一类不确定非线性系统的输出反馈动态面控制

Output Feedback Adaptive Dynamic Surface Control for a Class of Uncertain

Nonlinear Systems

Yang Jingyu, Wang Qiangde*

Department of Electrical Engineering and Automation, Qufu Normal University, Rizhao, China 276826

E-mail: *Corresponding author: qdwang70@http://doc.guandang.net

Abstract: In this paper, an adaptive output feedback controller was developed for a class of nonlinear systems with unkonwn parameters, uncertain disturbances and unmeasured states. Firstly estimated states was obtained by introducing an observer. Then a controller was developed based on the dynamic surface control approach. Explosion of terms can be avoided because this method no longer needs the repeated differentiations of the virtue control. Only two parameters instead of all the neural networks weights need to be updated.All the signals of closed-loop systems can be guaranteed semiglobally uniformly ultimately bounded through the analysis of this method.The tracking error can be arbitrarily small by choosing the proper design parameters. Simulation studies show the effectiveness of the proposed approach.

Key words: Dynamic Surface Control, Output Feedback, Adaptive Control, Neural Networks

1 INTRODUCTION dynamic surface control was developed in [10-12]. It is more convenient than backstepping control technique. In [13-14], a scheme was proposed for nonlinear time-delay systems based on the dynamic surface control and the Lyapunov-Krasovskii functionals. A control method was prestented for a class of nonlinear systems with unknown control gains by using neural network control and dynamic surface control in [8].

In this paper, we developed an adaptive dynamic surface neural network control scheme for uncertain nonlinear systems with unknown disturbances based on [8]. It only studies the aspect that the virtual control gains are one in [8]. And this paper extends to the systems with output nonlinear function gains and unknown disturbances. In this paper, a class of nonlinear systems with unknown functions, control gains and disturbances was considered. Unknown constants and neural networks weights aren't updated separately. Only two unknown parameters are updated by the coordinate transformation. It decreases the number of updated parameters and the complexity of the design with respect to the use of neural networks in [8].

Backstepping control technique has been widely used in the adaptive control of nonlinear systems in recent years.We can combine backstepping with neural network control method or fuzzy control method[1] for mismatched uncertain nonlinear system. This approach can change the model uncertainty problem into the problem of parameter estimation by choosing a suitable basis function or membership function. There are some good properties of this method in [2-6].Accordingly, backstepping control has been widely used in robotics, aerospace, civil engineering and other aspects.

However, the problem of explosion of terms was inherited in the backstepping control design because of repeated differentiations of the virtual controls, especially for high–order systems. To solve this disadvantage, D.Swaroop presented a dynamic surface control method in [7]. It avoids the differentiations of the virtual controls by introducing a first-order filtering. Subsequently, the dynamic surface control method has lots of research in the aspect of designing controller for nonlinear systems[7-9]. In [4], the output tracking problem of a class of uncertain nonlinear systems was solved with combination of the adaptive dynamic surface control and RBF neural networks. For stochastic nonlinear systems, the adaptive

2 PROBLEM FORMULATION

Consider the following unknown nonlinear system:

c2014IEEE978-1-4799-3708-0/14/$31.00

2329

1=h1(y)x2+f1(y)+p1(y)ω1,­x°x

°i=hi(y)xi+1+fi(y)+Φi(i)+pi(y)ωi,°

2≤i≤n 1 1 ®°x =bg(y)u+fn(y)+Φn(n)+pn(y)ωn°n°¯y=x1

",n 1, where i=[x1,x2,",xi]T∈Ri, i=1,

n=[x1,x2,",xn]T∈Rn is the state vector,u is the

l

ψiT(y)=ª¬ψi1(y),ψi2(y),",ψil(y)º¼∈R

i

T

i

is vector

-valued function, and ψij(y)is basis function which commonly selected as Gaussian function, i.e.

2ψij(y)=exp( (y μij)2ij),whereμijis the center of

the basis function and ηijis the width of the basis function. ψi (y) is the ideal constant weight. δi is the approximation error which satisfies i≤δm andδmis a known constant.

3.2 State observer design

control input.fi(y)is unknown nonlinear function.

hi(y)≠0, Φi(i) andg(y)≠0 are known smooth

nonlinear functions, pi(y) is known smooth bounded functions in C1, ωi∈Ris external interference. It is assumed that only output y can be measured, the aim of this paper is to develop a method for the above system such that all the signals of the closed loop system

are bounded and the output signal y(t)=x1 can track the desired trajectory yr with any prescribed small error.

To complete the controller design, we need to make some assumptions:

Assumption 1 Φi(i) satisfies

ij=2

We rewrite the plant (1) in the canonical form

=Ax+ψW+Bg(y)u+Φ(y)x+δ+P(y)ω­°x®T°¯y=e1x

(3)

T

where

W=[W1,",Wn]

T

,

δ=[δ1,",δn]

,

P(y)=diag{p1(y),",pn(y)},

ª0ºTB=«(n 1)×1»,ω=[ω1,",ωn],

Φi(i)=Φi(y,x2,",xi)=¦φ(i,j)(y)xj (2) ¬ b ¼

ª00

«0φ

(2,2)«

Φ(y)=«0φ(3,2)

«

#«#

«0φ(n,2)¬

00

"""%"

0º0»»0». »#»φ(n,n)»¼

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