# Adaptive Regulation of a Class of Uncertain Nonlinear Systems with Nonstrict Feedback Form

Authors: Jian-xiong Li; Chong-yi Gao; Yi-ming Fang; Cui Guo; Wen-bo Zhang
Journal: Mathematical Problems in Engineering
 (2016)

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## Abstract

This paper focuses on the semiglobal stabilization for a class of nonlinear systems with nonstrict feedback form. Based on a generalized scaling technique, an adaptive control algorithm with dynamic high gain is developed for a class of nonstrict feedback nonlinear systems. It can be proved that, under some appropriate design parameters, all signals of the resulting closed-loop system are bounded semiglobally, and the system state will be convergent to origin exponentially. Finally, a numerical simulation is provided to confirm the effectiveness of the proposed method.


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## Body

## 1. Introduction

In this paper, we consider a class of nonlinear systems as follows:(1)x˙i=xi+1+ϕit,x,i=1,…,n-1,x˙n=gt,xu+ϕnt,x,where x=x1,…,xnT is the system state and u is the control input. g(t,x):R×Rn→R, ϕi(t,x):R×Rn→R, i=1,…,n, are uncertain continuous functions with g(t,·)≠0 and ϕi(t,0)=0, i=1,…,n.System (1) can be stabilized by numerous adaptive controllers by means of backstepping technique [1–3] if (1) is of a strict feedback structure; that is, in the ith subsystem, the uncertain nonlinear function ϕi(t,x) is the function with respect to (w.r.t.) the first state variables x1,…,xi and time t. Furthermore, if the function ϕi(t,x) depends on whole state variables but its bounding function is assumed to be a function of x1,…,xi and t, thus system is called semistrict feedback systems, and the backstepping technique is also an alternative method to controller design [4–7]. However, when the subsystem functions contain whole state variables and the above assumption of semistrictness is removed, that is, system (1) is of a nonstrict feedback structure, the aforementioned control methodology would be invalidated.The control problem of nonstrict feedback nonlinear systems arises and attracts some researchers’ attention. In [8, 9], by utilizing the monotonically increasing property of the bounding functions, the authors developed the variables separation technique and by means of the approximation property of fuzzy logic systems and neural networks (NN) proposed adaptive fuzzy and NN control design methods, respectively. Subsequently, the design method was extended and applied to MIMO nonlinear systems [10], stochastic nonlinear systems [11, 12], and uncertain switched nonlinear systems [13]. Furthermore, [14] eliminated the assumption in [8–13] that the unknown nonlinear functions must satisfy the monotonically increasing property of the bounding functions.Different from [8–14], a semiglobal output feedback controller with high gain for a class of nontriangular nonlinear systems was proposed in [15], in which the authors introduced dilation δε(x)=x1,εx2,…,εn-1xn with constant ε≥1 and assumed that the nonlinear functions ϕi(t,x)’s should satisfy the following inequalities:(2)ϕit,δεx≤εi-1+σcix∑j=1nxj,i=1,…,n,where σ∈[0,1) and continuous function ci(x)≥0 were known. In [15], the up-bounds of some terms containing ci(x)’s, such as d2 and d3, should be known, and these up-bounds might be very large, so that a larger gain L would be required to guarantee the stability of the system. Consequently, the amplitude of the control input u would be very large. Besides, the large high gain would amplify the noise [16]. On the other hand, in fact, the high gain sometimes need not be larger than the up-bounds but be just larger than the terms. To determine the gain is not easy; a trial-and-error method is often adopted [17]. Different from the constant high gain used in controller design in [15], a dynamic high gain is employed in this paper. The dynamic high gain will increase until it is larger than the terms and the closed system is stable. It can be found that the dynamic high gain or the so-called adaptive high gain was employed in [7] for dealing with the unknown growth rate of nonlinear functions and was used to avoid amplifying the noise in Extended Kalman Filters in [16].In this paper, we focus on the stabilization scheme via state feedback for uncertain nonstrict feedback nonlinear system (1). Inspired by the generalized scaling technique of [18], a dynamic high-gain-based adaptive controller is designed for system (1), which can guarantee that all signals of the resulting closed-loop system are bounded semiglobally and the system state is convergent to the origin exponentially.The rest of this paper is organized as follows. Two assumptions imposed on system (1) and the problem statements are presented in Section 1. The main results and theoretical analysis are given in Section 2. Simulation examples are given in Section 3, followed by Section 4 that concludes the work.


## 2. Assumptions and Statements

Rewrite system (1) as the following form:(3)x˙=Ax+Φt,x+Bgt,xu,where Φ(t,x)=ϕ1(t,x),…,ϕn(t,x)T∈Rn, (4)A=010⋯0001⋯0⋮⋮⋮⋱⋮000⋯1000⋯0∈Rn×n,B=00⋮01∈R.And the following two assumptions are imposed on system (3).Assumption 1.
The functiong(t,x) is bounded and its sign is invariant and known. And there exist two known positive continuous functions gm(t,x) and gM(t,x), for all t≥0 and x∈R, such that(5)0<gmt,x≤gt,x≤gMt,x.
Without loss of generality, we suppose thatg(t,x)>0 for all t≥0 and x∈Rn.Assumption 2.
There exist constantsγi≥0 and qi>0 and bounded positive continuous function ci(x), i=1,…,n, such that, for t≥0, x∈Rn, and r(t)∈(1,∞), the following inequality holds:(6)ϕit,Trx≤rqi+γitcix∑j=1nXj,where Tr=diag(rq1,rq2,…,rqn)∈Rn×n, qi>γi, i=1,…,n.Assumption1 is given to ensure the controllability of system (3), and this assumption is generally satisfied.Assumption2 is about the nonlinear functions ϕi(t,x)’s. Compared with the corresponding assumption in [15] given by (2), the high gain r in (6) is a variable rather than a constant ε in (2), and, in the ith inequality, the power of high gain in the right-hand side of the inequality is qi in (6) instead of i-1 in (2). It can be seen that the assumption of this paper is more general, and (2) is a special case of (6) with qi=i-1, γi=σ, and r(t)=ε=L, where L is an appropriate positive constant. In addition, it should be pointed out that some terms containing functions ci(x)’s in [15] are assumed to be bounded with up-bounds d2 and d3, so a large enough L is required to guarantee the stability of the system. Too large gain will result in too large amplitude of the control input and will amplify the unavoidable noise [16]. Therefore, an appropriate high gain is required, and it is often achieved by using trial-and-error method [17]. In this paper, a dynamic high gain instead of constant gain will be employed in controller design.The main objective of this paper is to design a state feedback controlleru=fK(x) to stabilize the nonstrict feedback nonlinear system (3) under Assumptions 1 and 2.


## 3. Dynamic High-Gain-Based Control and Stability Analysis

In this section, the main result of this paper is given, an adaptive controller with dynamic high gain is presented for system (3), and the rigorous theoretical analysis is then provided.To achieve this, a scaling transformationz=Tr-1x is first introduced, and system (3) can be changed into the following:(7)z˙=Arz+Φ~t,x+Bvt,x,u-r˙rDqz,where Ar=AΛr, Φ~(t,x)=Tr-1Φ(t,x)=[ϕ~1(t,x),…,ϕ~n(t,x)]T, v(t,x,u)=γ-qng(t,x)u, Λr=diag(rq0,rq2-q1,…,rqn-qn-1), q0>0, and Dq=diag(q1,q2,…,qn).Notingx=Trz, inequality (6) in Assumption 2 imposed on system (1) can be written as follows:(8)ϕ~it,x=r-qiϕit,Trz≤rγiciz∑j=1nzj≤rγiciznz,i=1,…,n.The controllerv is designed as(9)vt,x,u=KΛrz,where K=[k1,…,kn].Proposition 3.
Under Assumption2, if there exist positive definite symmetric (PDS) matrices P, Q1, and Q2, that is, P=PT>0, Q1=Q1T>0, and Q2=Q2T≥0, such that (10)A+BKTP+PA+BK=-Q1,(11)DqP+PDq=Q2hold, and if we choose an appropriate gain r, then controller (9) can stabilize the state of system (7) exponentially.Proof.
Choose the positive definite functionV(x)=zTPz, and its time derivative along with (7) and (9) is derived as(12)V˙=zTΛrA+BKTP+PA+BKΛrz+2zTPΦ~t,x-r˙rzTDqP+PDqz.
By (8), the third term in the right-hand side of (12) satisfies(13)2zTPΦ~t,x≤rγmaxnPczz2,where γmax=max⁡γi, i=1,2,…,n, c(z)=[c1(z1),c2(z-2),…,cn(z)]T∈Rn.
Substituting (10), (11), and (13) into (12), it follows that(14)V˙≤-zTΛrQ1z+rγmaxnPczz2-r˙rzTQ2z≤-ηrρ-nλmaxPczλminQ1z2,where qmin=minq0,q2-q1,…,qn-qn-1, η=λmin(Q1)rγmax, ρ=qmin-γmax, and λmin(∗) and λmax(∗) denote the minimum eigenvalue and the maximum eigenvalue of the matrix ∗, respectively.
Letθ(z)=[nλmax(P)c(z)/λmin(Q1)]1/ρ and θε(z)=θ(z)+ε, where ε>0. We choose qi+1>qi and qmin>γmax, i=1,2,…,n-1, which means ρ>0. When we choose r≥θε(z), we have (15)V˙≤-ηερz2,where ερ≥θερ-θρ>0.
Therefore, the statez of system (7) will converge to origin exponentially. It should be pointed out that, in Assumption 2, ci(x)’s are supposed to be bounded, but we do not know the up-bounds of ci(x)’s, so the closed-loop system (7) and (9) is semiglobally stable. The proof is thus completed.The real controlleru can be directly obtained from (9) as the following form:(16)u=rqngt,xKΛrz.However, controller (16) is invalid because g(t,x) is unknown. Under Assumption 1, the bounds of g(t,x) could be used instead of g(t,x); the controller is designed as (17)u=rqngmt,xKΛrz.And it should be pointed out that, in Proposition3, the high gain r should be equal to or larger than θ-ε(z) to guarantee the convergence of z. If a constant gain r is chosen, for example, in [15], the bound of c(z) should be known, and a very large gain is required; meanwhile, with higher r, more noise is amplified [16]. And, from (17), it is easy to see that large r will bring about large control input u and may lead to the control input saturation due to physical limit in practice. To reduce this defect of large high gain, a dynamic high gain is employed in this paper, and the update law of the high gain r is given as follows:(18)r˙=αrθ-εz-rz12,r0≥1.From (18), it is obvious that γ is increasing until r≥θ-ε(z) or z1≡0.Theorem 4.
Under Assumptions1 and 2, if there exist PSD matrices P, Q1, and Q2, such that (10) and (11) hold, controller (17) with gain update law (18) can guarantee that all the signals of the closed-loop system are uniformly ultimately bounded and the states are convergent to origin exponentially.Proof.
Choose the positive definite functionV(x)=zTPz, and its time derivative along with (7) and (17) is derived as(19)V˙=zTΛrgA+BKTP+PA+BKΛrgz+2zTPTr-1Φt,x-r˙rzTDqP+PDqz,where Λrg=diag(1,…,1,gm-1g)Λr.
Substituting (10), (11), and (13) into (19) and noting gmg(t,x)≥1, we get(20)V˙≤-zTΛrgQ1z+rrmaxnPczz2-r˙rzTQ2z≤-ηrρ-nλmaxPczλminQ1z2.
Following the proof of Proposition3, when r≥θ-ε(z), we have V˙≤-ηερz2, which implies that the state z of system (7) will converge to origin exponentially.
The next step is to illustrate the boundedness of the high gainr.
Firstly, we supposez1≠0 at time t∈t0,tf, where t0 and tf are initial time and final time, respectively, and tf∈(t0,∞). In this case, we can get r˙≥0 when r≤θ-ε(z) and r˙<0 when r>θ-ε(z), which implies that r is bounded; that is, there exists an arbitrarily small positive constant ε′; the inequality r≤θ-ε(z)+ε′ holds.
In the case ofz1≡0, it follows that r˙=0 regardless of whether r≤θ-ε(z) or not, which also means r is bounded.
Then, we can obtain that the high gainr is bounded.
To sum up, we can conclude that all the signals of the closed-loop system of (6), (17), and (18) are bounded, and the system state x converges to origin exponentially. That completes the proof.


## 4. Numerical Simulation

To illustrate the effectiveness of the proposed method, a numerical simulation is presented in this section.We consider the following nonlinear system:(21)x˙1=x2+x12/3x21/3,x˙2=x3+x1x31/3ln⁡1+x22,x˙3=2+sin⁡x21+x12u+x1.It can be verified that system (21) satisfies Assumption 1 with gm=(1+x12) and gM=3(1+x12) and satisfies Assumption 2 with q0=1, q1=0, q2=1, q3=3, γ1=1/3, γ2=0.5, γ3=0, c1(z)=c3(z)=1, and c2(z)=z11/3(2+z22).The controller parameter of (17) and parameters of (18) are chosen as K=-4,-5,-3, α=0.5, and ε=0.01. Choosing Q1=diag(0.5,0.5,0.5) and solving (10), we get(22)P=0.2989-0.15690.0414-0.15690.3071-0.28740.0414-0.28741.8258.And, from (11), we have(23)Q2=1.1955-1.09820.3724-1.09823.0709-3.44910.3724-3.449125.5613.Furthermore, we can get λmax(P)=1.8814, λmin(Q1)=0.5, qmin=1, γmax=0.5, and ρ=0.5.The simulations are run under the initial conditionsx0=2,0,0T and three different values of r(0); that is, r(0)=1, 3, and 6.The simulation results are shown in Figures1–3. It is clearly shown that, under the action of the proposed control (17) and adaptive law (18), all the signals are bounded and the system states are convergent to origin.Figure 1
State response.
(a) (b) (c)Figure 2
Control inputu.
(a) (b)Figure 3
High gainr.Furthermore, it is easy to see that when largerr(0) is chosen, x(t) is driven faster to steady state (see Figure 1); correspondingly, u(t) and r(t) achieve steady state more quickly (see Figures 2 and 3).And, from the definition ofθε(z) and θ(z), it can be calculated that θε(z(0))=5.7213; according to (18), if r(0)<5.7213, r˙t=0>0; otherwise, r˙t=0≤0; therefore, in Figure 3, we can see that, at the beginning, r(t) increases when r(0)=1 and r(0)=3, while r(t) decreases when r(0)=6.In practice, the control input is often limited due to some physical limits. Too large high gain could lead to the control input saturation, which would degrade the performance of the control system and even cause system instability. To illustrate the above problem caused by too large high gain, some more simulations are given as follows.From Figure3, the high gain r can be approximately regarded as a constant, r=6. And if a constant high gain is employed in the controller design, to ensure the stability of the system, r=6 or a larger one would be chosen using trial-and-error strategy. So, the simulations are run with the same design parameters and the same initial condition of state and r(0)=6. And two cases of the control input saturation are considered; namely, u≤65 and u≤63. The simulation results are shown in Figures 4 and 5.Figure 4
State response with input saturation andr(0)=6.
(a)
u ≤ 65 (b)
u ≤ 63Figure 5
Control input with input saturation andr(0)=6.In comparison between the state responses withr(0)=6 in Figure 1 and the state responses in Figure 4, it can be found that the performance of the control system degrades when the control input is restricted by u≤65 but is still stable, while the system would be unstable when the control input is limited by u≤63.Under the control input limitu≤63, the controller with constant high gain r=6 or a larger one will be invalid. But a small gain cannot ensure the stability of the system. To solve the above problem, this paper gives an alternative method. We can find that if we choose r(0)=1, the input saturation does not occur (see Figure 2(a)). Therefore, a smaller r(0) will bring a smaller control input and will avoid the input saturation.


## 5. Conclusions

In this paper, an adaptive controller with dynamic high gain has been designed for a class of nonstrict feedback nonlinear systems. The main difference from the existing related works is that a dynamic high gain rather than a constant one is employed in the controller design. The advantage of the dynamic high gain is that it would not result in a large control input, and it is more likely to avoid the input saturation occurring which usually exists due to physical limits in practice. By Lyapunov stability theory method, a rigorous theoretical proof is provided to show that the proposed controller can guarantee the semiglobal stability of the system state. Finally, a numerical simulation is provided to illustrate the effectiveness of the proposed method.



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