# Global Asymptotic Stabilization Control for a Class of Nonlinear Systems with Dynamic Uncertainties

Authors: Jiangbo Yu; Jizhong Wang; Zhongcai Zhang
Journal: Mathematical Problems in Engineering
 (2015)

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## Abstract

This paper is concerned with the global asymptotic stabilization control problem for a class of nonlinear systems with input-to-state stable (ISS) dynamic uncertainties and uncertain time-varying control coefficients. Unlike the existing works, the ISS dynamic uncertainty is characterized by the uncertain supply rates. By using the backstepping control approach, a systematic controller design procedure is developed. The designed control law can guarantee that the system states are asymptotically regulated to the origin from any initial conditions and the other signals are bounded in closed-loop systems. Moreover, it is shown that, under some additional conditions, a linear control law can be designed by the proposed methodology. The simulation example demonstrates its effectiveness.


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## Body

## 1. Introduction

The nonlinear control theory is an active research direction in the control field because of its widespread applications in the real world. During the past two decades, various novel methodologies have been generated for the nonlinear feedback control; see the recent survey [1] and references therein for an interesting introduction to this area. One of the influential notions is the input-to-state stability (ISS) and its several variants. Since they are introduced by Sontag in [2, 3], the notion of ISS as well as its integral variant—integral ISS (iISS)—has become a foundational concept upon which much of modern nonlinear feedback analysis and design rest. As noted in [4], ISS provides a nonlinear generalization of finite gains with respect to supremum norms and also of finite L2 gains, and it plays a central role in recursive design, coprime factorizations, controllers for nonminimum phase systems, and many other areas. Based on the series of works on ISS, the nonlinear small-gain theorem was proposed in the state-space setting and is widely used in the stability analysis and control design for complex interconnected systems in [5]. The stochastic results can be found in [6, 7] and the references therein.It is noted that a unifying framework is presented in [8] for the global output feedback regulation control problem from ISS to iISS. The framework established in [8] extends many known classes of output feedback form systems. However, the system uncertainties investigated there depend only on the system output and the inverse system state. With unmeasured states dependent growth, in [9, 10], the problem of global stabilization by output/state feedback is investigated for a class of nonlinear systems with uncertain control coefficients. However, there is no dynamic uncertainty for the system under consideration. In [11], this work is further studied for a larger class of nonlinear uncertain systems, in which the observer gain is governed by a Riccati differential equation. Moreover, the output regulation problem is also considered in [12] for this class of nonlinear systems with iISS inverse dynamics. Later, in [13, 14], this work is further investigated for the nonlinear systems with uncertain nonlinearities dependent on all unmeasured states. However, the control coefficients in above results are required to be known a priori or unknown nonzero constants. In [15], the global set-point tracking control is investigated for a class of cascaded nonlinear systems with unknown control coefficients. However, a restrictive condition is that the control coefficients are required to have the same signs.In this paper, we will further investigate this problem for a class of nonlinear systems with more general nonlinear uncertainties. Unlike the existing works such as in [9, 12, 15–17], the studied system is with the uncertain control coefficients, which could be unknown time-varying functions. Another feature of this work is that the dynamic uncertainties are characterized by the uncertain ISS supply rates. This is different from the existing results reported in literatures where the ISS dynamic uncertainty is investigated under the hypothesis that supply rates are known a priori such as in [8, 11, 13, 15]. With the help of the backstepping approach [18], we design a robust adaptive controller which could achieve the system states convergent to the origin while the other signals are bounded. Moreover, it is of interest to note that a linear control law can be designed using the developed scheme if some stronger conditions are imposed on the nonlinear system.The rest of the paper is organized as follows. In Section2, we provide some mathematical preliminaries and state the problem. The controller design procedure is developed in Section 3, and the main result is presented in Section 4. Section 5 illustrates the obtained results by a numerical example. Section 6 concludes this paper.Notation. Let R(R+) denote the set of all (positive) real numbers and let Rn denote the real n-dimensional space. For a given vector or matrix X, XT denotes its transpose. For any column vector x=(x1,…,xn)T∈Rn, x¯i denotes the column vector consisting of the first i elements of x in the original order; that is, x¯i=(x1,…,xi)T. Specifically, for x=(x1,…,xn)T, x1=x¯1, x=x¯n. 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→∞. The notation β~(s)=O(β(s)) means that there exist two positive constants k and c such that β~(s)≤kβ(s), ∀s<c.


## 2. Problem Formulation

In this paper, we consider the following class of cascaded nonlinear systems with dynamic uncertainties:(1)η˙=qt,η,y,x˙1=d1tx2+g1t,η,x,⋮x˙n=dntu+gnt,η,x,y=x1,where u∈R is the control input, y∈R is the system output, x=(x1,…,xn)∈Rn are the system states, and η∈Rr is referred to as dynamic uncertainties, which is unmeasured and hence is not available for feedback design. The continuous functions dit(i=1,…,n) called the control coefficients are assumed to be unknown; particularly, di(t)≠0; the unmodeled (or uncertain) dynamics q· and gi·(i=1,…,n) are locally Lipschitz.The control objective in this paper is to find a smooth, dynamic, partial-state feedback law of the form(2)ξ˙=χξξ,x,u=χuξ,x,where χξ and χu are smooth functions such that all solutions (η(t),x(t),ξ(t)) in closed-loop system are bounded on [0,∞) and specially the system states (η(t),x(t)) asymptotically converge to the origin. Toward this end, throughout the paper, we make the following assumptions on system (1).Assumption 1.
Theη-subsystem is input-to-state stable (ISS) with state η and input y; that is, there exists a positive-definite and proper ISS-Lyapunov function V0, such that(3)α_η≤V0η≤α¯η,∂V0∂ηηqt,η,y≤-α0η+δ0γ0y,where α_(·),α¯(·),α0(·),γ0(·)∈K∞ and δ0>0 is an unknown constant.Remark 2.
According to [19], one knows that η-subsystem satisfying (3) is ISS, and the function pair (α0,δ0γ0) is viewed as the supply rates. Since δ0 in (3) is unknown, the dynamic uncertainty has uncertain ISS supply rates. This is different from the existing results reported in literatures, where the ISS dynamic uncertainty is investigated with the supply rates assumed to be known a priori, such as [8, 11, 13, 15].Assumption 3.
For the uncertain nonlinearitiesgit,η,x(i=1,…,n), there exist unknown positive constants pij(i=1,…,n;j=1,2) such that(4)git,η,x≤pi1ϕi1η+pi2ϕi2x1,…,xi,i=1,…,n,where ϕij· are known smooth functions and ϕij0=0(i=1,…,n;   j=1,2).Assumption 4.
There exist known positive constantsd_i and d¯i(i=1,…,n), such that(5)0<d_i≤dit≤d¯i,i=1,…,n.
To deal with the unmeasured stateη, we have the following lemma, which plays an important role in the coming feedback design and stability analysis.Lemma 5.
Consider theη-subsystem satisfying Assumption 1. Suppose(6)γ0s=Os2,and then we can choose a positive continuous function ρ(·), such that the function(7)V¯0η=∫0V0ηρsdsis another candidate ISS-Lyapunov function satisfying(8)∂V¯0∂ηηqt,η,y≤-1-ϵρ∘α_ηα0η+δγy,where ϵ(0<ϵ<1) is a small design constant, δ>0 is an unknown constant, and γ is a K∞-function with γ(s)=O(s2).Remark 6.
Ifγ(s)=O(s2), according to Lemma  2 in [19], there exists a smooth nondecreasing function γ¯(·) satisfying(9)γs≤γ¯ss2.Lemma 7.
For anyC1 function f(x1,…,xn), there exist continuous functions fix1,…,xn(1≤i≤n), such that(10)fx1,…,xn=f0,…,0+∑i=1nxifix1,…,xn.Remark 8.
According to Lemma7, from ϕi20=0(i=1,…,n) in Assumption 3, it is known that, for each ϕi2(·), there exist smooth functions ϕi2j·(1≤j≤i) satisfying(11)ϕi2x1,…,xi=∑j=1ixjϕi2jx1,…,xi,i=1,…,n.


## 3. Controller Design

In this section, we give the controller design procedure using the backstepping design method.Step 1.
Starting with thex1-subsystem x˙1=d1(t)x2+g1(t,η,x). We consider the variable x2 as the virtual control input. Let z1=x1 and z2=x2-ϑ1 where ϑ1 is the intermediate control input. Considering Lemma 5 and Remark 6, along solutions of (1), the time derivative of the function(12)V1=12z12+V¯0ηsatisfies(13)V˙1≤z1d1tϑ1+g1t,η,x+d1tz1z2-1-ϵρ∘α_ηα0η+δγ¯x1x12.According to Assumption 3 and the completion of squares, we have(14)z1g1t,η,x≤z1p11ϕ11η+p12z1ϕ121x1≤ϕ112η+14z12p112+z12p12ϕ121x1.Define p∗=maxδ,pi1,pi2,pi12,pi22∣i=1,…,n, and we get(15)z1g1t,η,x+δγ¯x1x12≤ϕ112η+14z12p112+z12p12ϕ121x1+δγ¯x1x12≤ϕ112η+p∗z1214+ϕ121x1+γ¯x1≤ϕ112η+z12ϕ^1x1p∗ with a new smooth function ϕ^1(x1)≥1/4+ϕ121(x1)+γ¯(x1)>0. As a result, there holds(16)V˙1≤-1-ϵρ∘α_ηα0η+d1tz1ϑ1+d1tz1z2+ϕ112η+z12ϕ^1x1p∗.Considering the unknown constant p∗ in (16), we use an adaptive signal p^ to estimate p∗. Consequently, we augment V1 with the parameter estimation error p~=p∗-p^, such as(17)V¯1=V1+12Υp~2,where Υ>0 is the design parameter. In view of (16) and p∗=p^+p~, a direct substitution leads to(18)V¯˙1≤-1-ϵρ∘α_ηα0η+d1tz1ϑ1+z12ϕ^1x1p^+d1tz1z2+ϕ112η+1Υp~Υϕ^1x1z12-p^˙.Considering Assumption 4, we take the virtual control(19)ϑ1=-1d_1ν1+ϕ^1x1p^z1,where ν1>0 is a design constant to be determined later. Let(20)τ1=Υϕ^1x1z12,and then we get(21)V¯˙1≤-1-ϵρ∘α_ηα0η-ν1z12+ϕ112η+d1tz1z2+1Υp~τ1-p^˙.Remark 9.
It is noted that, in (19), we assume that p^(t)≥0. In fact, from the updating law of p^˙ given later, this property can be guaranteed by choosing the initial condition p^(0)≥0. Alternatively, using the idea in [20], we also can apply the p^≤1+p^2 or p^≤1+p^2/2 instead of p^.Step 2.
Letz3=x3-ϑ2, where ϑ2 is the virtual control law. We consider the Lyapunov function(22)V2=V¯1+12z22. In view of (21), we have(23)V˙2≤-1-ϵρ∘α_ηα0η-ν1z12+ϕ112η+d2tz2z3+1Υp~τ1-p^˙+z2d2tϑ2+g2t,η,x-∂ϑ1∂x1g1t,η,x+d1tz1-∂ϑ1∂x1d1tx2-∂ϑ1∂p^p^˙. From (4) and (11), the following calculations hold:(24)z2g2t,η,x≤z2p21ϕ21η+z2p22x1ϕ121x¯2+z2p22z2+ϑ1ϕ122x¯2.Like the calculations in (14), by completing the squares, we have(25)-z2∂ϑ1∂x1g1t,η,x≤ϕ112η+z12+p∗z2214∂ϑ1∂x12+14∂ϑ1∂x12ϕ1212x1.Define ϕ^21(x1,p^)=1/4(∂ϑ1/∂x1)2(1+ϕ1212(x1)), and then we have(26)-z2∂ϑ1∂x1g1t,η,x≤ϕ112η+z12+z22ϕ^21x1,p^p∗.In the same manner, using the completion of squares again, it can be verified that(27)z2p21ϕ21η≤ϕ212η+14p212z22,z2p22z1ϕ121x¯2≤z12+14p222z22ϕ1212x¯2,z2p22z2+ϑ1ϕ122x¯2≤z22p22ϕ122x¯2+z2p22ϑ1ϕ122x¯2≤z22p22ϕ122x¯2+z2p22-1d_1ν1+ϕ^1x1p^z1ϕ122x¯2≤z22p22ϕ122x¯2+z12+14p222z221d_12ν1+ϕ^1x1p^2ϕ1222x¯2. As a result,(28)z2g2t,η,x≤ϕ212η+2z12+z22p∗14+14ϕ1212x¯2+ϕ122x¯2+141d_12ν1+ϕ^1x1p^2ϕ1222x¯2.Define ϕ^22(x¯2,p^)=1/4+1/4ϕ1212(x¯2)+ϕ122(x¯2)+1/41/d_12(ν1+ϕ^1(x1)p^)2ϕ1222(x¯2), and then(29)z2g2t,η,x≤ϕ212η+2z12+z22p∗ϕ^22x¯2,p^.Let ϕ^2(x¯2,p^)=ϕ^21(x1,p^)+ϕ^22(x¯2,p^)>0, and then we get(30)z2g2t,η,x-∂ϑ1∂x1g1t,η,x≤∑j=12ϕj12η+3z12+z22ϕ^2x¯2,p^p∗.From Assumption 4, it is deduced that(31)z2d1tz1-∂ϑ1∂x1d1tx2≤z2d1tz1+z2∂ϑ1∂x1d1tz2+ϑ1≤z12+d¯124z22+z22d¯121+∂ϑ1∂x12+z12+14z22∂ϑ1∂x12d¯12d_12ν1+ϕ^1x1p^2.Define ϕ¯21(x¯2,p^)=d¯12/4+1/4d¯12/d_12(∂ϑ1/∂x1)2(ν1+ϕ^1(x1)p^)2+d¯1/2(1+(∂ϑ1/∂x1)2), and there holds(32)z2d1tz1-∂ϑ1∂x1d1tx2≤2z12+z22ϕ¯21x¯2,p^.Take the following notation:(33)τ2=τ1+Υz22ϕ^2x¯2,p^,and furthermore, in view of (30) and (32), we obtain(34)V˙2≤-1-ϵρ∘α_ηα0η-ν1-5z12+2ϕ112η+ϕ212η+d2tz2z3+z2d2tϑ2+z2ϕ^2x¯2,p^p^+z2ϕ¯21x¯2,p^-∂ϑ1∂p^p^˙+1Υp~τ2-p^˙.Considering d2(t) is unknown, the term of -∂ϑ1/∂p^p^˙ can not be canceled. In fact, we express (34) as(35)V˙2≤-1-ϵρ∘α_ηα0η-ν1-5z12+2ϕ112η+ϕ212η+d2tz2z3+z2d2tϑ2+z2ϕ^2x¯2,p^p^+z2ϕ¯21x¯2,p^-∂ϑ1∂p^τ2+1Υp~+z2∂ϑ1∂p^τ2-p^˙. For the term of -z2∂ϑ1/∂p^τ2, according to (33), it can be dealt with as follows:(36)-z2∂ϑ1∂p^τ2=-z2∂ϑ1∂p^Υz12ϕ^1x1+z22ϕ^2x¯2,p^≤z12+14Υ2z22∂ϑ1∂p^2z12ϕ^12x1+z2214Υ1+z22ϕ^2x¯2,p^1+∂ϑ1∂p^2=z12+z2214Υ2∂ϑ1∂p^2z12ϕ^12x1+14Υ1+z22ϕ^2x¯2,p^1+∂ϑ1∂p^2.Define the following smooth function: ϕ¯22(x¯2,p^)=1/4Υ2z12(∂ϑ1/∂p^)2ϕ^12(x1)+1/4Υ(1+z22)ϕ^2(x¯2,p^)(1 + (∂ϑ1/∂p^)2), and we get(37)-z2∂ϑ1∂p^τ2≤z12+z22ϕ¯22x¯2,p^.Denote ϕ¯2(x¯2,p^)=ϕ¯21(x¯2,p^)+ϕ¯22(x¯2,p^)>0, and a direct substitution leads to(38)V˙2≤-1-ϵρ∘α_ηα0η-ν1-6z12+2ϕ112η+ϕ212η+d2tz2z3+z2d2tϑ2+z2ϕ^2x¯2,p^p^+z2ϕ¯2x¯2,p^+1Υp~+z2∂ϑ1∂p^τ2-p^˙.Take the virtual control(39)ϑ2=-1d_2ν2+ϕ^2x¯2,p^p^+ϕ¯2x¯2,p^z2,which is such that(40)V˙2≤-1-ϵρ∘α_ηα0η-ν1-6z12-ν2z22+2ϕ112η+ϕ212η+d2tz2z3+1Υp~+z2∂ϑ1∂p^τ2-p^˙.Step
i
(
3
≤
i
≤
n
). Assume that, in Step i-1, we have designed the virtual control ϑj(j=1,…,i-1) and the tuning function τj(j=1,…,i-1), such that the Lyapunov function(41)Vi-1=∑j=1i-112zj2+12Υp~2satisfies(42)V˙i-1≤-1-ϵρ∘α_ηα0η-ν1-6-∑j=3i-12j+1z12-ν2-7-∑j=4i-12jz22-⋯-νi-2-7zi-22-νi-1zi-12+1Υp~+∑j=2i-1zj∂ϑj-1∂p^τi-1-p^˙+di-1tzi-1zi+∑j=1i-1i-jϕj12η.In what follows, it will be shown that the property (42) also holds in Step i.Letϑi be the virtual control and zi+1=xi+1-ϑi. Consider the Lyapunov function(43)Vi=Vi-1+12zi2.To begin with, the dynamics of zi can be expressed as (44)z˙i=ditϑi+git,η,x-∑j=1i-1∂ϑi-1∂xjgjt,η,x-∑j=1i-1∂ϑi-1∂xjdjtxj+1-∂ϑi-1∂p^p^˙+ditzi+1. For notational convenience, denote ϑ0=0. From xk=zk+ϑk-1=zk-1/d_k-1νk-1+ϕ^k-1xk-1,p^p^+ϕ¯ixk-1,p^zk-1(k=1,…,i-1), like the calculations in (26), it can be verified that there exist smooth functions ϕ^ij·(j=1,…,i)>0, such that, for j=1,…,i-1,(45)-zi∂ϑi-1∂xjgjt,η,x≤ϕj12η+∑k=1j-12zk2+zj2+zi2ϕ^ijx¯j,p^p∗,zigit,η,x≤ϕi12η+∑k=1i-12zk2+zi2ϕ^iix¯i,p^p∗.Define ϕ^i(x¯i,p^)=∑j=1i‍ϕ^ij(x¯j,p^); from (45), it follows that(46)zigit,η,x-∑j=1i-1∂ϑi-1∂xjgjt,η,x≤∑j=1iϕj12η+∑j=1i-11+2i-jzj2+zi2ϕ^ix¯i,p^p∗.Similar to (31) and (32), there exists a smooth function ϕ¯i1(x¯i,p^) such that(47)zidi-1tzi-1-∑j=1i-1∂ϑi-1∂xjdjtxj+1≤z12+∑k=2i-22zk2+3zi-12+zi2ϕ¯i1x¯i,p^.Let(48)τi=τi-1+Υzi2ϕ^ix¯i,p^,and then, from (47) and (48), there holds(49)V˙i≤-1-ϵρ∘α_ηα0η-ν1-6-∑j=3i-12j+1-2iz12-ν2-7-∑j=4i-12j-1-2i-1z22-⋯-νi-2-7-7zi-22-νi-1-6zi-12+∑j=1ii-j+1ϕj12η+ditzizi+1+ziditϑi+ziϕ^ix¯i,p^p^+ziϕ¯i1x¯i,p^-∑j=2i-1zj∂ϑj-1∂p^·Υziϕ^ix¯i,p^-∂ϑi-1∂p^τi+1Υp~+∑j=2izj∂ϑj-1∂p^τi-p^˙.Remark 10.
In (49), we subtract two terms -∑j=2i-1‍zj∂ϑj-1/∂p^·Υziϕ^i(x¯i,p^) and -∂ϑi-1/∂p^τi in the brackets to generate the term (1/Υp~+∑j=2i‍zj∂ϑj-1/∂p^)(τi-p^˙).
However, from (49), it can be seen that, due to the unknown control coefficient di(t), the terms -∑j=2i-1‍zj∂ϑj-1/∂p^·ziϕ^i(x¯i,p^) and -∂ϑi-1/∂p^τi could not be directly canceled by the coming virtual control ϑi. We get around this burden by the following estimates:(50)-zi∑j=2i-1zj∂ϑj-1∂p^·Υziϕ^ix¯i,p^≤zi2121+Υ2∑j=2i-1zj∂ϑj-1∂p^2ϕ^i2x¯i,p^,-zi∂ϑi-1∂p^τi=-zi∂ϑi-1∂p^Υz12ϕ^1+⋯+zi2ϕ^i≤∑j=1i-1zj2+zi214Υ2∂ϑi-1∂p^2∑j=1izj2ϕ^j2x¯j,p^. Let ϕ¯i(x¯i,p^)=1/4Υ2(∂ϑi-1/∂p^)2∑j=1i‍zj2ϕ^j2(x¯j,p^)+1/2(1+Υ2(∑j=2i-1‍zj∂ϑj-1/∂p^)2ϕ^i2(x¯i,p^))+ϕ¯i1(x¯i, p^)>0, and then we get(51)ziziϕ¯i1x¯i,p^-∑j=2i-1zj∂ϑj-1∂p^·ziϕ^ix¯i,p^-∂ϑi-1∂p^τi≤∑j=1i-1zj2+zi2ϕ¯ix¯i,p^.By substituting (51) into (49), it follows that(52)V˙i≤-1-ϵρ∘α_ηα0η-ν1-6-∑j=3i2j+1z12-ν2-7-∑j=4i2jz22-⋯-νi-2-7-8zi-22-νi-1-7zi-12+ziditϑi+ziϕ^ix¯i,p^p^+ziϕ¯ix¯i,p^+1Υp~+∑j=2izj∂ϑj-1∂p^τi-p^˙+∑j=1ii-j+1ϕj12η+ditzizi+1.Take the virtual control(53)ϑi=-1d_iνi+ϕ^ix¯i,p^p^+ϕ¯ix¯i,p^zi,and then the following holds:(54)V˙i≤-1-ϵρ∘α_ηα0η-ν1-6-∑j=3i2j+1z12-ν2-7-∑j=4i2jz22-⋯-νi-1-7zi-12-νizi2+∑j=1ii-j+1ϕj12η+1Υp~+∑j=2izj∂ϑj-1∂p^τi-p^˙+ditzizi+1.
In particular, wheni=n, the actual control u appears, and we choose the controller u and updating law τn for p^(t) as follows:(55)u=-1d_nνn+ϕ^nx,p^p^+ϕ¯nx,p^zn,(56)p^˙=τn=∑j=1nΥzj2ϕ^jx¯j,p^, such that the Lyapunov function(57)Vn=∑j=1n12zj2+12Υp~2satisfies(58)V˙n≤-1-ϵρ∘α_ηα0η-ν1-6-∑j=3n2j+1z12-ν2-7-∑j=4n2jz22-⋯-νn-1-7zn-12-νnzn2+∑j=1nn-j+1ϕj12η.This completes the controller design procedure.


## 4. Main Results

After the above controller design procedure, we are now ready to state the main results.Theorem 11.
Suppose the investigated system (1) satisfies Assumptions 1, 3, and 4 together with the local conditions(59)ϕi12s=Oα0s,i=1,…,n.Then all the signals of the closed-loop system (1) with the controller (55) and updating law (56) are bounded on [0,+∞). Specifically, the following convergent property holds:(60)limt→∞⁡ηt+xt+ut=0.Proof.
From the local conditions (59), one can choose the smooth function ρ(·) such that(61)∑j=1nn-j+1ϕj12η≤1-ϵ2ρ∘α_ηα0η.One can choose positive constants νj(j=1,…,n) satisfying(62)ν1-6-∑j=3n2j+1≥1,ν2-7-∑j=4n2j≥1,⋮νn-1-7≥1,νn≥1.Then, from (58), (61), and (62), it follows that(63)V˙n≤-1-ϵ2ρ∘α_ηα0η-∑i=1nzi2.Now, assume that the maximal interval of existence of the solution of the closed-loop system starting from any given initial conditions is [0,tf) for some tf>0. In view of 0<ϵ<1, from (63), it can be concluded that Vn and hence the variables (η(t),z1(t),…,zn(t),p~(t)) are bounded on [0,tf). In terms of p~(t)=p∗-p^(t), we obtain the boundedness of p^(t). Considering (53), it can be derived that ϑi(i=1,…,n) are bounded. In view of zi=xi-ϑi-1(ϑ0=0), we further obtain that the states xi(i=1,…,n) are bounded on [0,tf).
So far all the closed-loop system signals are bounded on[0,tf). This guarantees that the finite time escape will not happen. Therefore, it is natural that tf can be maximized to +∞ by means of Theorem 3.3 in [21]. Next we will prove the convergence property of (60).
Again, according to (63), considering α_ and α0 are K∞-functions, a direct application of LaSalle’s invariance principle in [21] guarantees the convergence property of (η(t),z1(t),…,zn(t)); that is,(64)limt→∞⁡ηt=0,limt→∞⁡zit=0i=1,…,n.As a consequence, from (53) and (55), the following holds:(65)limt→∞⁡ϑit=0i=1,…,n.Particularly,(66)limt→∞⁡ut=0.In terms of (64), (65), and xi=zi+ϑi-1(i=1,…,n), we can obtain(67)limt→∞⁡xit=0i=1,…,n.This completes the proof.It is noted that, under some stronger conditions, the designed control law can be a linear controller. In fact, we have the following statement.Theorem 12.
Suppose that the conditions for Theorem11 are satisfied with δ0 known a priori and the following additional assumptions hold: (i)
The uncertain functionsϕi1·(i=1,…,n) satisfy(68)limsups→+∞⁡ϕi12sα0s<+∞.(ii)
There exist known constantspij>0(i=1,…,n;j=1,2), such that(69)git,η,x≤pi1ϕi1η+pi2x1+⋯+xi,i=1,…,n.(iii)
γ0(|s|)=ks2 where k is a positive constant.
Then, the proposed design method can result in a linear control law(70)u=-1d_nνnzn=-1d_nνnxn+νn-1xn-1+⋯+ν1x1,where νi(i=1,…,n) are some sufficiently large positive constants.Proof.
Under the above hypotheses (i)–(iii), it is known that the constantp∗ is known, and hence the estimation p^ for p∗ is no longer needed. Moreover, since conditions (59) and (68) are satisfied, the function ρ(·) in (7) can be chosen as a constant ρ>0. For i=1,…,n, we consider the following function:(71)V¯0iη=∫0V0ηρids,where ρi(i=1,…,n) are design constants. In view of (3), we can get(72)V¯˙0iη≤-ρiα0η+ρiδ0γ0y=-ϕi12η+ρ~iδ0γ0y+ϕi12η-ρiα0η-ρ~iδ0γ0y+ρiδ0γ0ywith some positive constants ρ~i(i=1,…,n). We will prove that if the constants ρi and ρ~i are chosen suitably, the following inequality holds:(73)ϕi12η-ρiα0η-ρ~iδ0γ0y+ρiδ0γ0y≤0,i=1,…,n.In fact, because of ϕi12(s)=O(α0(s)), there exist positive constants si>0, ci>0 satisfying ϕi12(s)≤ciα0(s), for s∈[0,si]. Take ρi≥ci+1, ρ~i=ρi+1, and then ϕi12(s)-ρiα0(s)≤0 and (ρi-ρ~i)δ0γ0(|y|)≤0. In view of limsups→∞ϕi12(s)/α0(s)<∞, there exist positive constants si′>0, ci′>0 satisfying ϕi12(s)≤ci′α0(s) if s∈[si′,∞]. Similarly, if we take ρi≥ci′+1 and ρ~i=ρi+1, then ϕi12(s)-ρiα0(s)≤0, (ρi-ρ~i)δ0γ0(|y|)≤0. In the finite closed interval [si,si′], let α0(s_) be the minimum value of α0(s) and let ϕi12(s¯) be the maximum value of ϕi12(s), respectively, and if we take ρi≥ϕi12(s¯)/α0(s_), ρ~i=ρi+1, then ϕi12(s)-ρiα0(s)≤0, (ρi-ρ~i)δ0γ0(y)≤0. According to the previous analysis, we choose ρi=ci+1+ci′+1+ϕi12(s¯)/α0(s_) and ρ~i=ρi+1, and then (72) holds. Therefore we get(74)V¯˙0iη≤-ϕi12η+ρ~iδ0γ0y,i=1,…,n.In view of γ0(s)=ks2, a direct substitution in (74) results in(75)V¯˙0iη≤-ϕi12η+ρ~iδ0ky2,i=1,…,n.To deal with the unmeasured dynamics η in this case, we can choose the candidate Lyapunov function as follows:(76)Vn=∑i=1n12zi2+∑i=1nn-i+2V¯0iη.Consequently, a modified version of the design procedure in Section 3 leads to the linear control law (77)u=-1d_nνnzn=-1d_nνnxn+νn-1xn-1+⋯+ν1x1 with some sufficiently large positive constants νi(i=1,…,n).


## 5. Simulation Example

In this section, we provide a simulation example to illustrate the proposed method in the paper. Consider the following nonlinear systems:(78)η˙=-η+δty2,x˙1=d1tx2+θ1x1+θ2x1x21+x22,x˙2=d2tu+θ3sin⁡tx1x2η,y=x1with q(t,η,y)=-η+δ(t)y2, δ(t)=1+e-t, g1(t,η,x)=θ1x1+θ2x1x2/1+x22, and g2(t,η,x)=θ3sin⁡(tx1x2)η. The inverse system η˙=-η+δ(t)y2 is ISS, and V0(η)=1/2η2 is a candidate ISS-Lyapunov function with the supply pair α0(η)=1/2η2, δtγ0x1=1+e-t1/4x14.Next, we use the proposed algorithm in Section3 to design the partial-state feedback controller.Step 1.
We consider the functionV1=1/2z12+1/2η2, whose time derivative satisfies(79)V˙1=z1d1tx2+θ1x1+θ2x1x21+x22+η-η+δty2.Like the calculations in (14), we have(80)z1θ1x1≤θ1z12,z1θ2x1x21+x22≤12θ2z12,ηδty2≤14η2+δ2tz14≤14η2+δ0z14,where δ0>0 satisfying δ2(t)≤δ0.
Definep∗=max{δ0,|θi|,θi2∣i=1,2,3}, and we get(81)z1θ1x1+z1θ2x1x21+x22+ηδty2≤14η2+p∗z121+12+z12≤14η2+z12ϕ^1x1p∗with a new smooth function ϕ^1(x1)≥1+1/2+z12>0.
Similar to (17), we augment V1 as follows:(82)V¯1=V1+12Υp~2,where Υ>0 is the design parameter. In view of (79) and p∗=p^+p~, a direct substitution leads to(83)V-˙1≤-1-14η2+d1tz1ϑ1+d1tz1z2+z12ϕ^1x1p^+1Υp~Υϕ^1x1z12-p^˙.We take the virtual control and the tuning function(84)ϑ1=-1d_1ν1+ϕ^1x1p^z1,ν1>0,τ1=Υϕ^1x1z12,and then we get(85)V¯˙1≤-1-14η2-ν1z12+d1tz1z2+1Υp~τ1-p^˙.Step 2.
To find the actual control lawu, we consider the Lyapunov function(86)V2=V¯1+12z22.In view of (78) and (85), we have(87)V˙2≤-1-14η2-ν1z12+d1tz1z2+1Υp~τ1-p^˙+z2d2tu+θ3sin⁡tx1x2η-∂ϑ1∂x1d1tx2+θ1x1+θ2x1x21+x22-∂ϑ1∂p^p^˙.As in (80), we have(88)d1tz1z2≤z12+d¯124z22,z2θ3sin⁡tx1x2η≤z2θ3sin⁡tx1x2η≤14η2+θ32z22,-z2∂ϑ1∂x1θ1x1≤z12+14θ12z22∂ϑ1∂x12,-z2∂ϑ1∂x1θ2x1x21+x22≤z12+14θ22z22∂ϑ1∂x1214,-z2∂ϑ1∂x1d1tx2≤z22d¯121+∂ϑ1∂x12+z12+14z22∂ϑ1∂x12d¯12d_12ν1+ϕ^1x1p^2. Consequently, in view of the definition of p∗, the following holds:(89)z2θ3sin⁡tx1x2η-z2∂ϑ1∂x1θ1x1-z2∂ϑ1∂x1θ2x1x21+x22≤14η2+2z12+z22ϕ^2x1,p^p∗, with ϕ^2(x1,p^)=1+1/4(∂ϑ1/∂x1)2+1/4∂ϑ1/∂x121/4. Define ϕ¯21(x1,p^)=d¯12/4+d¯1/2(1+(∂ϑ1/∂x1)2)+1/4d¯12/d_12(∂ϑ1/∂x1)2(ν1+ϕ^1(x1)p^)2, and one gets(90)z2d1tz1-∂ϑ1∂x1d1tx2≤2z12+z22ϕ¯21x1,p^.As a result, with τ2=τ1+Υz22ϕ^2(x1,p^), we have(91)V˙2≤-1-24η2-ν1-4z12+z22ϕ^2x1,p^p^+z22ϕ¯21x1,p^+1Υp~+z2∂ϑ1∂p^τ2-p^˙+z2d2tu-z2∂ϑ1∂p^τ2.For the term of -z2∂ϑ1/∂p^τ2, according to (36), it can be verified that(92)-z2∂ϑ1∂p^τ2≤z12+z2214Υ2∂ϑ1∂p^2z12ϕ^12x1+14Υ1+z22ϕ^2x1,p^1+∂ϑ1∂p^2.Define ϕ¯22(x¯2,p^)=1/4Υ2z12(∂ϑ1/∂p^)2ϕ^12(x1)+1/4Υ(1+z22)ϕ^2(x1,p^)(1+(∂ϑ1/∂p^)2), and we get(93)-z2∂ϑ1∂p^τ2≤z12+z22ϕ¯22x¯2,p^.Denote ϕ¯2(x¯2,p^)=ϕ¯21(x1,p^)+ϕ¯22(x¯2,p^)>0, and a direct substitution leads to(94)V˙2≤-1-24η2-ν1-5z12+z2d2tu+z2ϕ^2x¯2,p^p^+z2ϕ¯2x¯2,p^+1Υp~+z2∂ϑ1∂p^τ2-p^˙.
As in Step1, we design the following partial-state feedback controller:(95)u=-1d_2ν2+ϕ^2x¯2,p^p^+ϕ¯2x¯2,p^z2with the updating law for unknown parameter p∗(96)p^˙=Υϕ^1x1z12+Υϕ^2x1,x2,p^z22,which is such that(97)V˙2≤-12η2-ν1-5z12-ν2z22.The Lyapunov functionV2 can be made V˙2≤0 by choosing ν1>5, ν2>0, and the stability analysis can be done in the similar way to Theorem 11. The simulation plots shown in Figures 1 and 2 are performed by MATLAB with the following parameters: d1(t)=2-sin(t), d2(t)=2+sin⁡(t), d_1=1, d¯1=3, d_2=1, d¯2=3, ν1=6, ν2=1, Υ=1, the derived functions: ϕ11(η)=0, ϕ21(η)=|η|, ϕ12(x1)=|x1|+1/2|x1|, ϕ22(x1,x2)=0, ϕ^1(x1)=1+1/2+x12, ϕ^2(x1,x2,p^)=2+3/2(∂ϑ1/∂x1)2(ν1+p^)2, and the initial conditions: η(0)=0.5, x1(0)=1, z2(0)=0.1, p^(0)=1.Figure 1
The response of the closed-loop system.Figure 2
The control input of the closed-loop system.According to our results reported in Theorem11, the states (η,x1,x2) must asymptotically converge to the origin and the parameter estimate p^ is bounded on [0,∞). This fact can be verified from Figure 1, which plots the trajectories of these dynamic signals (η(t),x1(t),x2(t),p^(t)). It can be seen that, at about t=4.17s, t=0.68s, and t=0.59s, the states approach the origin, and at t=0.36s, the parameter estimate p^(t) is bounded near p∗=1.1. In addition, according to Theorem 11, the control input u is convergent to the origin. Figure 2 demonstrates this result, and it can be shown that, at about t=0.51s, the input signal u approaches the origin. As can be seen from Figures 1 and 2, our control scheme provides a fairly good asymptotic stabilization performance.


## 6. Conclusion

The state feedback stabilization problem is investigated for a class of nonlinear systems with dynamic uncertainties and uncertain control coefficients in this paper. The dynamic uncertainty is characterized by the uncertain ISS supply rates. A global asymptotic stabilization control scheme is proposed using the backstepping design scheme. The tuning function technique is applied in this procedure, which avoids the disadvantage of overparameterization. It is shown that, under some more restrictive conditions, a linear state feedback controller can be designed by the presented algorithm. The simulation example demonstrates the effectiveness of the proposed method.



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*Source: 206162-2015-10-11.xml*