# Global Output Control for a Class of Inherently Higher-Order Nonlinear Time-Delay Systems Based on Homogeneous  Domination Approach

Authors: Lin Chai
Journal: Discrete Dynamics in Nature and Society
 (2013)

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## Abstract

This paper addresses the problem of global output feedback stabilization for a class of inherently higher-order uncertain nonlinear systems subject to time-delay. By using the homogeneous domination approach, we construct a homogeneous output feedback controller with an adjustable scaling gain. With the aid of a homogeneous Lyapunov-Krasovskii functional, the scaling gain is adjusted to dominate the time-delay nonlinearities bounded by homogeneous growth conditions and render the closed-loop system globally asymptotically stable. In addition, we also show that the proposed approach is applicable for time-delay systems under nontriangular growth conditions.


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## Body

## 1. Introduction

This paper addresses the global stabilization problem for a class of uncertain systems with delay which is described by(1)x˙1(t)=x2(t)p1+ϕ1(t,x(t),x(t-τ),u(t)),0000000000000000⋮x˙n-1(t)=xn(t)pn-1+ϕn-1(t,x(t),x(t-τ),u(t)),x˙n(t)=u(t)+ϕn(t,x(t),x(t-τ),u(t)),y(t)=x1(t),x(t)=Ψ(t),-τ≤t≤0,
where x(t)=(x1(t),x2(t),…,xn(t))T∈ℝn are the system states, u(t)∈ℝ is the control input, y(t)∈ℝ is the system output with pi∈ℝ≥1odd, τ>0 is a given time-delay constant, and Ψ(t) is the initial function of the system state vector. The terms ϕi(·), i=1,…,n, represent nonlinear perturbations that are not guaranteed to be precisely known.It has been known that the problem of global output feedback stabilization for uncertain nonlinear systems is very challenging compared to the state feedback case. In the past decade, global stabilization by output feedback domination method has been proved to be achievable for a series of nonlinear systems. For the system of a five-spot pattern reservoir, a nonlinear reduced-order model is identified and an asymptotically stabilizing controller is proposed based on the circle criterion in [1]. With the help of linear feedback domination design [2], some interesting results have been established under a linear growth condition [2] and under a higher-order growth condition [3]. Recently, the homogeneous domination approach proposed in [3] has been used as a universal tool to solve the problem of global output feedback stabilization for inherently nonlinear systems. As a consequence, fruitful research results have been achieved in [3–9].However, the aforementioned results have not considered the time-delay effect which is actually very common in state, input, and output due to the time consumed in sensing, information transmitting, and controller computing. In the case when the nonlinearities contain time-delay, some interesting results have been obtained. For instance, in [10], the global asymptotic stability analysis problem is investigated for a class of stochastic bidirectional associative memory (BAM) networks with mixed time-delays and parameter uncertainties. The paper [11] investigated the state estimation problem for a class of discrete time-delay nonlinear complex networks with randomly occurring phenomena from the sensor measurements. In [12], an adaptive approach was employed to design a state feedback controller to globally stabilize a class of upper-triangular systems with time-delay. The work [13] relaxed the growth condition imposed in [12] by employing a dynamic gain. In [14], a state feedback stabilizer was constructed for a class of time-delay higher-order nonlinear systems. In [15], the problem of state-feedback stabilization for a class of lower-triangular stochastic time-delay nonlinear systems was investigated.In the case when only output is available, the problem of output feedback stabilization is more challenging and fewer results have been achieved for nonlinear systems with time-delay. For a linear system with time-delay in the input, the problem of output feedback stabilization of was solved in [16, 17]. For nonlinear systems (1) subject to time-delay and uncertainties, the problem of output feedback stabilization has not been extensively investigated.In this paper, we aim to tackle the problem by using the output feedback domination approach. First, based on homogeneous domination approach [3, 9], output feedback controllers are constructed to globally stabilize higher-order nonlinear time-delay systems whose nonlinearities are bounded by homogeneous growth conditions. Then, we construct a Lyapunov-Krasovskii functional and use it to choose an appropriate scaling gain in the output feedback controller to guarantee the closed-loop system globally asymptotic stability. In addition, homogeneous output stabilization controllers are extended to the nontriangular time-delay systems. The simulation results show the effectiveness of the proposed method.


## 2. Homogeneous Output Feedback Controllers of Inherently Nonlinear Systems

In this section, we show that under a lower-triangular homogeneous growth condition, the nonlinear time-delay system (1) can be globally stabilized by a homogeneous output feedback controller. First, we consider the nonlinear continuous functions ϕi:ℝ×ℝn×ℝn×ℝ→ℝ, i=1,2,…,n, which satisfy the following higher-order growth condition.Assumption 1.
Consider the following:(2)|ϕi(t,x(t),x(t-τ),u(t))|≤c1(|x1|(ri+m)/r1+⋯+|xi|(ri+m)/ri)+c2(|x1(t-τ)|(ri+m)/r1+⋯+|xi(t-τ)|(ri+m)/ri),
where i=1,…,n, r1=1, m≥0, ripi-1=m+ri-1, for i=2,…,n+1, and pn=1. Denoting κ1=0 and κi=(κi-1+1)/pi-1, for i=2,…,n+1. The objective of this note is to prove that the growth condition, namely, Assumption 1, guarantees the existence of a linear dynamic homogeneous output feedback controller based on observer:
(3)u(t)=-Lκn+1βn[1Lκnx^nμ/rn(t)+βn-1×(1Lκn-1x^n-1μ/rn-1(t)+⋯+β2(1Lκ2x^2μ/r2(t)+β1x^1μ/r1(t)1Lκn-1)⋯1Lκn-1)1Lκn](rn+m)/μ,(4)x^1=x1,η-˙2=-l1x^2p1,x^2=Lκ2[η-2+l1x^1]r2/r1,η-˙i=-li-1x^ipi-1,x^i=Lκi-(κi-1ri/ri-1)[η-i+li-1x^i-1]ri/ri-1,00000000000000000000000000000000000i=3,…,n,
where μ≥maxi=2,…,n+1{ripi-1} and li, i=1,…,n-1 are the gains selected by Theorem 3.1 in [5] renders system (7) globally asymptotically stable, with
(5)ln-1=14+1+α~+α-,li-1=mi-1[14+αi+1+α~+α-+gi+1(li)+hi+1(li)14],000000000i=n-1,…,3,l1=m2-1[14+α2+1+α~+α-+g3(l2)+h3(l2)],
where α~≥0, α-, and αi are constants, mi for i=2,…,n-1 are known powers of 2, and gi+1(li), hi+1(li) are continuous functions.Remark 2.
Whenm>0, (2) is a higher-order growth condition which is actually homogeneous (see Definition A.1 in the appendix) with the following dilation (For simplicity, in this paper we assume m=q/d with an even integer q and an odd integer d. Therefore, ri is a ratio of two odd integers.)
(6)r1=1,ripi-1=m+ri-1,i=2,…,n+1.
First, we construct a output feedback stabilizer for the following linear system(7)z˙i=zi+1pi,i=1,…,n-1,z˙n=v,y=z1,
where v∈ℝ, y∈ℝ are the control input and system output, respectively. Using the approach in [5], we can design for (7) a homogeneous output feedback stabilizer, which can be described in the following lemma.Lemma 3.
There exists a dynamic homogeneous output feedback controller based on observer:(8)u(t)=-βn[z^nμ/rn(t)+βn-1×(z^n-1μ/rn-1(t)+⋯+β2(z^2μ/r2(t)+β1z^1μ/r1(t))⋯β1z^1μ/r1(t))](rn+m)/μ,(9)z^1=x1,η˙2=-l1z^2p1,z^2=[η2+l1z1]r2/r1,η˙i=-li-1zipi-1,z^i=[ηi+li-1z^i-1]ri/ri-1,i=3,…,n,
where μ≥maxi=2,…,n+1{ripi-1}, and li, i=1,…,n-1 are the same with those in (4).It can be verified that the closed-loop system (7)–(9) is homogeneous according to Definition A.1 in the appendix. As a matter of fact, by defining the compact notation Z:=(z1,…,zn,η2,…,ηn)T, the closed-loop system (7)–(9) can be rewritten as the following compact form:
(10)Z˙=F(Z)=(z2p1,…,znpn-1,v,f1,…,fn-1)T,
where fi=η˙i+1, i=1,…,n-1. Moreover, by introducing the dilation weight
(11)Δ=(r1,r2,…,rn︸,r1,r2,…,rn-1)︸,forz1,…,zn,forη2,…,ηn,
system (10) is homogeneous of degree m according to Definition A.1 in the appendix. Since system (10) is globally stable by Lemma 3, therefore, by Lemmas A.2 and A.3 in the appendix, there exists a homogeneous Lyapunov function U with the degree 2rn+1-m such that
(12)U˙(Z)|(10)=∂U∂Z·F(Z)≤-c1∥Z∥Δ2μ,c1>0,
where ∥Z∥Δ=∑i=12n-1‍|Zi|2/ri. Moreover, by Lemma A.4 in the appendix, for i=1,…,n, the following holds:
(13)|∂U∂Zi|≤c2∥Z∥Δ2μ-m-ri,c2>0.Theorem 4.
Consider system (1) under Assumption 1. For any given delay constant τ, there exists a homogeneous output feedback controller (3) based on observer (4) such that system (1) can be globally stabilized.Proof.
The output feedback controller is constructed by introducing a scaling gain into the output feedback controller obtained in Lemma3. First, we define a change of coordinates as
(14)Lκizi=xi,Lκiz^i=x^i,i=1,…,n,Lκn+1v=u,η2=η-2,Lκi-1ηi=η-i,i=3,…,n,
where the constant gain L≥1 will be determined later. Under (14), system (1) can be rewritten as
(15)z˙1=Lz2p1+ϕ1,000000000000⋮z˙n-1=Lznpn-1+ϕn-1Lκn-1,z˙n=Lv+ϕnLκn.
The observer (9) can be rewritten as follows:
(16)η˙2=-Ll1z^2p1,z^2=[η2+l1z^1]r2/r1,η˙k=-Llk-1z^kpk-1,z^k=[ηk+lk-1z^k-1]rk/rk-1,00000000000000000000000.00k=3,…,n,v(t)=-βn(z^nμ/rn(t)+βn-1(z^n-1μ/rn-1(t)+⋯+β2(z^2μ/r2(t)+β1z^1μ/r1(t))⋯))(rn+m)/μ,
where the same βi, i=1,…,n as in (9). Now, the closed-loop system (15) and (16) can be rewritten as
(17)Z˙=LF(Z)+(ϕ1(·),1Lκ2ϕ2(·),…,1Lκnϕn(·),0,…,0)T,
where the vector field F(Z) has the exactly same construction of (10). Therefore, we adopt the same Lyapunov function U used in (12), whose derivative along (17) is
(18)U˙=∂U∂ZLF(Z)+∂U∂Z×(ϕ1(·),1Lκ2ϕ2(·),…,1Lκnϕn(·),0,…,0)T≤-Lc1∥Z∥Δ2μ+∑i=1n‍∂U∂ZiϕiLκi.
According to Assumption 1, there exists a constant li>0 such that
(19)|1Lκiϕi(t,x(t),x(t-τ),u(t))|≤c1(ri+mri|z1|(ri+m)/r1+|Lκ2z2|(ri+m)/r2+⋯+|Lκizi|(ri+m)/riri+mri)1Lκi+c2(|z1(t-τ)|(ri+m)/r1+|Lκ2z2(t-τ)|(ri+m)/r2+⋯+|Lκizi(t-τ)|(ri+m)/ri)1Lκi≤l-iL1-νi(|z1|(ri+m)/r1+⋯+|zi|(ri+m)/ri+|z1(t-τ)|(ri+m)/r1+⋯+|zi(t-τ)|(ri+m)/ri)≤l~iL1-νi(∥Z∥Δri+m+∥Z(t-τ)∥Δri+m),
for a constant νi>0, since it can be seen that by definition rj=mκj+1/(p1⋯pj-1), so
(20)κjri+1pirj-κi=κj(mκi+1/(p1⋯pi-1)+m)mκj+1/(p1⋯pi-1)-κi=mκj+κj/(p1⋯pi-1)-κi/(p1⋯pi-1)mκj+1/(p1⋯pi-1)≤mκjmκj+1/(p1⋯pi-1)<1,j=2,…,i,i=1,…,n.
Substituting (13) and (19) into (18) yields
(21)U˙≤-L(c1-c2∑i=1n‍l~iL-νi∥Z∥Δ2μ+c4∑i=1n‍l~iL-νi∥Z∥Δ2μ-ri-m·∥Z(t-τ)∥Δri+m).
By Lemma A.5 in the appendix, there exists a constant c3>0 such that
(22)c2∥Z∥Δ2μ-ri-m·∥Z(t-τ)∥Δri+m≤∥Z∥Δ2μ+c3∥Z(t-τ)∥Δ2μ,
which yields
(23)U˙≤-L(∑i=1n‍c1-(c2+1)∑i=1n‍l~iL-νi∥Z∥Δ2μ+c3∑i=1n‍l~iL-νi∥Z(t-τ)∥Δ2μ).
Construct a candidate Lyapunov functional as follows:
(24)W=U+∫t-τt∥Z(θ)∥Δ2μ‍Sdθ,
where S is a positive constant. Let S=c3∑i=1nl~iL1-νi, so it follows from (23) and (24) that
(25)W˙≤-L(c1-(1+c2+c3)∑i=1n‍l~iL-νi)∥Z∥Δ2μ.
Hence, by choosing a large enough L as L>max{(((1+c2+c3)/c1)∑i=1n‍l~i)1/ν,1}, where ν=mini=1,…,n{νi}, the right-hand side of (25) is negative definite, that is, there exists a constant ρ>0, such that
(26)W˙≤-ρL∥Z∥Δ2μ.
As a conclusion, we know that the system described by (1) under Assumption 1 can be globally stabilized by the output controller (8).


## 3. Extension to Nontriangular Systems

Assumption 5.
Consider the following(27)|1Lκiϕi(t,x(t),x(t-τ),u(t))|≤L1-νi(|xi(t-τ)Lκi|(ri+m)/ri|x1Lκ1|(ri+m)/r1+⋯+|xiLκi|(ri+m)/ri+|x1(t-τ)Lκ1|(ri+m)/r1+⋯+|xi(t-τ)Lκi|(ri+m)/ri).
By (19), it is apparent that Assumption 5 includes Assumption 1 as a special case. The next theorem is a more general result achieved under Assumption 5.Theorem 6.
Under Assumption5, there exist constants li and βi, i=1,…,n and L>1 such that the homogeneous output feedback controller (3) globally stabilizes system (1).Proof of Theorem6.
The proof is very similar to that of Theorem4. We can use the exactly same observer (4) and control law (3). Although the nonlinear function is not in the triangular form, Assumption 5 will lead directly to (19) by using the change of coordinates (14). Then, the global stabilization can be concluded with an appropriate choice of gain L, which is similar to that in (25). The detailed proof is omitted here for brevity.


## 4. Examples and Simulations

Consider the following inherently nonlinear time-delay system.Example 1.
Consider(28)x˙1(t)=x25/3(t)+0.25x12(t)sin(x2(t)),x˙2(t)=u(t)+0.125x25/3(t-0.2),y=x1(t).
Therefore, Assumption 1 holds for p1=5/3, m=2/3, r1=1, r2=1, and μ=5/3. By (3), the output controller can be constructed as
(29)η˙2=-L3/5[η2+l1y],u=-L8/5β2((η2+l1y)5/3+β1y5/3),
where we choose l1=0.55, β1=1.06, β2=1.93, and L=4. The computer simulation results of the closed-loop systems are given in Figures 1 and 2 with the following initial functions [x1(t)x2(t)]=[-5sin(t-0.2)-3sin(t-0.2)], η2(t)=0, for -0.2≤t≤0.Figure 1
State trajectories ofx(t)in (28).Figure 2
Time history ofu(t)in (29).


## 5. Conclusion

In this paper, we have studied the problem of global output feedback stabilization for a class of higher-order time-delay nonlinear systems under a homogeneous condition. First, homogeneous output feedback controllers have been constructed with adjustable scaling gains. Then, with the help of a homogeneous Lyapunov-Krasovskii functional, we’ve redesigned the homogeneous domination approach to tune the scaling gain for the overall stability of the closed-loop systems. The output feedback controllers proposed in this paper are memoryless and, therefore, can be easily implemented in practice.



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