Robust Adaptive Control Using a Universal Approximator for SISO Nonlinear Systems

Robust Adaptive Control Using a Universal Approximator for SISO Nonlinear Systems

Hyeongcheol Lee and Masayoshi Tomizuka

Abstract—This paper is concerned with the robust adaptive control of a class of nonlinear systems in the presence of para-metric uncertainties and dominant uncertain nonlinearities.The proposed controller utilizes the robust adaptive control(RAC) to guarantee uniform boundedness and convergence of tracking errors.In addition,an adaptive fuzzy logic system is used as a universal approximator to reduce the model uncertainties coming from uncertain nonlinearities and to improve tracking performance.The approach does not require the matching condition imposed on control systems by using the backstepping design procedure,and provides boundedness of tracking errors under poor parameter adaptation.The method can be applied to a class of single-input single-output(SISO)nonlinear systems, transformable to a parametric-strict-feedback form.

Index Terms—Backstepping design,robust adaptive control, smooth projection,stable adaptive fuzzy control,universal ap-proximator.

I.I NTRODUCTION

T HE focus of adaptive control research in the late1980's to early1990's was on performance properties and on ex-tending the results of the linear systems to a certain class of non-linear systems with unknown parameters.The adaptive control of nonlinear systems has undergone rapid development in this period[1]–[4]leading to global stability and tracking results for reasonably large classes of nonlinear systems.

By using the backstepping design procedure,Kanel-lakopoulos,et al.[1],[2],[5]have presented a systematic design of globally stable and asymptotically tracking adaptive controllers for a class of nonlinear systems transformable to a parametric-strict-feedback system.The robustness problem due to the integral type adaptation law and uncertain nonlinearities has been overcome by using a nonlinear damping[2],[6]or by combining backstepping adaptive control with conventional sliding-mode control[7],[8].

However,in the presence of uncertain nonlinearities,the con-troller guarantees that the output tracking error exponentially converges not to a point(asymptotic tracking),but to a ball-type error residual set(asymptotic bounding).As the size of the uncertain nonlinearities increases,the size of the residual set may also increase.Therefore,these control methods may not be allowable if the uncertain nonlinearities dominate the whole

Manuscript received March27,1998;revised March17,1999.

The authors are with the Department of Mechanical Engineering,University of California,Berkeley,CA94720USA.

Publisher Item Identifier S1063-6706(00)01620-9.system dynamics to the extent that the achievable residual set is beyond the allowable range.

The application of fuzzy set theory to control problems has been the focus of numerous studies[9]–[12].The motivation is often that the fuzzy set theory provides an alternative way to the traditional modeling and design of control systems when system knowledge and dynamic models in the traditional sense are uncertain and time varying.Although achieving many prac-tical successes,fuzzy control has not been viewed as rigorous due to the lack of formal synthesis techniques,which guarantee the basic requirements for control systems such as global sta-bility.Recently,some research has been focused on the use of the Lyapunov synthesis approach to construct stable adap-tive fuzzy controllers[13]–[18].Several adaptive neural con-trol schemes have been also developed based on the Lyapunov synthesis approach and applied to simple classes of nonlinear [18].Especially,based on the universal approximation theorem [16],an adaptive fuzzy logic control(AFLC)method can pro-vide stabilizing controllers(in Lyapunov sense)even for non-linear systems with dominant uncertain nonlinearities by using sufficiently complex approximation functions[15],[16]. However,AFLC is applicable only to some structurally restricted systems,i.e.,matched systems in which the control input and the uncertainties appear in the same equation.An-other drawback of AFLC is that the reference input can hardly satisfy the required persistent excitation(PE)condition because of an excessive number of unknown parameters.

In this paper,robust adaptive control using a universal approximator(RACUA)method is proposed in order to design a high-performance robust controller in the presence of para-metric uncertainties and dominant uncertain nonlinearities.The approach effectively combines the design techniques of robust adaptive control(RAC)by backstepping and adaptive fuzzy logic control(AFLC),and provides several attractive features. This is achieved by including a robustness term in the adaptive backstepping algorithm by approximating unknown nonlin-earities and by selecting parameter adaptation properly.The robustness term counteracts the effect of model uncertainties and parameter adaptation errors.To make the analysis general, only several required conditions on the robustness term are imposed instead of specifying them.The adaptive fuzzy logic system is used as a universal approximator to reduce the size of unknown nonlinearities.Thus,the size of the residual set of tracking errors can be reduced without using a high gain in the feedback loop.A simple smooth projection technique is used in the adaptation law to bound parameter estimation errors.

1063–6706/00$10.00?2000IEEE

RACUA is constructed for a class of SISO nonlinear systems, transformable to a parametric-strict-feedback form.

II.A DAPTIVE F UZZY L OGIC S YSTEM

There are many different interpretations for the fuzzy IF–THEN rules which result in different mappings of the fuzzy inference engine[10],[12].There are also many different types of fuzzifiers and defuzzifiers.Many combinations of these fuzzy inference engines,fuzzifiers,and defuzzifiers may constitute useful fuzzy logic https://www.360docs.net/doc/e53485962.html,putational efficiency and easiness of adaptation are important for real-time adaptive control problems.Therefore,we consider a fuzzy logic system (FLS)consisting of the product-inference rule,singleton fuzzifier,center average defuzzifier,and Gaussian membership function.

If the fuzzy rule base consists of a collection of fuzzy

IF–THEN

rules

and

is

are the input and output of the fuzzy logic system,

respectively,

,respectively,

and is the

number of rules in the rule base.

Then,the resulting fuzzy logic system can be represented as

[16]

(2)

where

(3)

where is a parameter vector.The pa-

rameter vector is the collection of the points at

which's

achieve their maximum

values,,

and's are the fuzzy basis functions(FBF's)defined

by

,

and are fixed parameters.

One of the most important advantages of FLS is that the FLS

has the capability to approximate nonlinear mappings.The fol-

lowing theorem shows that the FLS(3)with(5)is capable of

uniformly approximating any real continuous nonlinear func-

tion

over is compact.

Let

on a compact

set

III.C ONTROL OF F IRST-O RDER U NCERTAIN S YSTEM

The system equation considered in this section is the first-

order nonlinear system described

by

is a known

function,

is an unknown nonlinear function.Although the exact

values

of may not be known,their ranges can

often be predicted in advance.Thus,we can make the following

reasonable and practical assumptions

that are

bounded by some known parameters or known functions(see

the following).

Assumption

1:

is a known bounded function for

all

for two vectors is performed in terms of the

corresponding elements of the vectors

(e.g.,

and

(9)

where

.

Let be the desired output,which is assumed to be

bounded with bounded derivatives up to a sufficient order.The

control objective can be stated as that of designing a bounded

control law for the

input

LEE AND TOMIZUKA:ROBUST ADAPTIVE CONTROL USING UNIVERSAL APPROXIMATOR 97

the system is either globally uniformly bounded stable or

asymptotically stable and the

output

.An adaptive algorithm will be used to estimate the uncertain parameters in

vector

.There are several different types of modifications using dif-ferent projection functions [19],[20],[7],[21],[22].Since the derivatives of the control components are recursively required at each step to design a controller by the backstepping procedure

for

is a real-valued suf-ficiently smooth nondecreasing function with bounded deriva-tives up to

order

is known compact set

and

,

(13)

where

.This

function

(14)

where

remains bounded

for all

bounded

and

(15)

where is any bounded time-varying positive scalar,

i.e.,

is any known bounded function

satisfying

is

.

Applying (14)to (9),we obtain the error

equation

(20)

Now,we state the main theorem.

Theorem 2:With the control law (14),the adaptation law (20),and a robustness term satisfying Condition 1,the following results hold for the system described by (9)if Assumption 1is satisfied:

If

,then the bounded control input (14)guarantees global uniform boundedness of the tracking

error

,that is,from the universal approximation theorem,

the searching space for

and if the desired trajectory satisfies the persistent

excitation (PE)condition so that the estimated parameter converges to its true value,

i.e.,

when

when

98IEEE TRANSACTIONS ON FUZZY SYSTEMS,VOL.8,NO.1,FEBRUARY2000

Therefore

is a

bounded disturbance

and is also bounded for

all

decreases

whenever and,

hence,

,we can conclude that the

control input(14)is bounded.

Moreover,from

(22),.

The exponentially converging

rate to the compact

set

(24)

can be freely adjusted.

From

(13)

,noting(19),(20),and(25),the time derivative

of the positive definite

function

.From

(14),

is uniformly continuous.By Barbalat's lemma

[23],

.Thus,the PE condition will

guarantee

that

Four well known examples of the robustness

term

for

for

for

,thus eliminating

chattering[23].This achieves a tradeoff between tracking pre-

cision and robustness to unmodeled dynamics.The hyperbolic

tangent function is used to smoothen the saturation function at

the bound.This smooth modification is especially necessary in

the backstepping design procedure for general relative

degree

LEE AND TOMIZUKA:ROBUST ADAPTIVE CONTROL USING UNIVERSAL APPROXIMATOR

99

Fig.1.Robustness term

(z)).

IV.RACUA OF SISO NONLINEAR SYSTEMS

The RACUA technique can be generalized to a class of SISO nonlinear systems with arbitrary known relative degrees. This generalization is achieved by combining the RACUA design technique with the well-known adaptive backstepping algorithms[2].

Consider the SISO nonlinear system transformable to the fol-lowing parametric-strict-feedback

form

(31)

where

are known

functions,

is the vector of unknown constant parameters and

unknown nonlinear functions,respectively.We make the fol-

lowing assumptions

on

(32)

where

are known bounded

functions for

all

dynamics,

the

.In other words,only the feedback signals de-

termine the dynamics.Such a form is called parametric-strict-

feedback form and is studied in[2],[5].

From Theorem1,there exists a fuzzy logic

system

(33)

where for

arbitrary

be the desired output trajectory,which is assumed

to be bounded with bounded derivatives up

to

is designed

from by trajectory initialization

[2].

is also assumed to be known and bounded with bounded

derivatives up

to

such that under Assumption

2,the system is stable and the

output

.

We will start by adaptively stabilizing the first equation of

(34)

considering to be its control.At each subsequent step,

we will augment the designed subsystem by one equation.At

the th order subsystem is stabilized with respect to

a Lyapunov

function

are designed

at the final step.The design modifies the recursive backstep-

ping procedure in[1],[2],[5],and[7],which proceeds in the

following steps.

Step1:

The first equation of(34)can be rewritten

as

.From

(14),is given

by

and is the first robustness term.The

required conditions

for will be given in Condition2.