# Global Output Feedback Stabilization of Nonlinear Systems with a Time-Varying Power and Unknown Output Function

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

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## Abstract

This paper studies the problem of global output feedback stabilization for a class of nonlinear systems with a time-varying power and unknown output function. For nonlinear systems with a time-varying power and unknown continuous output function, by constructing a new nonlinear reduced-order observer together with adding a power integrator method, a new function to determine the maximal open sectorΩ of output function is given. As long as output function belongs to any closed sector included in Ω, it is shown that the equilibrium point of the closed-loop system can be guaranteed globally uniformly asymptotically stable by an output feedback controller.


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## Body

## 1. Introduction

Consider nonlinear systems with the unknown output function(1)x˙it=xi+1tpt+ϕit,x1t,…,xit,i=1,…,n-1,x˙nt=utpt+ϕnt,x1t,…,xnt,yt=hx1t,where x=(x1,…,xn)⊤∈Rn, u∈R, and y∈R are the unmeasurable state, control input, and output, respectively. For a real constant ϱ>0, [·]ϱ is defined as [·]ϱ=sgn(·)·ϱ. The time-varying power p(t):R+→R+ is a continuous bounded function satisfying 1≤p_≤p(t)≤p¯ with two constants p_ and p¯. For i=1,…,n, ϕi(·):R+×Ri→R are continuous in the first argument and locally Lipschitz with respect to the rest variables with ϕi(t,0)=0. Output function h(·) is an unknown continuous function with h(0)=0.Over the past decade, with the help of adding a power integrator method, homogeneous domination method, and recursive observer design, there exist some interesting results on state/output feedback design of high-order nonlinear systems, whose powers are known constant ratios of odd integers; see [1–16] and the references therein.In recent years, some interesting results have been achieved on output feedback design of nonlinear systems with known constant powers and unknown output function. For the nonlinear systems (1) with p(t)=1, when h(·) is a continuous differentiable function and its derivative with known upper and lower bounds, global output feedback stabilization and finite-time output feedback stabilization have been achieved in [17–19] and [20], respectively. When the derivative of h(·) is with unknown upper bound, [21] achieved semi-global output feedback control. Furthermore, when h(·) is extended to be only a continuous function, with the help of the given maximal sector region of output function, [22] achieved output feedback stabilization. Lately, in [23], a new design and analysis method for high-order nonlinear systems with unknown continuous output function was proposed based on adding a power integrator method and homogeneous domination method.According to some practice [24–26], it is well-known that the timely deteriorated performance of system often results in different running data, which usually identify the powers of system. Therefore, the powers of system are usually not fixed and can be varying with a suitable bound even in the same working condition. For example, as a practical second-order dynamic model of reduced-order boiler-turbine unit, two typical different powers p(t)=1.072 and p(t)=1.031 have been identified in [25] and [26], respectively. For nonlinear systems with a time-varying power, [27, 28] achieved global state feedback stabilization based on interval homogeneous domination approach. As far as we know, [29] is the first paper to study the output feedback stabilization of nonlinear systems (1) with h(x1)=x1 by the revamped method of adding a power integrator together with the recursive nonlinear observer design. Naturally, an interesting problem is put forward:For more general nonlinear systems (1) withh(x1)being an unknown continuous function, can we design an output feedback controller?In this paper, we make an attempt to handle this problem. Some essential technical difficulties in control design will be inevitably produced: (i) Compared with [29], since output function is unknown, we construct a new nonlinear reduced-order observer without using the unmeasurable state x1. (ii) Compared with [23], a new function to determine the maximal open sector Ω of output function is given since the power of system is time-varying. As long as output function belongs to any closed sector included in Ω, the equilibrium point of the closed-loop system is globally uniformly asymptotically stable under the constructed output feedback controller.This paper is organized as follows. Section2 gives some preliminaries. The design and analysis of output feedback controller are given in Section 3, following a simulation example in Section 4. Section 5 concludes this paper.


## 2. Mathematical Preliminaries

Some notations, definitions, and lemmas are to be used throughout this paper.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. For constant βi,i=1,…,n, let βi~j=∏k=ijβk,1≤i<j≤n, and βi~i=βi, βi~(i-1)=1, β0~i=0. For integers a,b and constant Λi, ∑i=abΛi=0 when a>b.Definition 1 (see [30]).
A functionh(·):R→R is said to belong to the sector [ρ1,ρ2] if (h(s)-ρ1s)(h(s)-ρ2s)≤0, where ρ1 and ρ2 are constants with ρ1<ρ2. If the inequality is strict, we write the sector as (ρ1,ρ2).The following lemmas will serve as the basis for the development of output feedback controller. Lemmas2–5 are used to enlarge inequalities. Lemmas 6 and 7 are Lyapunov stability theorem for the global uniformly asymptotically stable of the closed-loop system.Lemma 2 (see [31]).
Letp(t) be a real-valued function of t∈R+ satisfying 0<p(t)<+∞. For any xi∈R, i=1,…,n,(2)x1+⋯+xnpt≤max⁡npt-1,1x1pt+⋯+xnpt.Lemma 3 (see [32]).
Letp(t),q(t) be positive real-valued functions of t∈R+ and γ(x,y) be a positive real-valued function of x,y∈R. For any x,y∈R,(3)xptyqt≤ptpt+qtγx,yxpt+qt+qtpt+qtγ-pt/qtx,yypt+qt.Lemma 4 (see [33]).
Letp(t) be a real-valued function of t∈R+ satisfying p(t)≥1. For any x,y∈R,(4)xpt-ypt≤pt2pt-3+1x-ypt+x-yypt-1,where ypt-1≔0 if y=0 and p(t)=1.Lemma 5 (see [33]).
Letp(t) be a real-valued function of t∈R+ satisfying p(t)≥1. For any x,y∈R,(5)-x-yxpt-ypt≤-21-ptx-ypt+1.Lemma 6 (see [30]).
LetV:Rn→R be a continuous positive definite and radially unbounded function defined on Rn; then there exist class K∞ functions π1 and π2 defined on [0,+∞) such that π1(x)≤V(x)≤π2(x) for all x∈Rn.Lemma 7 (see [30]).
For the nonautonomous systemx˙=f(t,x), let x=0 be an equilibrium point of system and V:R+×Rn→R+ be a continuously differentiable function such that W1(x)≤V(t,x)≤W2(x) and ∂V/∂t+(∂V/∂x)f(t,x)≤-W3(x) hold for any t≥0 and x∈Rn, where W1(x),W2(x),W3(x) are continuous positive definite functions on Rn and W1(x) is radially unbounded. Then x=0 is globally uniformly asymptotically stable.


## 3. Output Feedback Controller Design and Stability Analysis

### 3.1. Control Objective of This Paper

The objective of this paper is to construct an output feedback controller for system (1) such that the equilibrium point of the closed-loop system is globally uniformly asymptotically stable when the maximal open sector of output function is given.Assumption 8.
There is a known constantc≥0 such that for i=1,…,n,(6)ϕit,x-ϕit,y≤c∑j=1N∑k=1ixk-ykak,jt∑q=1ixqbk,jt+yqbk,jt,for all x,y∈Rn, where N is an arbitrary positive integer and the real-valued functions ak,j(t) and bk,j(t) satisfy the following relations:(7)ak,jt+bk,jt=pt,a_k,j≤ak,jt≤a¯k,j,b_k,j≤bk,jt≤b¯k,j,for all j=1,…,N and k=1,…,i, with 0<a_k,j≤a¯k,j and 0<b_k,j≤b¯k,j. By Lemma 3 and Assumption 8, there exists a constant c~≥0 such that(8)ϕit,x1,…,xi≤c~x1pt+⋯+xipt,i=1,…,n.Remark 9.
As far as we know, under Assumption8, [29] is the first paper to study output feedback stabilization of nonlinear system (1) with h(x1)=x1. In this paper, we will consider system (1) with h(x1) being an unknown continuous function.


### 3.2. State Feedback Controller Design of System (1)

Step 1.
Takingξ1=x1, V1(x1)=(1/2)x12, it follows from (1) and (8) that(9)V˙1=x1x2pt+x1ϕ1≤x1x2pt-x2∗pt+x1x2∗pt+c~x1pt+1.Choose the virtual controller x2∗=-β1x1 with β1=(r1,1+c~)1/p_, where r1,1≥1 is a constant to be designed. Due to (r1,1+c~)p(t)/p_≥(r1,1+c~), (9) becomes(10)V˙1≤-r1,1ξ1pt+1+ξ1x2pt-x2∗pt.Inductive Step. Suppose that at step k-1(k≥2), there is a positive definite and radially unbounded Lyapunov function Vk-1(ξ1,…,ξk-1)=Vk-2(ξ1,…,ξk-2)+(1/2)ξk-12 and a set of virtual controllers x2∗,…,xk∗ defined by(11)ξ1=x1,xi∗=-βi-1ξi-1,ξi=xi-xi∗,i=2,…,k,such that(12)V˙k-1≤-∑i=1k-2ri,i-r~i+1,1-∑j=i+1k-1r¯j,iξipt+1-rk-1,k-1ξk-1pt+1+ξk-1xkpt-xk∗pt,where r~i+1,1,r¯j,i,j=i+1,…,k-1,i=1,…,k-2, are positive constants, r1,1,…,rk-1,k-1≥1 are constants to be designed, and β1,…,βk-1 are positive constants dependent on r1,1,…,rk-1,k-1. In what follows, we will show that (12) still holds at step k.
By (11), one has xk∗=-∑i=1k-1βi~(k-1)xi, which together with (1) lead to(13)ξ˙k=xk+1pt+ϕk+∑i=1k-1βi~k-1xi+1pt+ϕi.Constructing Vk(ξ1,…,ξk)=Vk-1(ξ1,…,ξk-1)+(1/2)ξk2, it follows from (12) and (13) that(14)V˙k≤-∑i=1k-2ri,i-r~i+1,1-∑j=i+1k-1r¯j,iξipt+1-rk-1,k-1ξk-1pt+1+ξk-1xkpt-xk∗pt+ξkxk+1pt+ϕk+∑i=1k-1βi~k-1xi+1pt+ϕi.By (8), (11), and Lemmas 2–4, we derive(15)ξk-1xkpt-xk∗pt≤ξk-1·p¯2p¯-3+1ξkpt+ξkxk∗pt-1≤r~k,1ξk-1pt+1+λk,1ξkpt+1,(16)ξkϕk+∑i=1k-1βi~k-1xi+1pt+ϕi≤ξkβ¯1ξ1pt+β¯2x2pt+⋯+β¯kxkpt≤∑i=1k-1r¯k,iξipt+1+λk,2ξkpt+1,where β¯i=β(i-1)~(k-1)+c~(1+∑j=ik-1βj~(k-1)), r~k,1,r¯k,1,…,r¯k,k-1 are positive constants, and λk,1 and λk,2 are positive constants dependent on r~k,1 and r¯k,1,…,r¯k,k-1, respectively. Substituting (15) and (16) into (14) leads to(17)V˙k≤-∑i=1k-1ri,i-r~i+1,1-∑j=i+1kr¯j,iξipt+1+ξkxk+1pt-xk+1∗pt+ξkxk+1∗pt+λk,1+λk,2ξkpt+1.Choose the virtual controller xk+1∗=-βkξk with βk=(rk,k+λk,1+λk,2)1/p_, where rk,k≥1 is a constant to be designed. Due to (rk,k+λk,1+λk,2)p(t)/p_≥(rk,k+λk,1+λk,2), (17) becomes(18)V˙k≤-∑i=1k-1ri,i-r~i+1,1-∑j=i+1kr¯j,iξipt+1-rk,kξkpt+1+ξkxk+1pt-xk+1∗pt.This completes the inductive step.Step n.
The Lyapunov functionVn(ξ)=Vn-1(ξ1,…,ξn-1)+(1/2)ξn2=∑i=1n(1/2)ξi2 and a positive constant λgive(19)V˙n≤-∑i=1n-1ri,i-r~i+1,1-∑j=i+1nr¯j,iξipt+1+λξnpt+1+ξnupt-xn+1∗pt+ξnxn+1∗pt,where ξi is defined as in (11) for i=1,…,n. Choose virtual controller xn+1∗=-βnξn with βn=(rn,n+λ)1/p_, where rn,n≥1 is a constant to be designed. Due to (rn,n+λ)p(t)/p_≥(rn,n+λ), (19) becomes(20)V˙n≤-∑i=1n-1ri,i-r~i+1,1-∑j=i+1nr¯j,iξipt+1-rn,nξnpt+1+ξnupt-xn+1∗pt.


### 3.3. Output Feedback Controller Design of System (1)

For system (1), since output function is unknown, the state x1 is exactly unknown. We construct the output-driven nonlinear reduced-order observer(21)z^˙i=-lix^ipt+ϕ^i-1,x^i=z^i+lix^i-1,i=2,…,n,where x^1=y, ϕ^i-1≔ϕi-1(t,x^1,…,x^i-1),i=2,…,n, and the observer gains li≥1,i=2,…,n, are constants to be determined.Remark 10.
Compared with the observer in [29], since the unknown output function h(·) causes x1 to be exactly unknown, a new output-driven nonlinear reduced-order observer (21) is designed without using x1 but y to rebuild unmeasured states. Moreover, not all but the first n-1 nonlinearities are used in the design of observer. For a second-order system Example 4.1 in [29], we can design an observer without using nonlinearity; see the simulation example in this paper for the details.Based onxn+1∗=-βnξn=-∑i=1nβi~nxi, an output feedback controller is designed as(22)u=-∑i=1nβi~nx^i=-β1~ny-∑i=2nβi~nx^i.Define the error(23)ei=zi-z^i,zi=xi-lixi-1,i=2,…,n.By (1), (21), and (23), one has(24)e˙i=-lixipt-x^ipt+xi+1pt+ϕi-liϕi-1-ϕ^i-1,i=2,…,n-1,e˙n=-lnxnpt-x^npt+upt+ϕn-lnϕn-1-ϕ^n-1.For the Lyapunov function U(e)=∑i=2n1/2ei2, it follows from (21), (23), and (24) that(25)U˙=-∑i=2nlieizi+lixi-1pt-z^i+lixi-1pt-∑i=2nlieiz^i+lixi-1pt-z^i+lix^i-1pt+∑i=2n-1eixi+1pt+enupt+∑i=2neiϕi-∑i=2nlieiϕi-1-ϕ^i-1.By (23), Lemma 5, and the fact that 1≤p_≤p(t)≤p¯, we obtain for i=2,…,n,(26)-lieizi+lixi-1pt-z^i+lixi-1pt=-lizi+lixi-1-z^i+lixi-1zi+lixi-1pt-z^i+lixi-1pt≤-21-p¯lieipt+1.Next, we give the estimate of the others on the right-hand side of (25) by Propositions 11–14, whose proofs are included in the Appendix.Proposition 11.
There is a positive constantb1, and positive constants c1,i,i=2,…,n, dependent on b1 such that(27)∑i=2n-1eixi+1pt+∑i=2neiϕi≤b1∑i=1nξipt+1+∑i=2nc1,ieipt+1.Proposition 12.
There is a positive constantb2, a positive constant αn dependent on b2, and positive constants αi,i=1,…,n-1, dependent on b2,li+1,…,ln such that(28)-∑i=2nlieiz^i+lixi-1pt-z^i+lix^i-1pt≤b2∑i=1nξipt+1+αnenpt+1+∑i=2n-1αili+1,…,lneipt+1+α1l2,…,lnx1-x^1pt+1.Proposition 13.
There is a positive constantb3, a positive constant γn dependent on b3, and positive constants γi,i=1,…,n-1, dependent on b3,li+1,…,ln such that(29)-∑i=2nlieiϕi-1-ϕ^i-1≤b3∑i=1nξipt+1+γnenpt+1+∑i=2n-1γili+1,…,lneipt+1+γ1l2,…,lnx1-x^1pt+1.Proposition 14.
There is a positive constantb4, a positive constant σn dependent on b4, and positive constants σi,i=1,…,n-1, dependent on li+1,…,ln such that(30)enupt≤b4ξnpt+1+σnenpt+1+∑i=2n-1σili+1,…,lneipt+1+σ1l2,…,lnx1-x^1pt+1.To estimate the termξn([u]p(t)-[xn+1∗]p(t)) in (20), we give the following proposition whose proof is also included in the Appendix.Proposition 15.
There is a positive constantb5, a positive constant ωn dependent on b5, and positive constants ωi,i=1,…,n-1, dependent on b5,li+1,…,ln such that(31)ξnupt-xn+1∗pt≤b5ξnpt+1+ωnenpt+1+∑i=2n-1ωili+1,…,lneipt+1+ω1l2,…,lnx1-x^1pt+1.DefineT(ξ,e)=Vn(ξ)+U(e)=∑i=1n(1/2)ξi2+∑i=2n(1/2)ei2. By (20), (25), (26), and Propositions 11–15, one obtains(32)T˙≤-∑i=1n-1ri,i-r~i+1,1-∑j=i+1nr¯j,i-b1+b2+b3ξipt+1-rn,n-b1+b2+b3+b4+b5ξnpt+1-21-p¯ln-θnenpt+1-∑i=2n-121-p¯li-θili+1,…,lneipt+1+θ1l2,…,lnx1-x^1pt+1,where θi=αi+γi+σi+ωi+c1,i with c1,1=0, i=1,…,n, it is easy to see that θn is a positive constant, and θi,i=1,…,n-1, are positive constants dependent on li+1,…,ln. For some positive constants b0,b~1,…,b~n, design r1,1,…,rn,n such that(33)r1,1≥max⁡r~2,1+∑j=2nr¯j,1+b1+b2+b3+b0+b~1,1,ri,i≥maxr~i+1,1+∑j=i+1nr¯j,i+b1+b2+b3+b~i,1,i=2,…,n-1,rn,n≥max⁡b1+b2+b3+b4+b5+b~n,1,and for some positive constants d~2,…,d~n, choose l2,…,ln as(34)ln≥max⁡2p¯-1d~n+θn,1,li≥max⁡2p¯-1d~i+θili+1,…,ln,1,i=n-1,…,2.By (33) and (34), (32) becomes(35)T˙≤-b0ξ1pt+1-∑i=1nb~iξipt+1-∑i=2nd~ieipt+1+θ1l2,…,lnx1-x^1pt+1.


### 3.4. Stability and Convergence Analysis

Theorem 16.
If Assumption8 holds for system (1), there is a maximal open sector Ω=(1-ρ¯,1+ρ¯) with ρ¯ being a positive constant, as long as the unknown output function h(·) belongs to any closed sector included in Ω, under the output feedback controller (21) and (22):
(i) the solutions of the closed-loop system (1), (21), (22) are well-defined on [0,+∞),
(ii) the equilibrium pointx=0 is globally uniformly asymptotically stable.Proof.
(i) We firstly give the choice ofr1,1,…,rn,n, β1,…,βn. At Step 1, for any given r~2,1,r¯2,1,…,r¯n,1,b1,b2,b3,b0,b~1, by (33), we can choose r1,1; then β1 can be calculated. At step 2, for any given r~3,1,r¯3,2,…,r¯n,2,b1,b2,b3,b~2, by (33), we can choose r2,2; then β2 can be calculated when λ2,1,λ2,2 have been obtained by (15) and (16). One by one, at Step n, for any given b1,b2,b3,b4,b5,b~n, by (33), we can choose rn,n; then βn can be calculated.
Secondly, we give the choice ofln,…,l2. For the given b1,b2,b3,b4,b5, we can get c1,n,αn,γn,σn,ωn. Then for any given d~n, by (34), we can choose ln. Next, for the given b1,b2,b3,b5, one get c1,n-1,αn-1(ln),γn-1(ln),σn-1(ln),ωn-1(ln). Then for any given d~n-1, ln-1 can be chosen by (34). In turn, we can get ln-2,…,l2.
Finally, we give the sector of output function. Denoteκ=(r~2,1,…,r~n,1,r¯2,1,…,r¯n,1,…,r¯n,n-2,r¯n,n-1,b1,…,b5,b0,b~1, …,b~n,d~2,…,d~n), Ωl=l=l2,…,ln∣r1,1≥maxr~2,1+∑j=2nr¯j,1+b1+b2+b3+b0+b~1,1,ri,i≥maxr~i+1,1+∑j=i+1nr¯j,i+b1+b2+b3+b~i,1,i=2,…,n-1,rn,n≥maxb1+b2+b3+b4+b5+b~n,1,ln≥max2p¯-1d~n+θn,1,li≥max2p¯-1d~i+θili+1,…,ln,1,i=n-1,…,2, and(36)ρ=ρl2,…,ln=min⁡b02θ1l2,…,ln1/p¯+1,b02θ1l2,…,ln1/p_+1.In fact, one can determine the supremum of ρ; that is, select constant(37)ρ¯=supκ∈R+,l∈Ωl⁡ρl2,…,ln,as the supremum of ρ. Hence, the maximal open sector of output function is Ω=(1-ρ¯,1+ρ¯). When output function h(x1) in (1) belongs to any closed sector [1-ρ¯+ϵ,1+ρ¯-ϵ]⊂Ω with 0<ϵ<ρ¯, by Definition 1,(38)hx1-1-ρ¯+ϵx1hx1-1+ρ¯-ϵx1≤0⇒hx1-1+ρ¯-ϵx1x1≤0≤hx1-1-ρ¯+ϵx1x1⇒1-ρ¯+ϵx12≤hx1x1≤1+ρ¯-ϵx12⇒-ρ¯-ϵx12≤hx1-x1x1≤ρ¯-ϵx12⇒x1-hx1≤ρ¯-ϵx1,from which one has(39)x1-x^1pt+1=ξ1-hξ1pt+1≤ρ¯-ϵpt+1ξ1pt+1.With the help of p_+1≤p(t)+1≤p¯+1, (36), and (37), we derive(40)ρ¯-ϵpt+1<ρpt+1≤ρp¯+1+ρp_+1≤b0θ1l2,…,ln.From (39) and (40), (35) becomes(41)T˙≤-∑i=1nb~iξipt+1-∑i=2nd~ieipt+1. Motivated by [29], with the help of p_+p(t)+1≤p_+p¯+1≤p(t)+1+p¯, one obtains(42)ξip_+p¯+1≤ξipt+1ξip_+ξip¯⇒ξipt+1≥ξip¯+11+ξip¯-p_,i=1,…,n,eip_+p¯+1≤eipt+1eip_+eip¯⇒eipt+1≥eip¯+11+ξip¯-p_,i=2,…,n,from which (41) becomes(43)T˙≤-∑i=1nb~iξip¯+11+ξip¯-p_+∑i=2nd~ieip¯+11+eip¯-p_≕-Wξ,e.It is easy to see that W(ξ,e) is a continuous and positive function with respect to (ξ,e).
By the transformations (11), system (1) can be transformed into a ξ-system:(44)ξ˙it=fit,ξt,ut,i=1,…,n,where fi(·) is a continuous function with fi(t,0)=0. Denote ϖ=(ξ1,…,ξn,e2,…,en)⊤, by the existence and continuation of the solution, the solution ϖ(t) of ϖ-system (24), (44) is defined on [0,tm) with 0<tm≤+∞.
Due toT(ϖ)=∑i=1n(1/2)ξi2+∑i=2n(1/2)ei2, it is obvious that T(ϖ) is a continuously differentiable, positive definite, and radially unbounded function. By Lemma 6, there are class K∞ functions π1(·) and π2(·) such that(45)π1ϖ≤Tϖ≤π2ϖ,∀ϖ∈R2n-1.Since π1(·) is a class K∞ function, then for any ε>0, one can always find a δ=δ(ε) with δ>ε>0 such that π2(ε)≤π1(δ). If ϖ(0)<ε, by (43) and (45), π1(ϖ(t))≤T(ϖ(t))≤T(ϖ(0))≤π2(ϖ(0))≤π2(ε)≤π1(δ) for all t∈[0,tm), which means that ϖ(t)≤δ for all t∈[0,tm). Hence, tm is not an escape time; that is, ϖ(t) is well-defined on [0,+∞), so is x(t).
(ii) Sincetm=+∞, from (43), (45), and Lemma 7, we know that the equilibrium point ϖ=0 of ϖ-system (24), (44) is globally uniformly asymptotically stable. By the continuity of xi∗(ξi-1) on ξi-1 and xi∗(0)=0, it is easy to recursively prove that the equilibrium point x=0 of the closed-loop system (1), (21), (22) is globally uniformly asymptotically stable.Remark 17.
Since the power of system is time-varying, a new function (36) is given to acquire its supremum, which is used to determine the maximal open sector of output function. Moreover, a rigorous analysis is given to establish the stability of the closed-loop system.


## 3.1. Control Objective of This Paper

The objective of this paper is to construct an output feedback controller for system (1) such that the equilibrium point of the closed-loop system is globally uniformly asymptotically stable when the maximal open sector of output function is given.Assumption 8.
There is a known constantc≥0 such that for i=1,…,n,(6)ϕit,x-ϕit,y≤c∑j=1N∑k=1ixk-ykak,jt∑q=1ixqbk,jt+yqbk,jt,for all x,y∈Rn, where N is an arbitrary positive integer and the real-valued functions ak,j(t) and bk,j(t) satisfy the following relations:(7)ak,jt+bk,jt=pt,a_k,j≤ak,jt≤a¯k,j,b_k,j≤bk,jt≤b¯k,j,for all j=1,…,N and k=1,…,i, with 0<a_k,j≤a¯k,j and 0<b_k,j≤b¯k,j. By Lemma 3 and Assumption 8, there exists a constant c~≥0 such that(8)ϕit,x1,…,xi≤c~x1pt+⋯+xipt,i=1,…,n.Remark 9.
As far as we know, under Assumption8, [29] is the first paper to study output feedback stabilization of nonlinear system (1) with h(x1)=x1. In this paper, we will consider system (1) with h(x1) being an unknown continuous function.


## 3.2. State Feedback Controller Design of System (1)

Step 1.
Takingξ1=x1, V1(x1)=(1/2)x12, it follows from (1) and (8) that(9)V˙1=x1x2pt+x1ϕ1≤x1x2pt-x2∗pt+x1x2∗pt+c~x1pt+1.Choose the virtual controller x2∗=-β1x1 with β1=(r1,1+c~)1/p_, where r1,1≥1 is a constant to be designed. Due to (r1,1+c~)p(t)/p_≥(r1,1+c~), (9) becomes(10)V˙1≤-r1,1ξ1pt+1+ξ1x2pt-x2∗pt.Inductive Step. Suppose that at step k-1(k≥2), there is a positive definite and radially unbounded Lyapunov function Vk-1(ξ1,…,ξk-1)=Vk-2(ξ1,…,ξk-2)+(1/2)ξk-12 and a set of virtual controllers x2∗,…,xk∗ defined by(11)ξ1=x1,xi∗=-βi-1ξi-1,ξi=xi-xi∗,i=2,…,k,such that(12)V˙k-1≤-∑i=1k-2ri,i-r~i+1,1-∑j=i+1k-1r¯j,iξipt+1-rk-1,k-1ξk-1pt+1+ξk-1xkpt-xk∗pt,where r~i+1,1,r¯j,i,j=i+1,…,k-1,i=1,…,k-2, are positive constants, r1,1,…,rk-1,k-1≥1 are constants to be designed, and β1,…,βk-1 are positive constants dependent on r1,1,…,rk-1,k-1. In what follows, we will show that (12) still holds at step k.
By (11), one has xk∗=-∑i=1k-1βi~(k-1)xi, which together with (1) lead to(13)ξ˙k=xk+1pt+ϕk+∑i=1k-1βi~k-1xi+1pt+ϕi.Constructing Vk(ξ1,…,ξk)=Vk-1(ξ1,…,ξk-1)+(1/2)ξk2, it follows from (12) and (13) that(14)V˙k≤-∑i=1k-2ri,i-r~i+1,1-∑j=i+1k-1r¯j,iξipt+1-rk-1,k-1ξk-1pt+1+ξk-1xkpt-xk∗pt+ξkxk+1pt+ϕk+∑i=1k-1βi~k-1xi+1pt+ϕi.By (8), (11), and Lemmas 2–4, we derive(15)ξk-1xkpt-xk∗pt≤ξk-1·p¯2p¯-3+1ξkpt+ξkxk∗pt-1≤r~k,1ξk-1pt+1+λk,1ξkpt+1,(16)ξkϕk+∑i=1k-1βi~k-1xi+1pt+ϕi≤ξkβ¯1ξ1pt+β¯2x2pt+⋯+β¯kxkpt≤∑i=1k-1r¯k,iξipt+1+λk,2ξkpt+1,where β¯i=β(i-1)~(k-1)+c~(1+∑j=ik-1βj~(k-1)), r~k,1,r¯k,1,…,r¯k,k-1 are positive constants, and λk,1 and λk,2 are positive constants dependent on r~k,1 and r¯k,1,…,r¯k,k-1, respectively. Substituting (15) and (16) into (14) leads to(17)V˙k≤-∑i=1k-1ri,i-r~i+1,1-∑j=i+1kr¯j,iξipt+1+ξkxk+1pt-xk+1∗pt+ξkxk+1∗pt+λk,1+λk,2ξkpt+1.Choose the virtual controller xk+1∗=-βkξk with βk=(rk,k+λk,1+λk,2)1/p_, where rk,k≥1 is a constant to be designed. Due to (rk,k+λk,1+λk,2)p(t)/p_≥(rk,k+λk,1+λk,2), (17) becomes(18)V˙k≤-∑i=1k-1ri,i-r~i+1,1-∑j=i+1kr¯j,iξipt+1-rk,kξkpt+1+ξkxk+1pt-xk+1∗pt.This completes the inductive step.Step n.
The Lyapunov functionVn(ξ)=Vn-1(ξ1,…,ξn-1)+(1/2)ξn2=∑i=1n(1/2)ξi2 and a positive constant λgive(19)V˙n≤-∑i=1n-1ri,i-r~i+1,1-∑j=i+1nr¯j,iξipt+1+λξnpt+1+ξnupt-xn+1∗pt+ξnxn+1∗pt,where ξi is defined as in (11) for i=1,…,n. Choose virtual controller xn+1∗=-βnξn with βn=(rn,n+λ)1/p_, where rn,n≥1 is a constant to be designed. Due to (rn,n+λ)p(t)/p_≥(rn,n+λ), (19) becomes(20)V˙n≤-∑i=1n-1ri,i-r~i+1,1-∑j=i+1nr¯j,iξipt+1-rn,nξnpt+1+ξnupt-xn+1∗pt.


## 3.3. Output Feedback Controller Design of System (1)

For system (1), since output function is unknown, the state x1 is exactly unknown. We construct the output-driven nonlinear reduced-order observer(21)z^˙i=-lix^ipt+ϕ^i-1,x^i=z^i+lix^i-1,i=2,…,n,where x^1=y, ϕ^i-1≔ϕi-1(t,x^1,…,x^i-1),i=2,…,n, and the observer gains li≥1,i=2,…,n, are constants to be determined.Remark 10.
Compared with the observer in [29], since the unknown output function h(·) causes x1 to be exactly unknown, a new output-driven nonlinear reduced-order observer (21) is designed without using x1 but y to rebuild unmeasured states. Moreover, not all but the first n-1 nonlinearities are used in the design of observer. For a second-order system Example 4.1 in [29], we can design an observer without using nonlinearity; see the simulation example in this paper for the details.Based onxn+1∗=-βnξn=-∑i=1nβi~nxi, an output feedback controller is designed as(22)u=-∑i=1nβi~nx^i=-β1~ny-∑i=2nβi~nx^i.Define the error(23)ei=zi-z^i,zi=xi-lixi-1,i=2,…,n.By (1), (21), and (23), one has(24)e˙i=-lixipt-x^ipt+xi+1pt+ϕi-liϕi-1-ϕ^i-1,i=2,…,n-1,e˙n=-lnxnpt-x^npt+upt+ϕn-lnϕn-1-ϕ^n-1.For the Lyapunov function U(e)=∑i=2n1/2ei2, it follows from (21), (23), and (24) that(25)U˙=-∑i=2nlieizi+lixi-1pt-z^i+lixi-1pt-∑i=2nlieiz^i+lixi-1pt-z^i+lix^i-1pt+∑i=2n-1eixi+1pt+enupt+∑i=2neiϕi-∑i=2nlieiϕi-1-ϕ^i-1.By (23), Lemma 5, and the fact that 1≤p_≤p(t)≤p¯, we obtain for i=2,…,n,(26)-lieizi+lixi-1pt-z^i+lixi-1pt=-lizi+lixi-1-z^i+lixi-1zi+lixi-1pt-z^i+lixi-1pt≤-21-p¯lieipt+1.Next, we give the estimate of the others on the right-hand side of (25) by Propositions 11–14, whose proofs are included in the Appendix.Proposition 11.
There is a positive constantb1, and positive constants c1,i,i=2,…,n, dependent on b1 such that(27)∑i=2n-1eixi+1pt+∑i=2neiϕi≤b1∑i=1nξipt+1+∑i=2nc1,ieipt+1.Proposition 12.
There is a positive constantb2, a positive constant αn dependent on b2, and positive constants αi,i=1,…,n-1, dependent on b2,li+1,…,ln such that(28)-∑i=2nlieiz^i+lixi-1pt-z^i+lix^i-1pt≤b2∑i=1nξipt+1+αnenpt+1+∑i=2n-1αili+1,…,lneipt+1+α1l2,…,lnx1-x^1pt+1.Proposition 13.
There is a positive constantb3, a positive constant γn dependent on b3, and positive constants γi,i=1,…,n-1, dependent on b3,li+1,…,ln such that(29)-∑i=2nlieiϕi-1-ϕ^i-1≤b3∑i=1nξipt+1+γnenpt+1+∑i=2n-1γili+1,…,lneipt+1+γ1l2,…,lnx1-x^1pt+1.Proposition 14.
There is a positive constantb4, a positive constant σn dependent on b4, and positive constants σi,i=1,…,n-1, dependent on li+1,…,ln such that(30)enupt≤b4ξnpt+1+σnenpt+1+∑i=2n-1σili+1,…,lneipt+1+σ1l2,…,lnx1-x^1pt+1.To estimate the termξn([u]p(t)-[xn+1∗]p(t)) in (20), we give the following proposition whose proof is also included in the Appendix.Proposition 15.
There is a positive constantb5, a positive constant ωn dependent on b5, and positive constants ωi,i=1,…,n-1, dependent on b5,li+1,…,ln such that(31)ξnupt-xn+1∗pt≤b5ξnpt+1+ωnenpt+1+∑i=2n-1ωili+1,…,lneipt+1+ω1l2,…,lnx1-x^1pt+1.DefineT(ξ,e)=Vn(ξ)+U(e)=∑i=1n(1/2)ξi2+∑i=2n(1/2)ei2. By (20), (25), (26), and Propositions 11–15, one obtains(32)T˙≤-∑i=1n-1ri,i-r~i+1,1-∑j=i+1nr¯j,i-b1+b2+b3ξipt+1-rn,n-b1+b2+b3+b4+b5ξnpt+1-21-p¯ln-θnenpt+1-∑i=2n-121-p¯li-θili+1,…,lneipt+1+θ1l2,…,lnx1-x^1pt+1,where θi=αi+γi+σi+ωi+c1,i with c1,1=0, i=1,…,n, it is easy to see that θn is a positive constant, and θi,i=1,…,n-1, are positive constants dependent on li+1,…,ln. For some positive constants b0,b~1,…,b~n, design r1,1,…,rn,n such that(33)r1,1≥max⁡r~2,1+∑j=2nr¯j,1+b1+b2+b3+b0+b~1,1,ri,i≥maxr~i+1,1+∑j=i+1nr¯j,i+b1+b2+b3+b~i,1,i=2,…,n-1,rn,n≥max⁡b1+b2+b3+b4+b5+b~n,1,and for some positive constants d~2,…,d~n, choose l2,…,ln as(34)ln≥max⁡2p¯-1d~n+θn,1,li≥max⁡2p¯-1d~i+θili+1,…,ln,1,i=n-1,…,2.By (33) and (34), (32) becomes(35)T˙≤-b0ξ1pt+1-∑i=1nb~iξipt+1-∑i=2nd~ieipt+1+θ1l2,…,lnx1-x^1pt+1.


## 3.4. Stability and Convergence Analysis

Theorem 16.
If Assumption8 holds for system (1), there is a maximal open sector Ω=(1-ρ¯,1+ρ¯) with ρ¯ being a positive constant, as long as the unknown output function h(·) belongs to any closed sector included in Ω, under the output feedback controller (21) and (22):
(i) the solutions of the closed-loop system (1), (21), (22) are well-defined on [0,+∞),
(ii) the equilibrium pointx=0 is globally uniformly asymptotically stable.Proof.
(i) We firstly give the choice ofr1,1,…,rn,n, β1,…,βn. At Step 1, for any given r~2,1,r¯2,1,…,r¯n,1,b1,b2,b3,b0,b~1, by (33), we can choose r1,1; then β1 can be calculated. At step 2, for any given r~3,1,r¯3,2,…,r¯n,2,b1,b2,b3,b~2, by (33), we can choose r2,2; then β2 can be calculated when λ2,1,λ2,2 have been obtained by (15) and (16). One by one, at Step n, for any given b1,b2,b3,b4,b5,b~n, by (33), we can choose rn,n; then βn can be calculated.
Secondly, we give the choice ofln,…,l2. For the given b1,b2,b3,b4,b5, we can get c1,n,αn,γn,σn,ωn. Then for any given d~n, by (34), we can choose ln. Next, for the given b1,b2,b3,b5, one get c1,n-1,αn-1(ln),γn-1(ln),σn-1(ln),ωn-1(ln). Then for any given d~n-1, ln-1 can be chosen by (34). In turn, we can get ln-2,…,l2.
Finally, we give the sector of output function. Denoteκ=(r~2,1,…,r~n,1,r¯2,1,…,r¯n,1,…,r¯n,n-2,r¯n,n-1,b1,…,b5,b0,b~1, …,b~n,d~2,…,d~n), Ωl=l=l2,…,ln∣r1,1≥maxr~2,1+∑j=2nr¯j,1+b1+b2+b3+b0+b~1,1,ri,i≥maxr~i+1,1+∑j=i+1nr¯j,i+b1+b2+b3+b~i,1,i=2,…,n-1,rn,n≥maxb1+b2+b3+b4+b5+b~n,1,ln≥max2p¯-1d~n+θn,1,li≥max2p¯-1d~i+θili+1,…,ln,1,i=n-1,…,2, and(36)ρ=ρl2,…,ln=min⁡b02θ1l2,…,ln1/p¯+1,b02θ1l2,…,ln1/p_+1.In fact, one can determine the supremum of ρ; that is, select constant(37)ρ¯=supκ∈R+,l∈Ωl⁡ρl2,…,ln,as the supremum of ρ. Hence, the maximal open sector of output function is Ω=(1-ρ¯,1+ρ¯). When output function h(x1) in (1) belongs to any closed sector [1-ρ¯+ϵ,1+ρ¯-ϵ]⊂Ω with 0<ϵ<ρ¯, by Definition 1,(38)hx1-1-ρ¯+ϵx1hx1-1+ρ¯-ϵx1≤0⇒hx1-1+ρ¯-ϵx1x1≤0≤hx1-1-ρ¯+ϵx1x1⇒1-ρ¯+ϵx12≤hx1x1≤1+ρ¯-ϵx12⇒-ρ¯-ϵx12≤hx1-x1x1≤ρ¯-ϵx12⇒x1-hx1≤ρ¯-ϵx1,from which one has(39)x1-x^1pt+1=ξ1-hξ1pt+1≤ρ¯-ϵpt+1ξ1pt+1.With the help of p_+1≤p(t)+1≤p¯+1, (36), and (37), we derive(40)ρ¯-ϵpt+1<ρpt+1≤ρp¯+1+ρp_+1≤b0θ1l2,…,ln.From (39) and (40), (35) becomes(41)T˙≤-∑i=1nb~iξipt+1-∑i=2nd~ieipt+1. Motivated by [29], with the help of p_+p(t)+1≤p_+p¯+1≤p(t)+1+p¯, one obtains(42)ξip_+p¯+1≤ξipt+1ξip_+ξip¯⇒ξipt+1≥ξip¯+11+ξip¯-p_,i=1,…,n,eip_+p¯+1≤eipt+1eip_+eip¯⇒eipt+1≥eip¯+11+ξip¯-p_,i=2,…,n,from which (41) becomes(43)T˙≤-∑i=1nb~iξip¯+11+ξip¯-p_+∑i=2nd~ieip¯+11+eip¯-p_≕-Wξ,e.It is easy to see that W(ξ,e) is a continuous and positive function with respect to (ξ,e).
By the transformations (11), system (1) can be transformed into a ξ-system:(44)ξ˙it=fit,ξt,ut,i=1,…,n,where fi(·) is a continuous function with fi(t,0)=0. Denote ϖ=(ξ1,…,ξn,e2,…,en)⊤, by the existence and continuation of the solution, the solution ϖ(t) of ϖ-system (24), (44) is defined on [0,tm) with 0<tm≤+∞.
Due toT(ϖ)=∑i=1n(1/2)ξi2+∑i=2n(1/2)ei2, it is obvious that T(ϖ) is a continuously differentiable, positive definite, and radially unbounded function. By Lemma 6, there are class K∞ functions π1(·) and π2(·) such that(45)π1ϖ≤Tϖ≤π2ϖ,∀ϖ∈R2n-1.Since π1(·) is a class K∞ function, then for any ε>0, one can always find a δ=δ(ε) with δ>ε>0 such that π2(ε)≤π1(δ). If ϖ(0)<ε, by (43) and (45), π1(ϖ(t))≤T(ϖ(t))≤T(ϖ(0))≤π2(ϖ(0))≤π2(ε)≤π1(δ) for all t∈[0,tm), which means that ϖ(t)≤δ for all t∈[0,tm). Hence, tm is not an escape time; that is, ϖ(t) is well-defined on [0,+∞), so is x(t).
(ii) Sincetm=+∞, from (43), (45), and Lemma 7, we know that the equilibrium point ϖ=0 of ϖ-system (24), (44) is globally uniformly asymptotically stable. By the continuity of xi∗(ξi-1) on ξi-1 and xi∗(0)=0, it is easy to recursively prove that the equilibrium point x=0 of the closed-loop system (1), (21), (22) is globally uniformly asymptotically stable.Remark 17.
Since the power of system is time-varying, a new function (36) is given to acquire its supremum, which is used to determine the maximal open sector of output function. Moreover, a rigorous analysis is given to establish the stability of the closed-loop system.


## 4. Simulation Example

Consider the second-order nonlinear system in [29](46)x˙1=x22+cost,x˙2=u2+cost+x11/3x2cos⁡t+5/3,y=hx1.As discussed in [29], Assumption 8 holds and p_=1, p¯=3. Following the design process as in Section 3 in this paper, we can design output feedback controller without using the nonlinearity x11/3x2cos(t)+5/3,(47)z^˙2=-l2x^2pt,x^2=z^2+l2y,u=-β1β2y-β2x^2.Choose κ=(r~2,1,r¯2,1,b1,b2,b4,b5,b0,b~1,b~2,d~2)=(0.1,0.1,0.1,0.1,0.1,0.1,0.4,0.1,0.1,0.4). By choosing r1,1=1, r2,2=1, we obtain β1=1, β2=9.63, and choose l=l2=12.Whenh(x1)=x1 as in [29], it means that we do not need to give the sector of the output function. In simulation, under the output feedback controller (47), by selecting the initial conditions x1(0)=-0.1, x2(0)=0.3, z^2(0)=0, Figure 1 demonstrates the effectiveness of control scheme.Figure 1
The responses of the closed-loop system (46), (47) with h(x1)=x1.Whenh(x1)≠x1, the design method in [29] is inapplicable. Following the design process as in Section 3 in this paper, we can get a sector of output function, that is, [1-ρ,1+ρ] with ρ=ρ(l2)=min{(b0/2θ1(l2))1/p¯+1,(b0/2θ1(l2))1/(p_+1)}=0.0131. In simulation, we choose h(x1)=x1+0.01x1 which is only continuous but not differentiable; h(x1) belongs to sector [0.9869,1.0131] obviously. By selecting the initial conditions x1(0)=0.1, x2(0)=-0.3, z^2(0)=0.01, Figure 2 illustrates the validness of control scheme.Figure 2
The responses of the closed-loop system (46), (47) with h(x1)=x1+0.01x1.


## 5. Conclusions

In this paper, output feedback stabilization of nonlinear systems with a time-varying power and unknown continuous output function is studied. By constructing a new nonlinear reduced-order observer together with adding a power integrator method, a new function to determine the maximal open sectorΩ of output function is given. As long as output function belongs to any closed sector included in Ω, an output feedback controller is constructed to guarantee that the equilibrium point of the closed-loop system is globally uniformly asymptotically stable.For the future work, an interesting problem is for more general nonlinear systems with time-varying powers, that is, nonlinear systems with different time-varying powers or under the weaker condition on nonlinearities, can we design a stable output feedback controller?



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