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