IEEE论文-An approach to adaptive control of fuzzy dynamic sy

268IEEE TRANSACTIONS ON FUZZY SYSTEMS,VOL.10,NO.2,APRIL2002

An Approach to Adaptive Control of Fuzzy Dynamic Systems

Gang Feng

Abstract—This paper discusses adaptive control for a class of fuzzy dynamic models.The adaptive control law is first designed in each local region and then constructed in global domain.It is shown that the resulting fuzzy adaptive control system is globally stable.Robustness issues of the adaptive control system are also addressed.A simulation example is given for demonstration of the application of the approach.

Index Terms—Adaptive control,fuzzy modeling,nonlinear sys-tems,stability.

I.I NTRODUCTION

S INCE the first paper on fuzzy sets[1]was published,fuzzy logic control has attracted a great attention from both the academic and industrial communities.Many people have de-voted a great deal of time and effort to both theoretical research and implementation techniques for fuzzy logic controllers. Much progress has been made in successfully applying FLC in industrial control systems[2]–[5].

During the past a couple of years,many systematic fuzzy con-troller design methods have been developed[6]–[12]based on the Takagi–Sugeno(T–S)model,or the fuzzy dynamic models. The basic idea of these methods is:i)to represent the complex nonlinear system in a family of local linear models,each linear model represents the dynamics of the complex system in one local region;ii)to construct a global nonlinear model by aggre-gating all the local models through the fuzzy membership func-tions.The primary advantage of this model is that the controller design can be mainly based on each local model,which is much easier than that for nonlinear systems in the global region,and then the global controller can be constructed from the local con-trollers.

It has also been shown that fuzzy systems can approximate any nonlinear functions over a convex compact region[13]. Based on this observation,a number of attempts have been made to use fuzzy logic for adaptive control of nonlinear systems [14]–[17].The basic idea of most of these works is to use fuzzy basis functions to approximate the unknown nonlinear functions and update the constant parameters of the function on line and then implement adaptive control using the conventional control technology.The author in[18]recently proposed a model-based fuzzy control as well as adaptive control method.

In this paper,we will develop an adaptive control design method for a class of fuzzy dynamic models.The basic idea is

Manuscript received April18,2001;revised August27,2001and October 3,2001.This work was supported by the City University of Hong Kong(SRG 7001245).

The author is with the Department of MEEM,City University of Hong Kong, Kowloon,Hong Kong(e-mail:megfeng@87b86e1bff00bed5b9f31d61.hk).

Publisher Item Identifier S1063-6706(02)02971-5.to design an adaptive controller in each local region and then construct the global adaptive controller by suitably integrating the local adaptive controllers together in such a way that the global closed-loop adaptive control system is stable.

The rest of the paper is organized as follows.Section II formu-lates the fuzzy system modeling,Section III presents the fuzzy adaptive control design and stability proof.Robustness issues are also discussed in the section.One example of simulation is presented in Section IV,which is followed by concluding re-marks in Section V.

II.F UZZY S YSTEM M ODELING

Many physical systems are very complex in practice so that their rigorous mathematical models can be very difficult to obtain if not impossible.However,many physical systems can indeed be expressed in some form of mathematical models locally,or those systems can be expressed as an aggregation of a set of mathematical models.Various fuzzy models have been proposed in the last few years,see,for example,[3], [6].Here,we consider using the following fuzzy model to represent a complex single-input–single-output(SISO)system that includes both fuzzy inference rules and local analytic linear

models:

AND

THEN

(2.1)

where-th fuzzy inference

rule,

are fuzzy

sets,

the system input

variable,the output of the

system,

th subsystem,

and

some measurable system variables.The model(2.1)can also be described in the state space form.

Let

..

.

(2.2) then,the model(2.1)

becomes

AND

THEN

(2.3)

1063-6706/02$17.00?2002IEEE

IEEE TRANSACTIONS ON FUZZY SYSTEMS,VOL.10,NO.2,APRIL2002269

where

..

.

Let

(2.4)

and

th fuzzy local dynamic model is a bi-

nary set defined as

FLDM

or represents the crisp input–output

relationship or dynamic properties of the system at the crisp

point

(2.6)

which can also be rewritten

as

are the same.

In this paper,we are going to discuss the design of control

systems for(2.7)when the coefficients of the plant are unknown.

It is assumed that the membership functions have been chosen

a priori based on the expert’s knowledge or the plant data.

The objective of the fuzzy adaptive control is to find an adap-

tive control law so that the output of the system tracks a given

bounded reference

signal

(3.1)

This is a regression form

with

as a regressor vector.It should be noted that the plant(3.1)is

in general nonlinear but it is linear with respect to its unknown

parameters.Therefore,all the parameter adaptation algorithms

developed for linear plants can be employed for the estimation

of the unknown parameters in(3.1).Here,we consider the fol-

lowing least squares

algorithm:

270IEEE TRANSACTIONS ON FUZZY SYSTEMS,VOL.10,NO.2,APRIL2002 troller design when the parameters of the fuzzy dynamic model

are known a priori.

B.Controller Design With Known Parameters

Define

then we can design the following local fuzzy control

law:

AND

THEN

(3.3)

The global control law can be obtained as

follows:

are coefficients of a stable polynomial defined

as

which specifies the desired output tracking error dynamics,

and

are assumed to be nonsingular.

Substituting the control law(3.4)into the fuzzy dynamic

model(2.7)leads to the following closed-loop

system

.

C.Adaptive Control Design

Based on the certainty equivalence principle,we choose the

following local adaptive control

law:

AND

THEN

(3.7)

where

Then,the global control law can be obtained as

follows:

has to be ensured.There are

a number of methods to achieve this in adaptive control commu-

nity such as projection to the known convex region[19]–[22].

Substituting the control law(3.8)into the fuzzy dynamic

model(2.7)leads to the following closed-loop

system:

(3.13)

where

..

.

has all its eigenvalues in

the left-hand side of

the

IEEE TRANSACTIONS ON FUZZY SYSTEMS,VOL.10,NO.2,APRIL2002271 the output tracking

error

and

,

(3.17)

Since

are bounded.Furthermore,it follows from the results

of Lemma1and the definition

of

that

as time goes to

infinity.

D.Robustness Issues

The fuzzy dynamic models discussed in the previous sec-

tions are assumed to be ideal,that is,the systems are assumed

to be modeled exactly by the fuzzy dynamic models.However

in practice,the systems are always subject to various kinds of

uncertainties such as unmodeled dynamics and/or bounded dis-

turbances.As shown for the linear adaptive control systems,

even a small uncertainty could lead to unstable adaptive control

system.As a result,various types of robust adaptation algorithm

and robust adaptive control algorithms have been developed to

cope with these various sorts of uncertainties.These techniques

include dead zones,relative dead

zones,

IF

is

THEN

(3.18)

where

for

the

uncertainties

(3.20)

where can be expressed

as

(3.21)

for two unknown small

constants

and.

Remark3.2:It is noted that the above assumption is in fact

weaker than the standard assumption as in the ordinary robust

adaptive control designs such as[19]and[20],where the two

small constants are also assumed to be known.

With the same definition of the regressor and the model pa-

rameter vector,the same certainty equivalence control law(3.7)

or equivalently(3.8),we obtain the following closed-loop con-

trol

system

272IEEE TRANSACTIONS ON FUZZY SYSTEMS,VOL.10,NO.2,APRIL2002

where,and the

term is a dead-zone func-

tion,which ensures that the parameter estimator is not disrupted

by small errors.The dead zone is defined as

follows:

otherwise

(3.25)

with

if

if

where,and is calculated

by

.

It should be noted

that will be always nonnegative

and nondecreasing.

Then,we have the following convergence properties for the

above robust parameter estimation algorithm.

Lemma2:The parameter update law(3.24)–(3.26),when

applied to the fuzzy dynamic model(3.19)has the following

properties.

E1)is continuous and bounded.

E2)are continuous and bounded,

and

converge to constants,

say,respectively.

E3)

Proof:Consider(3.23)and it follows that its autonomous

system,that

is,

,we

have

(3.30)

w h e r

e a n d(3.16)

h a s b e e n u s e d.

S i n c e t h e r i g h t-h a n d s i d e o f(3.30)i s m o n o t o n

c r e a s i n g,w e o b t a i

n

IEEE TRANSACTIONS ON FUZZY SYSTEMS,VOL.10,NO.2,APRIL2002273 Then,it follows from(3.16)and(3.31)that there exists some

constants

(3.32)

Since

is bounded.It implies

that,and

thus are

bounded.

Furthermore,

since

is uniformly continuous implies

that

approaches

infinity:

The proof is thus

completed.

If only bounded disturbances are present,then we can have

the following stronger results.

Corollary1:With the same conditions as in Theorem2ex-

cept that there is no unmodeled dynamics,

i.e.,,then the

adaptive control system with the simpler update law

for

(4.1)

where

m/s

is the mass of the cart,

and

kg,

being

around0

and

is a reference input,chosen

as

are chosen in this case.The initial

parameter

s.The result is shown in Fig.4.

It can be seen that the adaptive control algorithm can cope

with the variation of the plant dynamics well.Similarly for com-

parison,the response of the nonadaptive control system with the

same initial conditions are also obtained and shown in Fig.5

where the significant steady state tracking error can be observed.

274IEEE TRANSACTIONS ON FUZZY SYSTEMS,VOL.10,NO.2,APRIL

2002

Fig.3.Response with nonadaptive

control.

Fig.4.Response for adaptive control with m =2kg jumping to m =8kg.

It has been demonstrated through the simulations that the pro-posed fuzzy adaptive control schemes can be used for the control of unknown nonlinear pendulum-cart system.

V .C ONCLUSION

This paper presents a new fuzzy adaptive control system for a class of nonlinear systems represented by the fuzzy dynamic models.The basic idea of the approach is to design the local linear adaptive controller in each local region and construct the global fuzzy adaptive controller in such a way that the stability of the closed-loop adaptive control system is guaranteed.

This

Fig.5.

Response for nonadaptive control with m =2kg jumping to m =8kg.

paper only addresses a limited class of fuzzy dynamic models.However,it is believed that the idea can be extended to the more general cases,which though requires much more effort and will be our future research topics.

A CKNOWLEDGMENT

The author is grateful to the reviewers for a number of con-structive comments that have improved the presentation of this paper.

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