unknown constant and polynomial-of-output growth rates, has not yet been considered until now. In this paper, we make an attempt to handle this interesting problem by skillfully combining extended state observer, dynamic gain technique, and Lyapunov-Krasovskii theorem.Since there simultaneously exist three types of uncertainties in system (1) for the problem of disturbance attenuation, input matching uncertainty, two types of growth rates (unknown constant and polynomial-of-output growth rates), and unknown multiple time-varying delays, some essential technical difficulties to control design will be inevitably produced. (i) The observer in [5, 10] is inapplicable to systems of this paper due to the existence of input matching uncertainty, so a rather difficult work is how to construct a feasible observer. (ii) The analysis method in [16] is unsuitable due to the existence of unknown multiple time-varying delays; hence, another difficulty is how to give a new analysis method. This paper will focus on solving these two difficulties.This paper is organized as follows. Section2 gives preliminaries. In Section 3, the design and analysis of output feedback controller are presented, following a simulation example in Section 4. Section 5 concludes this paper.


## 2. Mathematical Preliminaries

In this paper, the argument of function will be omitted whenever no confusion can arise from the context.R, R+, and Rn denote the set of real numbers, the set of all nonnegative real numbers, and the real n-dimensional space, respectively. For any real vector or matrix A, A⊤ denotes its transpose; A>0 denotes that matrix A is a positive definite matrix; λmin(A) denotes the minimal eigenvalue of the symmetric matrix A. For any vector x, x1 and x denote its 1-norm and 2-norm, respectively. Clearly, x≤x1≤nx, where n is the dimension of x. diag{a1,⋯,an} denotes n×n diagonal matrix whose element (k,k) is ak and others are zero. In denotes the n-dimensional identity matrix. L2[0,T) and L∞[0,T) denote the appropriate dimension space of square integrable functions and the appropriate dimension space of uniformly bounded functions on [0,T), respectively, where 0<T≤+∞.Lemma 2 (see [12]).
For any positive real numberμ, there exist real number σ1>0, symmetric positive definite matrices P and Q, and column vectors a=[a1,⋯,an+1]⊤ and k=[k1,⋯,kn]⊤ satisfying the following set of inequalities: (3)A⊤P+PA≤-σ1In+1,Dn+1P+PDn+1≥μP,K⊤Q+QK≤-2σ1In,DnQ+QDn≥μQ,where Di=diag{μ,1+μ,⋯,i-1+μ},i=n,n+1, and (4)A=-a11⋯0⋮⋮⋱⋮-an0⋯1-an+10⋯0,K=01⋯0⋮⋮⋱⋮00⋯1-k1-k2⋯-kn.Lemma 3 (Young’s inequality).
Let real numbersp≥1 and q≥1 satisfy 1/p+1/q=1, then for any x,y∈R and any given positive number γ>0, xy≤γxp+1/q(pγ)-q/pyq.Lemma 4 (see [18]).
Forx,y∈R and p≥1 is a constant, then |x+y|p≤2p-1|xp+yp|.Lemma 5 (Barbalat’s lemma, see [19]).
For a continuously differentiable functionx(t):R+→R, if x(t),x˙(t)∈L∞ and x(t)∈Lp for some p∈[1,+∞), then limt→∞x(t)=0.


## 3. Design and Analysis of Output Feedback Controller

### 3.1. Control Objective of This Paper

The objective of this paper is to construct an output feedback controller for system (1) under Assumption 1 such that, by suitably choosing the design parameters,(i) whenω(t)=0 or ω(t)∈L2[0,+∞)∩L∞[0,+∞), all the states of the closed-loop system are bounded and the original system states and their corresponding observer states all converge to zero, and the estimation of the input matching uncertainty converges to its actual value.(ii) whenω(t)∈L2[0,+∞), for any pregiven small real number π>0, the system output y has the following property: (5)∫0ty2sds≤π2∫0tωs2ds+ρ·,where ρ(·) is a nonnegative bounded function.Remark 6.
Compared with the problem of disturbance attenuation of free-delay systems in [5], (6)ψi·≤θ1+ytp∑j=1ixjt,φi·≤θ1+ytp,where θ is a known constant, and compared with the problem of stabilization control of time-delay systems in [10], (7)ϕi·≤θ1+ytp∑j=1ixjt+θ1+yt-dtp∑j=1ixjt-dt,where d(t) is an unknown time-varying delay; it is worth mentioning that none of the systems in [5, 10] take into account the input match uncertainty. Furthermore, compared with the problem of stabilization control of free-delay systems in [16] with input matching uncertainty, (8)fi·≤θ1+ytp∑j=1ixjt,where θ is an unknown constant. This paper considers the problem of disturbance attenuation for the case in which there simultaneously exist input matching uncertainty, unknown polynomial-of-output growth rate, and unknown multiple time-varying delays; all these factors lead to some essential technical difficulties to control design of the more general systems in this paper.


### 3.2. The Design of Observer and Controller for System (1)

Motivated by [14–16], we design the following extended state observer to rebuild the unmeasured states and estimate the input matching uncertainty and construct a coupled controller: (9)x^˙i=x^i+1+aiLix1-x^1,i=1,⋯,n-1,x^˙n=u+x^n+1+anLnx1-x^1,x^˙n+1=an+1Ln+1x1-x^1,u=-Lnk1x^1-Ln-1k2x^2-⋯-Lknx^n-x^n+1,with L being a dynamic gain updated by (10)L˙t=max-ρ1L2t+ρ2Lt1+ytp2,L1-2μtx1t-x^1t2,L0=1,where x^=[x^1,⋯,x^n]⊤∈Rn and x^n+1∈R are the observer states. μ>0 is a design parameter and will be first selected such that 0<μ<1/2p, p is the same as in Assumption 1. Then, according to Lemma 2, a set (σ1,P,Q,a,k) can be determined to satisfy the inequalities in Lemma 2, and the vectors a and k are selected as the gains of the extended state observer and controller, respectively. ρ1 and ρ2 are positive design parameters to be determined. The dynamic gain L(t) has the following properties for all t≥0: (11)L˙t≥0,Lt≥1,Lt≥Lt-τit,i=1,⋯,n,L˙t≥-ρ1L2t+ρ2Lt1+ytp2,L˙t≥L1-2μtx1t-x^1t2.Remark 7.
Since the existence of the input matching uncertaintyν in system (1), the introduction of x^n+1 in the observer (9) is indispensable to compensate the input matching uncertainty ν. In Theorem 8, we will prove that x^n+1 ultimately converge to the actual value of the input matching uncertainty ν.Introduce coordinate transformation(12)εi=xi-x^iLi-1+μ,i=1,⋯,n+1,zi=x^iLi-1+μ,i=1,⋯,n.By (1), (9), and (12), we obtain (13)ε˙=LAε+F-L˙LDn+1ε,z˙=LKz+La∗ε1-L˙LDnz,where ε=[ε1,⋯,εn+1]⊤, z=[z1,⋯,zn]⊤, F=[f1+φ1⊤ω/Lμ,⋯,fn+φn⊤ω/Ln-1+μ,0]⊤, a∗=[a1,⋯,an]⊤. Let Vε=mε⊤Pε and Vz=z⊤Qz with m=1+Qa∗2/σ12. By (11), (13), Lemma 2, and the fact that λmin(P)ε2≤ε⊤Pε and λmin(Q)z2≤z⊤Qz, we have (14)V˙ε≤-σ1mLε2+2mε⊤PF+mρ1μλminPLε2-mρ2μλminP1+yp2ε2,(15)V˙z≤-2σ1Lz2+2Lz⊤Qa∗ε1+ρ1μλminQLz2-ρ2μλminQ1+yp2z2.By Lemma 3, it is easy to get (16)2Lz⊤Qa∗ε1≤σ1Lz2+Qa∗2σ1Lε12.By (2), (12), and the fact that L(t)≥1 is a monotone nondecreasing function for i=1,⋯,n, (17)fi+φi⊤ωLi-1+μ≤θ1+ypLi-1+μ∑j=1ixj+ω+θ1+yt-τitpLi-1+μ∑j=1ixjt-τit≤θ1+ypnε+z+θ1+ypω+θ1+yt-τitpnεt-τit+zt-τit.Using (17) and Lemma 3, it follows that (18)2mε⊤PF≤2mPε∑i=1nfi+φi⊤ωLi-1+μ≤1+yp2ε2+z2+d0ε2+g0ω2+∑i=1n1-κn1+yt-τitp2εt-τit2+zt-τit2,where d0=(2+2/1-κ)n3m2θ2P2 and g0=2n2m2θ2P2 are unknown positive constants related to θ and κ.Choose the Lyapunov-Krasovskii functional(19)V=Vε+Vz+∑i=1n∫t-τitt1n1+ysp2εs2+zs2ds,and select the design parameters ρ1 and ρ2 to satisfy (20)ρ1≤min⁡σ12mμλminP,σ12μλminQ,ρ2≥max⁡2mμλminP,2μλminQ.Then, using (14)-(16), (18)-(20), and τ˙i(t)≤κ<1,i=1,⋯,n, we arrive at (21)V˙≤-σ12Lε2+z2+d0ε2+g0ω2.


### 3.3. Stability and Convergence Analysis

We state the main result in this paper.Theorem 8.
Consider system (1) satisfying Assumption 1, and under the output feedback controller (9) and (10), the closed-loop system consisting of (1), (9), and (10) achieves global disturbance attenuation in the L2-gain sense. Moreover, if ω(t)=0 or ω(t)∈L2[0,+∞)∩L∞[0,+∞), then limt→+∞(x(t),x^(t),x^n+1,u(t))=(0,0,ν,-ν).Proof.
It is observed that the right-hand side of the closed-loop system consisting of (1), (9), and (10) is continuous and locally Lipschitz in (x,x^,x^n+1,u); hence, the closed-loop system has a unique solution on the maximal interval [0,tf) with 0<tf≤+∞. Next, we divide the proof into two steps.
Step I (the boundedness of L(t), ε(t), and z(t) on [0,tf))
(i) Boundedness of L ( t ) on [ 0 , t f ). We prove the boundedness of L(t) on [0,tf) by a contradiction argument. Suppose that L(t) is unbounded on [0,tf); note the monotone nondecreasing property of L(t); there holds limt→tfL(t)=+∞. Hence, there is a finite time t1∈[0,tf) such that L(t)≥max{1,4d0/σ1},∀t∈[t1,tf). Then, from (21), it follows that (22)V˙t≤-σ14Ltεt2-σ12Ltzt2+g0ωt2,∀t∈t1,tf,which, together with ω(t)∈L2[0,+∞), implies that ε(t) and z(t) are bounded on [t1,tf) and (23)σ14∫t1tfLsεs2ds≤Vt1+g0∫t1tfωs2ds<+∞.By (12), Lemma 4, and the fact that L(t)≥1, it is obvious that (24)-ρ1L2t+ρ2Lt1+ytp2≤-ρ1L2t+2ρ21+ε1t+z1t2pL1+2μpt.Then, by 0<μ<1/2p and the boundedness of ε(t) and z(t) on [t1,tf), there is a finite time t2∈[t1,tf), such that -ρ1L2(t)+ρ2L(t)(1+|y(t)|p)2≤0 on [t2,tf), which, together with (10) and (12), implies that (25)+∞=Ltf-Lt2=∫t2tfL˙sds=∫t2tfLsε12sds≤∫t1tfLsεs2ds.This contradicts with (23). Thus, L(t) is bounded on [0,tf) and suppose limt→tfL(t)=L- with L-≥1 being a constant.
(ii) Boundedness of z ( t ) on [ 0 , t f ). By (11)-(12), (15)-(16), and (20), we obtain (26)V˙z≤-σ12z2+Qa∗2σ1L˙-ρ2μλminQ1+yp2z2.Integrating both sides of (26) with L(t)≥1 being a monotone nondecreasing function and L(0)=1, limt→tfL(t)=L- leads to, on [0,tf), (27)λminQzt2≤Vzt≤Vz0-σ12∫0tzs2ds+Qa∗2σ1L--1,from which it follows that, ∀t∈[0,tf), (28)zt2≤1λminQVz0+Qa∗2σ1L-,∫0tzs2ds≤2σ1Vz0+Qa∗2σ1L-,which implies that z(t) and ∫0t‍z(s)2ds are bounded on [0,tf).
(iii) Boundedness of ε ( t ) on [ 0 , t f ). To prove the boundedness of ε(t) on [0,tf), we introduce a new change of coordinate (29)ϵi=εiL∗i-1+μ,i=1,⋯,n+1,where the constant is L∗≥1/m+2d1/σ1+1>1 with d1 being a positive constant to be defined. By (13) and (29), we arrive at (30)ϵ˙=LL∗Aϵ+LL∗aϵ1-LΓaϵ1+F∗-L˙LDn+1ϵ,where ϵ=[ϵ1,⋯,ϵn+1]⊤, Γ=diag{1,1/L∗,⋯,1/(L∗)n}, and F∗=[f1+φ1⊤ω/(LL∗)μ,⋯,fn+φn⊤ω/(LL∗)n-1+μ,0]⊤. Choose the Lyapunov function Vϵ=ϵ⊤Pϵ. By (11), (30), Lemma 2, and the fact that λmin(P)ϵ2≤ϵ⊤Pϵ, (31)V˙ϵ≤-σ1LL∗ϵ2+2LL∗ϵ⊤Paϵ1-2Lϵ⊤PΓaϵ1+2ϵ⊤PF∗+Lρ1μλminPϵ2-ρ2μλminP1+yp2ϵ2.Using (11), (12), (29), and Lemma 3, we obtain (32)2LL∗ϵ⊤Paϵ1≤σ1LL∗4ϵ2+4L∗1-2μPa2σ1L˙,-2Lϵ⊤PΓaϵ1≤σ1LL∗4ϵ2+4PΓa2σ1L∗1+2μL˙.Similar to the proof of (18) with L∗≥1, one gets (33)2ϵ⊤PF∗≤1+yp21mϵ2+z2+d1ϵ2+g1ω2+∑i=1n1-κn1+yt-τitp21mϵt-τit2+zt-τit2,where d1=n3θ2P2(2m+1+m+1/1-κ), g1=2mn2θ2P2 are unknown positive constants related to θ and κ.
Choose the Lyapunov-Krasovskii functional(34)V∗=Vϵ+Vz+∑i=1n∫t-τitt1n1+ysp21mϵs2+zs2ds.By (20), (26), (31)-(34), the definition of L∗, and the fact that L(t)≥1, we derive (35)V˙∗≤-σ12LL∗ϵ2+Lρ1μλminPϵ2-ρ2μλminP1+yp2ϵ2+2m1+yp2ε2+ηL˙+21+yp2z2+d1ϵ2+g1ω2-ρ2μλminQ1+yp2z2+σ12mLϵ2-σ12mLϵ2≤-σ12LL∗-1m-2d1σ1ϵ2+ηL˙+g1ω2≤-σ12ϵ2+ηL˙+g1ω2,where η=Qa∗2/σ1+4(L∗)1-2μPa2/σ1+4PΓa2/σ1(L∗)1+2μ is a positive constant. Similar to the proof of (27), integrating both sides of (35) yields, on [0,tf), (36)λminPϵt2≤V∗t≤V∗0-σ12∫0tϵs2ds+ηL--1+g1∫0tωs2ds,from which it follows that, ∀t∈[0,tf), (37)ϵt2≤1λminPV∗0+ηL-+g1∫0tωs2ds,∫0tϵs2ds≤2σ1V∗0+ηL-+g1∫0tωs2ds,which together with ω(t)∈L2[0,+∞) implies that ϵ(t) and ∫0t‍ϵ(s)2ds are bounded on [0,tf). From (29) with the fact that L∗ is a constant, we obtain ε(t) and ∫0t‍ε(s)2ds are bounded on [0,tf).
Step II (we prove that the closed-loop system consisting of (1), (9), and (10) achieves global disturbance attenuation in the L2-gain sense). From (12), it is easy to get (38)xi=Li-1+μεi+zi,i=1,⋯,n,x^i=Li-1+μzi,i=1,⋯,n,x^n+1=ν-Ln+μεn+1.When ω(t)∈L2[0,+∞), by the boundedness of L(t), z(t), ε(t) on [0,tf) and the fact that ν is a constant, it can be derived from (38) that x(t), x^(t), and x^n+1(t) are bounded on [0,tf). By (9), it follows that u(t) is also bounded on [0,tf).
Then, we prove thattf=+∞. The conclusion follows again by a contradiction argument. Suppose tf<+∞, then tf would be the finite-escape time of the closed-loop system, which means that at least one component of x(t), x^(t), x^n+1(t), and L(t) would tend to infinity at t=tf. However, x(t), x^(t), x^n+1(t), and L(t) are bounded on the maximal interval [0,tf) and hence also bounded at t=tf due to the continuity of the solution; this is a contradiction.
t f = + ∞ means that L(t), z(t), ε(t), ∫0t‍z(s)2ds, ∫0t‍ε(s)2ds are bounded on [0,+∞), which indicates that x(t), x^(t), x^n+1(t), and u(t) are bounded on [0,+∞), and ∫0+∞‍z(s)2ds<+∞, ∫0+∞‍ε(s)2ds<+∞. Meanwhile, from (38) with y=x1, we obtain ∫0+∞‍y2(s)ds<+∞.
Using (21), for any pregiven small real number π>0, (39)V˙+g0π2y2≤d0ε2+g0ω2+g0π2y2.Integrating both sides of (39) leads to, ∀t≥0, (40)∫0ty2sds≤π2∫0tωs2ds+π2g0V0+π2d0g0∫0tεs2ds+∫0ty2sds;choose ρ(·)=π2/g0V(0)+π2d0/g0∫0t‍ε(s)2ds+∫0t‍y2(s)ds. Obviously, ρ(·) is a nonnegative bounded function. Then the global disturbance attenuation of the closed-loop system is achieved in the L2-gain sense.
Moreover, ifω(t)=0 or ω(t)∈L2[0,+∞)∩L∞[0,+∞), by the boundedness of all signals on [0,+∞), from (13), it is obvious that ε˙(t) and z˙(t) are also bounded on [0,+∞). Then, by Lemma 5, we conclude that limt→+∞(ε(t),z(t))=(0,0). Therefore, from (9) and (38) with the boundedness of L(t), we deduce limt→+∞(x(t),x^(t),x^n+1(t),u(t))=(0,0,ν,-ν).


## 3.1. Control Objective of This Paper

The objective of this paper is to construct an output feedback controller for system (1) under Assumption 1 such that, by suitably choosing the design parameters,(i) whenω(t)=0 or ω(t)∈L2[0,+∞)∩L∞[0,+∞), all the states of the closed-loop system are bounded and the original system states and their corresponding observer states all converge to zero, and the estimation of the input matching uncertainty converges to its actual value.(ii) whenω(t)∈L2[0,+∞), for any pregiven small real number π>0, the system output y has the following property: (5)∫0ty2sds≤π2∫0tωs2ds+ρ·,where ρ(·) is a nonnegative bounded function.Remark 6.
Compared with the problem of disturbance attenuation of free-delay systems in [5], (6)ψi·≤θ1+ytp∑j=1ixjt,φi·≤θ1+ytp,where θ is a known constant, and compared with the problem of stabilization control of time-delay systems in [10], (7)ϕi·≤θ1+ytp∑j=1ixjt+θ1+yt-dtp∑j=1ixjt-dt,where d(t) is an unknown time-varying delay; it is worth mentioning that none of the systems in [5, 10] take into account the input match uncertainty. Furthermore, compared with the problem of stabilization control of free-delay systems in [16] with input matching uncertainty, (8)fi·≤θ1+ytp∑j=1ixjt,where θ is an unknown constant. This paper considers the problem of disturbance attenuation for the case in which there simultaneously exist input matching uncertainty, unknown polynomial-of-output growth rate, and unknown multiple time-varying delays; all these factors lead to some essential technical difficulties to control design of the more general systems in this paper.


## 3.2. The Design of Observer and Controller for System (1)

Motivated by [14–16], we design the following extended state observer to rebuild the unmeasured states and estimate the input matching uncertainty and construct a coupled controller: (9)x^˙i=x^i+1+aiLix1-x^1,i=1,⋯,n-1,x^˙n=u+x^n+1+anLnx1-x^1,x^˙n+1=an+1Ln+1x1-x^1,u=-Lnk1x^1-Ln-1k2x^2-⋯-Lknx^n-x^n+1,with L being a dynamic gain updated by (10)L˙t=max-ρ1L2t+ρ2Lt1+ytp2,L1-2μtx1t-x^1t2,L0=1,where x^=[x^1,⋯,x^n]⊤∈Rn and x^n+1∈R are the observer states. μ>0 is a design parameter and will be first selected such that 0<μ<1/2p, p is the same as in Assumption 1. Then, according to Lemma 2, a set (σ1,P,Q,a,k) can be determined to satisfy the inequalities in Lemma 2, and the vectors a and k are selected as the gains of the extended state observer and controller, respectively. ρ1 and ρ2 are positive design parameters to be determined. The dynamic gain L(t) has the following properties for all t≥0: (11)L˙t≥0,Lt≥1,Lt≥Lt-τit,i=1,⋯,n,L˙t≥-ρ1L2t+ρ2Lt1+ytp2,L˙t≥L1-2μtx1t-x^1t2.Remark 7.
Since the existence of the input matching uncertaintyν in system (1), the introduction of x^n+1 in the observer (9) is indispensable to compensate the input matching uncertainty ν. In Theorem 8, we will prove that x^n+1 ultimately converge to the actual value of the input matching uncertainty ν.Introduce coordinate transformation(12)εi=xi-x^iLi-1+μ,i=1,⋯,n+1,zi=x^iLi-1+μ,i=1,⋯,n.By (1), (9), and (12), we obtain (13)ε˙=LAε+F-L˙LDn+1ε,z˙=LKz+La∗ε1-L˙LDnz,where ε=[ε1,⋯,εn+1]⊤, z=[z1,⋯,zn]⊤, F=[f1+φ1⊤ω/Lμ,⋯,fn+φn⊤ω/Ln-1+μ,0]⊤, a∗=[a1,⋯,an]⊤. Let Vε=mε⊤Pε and Vz=z⊤Qz with m=1+Qa∗2/σ12. By (11), (13), Lemma 2, and the fact that λmin(P)ε2≤ε⊤Pε and λmin(Q)z2≤z⊤Qz, we have (14)V˙ε≤-σ1mLε2+2mε⊤PF+mρ1μλminPLε2-mρ2μλminP1+yp2ε2,(15)V˙z≤-2σ1Lz2+2Lz⊤Qa∗ε1+ρ1μλminQLz2-ρ2μλminQ1+yp2z2.By Lemma 3, it is easy to get (16)2Lz⊤Qa∗ε1≤σ1Lz2+Qa∗2σ1Lε12.By (2), (12), and the fact that L(t)≥1 is a monotone nondecreasing function for i=1,⋯,n, (17)fi+φi⊤ωLi-1+μ≤θ1+ypLi-1+μ∑j=1ixj+ω+θ1+yt-τitpLi-1+μ∑j=1ixjt-τit≤θ1+ypnε+z+θ1+ypω+θ1+yt-τitpnεt-τit+zt-τit.Using (17) and Lemma 3, it follows that (18)2mε⊤PF≤2mPε∑i=1nfi+φi⊤ωLi-1+μ≤1+yp2ε2+z2+d0ε2+g0ω2+∑i=1n1-κn1+yt-τitp2εt-τit2+zt-τit2,where d0=(2+2/1-κ)n3m2θ2P2 and g0=2n2m2θ2P2 are unknown positive constants related to θ and κ.Choose the Lyapunov-Krasovskii functional(19)V=Vε+Vz+∑i=1n∫t-τitt1n1+ysp2εs2+zs2ds,and select the design parameters ρ1 and ρ2 to satisfy (20)ρ1≤min⁡σ12mμλminP,σ12μλminQ,ρ2≥max⁡2mμλminP,2μλminQ.Then, using (14)-(16), (18)-(20), and τ˙i(t)≤κ<1,i=1,⋯,n, we arrive at (21)V˙≤-σ12Lε2+z2+d0ε2+g0ω2.


## 3.3. Stability and Convergence Analysis

We state the main result in this paper.Theorem 8.
Consider system (1) satisfying Assumption 1, and under the output feedback controller (9) and (10), the closed-loop system consisting of (1), (9), and (10) achieves global disturbance attenuation in the L2-gain sense. Moreover, if ω(t)=0 or ω(t)∈L2[0,+∞)∩L∞[0,+∞), then limt→+∞(x(t),x^(t),x^n+1,u(t))=(0,0,ν,-ν).Proof.
It is observed that the right-hand side of the closed-loop system consisting of (1), (9), and (10) is continuous and locally Lipschitz in (x,x^,x^n+1,u); hence, the closed-loop system has a unique solution on the maximal interval [0,tf) with 0<tf≤+∞. Next, we divide the proof into two steps.
Step I (the boundedness of L(t), ε(t), and z(t) on [0,tf))
(i) Boundedness of L ( t ) on [ 0 , t f ). We prove the boundedness of L(t) on [0,tf) by a contradiction argument. Suppose that L(t) is unbounded on [0,tf); note the monotone nondecreasing property of L(t); there holds limt→tfL(t)=+∞. Hence, there is a finite time t1∈[0,tf) such that L(t)≥max{1,4d0/σ1},∀t∈[t1,tf). Then, from (21), it follows that (22)V˙t≤-σ14Ltεt2-σ12Ltzt2+g0ωt2,∀t∈t1,tf,which, together with ω(t)∈L2[0,+∞), implies that ε(t) and z(t) are bounded on [t1,tf) and (23)σ14∫t1tfLsεs2ds≤Vt1+g0∫t1tfωs2ds<+∞.By (12), Lemma 4, and the fact that L(t)≥1, it is obvious that (24)-ρ1L2t+ρ2Lt1+ytp2≤-ρ1L2t+2ρ21+ε1t+z1t2pL1+2μpt.Then, by 0<μ<1/2p and the boundedness of ε(t) and z(t) on [t1,tf), there is a finite time t2∈[t1,tf), such that -ρ1L2(t)+ρ2L(t)(1+|y(t)|p)2≤0 on [t2,tf), which, together with (10) and (12), implies that (25)+∞=Ltf-Lt2=∫t2tfL˙sds=∫t2tfLsε12sds≤∫t1tfLsεs2ds.This contradicts with (23). Thus, L(t) is bounded on [0,tf) and suppose limt→tfL(t)=L- with L-≥1 being a constant.
(ii) Boundedness of z ( t ) on [ 0 , t f ). By (11)-(12), (15)-(16), and (20), we obtain (26)V˙z≤-σ12z2+Qa∗2σ1L˙-ρ2μλminQ1+yp2z2.Integrating both sides of (26) with L(t)≥1 being a monotone nondecreasing function and L(0)=1, limt→tfL(t)=L- leads to, on [0,tf), (27)λminQzt2≤Vzt≤Vz0-σ12∫0tzs2ds+Qa∗2σ1L--1,from which it follows that, ∀t∈[0,tf), (28)zt2≤1λminQVz0+Qa∗2σ1L-,∫0tzs2ds≤2σ1Vz0+Qa∗2σ1L-,which implies that z(t) and ∫0t‍z(s)2ds are bounded on [0,tf).
(iii) Boundedness of ε ( t ) on [ 0 , t f ). To prove the boundedness of ε(t) on [0,tf), we introduce a new change of coordinate (29)ϵi=εiL∗i-1+μ,i=1,⋯,n+1,where the constant is L∗≥1/m+2d1/σ1+1>1 with d1 being a positive constant to be defined. By (13) and (29), we arrive at (30)ϵ˙=LL∗Aϵ+LL∗aϵ1-LΓaϵ1+F∗-L˙LDn+1ϵ,where ϵ=[ϵ1,⋯,ϵn+1]⊤, Γ=diag{1,1/L∗,⋯,1/(L∗)n}, and F∗=[f1+φ1⊤ω/(LL∗)μ,⋯,fn+φn⊤ω/(LL∗)n-1+μ,0]⊤. Choose the Lyapunov function Vϵ=ϵ⊤Pϵ. By (11), (30), Lemma 2, and the fact that λmin(P)ϵ2≤ϵ⊤Pϵ, (31)V˙ϵ≤-σ1LL∗ϵ2+2LL∗ϵ⊤Paϵ1-2Lϵ⊤PΓaϵ1+2ϵ⊤PF∗+Lρ1μλminPϵ2-ρ2μλminP1+yp2ϵ2.Using (11), (12), (29), and Lemma 3, we obtain (32)2LL∗ϵ⊤Paϵ1≤σ1LL∗4ϵ2+4L∗1-2μPa2σ1L˙,-2Lϵ⊤PΓaϵ1≤σ1LL∗4ϵ2+4PΓa2σ1L∗1+2μL˙.Similar to the proof of (18) with L∗≥1, one gets (33)2ϵ⊤PF∗≤1+yp21mϵ2+z2+d1ϵ2+g1ω2+∑i=1n1-κn1+yt-τitp21mϵt-τit2+zt-τit2,where d1=n3θ2P2(2m+1+m+1/1-κ), g1=2mn2θ2P2 are unknown positive constants related to θ and κ.
Choose the Lyapunov-Krasovskii functional(34)V∗=Vϵ+Vz+∑i=1n∫t-τitt1n1+ysp21mϵs2+zs2ds.By (20), (26), (31)-(34), the definition of L∗, and the fact that L(t)≥1, we derive (35)V˙∗≤-σ12LL∗ϵ2+Lρ1μλminPϵ2-ρ2μλminP1+yp2ϵ2+2m1+yp2ε2+ηL˙+21+yp2z2+d1ϵ2+g1ω2-ρ2μλminQ1+yp2z2+σ12mLϵ2-σ12mLϵ2≤-σ12LL∗-1m-2d1σ1ϵ2+ηL˙+g1ω2≤-σ12ϵ2+ηL˙+g1ω2,where η=Qa∗2/σ1+4(L∗)1-2μPa2/σ1+4PΓa2/σ1(L∗)1+2μ is a positive constant. Similar to the proof of (27), integrating both sides of (35) yields, on [0,tf), (36)λminPϵt2≤V∗t≤V∗0-σ12∫0tϵs2ds+ηL--1+g1∫0tωs2ds,from which it follows that, ∀t∈[0,tf), (37)ϵt2≤1λminPV∗0+ηL-+g1∫0tωs2ds,∫0tϵs2ds≤2σ1V∗0+ηL-+g1∫0tωs2ds,which together with ω(t)∈L2[0,+∞) implies that ϵ(t) and ∫0t‍ϵ(s)2ds are bounded on [0,tf). From (29) with the fact that L∗ is a constant, we obtain ε(t) and ∫0t‍ε(s)2ds are bounded on [0,tf).
Step II (we prove that the closed-loop system consisting of (1), (9), and (10) achieves global disturbance attenuation in the L2-gain sense). From (12), it is easy to get (38)xi=Li-1+μεi+zi,i=1,⋯,n,x^i=Li-1+μzi,i=1,⋯,n,x^n+1=ν-Ln+μεn+1.When ω(t)∈L2[0,+∞), by the boundedness of L(t), z(t), ε(t) on [0,tf) and the fact that ν is a constant, it can be derived from (38) that x(t), x^(t), and x^n+1(t) are bounded on [0,tf). By (9), it follows that u(t) is also bounded on [0,tf).
Then, we prove thattf=+∞. The conclusion follows again by a contradiction argument. Suppose tf<+∞, then tf would be the finite-escape time of the closed-loop system, which means that at least one component of x(t), x^(t), x^n+1(t), and L(t) would tend to infinity at t=tf. However, x(t), x^(t), x^n+1(t), and L(t) are bounded on the maximal interval [0,tf) and hence also bounded at t=tf due to the continuity of the solution; this is a contradiction.
t f = + ∞ means that L(t), z(t), ε(t), ∫0t‍z(s)2ds, ∫0t‍ε(s)2ds are bounded on [0,+∞), which indicates that x(t), x^(t), x^n+1(t), and u(t) are bounded on [0,+∞), and ∫0+∞‍z(s)2ds<+∞, ∫0+∞‍ε(s)2ds<+∞. Meanwhile, from (38) with y=x1, we obtain ∫0+∞‍y2(s)ds<+∞.
Using (21), for any pregiven small real number π>0, (39)V˙+g0π2y2≤d0ε2+g0ω2+g0π2y2.Integrating both sides of (39) leads to, ∀t≥0, (40)∫0ty2sds≤π2∫0tωs2ds+π2g0V0+π2d0g0∫0tεs2ds+∫0ty2sds;choose ρ(·)=π2/g0V(0)+π2d0/g0∫0t‍ε(s)2ds+∫0t‍y2(s)ds. Obviously, ρ(·) is a nonnegative bounded function. Then the global disturbance attenuation of the closed-loop system is achieved in the L2-gain sense.
Moreover, ifω(t)=0 or ω(t)∈L2[0,+∞)∩L∞[0,+∞), by the boundedness of all signals on [0,+∞), from (13), it is obvious that ε˙(t) and z˙(t) are also bounded on [0,+∞). Then, by Lemma 5, we conclude that limt→+∞(ε(t),z(t))=(0,0). Therefore, from (9) and (38) with the boundedness of L(t), we deduce limt→+∞(x(t),x^(t),x^n+1(t),u(t))=(0,0,ν,-ν).


## 4. Simulation Example

Consider a simple system(41)x˙1=x2+c1x13t-τ1t+c2ln⁡1+x12ω,x˙2=u+ν+c3x12x2+c4ln⁡1+x12t-τ2tx2t-τ2t+c5ω,y=x1,where ν is an unknown constant representing the input matching uncertainty, and let x3=ν. ci,i=1,⋯,5, are unknown constants, and τ1(t)=1+l1sin⁡(t) and τ2(t)=1/2l2cos2(t) represent unknown time-varying delays with l1<1, 0<l2<1 being unknown constants. The system disturbance is ω(t)=1/1+t; obviously, ω(t)∈L2[0,+∞). Since ln⁡(1+x12)≤1+x12, Assumption 1 holds with θ=max⁡{ci,i=1,⋯,5}, p=2.From Lemma2, we choose μ=0.2<1/4, a=[3,3,1]⊤, k=[3,3]⊤, σ1=18. Then, by (20), select ρ1=0.9 and ρ2=12.8. According to Section 3, we get the output feedback controller (42)x^˙1=x^2+3Lx1-x^1,x^˙2=x^3+u+3L2x1-x^1,x^˙3=L3x1-x^1,u=-3L2x^1-3Lx^2-x^3,L˙=max-0.9L2+12.8L1+y22,L0.6x1-x^12,L0=1.In the simulation, we chooseci=1,i=1,⋯,5, ν=-8, l1=l2=0.5 and the initial values [x1(t0),x2(t0)]=[0.5,0.8] for t0∈[-1.5,0], [x^1(0),x^2(0),x^3(0)]=[0.2,0.5,4]. Figure 1 demonstrates the effectiveness of the control scheme.Figure 1
The responses of the closed-loop system (41)-(42).


## 5. Conclusions

By skillfully combining extended state observer, dynamic gain technique, and Lyapunov-Krasovskii theorem, the problem of output feedback disturbance attenuation for nonlinear systems with input matching uncertainty and unknown multiple time-varying delays is solved in this paper based on only one dynamic gain.Some interesting problems still remained; e.g., (i) for system (1) with the unknown output function [20–24], can we design an output feedback controller? (ii) Another work is to consider more general input matching uncertainty such as an uncertain harmonic signal.



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*Source: 2870496-2018-07-05.xml*