# Global Output Feedback Stabilization for a Class of Nonlinear Cascade Systems

Authors: Cai-Yun Liu; Zong-Yao Sun; Qing-Hua Meng; Chih-Chiang Chen; Bin Cai; Yu Shao
Journal: Mathematical Problems in Engineering
 (2018)

---

## Abstract

This paper focuses on the problem of global output feedback stabilization for a class of nonlinear cascade systems with time-varying output function. By using double-domination approach, an output feedback controller is developed to guarantee the global asymptotic stability of closed-loop system. The novel control strategy successfully constructs a unified Lyapunov function, which is suitable for both upper-triangular and lower-triangular systems. Finally, two numerical examples are provided to illustrate the effectiveness of a control strategy.


---

## Body

## 1. Introduction

It is well known that global output feedback stabilization is viewed as one of the most challenging fields of nonlinear control. Researchers have not yet found any unified way to handle the problem of global output feedback stabilization because the measure of states is difficult. Fortunately, with the help of nonseparation principle [1], homogeneous domination approach [2], and backstepping method, many interesting results such as [3–11] have been achieved.It is worth pointing out that the structures of system output and nonlinear functions determine the possible forms of observer and controller. More specifically, the uncertainty of nonlinearities has led to the emergence of many kinds of observers, including high-gain observer, homogeneous observer, and time-varying observer. For example, [12] solved the problem of global output feedback stabilization based on linear high-gain observer for a class of uncertain nonlinear systems, where controller is independent of higher-order nonlinearities. Under uncertain linear growth condition in [13], a dynamic high-gain observer is proposed without requiring precise information of output function. References [14, 15] achieved system global stabilization by using time-varying observer, which uses the appropriate functions of time, rather than the dynamic compensator. Since some nonlinear functions satisfy neither the linear growth nor Lipschitz condition in practice, the existing approaches are not suitable. Therefore, [16–19] proposed homogeneous domination method to overcome this obstacle. Based on the existing results, some special observers are proposed, such as dual-observer [20] and reduced-observer [21]. In practice, complex systems are usually composed of simple subsystems. Therefore, cascade systems have become one of the most interesting topics of nonlinear systems. A great deal of research has been devoted to this subject over the last decades, as evidenced by the comprehensive books of [22, 23]. However, when zero-dynamics exist and obey mild conditions, the tracking problem cannot be solved by trivially extending the corresponding results without zero-dynamics; that is, there do not exist appropriate observers to tracking states of cascade systems. As further investigation, researchers now consider cascade connections in which the nonlinear systems are globally stable, but the input subsystem is more complex than just an integrator; for instance, [24–26] successfully investigated output feedback stabilization for uncertain cascade systems under growth condition. Regrettably, their approaches are only suitable for lower-triangular cascade systems. On the other hand, some literatures [27] achieved global output feedback stabilization when output function depends only on a state. References [28, 29] required that output function be continuous differentiability and initial value equals zero when output is unknown. The above conditions are restrictive; researchers turned to study time-varying output function. For instance, [30] further investigated the problem of global output feedback stabilization for a class of nonlinear systems with unknown measurement sensitivity. Meanwhile, a new method, namely, dual-domination approach, is proposed in [30].In view of the above argument, an interesting question is proposed simultaneously:Is it possible to find a new approach to solve the problem of global output feedback stabilization for nonlinear cascade systems with unknown time-varying output function, which is suitable for both upper-triangular and lower-triangular systems? Based on above analysis and references, we will solve aforementioned question and provide satisfactory answer. There are three troublesome difficulties throughout the paper. The first is to find the appropriate Lyapunov function that is independent of the derivative of output function, since output function is unknown and does not satisfy differentiability. The second is to choose allowable sensitivity error, since it appears in the construction of controller. The third is to design rational observers to successfully track states, since nonlinearities and output function are unknown. A novel observer is proposed, which is different from the existing results [24, 25].The main contributions of this paper are divided into three aspects: (i) double-domination approach is provided to handle time-varying sensitivity and uncertain nonlinearities, which is suitable for both upper-triangular and lower-triangular systems; (ii) linear observer does not rely on precise information of nonlinearities and output function; (iii) the construction of Lyapunov function avoids the use of the differentiability of output function.


## 2. Preliminaries and Problem Formulation

### 2.1. Preliminaries

We will adopt the following notations throughout this paper.R denotes the set of real numbers, R+ denotes the set of all nonnegative real numbers, and Rn denotes Euclidean space with dimension n. For any real vector x=[x1,…,xn]T∈Rn, the norm xp is defined by xp=∑i=1nxip1/p, 1≤p<∞. For matrix A=aijn×n, i=1,…,n, j=1,…,n, A denotes the norm A=λmax(ATA), where λmax(ATA) denotes maximum eigenvalue of square matrix ATA. Ci denotes the set of all functions with continuous ith partial derivatives. A continuous function α:[0,a)→[0,∞) is said to belong to class K if it is strictly increasing and α(0)=0. It is said to belong to class K∞ if a=∞ and α(r)=∞ as r→∞. For a continuously differentiable function V:Rn→R+, it is positive definite if V(x)≥0 and V(x)=0 if and only if x=0; it is radially unbounded if V(x)→∞ as x→∞. The arguments of functions are sometimes simplified; for instance, a function f(x(t)) is denoted by f(x) or f and x2 is denoted by x.In the following, we list three lemmas that play an important role in proving the main results, and their proofs can be found in [31–33].Lemma 1 (see [31]).
Letc and d be positive constants; given any positive real-valued function γ(x,y), the following inequality holds:(1)xcyd≤cc+dγx,yxc+d+dc+dγx,y-c/dyc+d.Lemma 2 (see [32]).
For anyx=x1,…,xnT∈Rn, the following inequality holds: (2)x2≤x1≤nx2.Lemma 3 (see [33]).
Forx∈R and y∈R, p≥1 is an integer, and the following inequality holds:(3)x+yp≤2p-1xp+yp.


### 2.2. Problem Formulation

This paper investigates the nonlinear cascade system described by(4)z˙=f0t,z,x,u,x˙i=xi+1+fit,z,x,u,i=1,…,n-1,x˙n=u+fnt,z,x,u,yt=θtx1t,where x(t)=x1t,…,xntT∈Rn and z(t)∈Rm are system states with the initial values x(0)=x0, z(0)=z0, u(t)∈R, y(t)∈R being control input and output, respectively. f0:R+×Rm×Rn×R→Rm is continuous function with f0(t,0,0,0)=0 and globally Lipschitz with respect to x; fi:R+×Rm×Rn×R→R, i=1,…,n, are continuous functions with fi(t,0,0,0)=0. θ(t):R+→R is a continuous function that represents time-varying sensitivity.The following assumptions are needed.Assumption 4.
For the continuous functionθ(t), there is a known positive parameter θ¯ satisfying 1-θt≤θ¯<θ~<1, where θ¯ is an allowable sensitivity error and θ~ is the upper bound of the allowable sensitivity error.Assumption 5.
There exists a positive-definite and radially unbounded functionU(z)∈C2 such that(5)∂U∂zf0t,z,0,u≤-c1z2,∂U∂z≤c2z,where c1∈(c1∗,∞), c2∈0,c2∗, and c1∗ and c2∗ are positive constants.Assumption 6.
There exists a constantc≥0 such that(6)fi·≤cz+xi+2+⋯+xn,i=1,…,n-2,fn-1·≤cz,fn·≤cz.Remark 7.
Since output function contains unknown parameter, it implies that the scope of this paper is more general than [1, 2, 12, 16] whose output function is equal to x1.Remark 8.
In terms of the appearance ofθ(t), two obstacles will be encountered. The first is to find an appropriate observer, which does not use the information of output function. The second is to find the feasible range of θ(t), because the information of θ(t) will be used in the design of controller.


## 2.1. Preliminaries

We will adopt the following notations throughout this paper.R denotes the set of real numbers, R+ denotes the set of all nonnegative real numbers, and Rn denotes Euclidean space with dimension n. For any real vector x=[x1,…,xn]T∈Rn, the norm xp is defined by xp=∑i=1nxip1/p, 1≤p<∞. For matrix A=aijn×n, i=1,…,n, j=1,…,n, A denotes the norm A=λmax(ATA), where λmax(ATA) denotes maximum eigenvalue of square matrix ATA. Ci denotes the set of all functions with continuous ith partial derivatives. A continuous function α:[0,a)→[0,∞) is said to belong to class K if it is strictly increasing and α(0)=0. It is said to belong to class K∞ if a=∞ and α(r)=∞ as r→∞. For a continuously differentiable function V:Rn→R+, it is positive definite if V(x)≥0 and V(x)=0 if and only if x=0; it is radially unbounded if V(x)→∞ as x→∞. The arguments of functions are sometimes simplified; for instance, a function f(x(t)) is denoted by f(x) or f and x2 is denoted by x.In the following, we list three lemmas that play an important role in proving the main results, and their proofs can be found in [31–33].Lemma 1 (see [31]).
Letc and d be positive constants; given any positive real-valued function γ(x,y), the following inequality holds:(1)xcyd≤cc+dγx,yxc+d+dc+dγx,y-c/dyc+d.Lemma 2 (see [32]).
For anyx=x1,…,xnT∈Rn, the following inequality holds: (2)x2≤x1≤nx2.Lemma 3 (see [33]).
Forx∈R and y∈R, p≥1 is an integer, and the following inequality holds:(3)x+yp≤2p-1xp+yp.


## 2.2. Problem Formulation

This paper investigates the nonlinear cascade system described by(4)z˙=f0t,z,x,u,x˙i=xi+1+fit,z,x,u,i=1,…,n-1,x˙n=u+fnt,z,x,u,yt=θtx1t,where x(t)=x1t,…,xntT∈Rn and z(t)∈Rm are system states with the initial values x(0)=x0, z(0)=z0, u(t)∈R, y(t)∈R being control input and output, respectively. f0:R+×Rm×Rn×R→Rm is continuous function with f0(t,0,0,0)=0 and globally Lipschitz with respect to x; fi:R+×Rm×Rn×R→R, i=1,…,n, are continuous functions with fi(t,0,0,0)=0. θ(t):R+→R is a continuous function that represents time-varying sensitivity.The following assumptions are needed.Assumption 4.
For the continuous functionθ(t), there is a known positive parameter θ¯ satisfying 1-θt≤θ¯<θ~<1, where θ¯ is an allowable sensitivity error and θ~ is the upper bound of the allowable sensitivity error.Assumption 5.
There exists a positive-definite and radially unbounded functionU(z)∈C2 such that(5)∂U∂zf0t,z,0,u≤-c1z2,∂U∂z≤c2z,where c1∈(c1∗,∞), c2∈0,c2∗, and c1∗ and c2∗ are positive constants.Assumption 6.
There exists a constantc≥0 such that(6)fi·≤cz+xi+2+⋯+xn,i=1,…,n-2,fn-1·≤cz,fn·≤cz.Remark 7.
Since output function contains unknown parameter, it implies that the scope of this paper is more general than [1, 2, 12, 16] whose output function is equal to x1.Remark 8.
In terms of the appearance ofθ(t), two obstacles will be encountered. The first is to find an appropriate observer, which does not use the information of output function. The second is to find the feasible range of θ(t), because the information of θ(t) will be used in the design of controller.


## 3. Main Results

### 3.1. Output Feedback Controller Design for Upper-Triangular Case

We now summarise main results of this paper.Theorem 9.
For system (4) under Assumptions 4–6, there exists an output feedback controller such that states of the closed-loop system are uniformly bounded over [0,+∞) and limt→+∞x(t)=0.Proof.
The proof is in four parts. At first, a linear observer with a domination gain is introduced to reconstruct all the states. Secondly, an output feedback controller composed of another domination gain is constructed to counteract the destabilized terms. Finally, a delicate selection of gains is provided and strict analysis is performed to guarantee that the closed-loop systems are globally asymptotically stable.
Part I: Design of an Observer. We construct the following linear observer:(7)x^˙i=x^i+1-aiLix^1,i=1,…,n-1,x^˙n=u-anLnx^1,where ai>0 are coefficients of Hurwitz polynomial p1(s)=sn+a1sn-1+⋯+an-1s+an and the domination gain L≥1 will be determined later. Define the estimation error as follows:(8)εi=Li-1xi-x^i,i=1,…,n.Then, the error equation can be rewritten as(9)ε˙=1LAεε+1LDεx1+F,where(10)ε=ε1ε2⋮εn,Aε=-a11⋯0⋮⋮⋱⋮-an-10⋯1-an0⋯0,Dε=a1a2⋮an,F=f1·Lf2·⋮Ln-1fn·.Since Aε is Hurwitz, there exists a symmetric positive-definite matrix P1∈Rn×n such that AεTP1+P1Aε=-I, where I is an identity matrix of Rn×n. Consider positive-definite and radially unbounded function Vε(ε)=εTP1ε; a direct calculation gives(11)V˙ε=-1LεTε+21LεTP1Dεx1+2εTP1F≤-1Lε2+21Lε·Dε·P1·x1+2ε·P1·F.According to Assumption 6 and Lemma 2, we obtain(12)F=f12+L2f22+⋯+L2n-1fn2≤f1+Lf2+⋯+Ln-1fn≤cLn-1nz+∑i=3ncLi-3i-2xi.Substituting (12) into (11), one has(13)V˙ε≤-1Lε2+2cP1·εLn-1nz+∑i=3nLi-3i-2xi+2LP1·ε·Dε·x1.In addition, Lemma 1 shows that(14)2LP1·ε·Dε·x1≤12Lε2+1LK2x12,and(15)2cP1·εLn-1nz+∑i=3nLi-3i-2xi≤∑i=3nKiL2i-2xi2+K1L2ε2+12cnL2nP1·z2,where K1=c(2n+(n-1)(n-2))P1≥0, K2=2Dε2P12>0, and Ki=1/2c(i-2)P1≥0,i=3,…,n, are independent of a domination gain L. Therefore, substituting (14) and (15) into (13) yields(16)V˙ε≤-12Lε2+K1L2ε2+1LK2x12+∑i=3nKiL2i-2xi2+12cnL2nP1·z2.
Part II: Construction of a Controller. Consider the following system:(17)x˙1=x2+f1t,z,x,u,x^˙i=x^i+1+aiLiε1-x1,i=2,…,n-1,x^˙n=u+anLnε1-x1.Define the following change of coordinates:(18)ξ1=x1,ξi=Li-1Hi-1x^i,υ=LnHnu,i=2,…,n,where H≥1 is a domination gain that will be determined later. Using above coordinate transform, (17) can be rewritten as(19)ξ˙1=HLξ2+1Lε2+f1t,z,x,u,ξ˙i=HLξi+1+aiLHi-1ε1-ξ1,i=2,…,n-1,ξ˙n=HLυ+anLHn-1ε1-ξ1.Design the output feedback control law as follows:(20)υ=-bny-bn-1ξ2-⋯-b2ξn-1-b1ξn,where bi>0, i=1,…,n, are coefficients of Hurwitz polynomial p2(s)=sn+b1sn-1+⋯+bn-1s+bn. Substituting (20) into (19), we have(21)ξ˙=HLAξξ+HLDξbn1-θtξ1+B2Lε2+B1LHε1-ξ1+F¯,where(22)Aξ=01⋯0⋮⋮⋱⋮00⋯1-bn-bn-1⋯-b1,B1=0a2a3H⋮anHn-2,B2=10⋮0,Dξ=0⋮01,F¯=f1·0⋮0.Aξ is a Hurwitz matrix that shows that there exists a symmetric positive-definite matrix P2 such that AξTP2+P2Aξ=-I. Choose scalar function Vξ(ξ)=ξTP2ξ, which is positive-definite and radially bounded. Noting that B2=Dξ=1, the time derivative of Vξ(ξ) along the trajectories of (21) is(23)V˙ξ=-HLξTξ+2HLξTP2Dξbn1-θtξ1+2ξTP2F¯+2ξTP2B1LHε1+B2Lε2-B1LHξ1≤-HLξ2+2bn1-θtHLP2·ξ2+2ξ·P2·f1+2ξ·P21Lε+B1LHε+2LHP2·B1·ξ2.Firstly, by virtue of Assumption 6 and Lemma 1, it holds that(24)2ξ·P2·f1≤2cξ·P2z+x3+⋯+xn≤cn-1L2P2·ξ2+cP2L2z2+x32+⋯+xn2≜K¯1L2ξ2+K¯2L2∑i=3nxi2+K¯2L2z2,where K¯1=c(n-1)P2≥0 and K¯2=cP2≥0. Furthermore, it is deduced from Lemma 1 that(25)2Lξ·P2·ε≤18Lε2+8LP22·ξ2,2γLHξ·P2·ε≤18Lε2+8γ2LH2P22·ξ2,and because of H>1, one can conclude that(26)B1=∑i=2n1H2i-2ai21/2≤∑i=2nai21/2=Δγ.Consequently, one obtains(27)V˙ξ≤-HL1-2bn1-θt·P2ξ2+14Lε2+8γ2LH2P22·ξ2+8LP22·ξ2+2γLHP2·ξ2+K¯1L2ξ2+K¯2L2∑i=3nxi2+K¯2L2z2≤-HL1-2bn1-θt·P2ξ2+K¯2L2∑i=3nxi2+K¯2L2z2+14Lε2+HL8γ2HP22+8HP22+2γHP2+K¯1Lξ2≜-HL1-2bn1-θt·P2ξ2+K¯2L2∑i=3nxi2+K¯2L2z2+14Lε2+HLK3¯H+K¯1Lξ2,where K¯3=2γ1+4γP2P2+8P22>0 is independent of the domination gains L and H.
Part III: Determination of Domination Gains. According to above arguments, it follows that(28)x1=ξ1,xi=1Li-1εi+Hi-1Li-1ξi,i=2,…,n.Using Lemma 3, there is(29)xi2=1Li-1εi+Hi-1Li-1ξi2≤2L2i-1ε2+2H2i-1L2i-1ξ2,and putting (29) together with (27), we have(30)V˙ξ≤-HL1-2bn1-θt·P2ξ2+14Lε2+HLK3¯H+K¯1Lξ2+2K¯2∑i=3n1L2i-2ε2+2K¯2∑i=3nH2i-1L2i-2ξ2+K¯2L2z2≤-HL1-2bn1-θt·P2ξ2+HLK^3H+K^1Lξ2+1L2K^4ε2+K^2L2z2+14Lε2,where K^1=K¯1+2K¯2(H3+⋯+H2n-3)≥0,K^2=K¯2≥0,K^3=K¯3>0,K^4=2(n-2)K¯2≥0. Now, we choose the allowable sensitivity error θ¯ as(31)θ¯<θ~=12bnP2.Because of 1-θ≤θ¯, one has(32)1-2bn1-θ·P2≥1-2bnθ¯P2≜β.Obviously, 0<β<1 and (30) can be rewritten as(33)V˙ξ≤-HLβξ2+HLK^3H+K^1Lξ2+1L2K^4ε2+K^2L2z2+14Lε2.Similarly, substituting (29) into (16) yields(34)V˙ε≤-12Lε2+K2Lξ2+K1L2ε2+12cnL2nP1z2+2∑i=3nKiL2ε2+2∑i=3nKiH2i-1L2ξ2≤-12Lε2+1L2K^5ε2+1LK^6ξ2+HK^7L2ξ2+K^8L2nz2,where K^5=K1+2(K3+⋯+Kn)≥0,K^6=K2>0,K^7=2(K3H3+⋯+KnH2n-3)≥0,K^8=1/2cnP1≥0. Applying Assumption 5 and the property of Lipschitz function, one has(35)∂U∂zf0t,z,x,u=∂U∂zf0t,z,x,u-∂U∂zf0t,z,0,u+∂U∂zf0t,z,0,u≤∂U∂z·f0t,z,x,u-f0t,z,0,u+∂U∂zf0t,z,0,u≤c¯2z·x-c1z2,where c¯2 is a positive constant. By Lemma 1, it is easy to obtain that x≤x1+⋯+xn, and, according to Lemma 2, the following inequalities hold:(36)c¯2z·x2+⋯+xn≤c12z2+c¯222c1x22+⋯+xn2,c¯2z·x1≤c¯222Lx12+L2z2.Substituting (36) into (35), one has(37)∂U∂zf0t,z,x,u≤-c12-L2z2+c¯222c1x22+⋯+xn2+c¯222Lx12.According to (29) and the definition of norms, it is easy to deduce that(38)x12≤ξ2,x22+x32+⋯+xn2≤2n-1L2ε2+HL22K^9ξ2,where K^9=H+H3+⋯+H2n-3; therefore (37) can be further expressed as(39)∂U∂zf0t,z,x,u≤-c12-L2z2+c¯22n-1c1L2ε2+c¯22K^9Hc1L2+c¯222Lξ2.Consider positive-definite and radially unbounded function as follows:(40)V=U+Vε+Vξ,and substituting (33), (34), and (39) into (40), there is(41)V˙≤-c12-L2-K^8L2n-K^2L2z2-1L14-K^5+K^4L-2c¯22n-1Lε2-HLβ-K^6+K^3H-K^1+K^7L-c¯222H-c¯22K^9Lξ2.In order to ensure that V˙≤0, we choose the domination gains L and H as follows. Firstly,(42)β-K^6+K^3H-c¯222H≥β2,implying that(43)H≥2K^3+2K^6+c¯22β,and, with H≥1 in hand, one deduces(44)H≥max⁡1,2K^3+2K^6+c¯22β.As consequence, (41) can be rewritten as(45)V˙≤-c12-L2-K^8L2n-K^2L2z2-1L14-K^5+K^4L-2c¯22n-1Lε2-HLβ2-K^1+K^7L-c¯22K^9Lξ2.Secondly, we can selectL to satisfy the following inequalities:(46)14-K^5+K^4L-2c¯22n-1L≥β4,β2-K^1+K^7L-c¯22K^9L≥β4,and, in light of L≥1, it is straightforward to show that(47)L≥max⁡1,4K^1+4K^7+4c¯22K^9β,4K^4+4K^5+8c¯22n-11-β,and c1 satisfies(48)c1=β4+c1∗,c1∗≥2K^8L2n+K^2L2+L.Under above choice of domination gains, it is clear that(49)V˙≤-β4z2-β4Lξ2-β4Lε2.
Part IV: Stability Analysis. Consider transformed systems (9), (20), and (21). By the existence and continuity of solution, the closed-loop system state composed of Y(t)=zT,xT,x^TT can be defined on a time interval 0,tm, where tm>0 may be a finite constant or infinity.
(i)  For the Boundedness ofY(t).Due to the fact that ν1(z)≤U(z)≤ν2(z), λmin(P)x2≤xTPx≤λmax(P)x2, where ν1 and ν2 are K∞ functions; we obtain(50)V=U+Vε+Vξ≥ν1z+λminP1ε2+λminP2ξ2≥ν1z+minλminP1,λminP2ωt2≜φ1ϖt,and(51)V=U+Vε+Vξ≤ν2z+λmaxP1ε2+λmaxP2ξ2≤ν2z+maxλmaxP1,λmaxP2ωt2≜φ2ϖt,where ω(t)=[εT(t),ξT(t)]T, ϖ(t)=[zT,εT(t),ξT(t)]T, and φ1,φ2 are K∞ functions.
For anyϵ>0, noting that lims→+∞φ1(s)=+∞, one can always find a constant β=β(ϵ)>ϵ>0, such that φ2(ϵ)≤φ1(β). From (49), it follows that Vϖt≤V(ϖ(0)), ∀t∈[0,tm). Then, if ϖ(0)<ϵ, it holds that(52)φ1ϖt≤Vϖt≤Vϖ0≤φ2ϖ0<φ2ϵ≤φ1β,∀t∈0,tm,which implies that ϖ(t)<β, ∀t∈0,tm; that is, ϖ(t) is bounded. Therefore, by virtue of estimation errors and coordinates transform, it is easy to get that the state Y(t)=zT,xT,x^TT is bounded on 0,tm as well.
(ii)tm=+∞.This claim can be shown by contradiction; if tm is finite, then tm would be a finite escape time; that is, the state Y(t) would tend to ∞ as t=tm. However, the continuity of solution guarantees the boundedness of Y(t) at t=tm, since Y(t) is bounded on [0,tm). This is clearly a contradiction. Therefore, the state of the closed-loop system is uniformly bounded over [0,+∞).
(iii)limt→+∞x(t)=0 and limt→+∞z(t)=0.Above all, it is clear that(53)0≤∫0tβ4z2+β4Lε2+β4Lξ2ds≤-∫0tV˙ϖsds<Vϖ0<+∞,∀t≥0,which implies that ∫0∞β/4z2+β/4Lε2+β/4Lξ2ds exists and is finite. On the other hand, the boundedness of ϖ(t) over [0,+∞) means that ϖ˙(t) is uniformly bounded in t over [0,+∞). Thus, ϖ(t) is uniformly continuous in t over [0,+∞) and so is ϖ2(t). Using well-known Barbalat’s Lemma in [22], one obtains limt→+∞zi2(t)=0, limt→+∞ξi2(t)=0, and limt→+∞εi2(t)=0, which shows that limt→+∞z(t)=0, limt→+∞ξ(t)=0, and limt→+∞ε(t)=0. Finally, from (29), it is easy to see that limt→+∞x(t)=0. This completes the proof.Remark 10.
A double-domination method is proposed to handle the time-varying output function and nonlinearities in the proof of Theorem9; that is, two domination gains H and L are used to dominate time-varying function θ(t) and nonlinearities fi, respectively.Remark 11.
It should be noted fromθ~=1/2bnP2 that the upper bound θ~ depends on coefficients b1,…,bn for the Hurwitz polynomial p2(s). When b1,…,bn are specified, the corresponding matrix P2 as well as the upper bound θ~ can be computed. For example, θ~=0.4420, for n=2; θ~=0.1501, for n=3.


### 3.2. Extension to Lower-Triangular Case

Some subsystems do not satisfy upper-triangular structure in practical application, so we extend the subsystems to lower-triangular form and impose following assumptions on system (4).Assumption 12.
Fori=1,…,n, there exists a constant c≥0 such that (54)fi≤cz+x1+⋯+xi.Theorem 13.
For a class of nonlinear cascade system (4) under Assumptions 4, 5, and 12, there exists an output feedback controller, such that states of the closed-loop system are uniformly bounded over [0,+∞) and limt→+∞x(t)=0.Proof.
The proof is analogous to the proof of Theorem9 with an obvious modification. To facilitate comparison, we select same notations as Theorem 9 and many similarities will be omitted.
Firstly, we construct the similar observer as (7)(55)x^˙i=x^i+1-Liaix^1,i=1,…,n-1,x^˙n=u-Lnanx^1,where ai>0 and L>1. With the help of the previous process, define the estimation error(56)εi=xi-x^iLi-1,i=1,…,n,and it is straightforward to show that(57)ε˙=LAεε+LGεx1+F,and since the definitions of the associated symbol are same as (9), we just give different symbol in the following paper: F=f1(·),f2(·)/L,…,fn(·)/Ln-1T∈Rn×1. Consider same scalar function Vε(ε)=εTP1ε, which is proper and radially unbounded. Evidently, following (16), we arrive at(58)V˙ε≤LK2+K3x12+K4L12x22+⋯+Kn+2L12n-2xn2-L2ε2+K1ε2+K3z2,where K1=cn(n+3)P1≥0,K2=2Gε2P12>0,Ki=1/2(n-i+3)cP1≥0,i=3,…,n+2, are independent of a domination gain L.
Secondly, consider system described by(59)x˙1=x2+f1t,z,x,u,x^˙i=x^i+1+Liaiε1-x1,i=2,…,n-1,x^˙n=u+Lnanε1-x1.Then, introduce the following transformations:(60)ξ1=x1,ξi=x^iLHi-1,υ=uLHn,i=2,…,n,where H≥1 is a domination gain to be determined later. With the help of (60), it is easy to see that system (59) can be rewritten as(61)ξ˙1=LHξ2+Lε2+f1t,z,x,u,ξ˙i=LHξi+1+LaiHi-1ε1-ξ1,i=2,…,n-1,ξ˙n=LHυ+LanHn-1ε1-ξ1.If the control law is designed as (20), then substituting (20) into (61) leads to(62)ξ˙=LHAξξ+LHGξbn1-θtξ1+LB2ε2+LHB1ε1-ξ1+F¯,where the definitions of notations are the same as (21). Consider the same quadratic function(63)Vξξ=ξTP2ξ,and, after calculations, we arrive at(64)V˙ξ≤-LHβξ2+L4ε2+LHK¯1H+K¯2Lξ2+K¯3z2,where K¯1=2γ1+4γP2P2+8P22>0,K¯2=3cP2≥0,K¯3=cP2≥0 are independent of the domination gains L and H. It follows from (56) and (60) that(65)x1=ξ1,xi=Li-1εi+x^i=Li-1εi+LHi-1ξi,i=2,…,n.Hence, by Lemma 2, it is easy to obtain the following inequality:(66)1L2i-1xi2=1L2i-1Li-1εi+LHi-1ξi2≤1L2i-12L2i-1εi2+LH2i-1ξi2≤2ε2+2H2i-1ξ2,i=2,…,n.Putting together (66) and (58), there would always hold(67)V˙εε≤-L2ε2+K^1ε2+LK^2+HK^4ξ2+K^3z2,where K^1=K1+2(K4+⋯+Kn+2)≥0,K^2=K2+K3>0,K^3=K3≥0,K^4=2(K4H+⋯+Kn+2H2n-3)≥0. Furthermore, following (39), we arrive at(68)∂U∂zf0≤-c12-L2+⋯+L2n-12z2+HK^5ξ2+n-1c¯22ε2,where K^5=c¯22(H+H3+⋯+H2n-3). Finally, let(69)V=U+Vε+Vξ,and a direct calculation yields(70)V˙≤-c12-L2+⋯+L2n-12-K^3-K¯3z2-L14-K^1+c¯22n-1Lε2-LHβ-K¯1+K^2H-K¯2+K^4+K^5Lξ2.We choose domination gains L and H as(71)H≥max⁡1,2K¯1+2K^2β,L≥max1,4K^1+4c¯22n-11-β,4K¯2+4K^4+4K^5β,where c1 satisfies(72)c1=c1∗+β4,c1∗≥2K¯3+2K^3+L2+⋯+L2n-3,and c¯2 is a positive constant; there is(73)V˙≤-β4z2-β4ε2-β4ξ2.
The process of stability analysis is analogous to Theorem9 and is omitted for the sake of space. This completes the proof.Remark 14.
The process of Theorems9 and 13 means that Assumptions 4 and 5 are suitable for two cases of triangular systems. The contribution of these assumptions is that f0 contains all of states of nonlinear cascade systems. Meanwhile, a unified Lyapunov function is successfully constructed and we proposed a novel observer that is different from [24, 26].


## 3.1. Output Feedback Controller Design for Upper-Triangular Case

We now summarise main results of this paper.Theorem 9.
For system (4) under Assumptions 4–6, there exists an output feedback controller such that states of the closed-loop system are uniformly bounded over [0,+∞) and limt→+∞x(t)=0.Proof.
The proof is in four parts. At first, a linear observer with a domination gain is introduced to reconstruct all the states. Secondly, an output feedback controller composed of another domination gain is constructed to counteract the destabilized terms. Finally, a delicate selection of gains is provided and strict analysis is performed to guarantee that the closed-loop systems are globally asymptotically stable.
Part I: Design of an Observer. We construct the following linear observer:(7)x^˙i=x^i+1-aiLix^1,i=1,…,n-1,x^˙n=u-anLnx^1,where ai>0 are coefficients of Hurwitz polynomial p1(s)=sn+a1sn-1+⋯+an-1s+an and the domination gain L≥1 will be determined later. Define the estimation error as follows:(8)εi=Li-1xi-x^i,i=1,…,n.Then, the error equation can be rewritten as(9)ε˙=1LAεε+1LDεx1+F,where(10)ε=ε1ε2⋮εn,Aε=-a11⋯0⋮⋮⋱⋮-an-10⋯1-an0⋯0,Dε=a1a2⋮an,F=f1·Lf2·⋮Ln-1fn·.Since Aε is Hurwitz, there exists a symmetric positive-definite matrix P1∈Rn×n such that AεTP1+P1Aε=-I, where I is an identity matrix of Rn×n. Consider positive-definite and radially unbounded function Vε(ε)=εTP1ε; a direct calculation gives(11)V˙ε=-1LεTε+21LεTP1Dεx1+2εTP1F≤-1Lε2+21Lε·Dε·P1·x1+2ε·P1·F.According to Assumption 6 and Lemma 2, we obtain(12)F=f12+L2f22+⋯+L2n-1fn2≤f1+Lf2+⋯+Ln-1fn≤cLn-1nz+∑i=3ncLi-3i-2xi.Substituting (12) into (11), one has(13)V˙ε≤-1Lε2+2cP1·εLn-1nz+∑i=3nLi-3i-2xi+2LP1·ε·Dε·x1.In addition, Lemma 1 shows that(14)2LP1·ε·Dε·x1≤12Lε2+1LK2x12,and(15)2cP1·εLn-1nz+∑i=3nLi-3i-2xi≤∑i=3nKiL2i-2xi2+K1L2ε2+12cnL2nP1·z2,where K1=c(2n+(n-1)(n-2))P1≥0, K2=2Dε2P12>0, and Ki=1/2c(i-2)P1≥0,i=3,…,n, are independent of a domination gain L. Therefore, substituting (14) and (15) into (13) yields(16)V˙ε≤-12Lε2+K1L2ε2+1LK2x12+∑i=3nKiL2i-2xi2+12cnL2nP1·z2.
Part II: Construction of a Controller. Consider the following system:(17)x˙1=x2+f1t,z,x,u,x^˙i=x^i+1+aiLiε1-x1,i=2,…,n-1,x^˙n=u+anLnε1-x1.Define the following change of coordinates:(18)ξ1=x1,ξi=Li-1Hi-1x^i,υ=LnHnu,i=2,…,n,where H≥1 is a domination gain that will be determined later. Using above coordinate transform, (17) can be rewritten as(19)ξ˙1=HLξ2+1Lε2+f1t,z,x,u,ξ˙i=HLξi+1+aiLHi-1ε1-ξ1,i=2,…,n-1,ξ˙n=HLυ+anLHn-1ε1-ξ1.Design the output feedback control law as follows:(20)υ=-bny-bn-1ξ2-⋯-b2ξn-1-b1ξn,where bi>0, i=1,…,n, are coefficients of Hurwitz polynomial p2(s)=sn+b1sn-1+⋯+bn-1s+bn. Substituting (20) into (19), we have(21)ξ˙=HLAξξ+HLDξbn1-θtξ1+B2Lε2+B1LHε1-ξ1+F¯,where(22)Aξ=01⋯0⋮⋮⋱⋮00⋯1-bn-bn-1⋯-b1,B1=0a2a3H⋮anHn-2,B2=10⋮0,Dξ=0⋮01,F¯=f1·0⋮0.Aξ is a Hurwitz matrix that shows that there exists a symmetric positive-definite matrix P2 such that AξTP2+P2Aξ=-I. Choose scalar function Vξ(ξ)=ξTP2ξ, which is positive-definite and radially bounded. Noting that B2=Dξ=1, the time derivative of Vξ(ξ) along the trajectories of (21) is(23)V˙ξ=-HLξTξ+2HLξTP2Dξbn1-θtξ1+2ξTP2F¯+2ξTP2B1LHε1+B2Lε2-B1LHξ1≤-HLξ2+2bn1-θtHLP2·ξ2+2ξ·P2·f1+2ξ·P21Lε+B1LHε+2LHP2·B1·ξ2.Firstly, by virtue of Assumption 6 and Lemma 1, it holds that(24)2ξ·P2·f1≤2cξ·P2z+x3+⋯+xn≤cn-1L2P2·ξ2+cP2L2z2+x32+⋯+xn2≜K¯1L2ξ2+K¯2L2∑i=3nxi2+K¯2L2z2,where K¯1=c(n-1)P2≥0 and K¯2=cP2≥0. Furthermore, it is deduced from Lemma 1 that(25)2Lξ·P2·ε≤18Lε2+8LP22·ξ2,2γLHξ·P2·ε≤18Lε2+8γ2LH2P22·ξ2,and because of H>1, one can conclude that(26)B1=∑i=2n1H2i-2ai21/2≤∑i=2nai21/2=Δγ.Consequently, one obtains(27)V˙ξ≤-HL1-2bn1-θt·P2ξ2+14Lε2+8γ2LH2P22·ξ2+8LP22·ξ2+2γLHP2·ξ2+K¯1L2ξ2+K¯2L2∑i=3nxi2+K¯2L2z2≤-HL1-2bn1-θt·P2ξ2+K¯2L2∑i=3nxi2+K¯2L2z2+14Lε2+HL8γ2HP22+8HP22+2γHP2+K¯1Lξ2≜-HL1-2bn1-θt·P2ξ2+K¯2L2∑i=3nxi2+K¯2L2z2+14Lε2+HLK3¯H+K¯1Lξ2,where K¯3=2γ1+4γP2P2+8P22>0 is independent of the domination gains L and H.
Part III: Determination of Domination Gains. According to above arguments, it follows that(28)x1=ξ1,xi=1Li-1εi+Hi-1Li-1ξi,i=2,…,n.Using Lemma 3, there is(29)xi2=1Li-1εi+Hi-1Li-1ξi2≤2L2i-1ε2+2H2i-1L2i-1ξ2,and putting (29) together with (27), we have(30)V˙ξ≤-HL1-2bn1-θt·P2ξ2+14Lε2+HLK3¯H+K¯1Lξ2+2K¯2∑i=3n1L2i-2ε2+2K¯2∑i=3nH2i-1L2i-2ξ2+K¯2L2z2≤-HL1-2bn1-θt·P2ξ2+HLK^3H+K^1Lξ2+1L2K^4ε2+K^2L2z2+14Lε2,where K^1=K¯1+2K¯2(H3+⋯+H2n-3)≥0,K^2=K¯2≥0,K^3=K¯3>0,K^4=2(n-2)K¯2≥0. Now, we choose the allowable sensitivity error θ¯ as(31)θ¯<θ~=12bnP2.Because of 1-θ≤θ¯, one has(32)1-2bn1-θ·P2≥1-2bnθ¯P2≜β.Obviously, 0<β<1 and (30) can be rewritten as(33)V˙ξ≤-HLβξ2+HLK^3H+K^1Lξ2+1L2K^4ε2+K^2L2z2+14Lε2.Similarly, substituting (29) into (16) yields(34)V˙ε≤-12Lε2+K2Lξ2+K1L2ε2+12cnL2nP1z2+2∑i=3nKiL2ε2+2∑i=3nKiH2i-1L2ξ2≤-12Lε2+1L2K^5ε2+1LK^6ξ2+HK^7L2ξ2+K^8L2nz2,where K^5=K1+2(K3+⋯+Kn)≥0,K^6=K2>0,K^7=2(K3H3+⋯+KnH2n-3)≥0,K^8=1/2cnP1≥0. Applying Assumption 5 and the property of Lipschitz function, one has(35)∂U∂zf0t,z,x,u=∂U∂zf0t,z,x,u-∂U∂zf0t,z,0,u+∂U∂zf0t,z,0,u≤∂U∂z·f0t,z,x,u-f0t,z,0,u+∂U∂zf0t,z,0,u≤c¯2z·x-c1z2,where c¯2 is a positive constant. By Lemma 1, it is easy to obtain that x≤x1+⋯+xn, and, according to Lemma 2, the following inequalities hold:(36)c¯2z·x2+⋯+xn≤c12z2+c¯222c1x22+⋯+xn2,c¯2z·x1≤c¯222Lx12+L2z2.Substituting (36) into (35), one has(37)∂U∂zf0t,z,x,u≤-c12-L2z2+c¯222c1x22+⋯+xn2+c¯222Lx12.According to (29) and the definition of norms, it is easy to deduce that(38)x12≤ξ2,x22+x32+⋯+xn2≤2n-1L2ε2+HL22K^9ξ2,where K^9=H+H3+⋯+H2n-3; therefore (37) can be further expressed as(39)∂U∂zf0t,z,x,u≤-c12-L2z2+c¯22n-1c1L2ε2+c¯22K^9Hc1L2+c¯222Lξ2.Consider positive-definite and radially unbounded function as follows:(40)V=U+Vε+Vξ,and substituting (33), (34), and (39) into (40), there is(41)V˙≤-c12-L2-K^8L2n-K^2L2z2-1L14-K^5+K^4L-2c¯22n-1Lε2-HLβ-K^6+K^3H-K^1+K^7L-c¯222H-c¯22K^9Lξ2.In order to ensure that V˙≤0, we choose the domination gains L and H as follows. Firstly,(42)β-K^6+K^3H-c¯222H≥β2,implying that(43)H≥2K^3+2K^6+c¯22β,and, with H≥1 in hand, one deduces(44)H≥max⁡1,2K^3+2K^6+c¯22β.As consequence, (41) can be rewritten as(45)V˙≤-c12-L2-K^8L2n-K^2L2z2-1L14-K^5+K^4L-2c¯22n-1Lε2-HLβ2-K^1+K^7L-c¯22K^9Lξ2.Secondly, we can selectL to satisfy the following inequalities:(46)14-K^5+K^4L-2c¯22n-1L≥β4,β2-K^1+K^7L-c¯22K^9L≥β4,and, in light of L≥1, it is straightforward to show that(47)L≥max⁡1,4K^1+4K^7+4c¯22K^9β,4K^4+4K^5+8c¯22n-11-β,and c1 satisfies(48)c1=β4+c1∗,c1∗≥2K^8L2n+K^2L2+L.Under above choice of domination gains, it is clear that(49)V˙≤-β4z2-β4Lξ2-β4Lε2.
Part IV: Stability Analysis. Consider transformed systems (9), (20), and (21). By the existence and continuity of solution, the closed-loop system state composed of Y(t)=zT,xT,x^TT can be defined on a time interval 0,tm, where tm>0 may be a finite constant or infinity.
(i)  For the Boundedness ofY(t).Due to the fact that ν1(z)≤U(z)≤ν2(z), λmin(P)x2≤xTPx≤λmax(P)x2, where ν1 and ν2 are K∞ functions; we obtain(50)V=U+Vε+Vξ≥ν1z+λminP1ε2+λminP2ξ2≥ν1z+minλminP1,λminP2ωt2≜φ1ϖt,and(51)V=U+Vε+Vξ≤ν2z+λmaxP1ε2+λmaxP2ξ2≤ν2z+maxλmaxP1,λmaxP2ωt2≜φ2ϖt,where ω(t)=[εT(t),ξT(t)]T, ϖ(t)=[zT,εT(t),ξT(t)]T, and φ1,φ2 are K∞ functions.
For anyϵ>0, noting that lims→+∞φ1(s)=+∞, one can always find a constant β=β(ϵ)>ϵ>0, such that φ2(ϵ)≤φ1(β). From (49), it follows that Vϖt≤V(ϖ(0)), ∀t∈[0,tm). Then, if ϖ(0)<ϵ, it holds that(52)φ1ϖt≤Vϖt≤Vϖ0≤φ2ϖ0<φ2ϵ≤φ1β,∀t∈0,tm,which implies that ϖ(t)<β, ∀t∈0,tm; that is, ϖ(t) is bounded. Therefore, by virtue of estimation errors and coordinates transform, it is easy to get that the state Y(t)=zT,xT,x^TT is bounded on 0,tm as well.
(ii)tm=+∞.This claim can be shown by contradiction; if tm is finite, then tm would be a finite escape time; that is, the state Y(t) would tend to ∞ as t=tm. However, the continuity of solution guarantees the boundedness of Y(t) at t=tm, since Y(t) is bounded on [0,tm). This is clearly a contradiction. Therefore, the state of the closed-loop system is uniformly bounded over [0,+∞).
(iii)limt→+∞x(t)=0 and limt→+∞z(t)=0.Above all, it is clear that(53)0≤∫0tβ4z2+β4Lε2+β4Lξ2ds≤-∫0tV˙ϖsds<Vϖ0<+∞,∀t≥0,which implies that ∫0∞β/4z2+β/4Lε2+β/4Lξ2ds exists and is finite. On the other hand, the boundedness of ϖ(t) over [0,+∞) means that ϖ˙(t) is uniformly bounded in t over [0,+∞). Thus, ϖ(t) is uniformly continuous in t over [0,+∞) and so is ϖ2(t). Using well-known Barbalat’s Lemma in [22], one obtains limt→+∞zi2(t)=0, limt→+∞ξi2(t)=0, and limt→+∞εi2(t)=0, which shows that limt→+∞z(t)=0, limt→+∞ξ(t)=0, and limt→+∞ε(t)=0. Finally, from (29), it is easy to see that limt→+∞x(t)=0. This completes the proof.Remark 10.
A double-domination method is proposed to handle the time-varying output function and nonlinearities in the proof of Theorem9; that is, two domination gains H and L are used to dominate time-varying function θ(t) and nonlinearities fi, respectively.Remark 11.
It should be noted fromθ~=1/2bnP2 that the upper bound θ~ depends on coefficients b1,…,bn for the Hurwitz polynomial p2(s). When b1,…,bn are specified, the corresponding matrix P2 as well as the upper bound θ~ can be computed. For example, θ~=0.4420, for n=2; θ~=0.1501, for n=3.


## 3.2. Extension to Lower-Triangular Case

Some subsystems do not satisfy upper-triangular structure in practical application, so we extend the subsystems to lower-triangular form and impose following assumptions on system (4).Assumption 12.
Fori=1,…,n, there exists a constant c≥0 such that (54)fi≤cz+x1+⋯+xi.Theorem 13.
For a class of nonlinear cascade system (4) under Assumptions 4, 5, and 12, there exists an output feedback controller, such that states of the closed-loop system are uniformly bounded over [0,+∞) and limt→+∞x(t)=0.Proof.
The proof is analogous to the proof of Theorem9 with an obvious modification. To facilitate comparison, we select same notations as Theorem 9 and many similarities will be omitted.
Firstly, we construct the similar observer as (7)(55)x^˙i=x^i+1-Liaix^1,i=1,…,n-1,x^˙n=u-Lnanx^1,where ai>0 and L>1. With the help of the previous process, define the estimation error(56)εi=xi-x^iLi-1,i=1,…,n,and it is straightforward to show that(57)ε˙=LAεε+LGεx1+F,and since the definitions of the associated symbol are same as (9), we just give different symbol in the following paper: F=f1(·),f2(·)/L,…,fn(·)/Ln-1T∈Rn×1. Consider same scalar function Vε(ε)=εTP1ε, which is proper and radially unbounded. Evidently, following (16), we arrive at(58)V˙ε≤LK2+K3x12+K4L12x22+⋯+Kn+2L12n-2xn2-L2ε2+K1ε2+K3z2,where K1=cn(n+3)P1≥0,K2=2Gε2P12>0,Ki=1/2(n-i+3)cP1≥0,i=3,…,n+2, are independent of a domination gain L.
Secondly, consider system described by(59)x˙1=x2+f1t,z,x,u,x^˙i=x^i+1+Liaiε1-x1,i=2,…,n-1,x^˙n=u+Lnanε1-x1.Then, introduce the following transformations:(60)ξ1=x1,ξi=x^iLHi-1,υ=uLHn,i=2,…,n,where H≥1 is a domination gain to be determined later. With the help of (60), it is easy to see that system (59) can be rewritten as(61)ξ˙1=LHξ2+Lε2+f1t,z,x,u,ξ˙i=LHξi+1+LaiHi-1ε1-ξ1,i=2,…,n-1,ξ˙n=LHυ+LanHn-1ε1-ξ1.If the control law is designed as (20), then substituting (20) into (61) leads to(62)ξ˙=LHAξξ+LHGξbn1-θtξ1+LB2ε2+LHB1ε1-ξ1+F¯,where the definitions of notations are the same as (21). Consider the same quadratic function(63)Vξξ=ξTP2ξ,and, after calculations, we arrive at(64)V˙ξ≤-LHβξ2+L4ε2+LHK¯1H+K¯2Lξ2+K¯3z2,where K¯1=2γ1+4γP2P2+8P22>0,K¯2=3cP2≥0,K¯3=cP2≥0 are independent of the domination gains L and H. It follows from (56) and (60) that(65)x1=ξ1,xi=Li-1εi+x^i=Li-1εi+LHi-1ξi,i=2,…,n.Hence, by Lemma 2, it is easy to obtain the following inequality:(66)1L2i-1xi2=1L2i-1Li-1εi+LHi-1ξi2≤1L2i-12L2i-1εi2+LH2i-1ξi2≤2ε2+2H2i-1ξ2,i=2,…,n.Putting together (66) and (58), there would always hold(67)V˙εε≤-L2ε2+K^1ε2+LK^2+HK^4ξ2+K^3z2,where K^1=K1+2(K4+⋯+Kn+2)≥0,K^2=K2+K3>0,K^3=K3≥0,K^4=2(K4H+⋯+Kn+2H2n-3)≥0. Furthermore, following (39), we arrive at(68)∂U∂zf0≤-c12-L2+⋯+L2n-12z2+HK^5ξ2+n-1c¯22ε2,where K^5=c¯22(H+H3+⋯+H2n-3). Finally, let(69)V=U+Vε+Vξ,and a direct calculation yields(70)V˙≤-c12-L2+⋯+L2n-12-K^3-K¯3z2-L14-K^1+c¯22n-1Lε2-LHβ-K¯1+K^2H-K¯2+K^4+K^5Lξ2.We choose domination gains L and H as(71)H≥max⁡1,2K¯1+2K^2β,L≥max1,4K^1+4c¯22n-11-β,4K¯2+4K^4+4K^5β,where c1 satisfies(72)c1=c1∗+β4,c1∗≥2K¯3+2K^3+L2+⋯+L2n-3,and c¯2 is a positive constant; there is(73)V˙≤-β4z2-β4ε2-β4ξ2.
The process of stability analysis is analogous to Theorem9 and is omitted for the sake of space. This completes the proof.Remark 14.
The process of Theorems9 and 13 means that Assumptions 4 and 5 are suitable for two cases of triangular systems. The contribution of these assumptions is that f0 contains all of states of nonlinear cascade systems. Meanwhile, a unified Lyapunov function is successfully constructed and we proposed a novel observer that is different from [24, 26].


## 4. Simulation Example

As application of the design method, two examples are provided as follows.Example 1.
Consider nonlinear cascade system:(74)z˙t=-2zt-x2t,x˙1t=x2t+x3t-sin⁡x1zt,x˙2t=x3t+zt,x˙3t=ut+zt,yt=θtx1t,where θ(t)=1+0.15sin6t. Obviously, systems satisfy Assumptions 4–6 with c=1, c1=204, and c2=20. And we choose p2(s)=s+0.43; that is, b1=1.2, b2=0.48, b3=0.064, and θ~=0.1501. Besides, for the observer design, we select θ¯=0.15<θ~, a1=4, a2=3, and a3=1. In consequence, the control law is given as follows:(75)ut=-b3H3L3y-b2H2L2x^2-b1HLx^3,x^˙1t=x^2t-a1Lx^1t,x^˙2t=x^3t-a2L2x^1t,x^˙3t=ut-a3L3x^1t,where L=2 and H=7. In simulation, initial values are chosen as(76)z,x10,x20,x30,x^10,x^20,x^30T=-1,-2,1,4,4,-2,-1T,and one gets Figures1–4, which illustrate that the control law (20) is effective.Figure 1
The curves of statesx1 and x^1.Figure 2
The curves of statesx2 and x^2.Figure 3
The curves of statesx3 and x^3.Figure 4
The curve of statez.Example 2.
Consider the following nonlinear cascade system:(77)z˙t=-zt+x1t,x˙1t=x2t+sin⁡x1z,x˙2t=ut+x1t,yt=θtx1t,where θ(t)=1+0.44sin6t. Evidently, Assumptions 4, 5, and 12 are satisfied with c=1, c1=18, and c2=1. And we choose p2(s)=s+0.42; that is, b1=0.8 and b2=0.16. After sample calculation, θ~=0.4420 and select θ¯=0.44<θ~, a1=5, and a2=1; we construct an output feedback controller as follows:(78)ut=-b2LH2y-b1LHx^2,x^˙1t=x^2t-a1Lx^1t,x^˙2t=ut-a2L2x^1t,where L=4 and H=5. In the simulation, we choose the initial values as z0,x10,x10,x^10,x^20T=3,-2,4,-1,1.5T; one can obtain the simulation results as shown in Figures 5–8, which exhibit the effectiveness of the control scheme.Figure 5
The curves of statesx1 and x^1.Figure 6
The curves of statesx2 and x^2.Figure 7
The curve of statez.Figure 8
The curve of controlu.


## 5. Conclusions

This paper solves the problem of global output feedback stabilization for the nonlinear cascade systems with time-varying output function. The construction of the output feedback controller is based on the double-domination method. There still exist some problems to be investigated. For example, (I) [34–36] solved the problem of finite-time stabilization; it is unclear whether scheme can be applied to solve the finite-time stabilization for nonlinear systems with time-varying output function. (II) References [37, 38] proposed the output feedback controller that ensures that the equilibrium is globally asymptotically stable in probability. Then, is it possible to achieve the stabilization of nonlinear stochastic systems by the proposed strategy of this paper? (III) References [39–45] advanced the solution to the stabilization problem of time-delay systems. However, it is unclear whether this method could be used to address the stabilization of time-delay nonlinear systems. (IV) References [46–48] focus on global adaptive state-feedback stabilization for a class of high-order uncertain nonlinear systems. Whenc in Assumption 6 is unknown, can the approach of this paper be used to solve the problem of adaptive stabilization?



---

*Source: 5185394-2018-07-02.xml*