PARAMETER CONVERGENCE IN ADAPTIVE FUZZY CONTROL

Domenico Bellomo, David Naso

Dipartimento di Elettronica ed Elettrotecnica, Politecnico di Bari

Via Re David 200, 70125 Bari, Italy

Robert Babu

ska

Delft Center for Systems and Control, Delft University of Technology

Mekelweg 2, 2628 CD Delft, the Netherlands

Keywords:

Adaptive fuzzy control, tracking control, identiﬁcation, feedback linearization, interactions.

Abstract:

In this paper, the convergence of parameter estimates and the interactions among the two adaptive fuzzy

systems constituting an indirect adaptive fuzzy controller are studied, both analytically and by means of sim-

ulations with a second-order nonlinear system. The analytical results and the simulations, performed with

various initial conditions and learning rates, highlight how the interactions affect the behavior of the adaptive

control scheme with regard to the control performance in terms of a tracking error, accuracy and relevance of

the identiﬁed fuzzy models.

1 INTRODUCTION

The main goals of adaptive control are (i) to adjust

the on-line controller such that the required closed-

loop performance (stability in the ﬁrst place) is pre-

served in the presence of unforeseen parameter vari-

ations and/or (ii) to learn a suitable control law when

a priori information on the controlled plant is lacking

(e.g., theplant parameters are partly or completely un-

known).

Adaptive fuzzy control (AFC) combines results

from modern control theory, fuzzy systems and adap-

tation techniques. The most common stable AFC

schemes are based on feedback linearization, employ-

ing fuzzy systems (mostly of the singleton type) as

universal function approximators. They can be used

either to approximate the unknown plant dynamics

(indirect schemes) or directly the control law (direct

schemes). With respect to other universal interpola-

tors, such as neural networks, fuzzy systems offer the

possibility to interpret the input-output relations by

means of linguistic rules. This feature allows one to

incorporate a priori knowledge in the initial model

and/or control law and, at least in principle, to gather

useful insights about the controlled process at the end

of the learning stage.

The parameter learning laws of AFC are often de-

rived by using Lyapunov synthesis (Wang, 1994) and

are basically guided by the tracking error with respect

to some reference trajectory. The earliest controllers

of this type have been introduced for SISO systems in

the controllable canonical form (Wang, 1993; Wang,

1996). Since then, considerable efforts have been de-

voted to improving the performance and extending

the applicability to wider classes of systems. For

instance, extensions to systems with unmeasurable

states are proposed in (Tong et al., 2000), MIMO sys-

tems are considered in (Ordonez and Passino, 1999;

Tong and Chai, 1999), while in (Wang et al., 2002;

Tsay et al., 1999; Spooner and Passino, 1996) ﬁrst-

order Takagi-Sugeno fuzzy systems are used as ap-

proximators. Adaptive fuzzy control in the presence

of uncertainties is realized by adding a sliding mode

term to the control law (Su and Stepanenko, 1994;

Han et al., 2001; Fishle and Schroder, 1999), or by

using H

∞

performance indices (Chen et al., 1996;

S. Tong and Wang, 2000).

Despite numerous studies on AFC, some basic

problems arising even in the case of simple nonlinear

SISO systems have not been addressed in the litera-

ture. Examples of these problems are:

• dependence of the AFC performance (learning

time, stability of the adapted parameters and qual-

ity of the identiﬁed models) on user-deﬁned param-

eters;

• inherent conﬂict between the control goal (reduc-

tion of the tracking error) and the identiﬁcation

goal (reduction of the identiﬁcation error with re-

spect to the unknown system dynamics);

135

Bellomo D., Naso D. and Babuška R. (2004).

PARAMETER CONVERGENCE IN ADAPTIVE FUZZY CONTROL.

In Proceedings of the First International Conference on Informatics in Control, Automation and Robotics, pages 135-142

DOI: 10.5220/0001128701350142

 SciTePress

• interactions between the fuzzy systems’ parame-

ters and their convergence during the learning pro-

cess.

The lack of a proper understanding of these basic as-

pects makes it difﬁcult to properly exploit the trans-

parency and interpretability of fuzzy systems, which

are in fact neglected in a considerable part of the re-

cent literature on AFC.

In this paper, we address the issue of interaction

between the fuzzy systems’ parameters and their con-

vergence in the case of indirect AFC based on feed-

back linearization. First, we determine analytical re-

lations between the rates of change of the tracking er-

ror and the parameter errors. This analysis highlights

the occurrence of conﬂicts between the identiﬁcation

of unknown system dynamics and the tracking con-

trol. Then, we show, by means of simulations with a

second-order system, that the AFC performance criti-

cally depends on user-deﬁned parameters such as the

learning rates or initial values of the adapted parame-

ters. We consider the scheduling of learning rates as

a possible solution to overcome the above-mentioned

problems.

The remainder of this paper is structured as fol-

lows. In Section 2, the basic elements of indirect

adaptive fuzzy control are described. In Section 3,

an analysis of interactions is carried out. Section 4

describes a case-study with a second-order nonlinear

system. The scheduling of the learning rates and its

effects on the performance are discussed in Section 5.

Finally, in Section 6, conclusions are given.

2 INDIRECT AFC

2.1 Structure of the Controller

Indirect adaptive fuzzy control (Wang, 1996) is suit-

able for systems in controllable canonical form

(n)

= f(x)+g(x)u (1)

y = x. (2)

where x =



x, ˙x,...,x

(n−1)



∈ R

is the state

vector. The control goal is to track a desired trajec-

tory y

while keeping all the signals bounded in the

closed-loop. Deﬁne the tracking error e = y

− y

as the difference between the reference trajectory and

the output of the system. Further, introduce the vec-

tor of the tracking error and its n − 1 derivatives

e =



e, ˙e,...,e

(n−1)



and the vector of the feed-

back gains k =[k

,...,k

]

. If the functions f(x)

and g(x) are known and if the gains k

are cho-

sen such that the roots of the polynomial h(s)=

+ k

n−1

+ ...+ k

are in the open left-half com-

plex plane, the feedback linearizing control law

∗

g(x)



−f(x)+y

(n)

+ k



(3)

produces the desired linear error dynamic:

(n)

+ k

e =0 (4)

or equivalently

e = Λ

e (5)

where the matrix Λ

∈ R

nxn

is given by

⎡

⎢

⎣

010... 0

001

00... 01

−k

n−1

... ... −k

⎤

⎥

⎦

. (6)

The ideal control law (3) guarantees that

lim

t→∞

e(t)=0. The basic idea of Wang’s

indirect adaptive control is to approximate the

unknown functions f(x) and g(x) in the control law

(3) by using two singleton fuzzy systems:

f(x)=θ

(x) (7)

ˆg(x)=θ

(x) . (8)

where θ

and θ

are the consequent parameters to be

adapted, ξ

(x) and ξ

(x) are the normalized degrees

of fulﬁllment of the (ﬁxed) fuzzy rules. If we replace

the functions f (x) and g(x) in (3) with their fuzzy

approximations, we have a control law

u =

ˆg(x)



−

f(x)+y

(n)

+ k



(9)

that is an approximation of the ideal law (3). To

properly isolate the effects of interactions between

the learning systems from other phenomena related

to the minimal approximation error, we choose f(x)

and g(x) that can be exactly approximated by the two

fuzzy systems

f(x) and ˆg(x): f (x)=θ

∗

ξ(x) and

g(x)=θ

∗

ξ(x) (where θ

∗

and θ

∗

represent the op-

timal parameters). Adding and subtracting ˆg(x)u at

the right-hand side of (1), substituting the control law

(9) only in the term ˆg(x)u and substituting equations

(7), (8) with the actual and optimal parameters, we

have, after some manipulations, the error dynamic

e = Λ

e + b



(x)+φ

(x)u



(10)

where b

=[0, ..., 0, 1]

; the parameter error φ

− θ

∗

is the difference between the actual param-

eters θ

of the fuzzy system

f(x) and the optimal

parameters θ

∗

; the same holds for the parameter error

= θ

− θ

∗

ICINCO 2004 - INTELLIGENT CONTROL SYSTEMS AND OPTIMIZATION

136

2.2 Adaptive Laws

The adaptive laws are derived using Lyapunov syn-

thesis. The considered Lyapunov function

V =

Pe +

2γ

= V

+ V

. (11)

is the sum of three contributions depending respec-

tively on the tracking error e and the parameter errors

and φ

. Matrices P ∈ R

nxn

and Q ∈ R

nxn

are positive-deﬁnite matrices that fulﬁll the Lyapunov

equation

P + PΛ

= −Q . (12)

This choice for V is made with the goal of guaran-

teeing simultaneously the boundedness of tracking er-

ror and of parameter errors. The time-derivative of V

is obtained by differentiating (11), substituting for

from (10) and using (12)

V = −

Qe +



+ γ

(x)





+ γ

(x) u



(13)

where γ

and γ

are the learning rates, e

= e

and p

is the last column of P. If the parameters θ

and θ

are adapted according to the following laws

= −γ

(x) (14)

= −γ

(x) u (15)

the terms in the brackets in (13) are zero and the

derivative

V is negative-deﬁnite. This guarantees that

V is decreasing and asymptotically converges to zero.

3 ANALYSIS OF INTERACTIONS

This section focuses on the interactions between the

two adaptive fuzzy systems (7) and (8) and the effects

of these interactions on the evolution of the tracking

error and the parameter errors. In (13) we can separate

the time derivatives of the terms V

, V

and V

(16)

(17)

= −

Qe + e

(x)+e

(x)u (18)

If we substitute the expression of

given by (14) in

(16), remembering the deﬁnition of φ

we get

= −e

(x)=−e



f(x) − f (x)



. (19)

In the same way, substituting

from (15) into (17)

and recalling the deﬁnition of φ

, we obtain

= −e

(x)u = −e

(ˆg(x) − g(x)) u. (20)

Substituting the control law (9) into ˆg(x)u and deriv-

ing from (1) the product g(x)u, we can recast the last

equation in the following form (after some manipula-

tions):

= −

− e



(n)

+ k



. (21)

In the same way, we can prove that

= −

Qe + e



(n)

+ k



. (22)

Equations (19), (21) and (22) represent the contribu-

tions to

V due to the parameter errors and the tracking

error, respectively. They highlight how the adaptive

systems interact and how such interactions affect the

identiﬁcation task (reduction of the parameter errors)

and the control task (reduction of the tracking error).

The term in square brackets in equations (21) and (22)

represents the error dynamic if the control law is the

ideal feedback linearizing control law u

∗

. Hence, in

the ideal case, it is zero. Since we only have an ap-

proximation of this ideal control law, this term is not

zero. However, we can expect that if the approxi-

mation of the ideal control law is good enough, the

error will approximatively fulﬁll the ideal error dy-

namic and the term in square brackets will be small

compared to −

(see the simulations results in Sec-

tion 4). As a result, we can see from equations (19)

and (21) that, the variations of V

and V

would be

of opposite sign: an improvement in the identiﬁcation

of f(x) would be compensated by a deterioration in

the identiﬁcation of g(x) and vice-versa. Equation

(22), on the other hand, shows the interference be-

tween identiﬁcation and control tasks.

Remark: Note that we have not considered an ap-

proximation error. In the presence of a minimal ap-

proximation error, an additional term appears in the

time-derivative of the Lyapunov equation, but the re-

lationship between

and

remains unchanged.

4 SIMULATION EXAMPLE

Consider the following second-order nonlinear sys-

tem:

˙x

= x

(23)

˙x

= −x

+ g(x

) u (24)

y = x

. (25)

The functions f(x

)=−x

and g(x

) are chosen

such that they can be exactly represented by fuzzy

PARAMETER CONVERGENCE IN ADAPTIVE FUZZY CONTROL

137

−1.5 −1 −0.5 0 0.5 1 1

−1.5

−1

−0.5

0.5

1.5

f(x

)

(a) Final approximation for f(x

)

0 1 2 3 4 5

g(x

)

(b) Final approximation for g(x

)

Figure 1: Final approximations for f (x

) and g(x

): ’o’

represent the true parameters; ’*’ represent the actual pa-

rameters.

systems with a ﬁnite number of rules. This means

that there is no minimal approximation error. In fact,

f(x

) is linear and g(x

) is a singleton fuzzy sys-

tem (with four singleton consequents and four an-

tecedent triangular membership functions centered at

=0, 2, 4, 6). The consequents are the values of

the parabolic function x

+1 for x

=0, 2, 4, 6. The

fuzzy system ˆg(x

) has the same structure as g(x

)

and hence it can exactly approximate g(x

). Another

distinct feature of the above system is that f(x) and

g(x) only depend on one component of the state vec-

tor each. This makes it possible to study the inter-

actions between adaptive fuzzy systems by using two

simple fuzzy systems with one premise variable.

The simulations have been performed with

the Matlab/Simulink implementation of the ode45

Dormand-Prince method and the relative tolerance of

0.001. Thanks to the absence of inherent approxi-

mation error in our case-study, supervisory control

is not necessary for stability assurance and is there-

fore omitted. The fuzzy systems

f(x

) and ˆg(x

)

have four triangular membership functions equally

distributed in their respective domains, [−1.5, 1.5]

and [0, 6]. The reference model is a second order

linear system with natural frequency ω

=1rads

−1

and damping ratio δ =0.9. The reference signal r

is represented by a repeating sequence with values in

the range [0, 6].

The feedback gains k

were set to k

=1.8 and

=1. The positive-deﬁnite symmetric matrices P

and Q satisfying equation (12) were obtained by nu-

merically solving this linear matrix inequality (LMI).

4.1 Standard Setting

An attempt to train

f(x

) and ˆg(x

) simultaneously

(with equal learning rates γ

= γ

= 100) results

in poor approximations of the true functions f(x

)

and g(x

), see Fig. 1. This is despite the fact that an

important piece of prior knowledge was used, namely

the fact that each of the functions only depends on one

component of the state.

It can also be seen in Fig. 2 that in this case the

parameters are not converging at all. The analysis of

0 200 400 600 800 1000 1200 1400 1600 180

−2

−1

t [s]

(a) Filtered singletons for f(x

)

0 200 400 600 800 1000 1200 1400 1600 1800

t [s]

(b) Singletons for g(x

)

Figure 2: Singletons for f(x

) and g(x

interactions carried out in Section 3 offers an expla-

nation of this simulation results. In Fig. 3, we can

see that γ

and γ

exhibit similar complemen-

tary variations (the interplay between the two iden-

tiﬁcation tasks): if γ

decreases, γ

increases

and vice-versa. The simultaneous identiﬁcation of the

unknown system dynamics cannot be accomplished

most of the time. Moreover, in Fig. 4 we can see

the interplay between identiﬁcation and control: an

improvement in the parameter errors corresponds to

ICINCO 2004 - INTELLIGENT CONTROL SYSTEMS AND OPTIMIZATION

138

higher tracking error and vice-versa. Hence, we can

see that the adaptation is negatively affected by these

interactions, in terms of learning time, oscillation of

the parameters and accuracy of the ﬁnal fuzzy models.

Remark 1: For sake of clear presentation, the pa-

rameters shown in Fig. 2(a) were ﬁltered by a low-

pass Butterworth ﬁlter with the cut-off frequency of

0.3 rad s

−1

. Without this ﬁltering, the plot is rather

confusing due to large overlapping parameter oscilla-

tions.

Remark 2: We have plotted γ

and γ

rather

than V

and V

because we are interested in the evolu-

tion over time of the parameter errors and the learning

rates represent just scaling factors.

0 20 40 60 80 10

t [s]

(a) γ

0 20 40 60 80 10

1550

1600

1650

1700

t [s]

(b) γ

Figure 3: Zoom on the evolution over time of γ

and

4.2 Alternative Settings

The simulations have been also carried out in differ-

ent settings, in order to see how the phenomena high-

lighted in the previous section depend on the particu-

lar settings and to what extent they can be generalized.

First, one of the two functions f(x

) and g(x

) is

assumed to be known and the other one is learnt. This

allows us to study what happens when there are no

interactions between the learning processes. Second,

f(x

) and g(x

) are learnt simultaneously, but in this

case with different learning rates and with different

0 200 400 600 800 1000 1200 1400 1600 180

0.05

0.1

0.15

0.2

0.25

t [s]

(a) V

0 200 400 600 800 1000 1200 1400 1600 180

900

1000

1100

1200

1300

1400

1500

1600

1700

t [s]

Vf+ γ

(b) γ

+ γ

Figure 4: Evolution over time of V

and γ

+ γ

initial conditions for their parameters. Finally, simu-

lations have been carried out in which the two fuzzy

systems have both states as premise variables (less

prior knowledge on the system structure is used).

4.2.1 Separate Adaptation of

f(x

) and ˆg(x

)

First, we adapt only f (x

), assuming that g(x

) is

known. The learning rate is γ

= 100. At the

end of the learning, after 1800s, we get a good ap-

proximation of f (x

) and the output tracks the refer-

ence with a very small tracking error. The parameters

show some oscillations but their amplitude is quite

small (Fig. 5). With no interacting fuzzy systems, the

AFC works more efﬁciently and the learning time de-

creases.

When adapting only g(x

), assuming that f(x

) is

known, one obtains similar results: the ﬁnal approxi-

mation for the unknown function g(x

) is good and

the convergence of the adapted parameters is good

with only small oscillations.

4.2.2 Different Learning Rates

If we set γ

=1(one hundredth of γ

), the pa-

rameters get close to their optimal values and show

only reasonable oscillations (Fig. 6). Also the ﬁnal

approximations are very good. Basically, what we

have done is decreasing the level of interactions of

PARAMETER CONVERGENCE IN ADAPTIVE FUZZY CONTROL

139

0 200 400 600 800 1000 1200 1400 1600 180

−2

−1

t [s]

Figure 5: Singletons for f(x

) when g(x

) is known.

the fuzzy systems. The adaptive processes become

decoupled and do not hamper each other.

0 200 400 600 800 1000 1200 1400 1600 180

−2

−1

t [s]

(a) Final approximation for f

0 200 400 600 800 1000 1200 1400 1600 1800

t [s]

(b) Final approximation for g

Figure 6: Singletons for f (x

) and g(x

) with different

learning rates (γ

=1and γ

= 100).

4.2.3 Initial Conditions

In all previous experiments, the singletons of f(x)

were initialized to 0 and the singletons of g(x) to 0.1.

The adaptive scheme should work whatever the initial

conditions are. However, the initial values for the sin-

gletons of ˆg(x

) are quite far from the true values. Of

course, this makes the learning more difﬁcult. Further

simulations indicate that if the initial values of the sin-

gletons of both the functions are close to their optimal

positions, the learning works properly and the param-

eters converge (this is equivalent to say that some kind

of prior knowledge is embedded). If the initial sin-

gletons of ˆg(x

) are far from the optimal values and

the initial parameters for

f(x

) are exactly optimal,

we have an undesirable phenomenon: the adaptation

initially changes the parameters of

f(x

) and after a

while it recovers the optimal parameters. The adapta-

tion is making a kind of bootstrapping: in the attempt

of learning the control law it changes the parameters

f(x

) for compensating the error on ˆg(x

). Gener-

ally speaking, an important requirement for learning

systems is the monotonicity of the learning process.

We would like to have a kind of smooth adaptation

that goes straight to the solution without wandering

around it. In this particular case, the learning is evi-

dently non-monotonous.

4.2.4 Two Premise Variables

If we assume that no prior knowledge is available on

the functions f(x) and g(x), we have to consider

fuzzy interpolators with two premise variables (both

state variables). In this case, the above mentioned is-

sues (interactions of fuzzy systems, conﬂict between

identiﬁcation and control) are still present. However,

it should be noted that in this case, there are more

parameters involved and the learning task is more dif-

ﬁcult. The best performing adaptive fuzzy systems

(obtained with learning rates γ

=10and γ

= 100)

are thus not able to approximate the unknown func-

tions with the same degree of accuracy as we have

fuzzy systems with only one premise variable.

5 SCHEDULING OF LEARNING

RATES

The analytical developments and the simulation re-

sults with regard to the interactions of adaptive fuzzy

systems suggest that it maybe useful to decouple the

adaptation processes of the fuzzy systems. One way

to accomplish this is to schedule the learning rates,

thus allocating a time slot to the learning of f(x) and

another slot to the learning of g(x). Several differ-

ent values of the learning rates were used: 100/10,

100/1, 100/0 and vice-versa. Moreover, also dif-

ferent scheduling times (T = 200s, 100s, 50s, 10s)

were tested. The best setting found allocates 200s

(equivalent to four periods of the reference signal) al-

ternatively to the learning of one of the two functions

and assigns the values 1 and 100 to the learning rates.

Although the adaptation is not completely switched

off for any of the two adaptive systems, the signiﬁ-

cantly different learning rates ensure a reduced inter-

ference between the adaptive systems.

ICINCO 2004 - INTELLIGENT CONTROL SYSTEMS AND OPTIMIZATION

140

One can see in Fig. 7 that the parameter errors

quickly converge to zero. Also the tracking error con-

verges to zero as it can be seen in Fig. 8, but in the

ﬁrst phase of the adaptation it is higher than in the

previous cases. We can say that the price for better

identiﬁcation is a worse control performance in terms

of tracking error, at least in the initial phase.

0 200 400 600 800 1000 1200 1400 1600 180

t [s]

(a) γ

0 200 400 600 800 1000 1200 1400 1600 180

500

1000

1500

t [s]

(b) γ

Figure 7: Evolution over time of the γ

Vf and γ

Vgwith

scheduling of the learning rates.

6 DISCUSSION

From the mathematical analysis and from the sim-

ulations results presented, one can conclude that in

an indirect AFC scheme with adaptation laws derived

through Lyapunov synthesis, the tracking control and

the identiﬁcation of the system dynamics are most of

the time conﬂicting goals. There are strong interac-

tions between the two fuzzy systems. As a conse-

quence, the adaptation works through successive ad-

justments: the changes in one of the fuzzy system try

to compensate the changes in the other. This interplay

can hamper the learning process, which becomes non-

monotonous. Some works related to this topic can

also be found in the standard system-identiﬁcation lit-

erature.

In the context of ‘closed-loop system identiﬁca-

tion’, and ‘identiﬁcation for control’ (Landau, 1999;

0 200 400 600 800 1000 1200 1400 1600 180

0.5

1.5

t [s]

Figure 8: Evolution of Ve.

Hof and Schrama, 1995) it is stated that with stan-

dard identiﬁcation and control design methods it is

not possible to simultaneously optimize both the sys-

tem model and the controller. Hence, in identiﬁcation

for control, system identiﬁcation and control design

are not simultaneous; instead they are temporally sep-

arated. System dynamics are identiﬁed in closed-loop

in the presence of a ﬁxed controller and then, based

on the identiﬁed model, a new controller is designed.

These steps can be repeated. The analysis carried out

in this paper suggests that the separation of identiﬁ-

cation and control may be beneﬁcial in the context of

AFC. In (Hojati and Gazor, 2002), it is proven that

adaptive laws driven by two sources of information,

namely tracking error and prediction error (deﬁned

with regard to a series-parallel identiﬁcation model)

outperform adaptive laws based only on tracking er-

ror.

In this paper, the learning rates are abruptly

switched. With regard to stability, it may be better to

have a smooth transition of the learning rates. More-

over, if the plant parameters are unknown but ﬁxed,

the learning rate should decrease over time (only

plants with time-varying parameters require constant

learning rates). In the literature on reinforcement

learning and neural networks, some heuristics for

the choice of learning rates are provided (Sutton and

Barto, 1998; Jacobs, 1988).

7 CONCLUSIONS

The main contribution of this paper is the analysis

of interactions between parameter updates of the two

fuzzy systems constituting an indirect adaptive fuzzy

controller based on feedback linearization. First, it

has been shown analytically what are the relation-

ships among the time-derivative of the norm of the

tracking error and of the parameter errors with re-

gard to the unknown system dynamics. The analy-

sis highlights the existence of conﬂicting goals: the

PARAMETER CONVERGENCE IN ADAPTIVE FUZZY CONTROL

141

identiﬁcation of the unknown system dynamics and

tracking control cannot be simultaneously optimized.

Then by means of simulations with a second order

system, under different scenarios, the mathematical

developments are validated. The links with the re-

lated literature have been explored and ﬁnally some

possible improvements were suggested. In particu-

lar, we propose the scheduling of the learning rates as

possible means to overcome some parameter conver-

gence problems, simultaneously achieving the control

goal while performing a proper identiﬁcation of fuzzy

models, which are fully transparent and amenable to

off-line interpretation.

REFERENCES

Chen, B. S., Lee, C. H., and Chang, Y. C. (1996). H

∞

track-

ing design of uncertain nonlinear siso systems: Adap-

tive fuzzy approach. IEEE Trans. on Fuzzy Systems,

4(1):32–43.

Fishle, K. and Schroder, D. (1999). An improved adaptive

fuzzy control method. IEEE Trans. on Fuzzy Systems,

7(1):27–40.

Han, H., Su, C. Y., and Stepanenko, Y. (2001). Adap-

tive control of a class of nonlinear systems with non-

linearly parameterized fuzzy approximators. IEEE

Trans. on Fuzzy Systems, 9(2):315–323.

Hof, P. M. J. V. D. and Schrama, R. J. P. (1995). Identi-

ﬁcation and control-closed-loop issues. Automatica,

31(12):1751–1770.

Hojati, M. and Gazor, S. (2002). Hybrid adaptive fuzzy

identiﬁcation and control of nonlinear systems. IEEE

Trans. on Fuzzy Systems, 10(2):198–210.

Jacobs, R. A. (1988). Increased rates of convergence

through learning rate adaptation. Neural Networks,

1:295–307.

Landau, I. D. (1999). From robust control to adaptive con-

trol. Control Engineering Practice, (7):1113–1124.

Ordonez, R. and Passino, K. M. (1999). Stable multi-input

multi-output adaptive fuzzy/neural control. IEEE

Trans. on Fuzzy Systems, 7(3):345–353.

S. Tong, J. T. and Wang, T. (2000). Fuzzy adaptive control

of multivariable nonlinear systems. Fuzzy Sets and

Systems, (111):153–167.

Spooner, J. T. and Passino, K. M. (1996). Stable adap-

tive control using fuzzy systems and neural networks.

IEEE Trans. on Fuzzy Systems, 4(3):339–359.

Su, C. Y. and Stepanenko, Y. (1994). Adaptive control of

a class of nonlinear systems with fuzzy logic. IEEE

Trans. on Fuzzy Systems, 2(4):285–294.

Sutton, R. S. and Barto, A. G. (1998). Reinforcement Learn-

ing: an Introduction. MIT Press.

Tong, S. and Chai, T. (1999). Indirect adaptive control and

robust analysis for unknown multivariable nonlinear

systems with fuzzy logic systems. Fuzzy Sets and Sys-

tems, (106):309–319.

Tong, S., Wang, T., and Tang, J. T. (2000). Fuzzy adaptive

output tracking control of nonlinear systems. Fuzzy

Sets and Systems, (111):169–182.

Tsay, D. L., Chung, H. Y., and Lee, C. J. (1999). The adap-

tive control of nonlinear systems using the sugeno-

type of fuzzy logic. IEEE Trans. on Fuzzy Systems,

7(2):225–229.

Wang, C. H., Liu, H. L., and Lin, T. C. (2002). Direct adap-

tive fuzzy-neural control with state observer and su-

pervisory controller for unkown nonlinear dynamical

systems. IEEE Trans. on Fuzzy Systems, 10(1):39–49.

Wang, L. X. (1993). Stable adaptive fuzzy control of nonlin-

ear systems. IEEE Trans. on Fuzzy Systems, 1(2):146–

154.

Wang, L. X. (1994). Adaptive Fuzzy Systems and Control:

Design and Stability Analysis. Prentice-Hall.

Wang, L. X. (1996). Stable adaptive fuzzy controllers

with application to inverted pendulum tracking. IEEE

Trans. on Systems, Man, and Cybernetics, 26(5):677–

691.

ICINCO 2004 - INTELLIGENT CONTROL SYSTEMS AND OPTIMIZATION

142