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8th International IFAC Symposium onDynamics and Control of Process Systems

STABLE INDIRECT ADAPTIVE FUZZY CONTROL BASED ON TAKAGI-SUGENO MODEL

Ruiyun Qia,*, Mietek A. Brdys

a

aDepartment of Electronic Electrical & Computer Engineering, University of Birmingham, Edgbaston,Birmingham B15 2TT, UK

Abstract: This paper presents an indirect adaptive fuzzy control scheme for nonlinear

uncertain stable plants with unmeasurable states. A discrete-time T-S fuzzy model isemployed as a dynamic model of an unknown plant. Based on this model, a feedbacklinearization controller is designed and applied to both the model and the plant.

Parameters of the model are updated on-line to allow for partially unknown and time-varying plants. Stability analysis shows that the adaptive controller guarantees theboundedness of all the closed-loop signals and achieves bounded tracking error. In the

ideal case where there is no modelling error and the signal for parameter learning ispersistently exciting, perfect tracking is ensured. The effectiveness of the method isverified by simulation examples. Copyright 2007 IFAC

Keywords: nonlinear control, adaptive control, fuzzy control,.

1. INTRODUCTION

Adaptive control has become very popular in manyfields of control engineering and attracted much

attention in developing advanced applications.Adaptive control is based on feedback of signals in acontrolled system for control adaptation to

effectively handle system uncertainties (Tao, 2003).

The adaptive control can be divided into two classes:direct adaptive control and indirect adaptive control.The former tunes the parameters of the controllerwhile the latter updates the parameters of the model.

This paper focuses on indirect adaptive control,which means the T-S fuzzy model is adapted on-line.Design of provably stable indirect adaptive fuzzycontroller posed more difficult theoretical questions.

Some of the early adaptive control schemes usingcontinuous-time T-S fuzzy model (Takagi &Sugeno,

1985) were presented in the papers (Wang, 1994;*

Corresponding author. Tel: 44-1214143150; fax: 44-1214144291.

E-mail address: [email protected] (Ruiyun Qi),[email protected](M.A. Brdys)

Spooner 1996; Yoo, 1998) with stability results.Wangs control law is composed of a certaintyequivalence control and a supervisory controllerwhich is used to force the tracking error not to

exceed certain error ball. Spooner and Passinosapproach consists of a certainty equivalence controlterm, a bounding control term used to restrict the

plant output trajectory, and a sliding mode control

term to compensate the approximation errors. Theirparameter adaptation algorithms are chosen to makethe derivative of a certain Lyapunov equationnonpositive. Recent adaptive fuzzy control schemes

for continuous-time system can be found in the

papers (Golea2003; Park & Cho, 2004). In Park

and Cho (2004), an adaptive law for updatingparameters is designed based the Lyapunov theory

and with the on-line parameter estimator, any type offuzzy controllers works adaptively to the parameterperturbations.All the above adaptive control schemes assume thatfull states of the controlled plant are available for

measurement. Adaptive output feedback control fornonlinear systems when full states are notmeasurable remains one of the outstanding problems.Kulawski and Brdys (2000) present an adaptive

Preprints Vol.2, June 6-8, 2007, Cancn, Mexico

385
mailto:[email protected]:[email protected]:[email protected]:[email protected]


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control scheme for uncertain nonlinear plants withunmeasurable states based on dynamic neuralnetworks. In their approach, a recurrent neural

network is employed as a dynamical model of theplant and the control law is designed by using thestates of the neural network instead of the plant. Our

proposal in this paper is inspired by the aboveconsideration and applies it to discrete-time adaptivefuzzy control system. A discrete-time T-S fuzzy

model is trained to approximate the input-outputbehaviour of an unknown continuous-time plant byusing I/O data from the plant. Then an appropriatecontrol law is calculated completely based on themodel, using the model structure, parameters andstates. The control input is applied to both the plant

and the model. The parameters of the model areupdated on-line by recursive least square estimation(RLSE) method, which is fast and is able to ensure

the boundedness of parameters (Tao, 2003). Withthis approach, at least the model can be stabilizedand provides bounded control input without extrasupervisory controller. The sufficient conditions for

the convergence of tracking error are derived forstable plant.The paper is organized as follows. The controlalgorithm including parameter adaptation law ispresented in Section 2. Stability of the closed-loop

system and convergence of the tracking error areproven in Section 3. Section 4 shows the simulation

results on control of a single-link robot manipulatorwith parameter disturbance. Finally, Section 5concludes the paper.

2. CONTROL ALGORITHM

2.1 Modelling for adaptive controlIt has been shown that T-S fuzzy systems with

piecewise linear rule consequents are universalapproximators to approximate any continuousnonlinear system to an arbitrary accuracy (Ying,

1998 & 1999) if all fuzzy membership functions areGaussian and the t-norm is the multiplication. In thispaper, the T-S fuzzy model is used to express a real

nonlinear plant. That is, the nonlinear plant can berepresented by the following fuzzy rules:

1 1

0 1 1 1

: If ( ) is and and ( ) is , then

( 1) ( ) ( ) ( )

i i i

n n

i i i i i

n n n n

R x k M x k M

x k x k x k u kq q q q ++ = + + + +

L

L

(1)

where iR ( 1, 2, ,i N= L ) denotes the ith fuzzy rule

andNis the number of rules.1( ), , ( )

nx k x kL are the

premise variables which are the past plant output

measurements and ( )u k is the control input. iM

( 1, ,i n= L ) are fuzzy sets which are described byGaussian functions

2

2

( )( | , ) exp

2( )

i

j ji i i

j j j j i

j

x cM x c s

s

- = -

(2)

The model parameter values ij

q ,i

jc andis ,

1, ,i N= L , 1, ,j n= L are unknown. The T-S fuzzy

model approximates a nonlinear system bycombining a group of local affine models. The

premise of a fuzzy rule defines a local operatingregion and each consequent describes a local input-

output relation in the local operating region. Let

1 2( ) [ ( ), ( ), , ( )] [ ( 1), ,T

nk x k x k x k y k n= = - +L Lx

( 1), ( )]Ty k y k- , then the overall T-S fuzzy model is

calculated as

1

( 1) ( | , ) 1 ( ) ( )

( ( ) | , )

( ) ( )

NT

i i T

n

i

T

n

x k k u k

k

y k x k

w=

+ =

= Y Q

=

i ix c x

x c (3)

where

{ }{ }

( ) ( )

1

1

1

0 1

1 2

( | , ) ( | , ),

( | , ) ( | , ) ( | , ) ,

| 1, , ; 1, ,

| 1, , ; 1, ,

, ,

( ) 1 ( ) ( ) ,

( | , ) .

ni i i i

j j j j

jN

i i i

ii

j

i

j

TT TT

i i i N

n

TTe

TT T T N T

e e e

M x c

c i N j n

i N j n

k k u k

w s

w w w

s

q q

w w w

=

=

+

=

=

= = =

= = =

= Q =

=

Y =

L L

L L

L L

L

i i

i i

x c

x c x c x c

c

x x

x c x x x

(4)

An assumption is made here.

Assumption 1. The modelling error is negligible and

there exist parameter values ( , )c and Q that make

the model (3) to become a perfect representation ofthe real plant.From now on, the nonlinear plant is completelyrepresented by (3). Since all the parameters in (3) are

unknown, we define an estimation model as

1

( 1) ( | , ) 1 ( ) ( )

( ( ) | ( ), ( )) ( )

( ) ( )

NT

i i T

ni

T

n

x k k u k

k k k k

y k x k

w= + =

= Y Q

=

i i

x c

x

x c (5)

where ( ( ), ( ))k kc and ( )kQ are the estimates of

( , )c and Q at time k. ( ) nk x is the model state

vector.

2.2Control synthesis

( 1)ry k+( 1)y k+

( 1)y k+

( )u k

( 1)e k+

( )kx

( )k

M

1-

( )ke

( )v k

Tk

1-

1-

Fig. 1 Adaptive control based on T-S fuzzy model

The control objective is to make the output of the

nonlinear plant (3) tracking a specified referencetrajectoryr

y .

Assumption2. The reference ( )r

y k satisfies

( )r k Uy (6)

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where [ ]( ) ( 1) ( )T

r r rk y k n y k = - +y L and U is

a known bound.The overall control structure is shown in Fig. 1. Itconsists of three parts: a nonlinear uncertain plant, an

adaptive T-S fuzzy model and a model basedfeedback linearization controller.

The control law is designed based on the model (5)whose evolution of output can be expressed as

1

1

1

( 1) ( ( ) | , ) ( ) 1 ( ) ( )

( ( ) | , ) ( ) 1 ( )

( ( ) | , ) ( ) ( )

( ( ) | ( )) ( ( ) | ( )) ( )

NT

i i T

i

NT

i i T

f

i

Ni i

g

i

y k k k k u k

k k k

k k u k

f k k g k k u k

w

w

w

=

=

=

+ =

=

+

= +

i i

i i

i i

x c x

x c x

x c

x p x p

(7)where

{ }

0 1

1

1

1

( ) ( ) ( ) ( ) ,

( ) ( ), ( ) ( ), ( ), ( )

( ( ) | ( )) ( ( )) ( ) 1 ( ) ,

( ( ) | ( )) ( ( )) ( ).

Ti i i i

f n

i i

g n

NT

i i T

f

i

Ni i

g

i

k k k k

k k k k k k

f k k k k k

g k k k k

q q q

q

w

w

+

=

=

=

= = Q

=

=

L

p c

x p x x

x p x

(8)

To apply feedback linearization technique to design

the control law ( ( ) | ( ))g k kx p is required to be

invertible.

Assumption 3. ( ( ) | ( ))g k k e>x p where e is a

small real positive number, which implies therelative degree of the T-S fuzzy model is equal toone.With the assumption 3, the control law can becalculated as

1 ( ) [ ( | ( )) ( )] ( | ( ))

u k f k v k g k

= - +x px p

(9)

where ( )v k is a new input chosen as

( ) ( 1) ( | ( ))Tr

v k y k k k = + + k e p (10)

where [ ( | ( )) ( 1| ( 1))k k e k n k n= - + - + Le p p

]( | ( ))T

e k kp and ( | ( ))e k kp is the model tracking

error ( | ( )) ( ) ( | ( ))

re k k y k y k k = -p p (11)

and1

( , , )Tn

k k= L k be such that the zeros of the

polynomial 11

( ) n nn

h z z k z k -= + + +L lie within the

unit circle centered at the origin of the plane.The control law (9) applied to (7) results in it beingdecoupled and linear with respect to the new input

( )v k

( 1) ( )y k v k+ = (12)

2.3Online parameter adaptation

The initial T-S fuzzy model can be obtained by eitheroff-line or on-line identification methods. In thispaper, the initial model is obtained by the on-line

identification method proposed in ((Qi and Brdys,

2005). The initial model is good enough for themodel based controller to produce proper controlsignal in the beginning of closed-loop control.

However, the plant may have external disturbancesand parameters disturbances so that the initial modelneeds to be updated. Therefore, the model should be

adaptive when plant changes. The parameters of themodel can be adjusted on-line to provide an adaptivemodel.

For fixed parameters

( ( ), ( ))k kc

, Q can beestimated by recursive least square estimation(RLSE).

( ) ( ) ( 1) ( 1) ( )

1 ( ) ( ) ( )TS k k k

k kk S k k

eY +Q + = Q -

+Y Y(13)

( ) ( ) ( ) ( )( 1) ( )

1 ( ) ( ) ( )

T

T

S k k k S k S k S k

k S k k

Y Y+ = -

+Y Y(14)

where ( )kY denotes ( | ( ), ( ))k kY x c and

( 1) ( 1) ( 1) ( ) ( ) ( 1)Tk y k y k k k y k e + = + - + = Y Q - + .

The initial conditions are chosen as

1 2 (0) (0) (0) (0) , (0)

TT T T

N S Iq q q Q = = W L

where1 2

(0) (0) (0)T T TN

q q qL are the parameters

of the initial T-S model and W is a large positiveconstant. Large W can speed up parameterconvergence, but too large may cause instability. It

needs to be chosen properly during on-lineadaptation.

3. STABILITY ANALYSIS

3.1Convergence of model tracking error

Applying (9) to (7) , we get

( 1) ( 1) ( | ( ))Tr

y k y k k k+ = + + k e p (15)

Since ( 1| ( )) ( 1) ( 1| ( 1))r

e k k y k y k k + = + - + +p p ,

we have the following error dynamics

2 1

( 1| ( 1)) ( 1 | ( 1))

( 1| ( 1)) ( | ( ))n

e k k k e k n k n

k e k k k e k k

+ + = - - + - +- - - - -Lp p

p p(16)

which can be formulated in matrix form

( 1| ( 1)) ( | ( ))k k k k + + = Le ep p (17)

where

1 1

0 1 0 0 0 0

0 0 1 0 0 0

,

0 0 0 0 0 1

n nk k k-

L = - - -

L

L

M M M M O M M

L

L L L L

(18)

Since L is a stable matrix

( 11

n n

nI z k z k--L = + + +L is stable), the system

(17) is asymptotically stable, which implies

( | ( )) 0k k e p when k . The model tracking

error will converge to zero.

3.2Boundedness of parameters and control signalAssumption 4. In order to deal with the T-S model

bring linear in parameters when performing tracking

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error boundedness analysis, we shall assume that the

premise parameters of the model ( , )c are the same

as the plant parameters ( , )c .

However, updates of all paramters will be carried

during the control simulation studies.

Lemma 1. The adaptive algorithm (13)-(14)

guarantees that ( )kQ% and ( )kQ are bounded.

Lemma 2.If the signal ( )kY is persistently exciting,then the least-squares algorithm (13)-(14) ensures

that lim ( ) 0kk

Q =

% .

These follow from standard properties of the least-square estimation methods applied to linearlyparameterized models.

Applying (9) to (7) yields

( 1) ( 1) ( )Tr

y k y k k+ = + + k e (19)

From assumption 2, ( 1)r

y k+ is bounded and since

( )ke is bounded. Thus, equation (19) implies theboundedness of ( 1)y k+ . Since ( )kx is a vector

consisting of past outputs, ( )kx is bounded.

Bounded ( )kx , bounded parameters ( )kQ and

bounded ( ( ))i kw x imply bounded ( ( ))f kx and

bounded ( ( ))g kx in (8). Thus, from (9) we obtain

the boundedness of ( )u k .

3.3Boundedness of plant tracking error

From (3), we have the plant output evolution.

1

1 1

( 1) ( ( )) 1 ( ) ( )

( ) 1 ( ) ( ) ( ) ( )

( ( )) ( ( )) ( )

N

Ti i T

i

N NT

i i T i i

f g

i i

y k k k u k

k k u k

f k g k u k

w

w w

=

= =

+ =

= +

= +

x x

x x x

x x

(20)Applying (9) to (20) and after some manipulations,we have

( 1) ( ( )) ( ( )) ( )

( ) ( ) ( ) ( ( ) ( ) ( ))

( 1) ( )Tr

y k f k g k u k

f g u k f g u k

y k k

+ = +

= + - +

+ + +

x x

x x x x

k e

(21)

Moving ( 1)r

y k+ to the left side of (21) and after

some manipulation, we get

( 1) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

T T T

T T T

e k k k k k

k k k k

+ = Y Q -Y Q-

= Y Q +Y Q-% %

k e

k e

(22)

where ( )kY denotes ( ( ) | , )kY x c and ( )kY

denotes ( ( ) | ( ), ( ))k k kY x c .

Considering

1 1

( ) ( ) ( )

( ) ( ) ( ) ( )

T T T

N Ni iT i iT

e ei i

k k k

k kw w= =

Y Q = Y Q-Y Q

= -

%

x

x x

x

( )1

1 1

( ) ( ) ( )

( ) ( ) ( ) ( )

Ni i iT

e

iN N

i iT i iT

x x

i i

k

k k

w w

w w

=

= =

= -

- +

x x x

x e x e

(23)

where1 2

Ti i i i

x nq q q = L .

Substituting (23) into (22), the plant tracking error

dynamics can be obtained as follows

( )1

1 1

1

( 1) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

NT i i iT

e

iN N

i iT i iT T

x x

i iN

i iT

x

i

e k k k k

k k k

k k

w w

w w

w

=

= =

=

+ = Y Q + -

- + -

= +G

%x x x

x e x e k e

x e

(24)where

( )

1

1

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

NT i iT T

x

i

Ni i iT

e

i

k k k k k

k

w

w w

=

=

G = Y Q - -

+ -

% x e k e

x x x

(25)

The error dynamics (24) can also be written into a

matrix form

1

( 1) ( ) ( ) ( )N

i i

i

k A k k w=

+ = +Ge x e (26)

where

1 2

0 1 0 0 0 0

0 0 1 0 0 0

, ( ) .( )

0 0 0 0 0 1

i

i i i

n

A kk

q q q

= G = G

0

L

L

M M M M O M M

L

L L L L

(27)

Assumption 5. There exists a positive constant a and a common positive definite symmetric matrix

Pfor all the matrices ( 1, , )iA i N= L such that

0 for 1, ,iT iA PA P aI i N- - < = L (28)

Theorem 1. Consider the closed-loop systemconsisting of plant(3), controller(9) and parameter

adaptation law (13)-(14). If the assumptions 1-5 hold,then

(i) the plant tracking error ( )ke is bounded;(ii) if in addition, the signal ( )kY is

persistently exciting, ( )ke converges tozero.

Proof. Consider the following possible Lyapunovfunction

( ( )) ( ) ( )TV k k P k =e e e (29)

Then the difference of the Lyapunov function alongthe plant tracking error trajectory (26) is calculated as

2

1

1 1

1

( 1) ( 1) ( ) ( )

( ( )) ( ) ( ) ( )

( ) ( 2 ) ( )

2 ( ) ( ) ( ) ( ) ( )

T T

Ni T iT i

iN N

i j T iT j jT i

i j i

NT i i T

i

V k P k k P k

k A PA P k

k A PA A PA P k

k P A k k P k

w

w w

w

=

= = +

=

D = + + -

= - +

+ + -

+ G +G G

e e e e

x e e

e e

x e

(30)

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With assumption 5, we have

2 2 for all andiT j jT iA PA A PA P aI i j+ - - (31)

Applying (31) in (30), the following inequality canbe obtained

1 1

1

2 22

max

( ) ( ) ( ) ( )

2 ( ) ( ) ( ) ( ) ( )

1 2( ) ( ) ( )2

N NT i j

i j

NT i i T

i

V k a k

k P A k k P k

Ma k P k a

w w

w

l

= =

=

D -

+ G +G G

- + + G

e x x e

x e

e

(32)

where

{ }max1

( ) max , 1, ( ( ) )N

i i i

ii

M P A i N P x Al w=

= = L

Consider

1

1

( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( )

NT i iT T

x

i

Ni i i

e

i

k k

k k k

k

w

w w

=

=

G = G

Y Q + +

+ -

% x k e

x x x

(33)

Since all the elements in (33) are bounded, ( )kG is

bounded. Hence there exists positive constant rG

such that

( )k rGG (34)

We can conclude outside the ball

2

min4 2 ( )

( ) : ( )M a P

k ka

le rG

+ < =

e e (35)

the Lyapunov function difference can be bounded by

( ( )) 0 ( )V k k eD " e e (36)

The hypothesis (i) has now been proven.

With Lemma 2, if the signal ( )kY is persistentlyexciting, then the least-squares algorithm (13)-(14)

ensures that lim ( ) 0k

k

Q % . With the assumption 4,

lim ( ) 0k

k

Q % means the model become a perfect

representation of the plant, which will make the

model state ( )kx become the same as the plant state

( )kx . Thus we have lim ( ) ( ) 0i ik

w w

- =x x . With

the control law (9), we have proven lim ( ) 0k

k

e .

Therefore, when k , ( )kG in (33) converges

to zero and (32) becomes

21( ( )) ( )

2V k a k k D - e e (37)

we obtain ( ) 0k e as k . The hypothesis (ii)

and the Theorem 1 have now been proven.

5. SIMULATION RESULTSIn this example, the validity and effectiveness of theproposed control scheme are illustrated through the

tracking control of a single-link robot arm described

by1 2

2 1 1 2 2

( ) ( )

( ) sin[ ( )] ( ) ( )

q t q t

q t q t q t bu t a a

=

= - - +

&

& (38)

where1

( ) ( )y t q t= is the arm position which is the

measured output,2( )q t is the angular velocity which

is unmeasurable and ( )u t is the input torque. The

parameters1

a and2

a depend on the mass and

length of the arm. The values of1

a ,2

a and b are

chosen as1 2

1ba a= = = . The premise variables are

selected as1( ) [( 1) )]x kT y k T= - and

2( ) ( )x kT y kT= .

The sampling time 0.01secT= . The controlobjective is to make1

q track a bounded reference

signal. The initial T-S fuzzy model for closed-loopcontrol is an input-output model obtained by the on-

line identification method proposed in Qi and Brdys(2005) which follows the structure of (38).

0 10 20 30 40 50 60

-1

0

1

2

Time (sec)

Outputs

plant output

reference

0 10 20 30 40 50 60-0.05

0

0.05

0.1

0.15

0.2

Time (sec)

PlantTrackingError

Fig. 2. Plant tracking response with parameter

disturbance (2

a changes from 1 to 1.5 at 5t s= )

0 20 40 60-15

-10

-5

0

5x 10-3

Time (sec)0 20 40 601.99

1.9952

2.0052.01

Time (sec)

0 20 40 60-1-0.998-0.996-0.994-0.992

-0.99

Time (sec)0 20 40 60-1

0

1

23x 10

-3

Time (sec)

a30 a31

a32 a33

Fig. 3. Consequent parameter adaptation of rule 3

0 5 10 15 20 25 30 35 40-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

Time (sec)

Controlinput

Fig. 4. Control signal

Fig. 2 illustrates the plant tracking response withdisturbance and tracking error and Fig. 4 shows the

control signal. The plant parameter2

a changes from

1 to 1.5 at 5t s= , which causes the tracking error to

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increase after 5t s= . However, the controller is ableto adapt to the change and regain good tracking

performance. The tracking error decreases graduallyas the model is learning to become a goodapproximation of the plant. Parameter adaptation of

rule 3 is shown in Fig. 3. The learning rate W is

chosen as 61.0 10W = .

The reference trajectory in Fig. 5 is obtained by

passing a piecewise constant through a first-orderstable linear filter. A disturbance in

1a was

introduced at 44t s= . As a result, it can be seen thatthe tracking error increases after 44s . The parameter

adaptations are shown in Fig. 6.The control law canproduce control signal that adapts to the change and

regain good tracking performance.

0 10 20 30 40 50 60 70 80 90 100

-1

0

1

2

Time (sec)

Outputs

0 10 20 30 40 50 60 70 80 90 100

-0.2

0

0.2

0.4

0.6

0.8

Time (sec)

PlantTrackingError

plant output

reference

Fig. 5. Plant tracking response with parameter

disturbance (1

a changes from 1 to 1.5 at 44t s= )

0 50 100-0.1

-0.05

0

0.05

0.1

Time (sec)

0 50 1001.99

1.995

2

2.005

Time (sec)

0 50 100-1

-0.98

-0.96

-0.94

Time (sec)

0 50 1000

1

x 10-4

Time (sec)

30

^33

^31

32

Fig. 6. Consequent parameter adaptation of rule 2

6. CONCLUSION

The indirect adaptive fuzzy control algorithm for

uncertain nonlinear stable plants is presented in thispaper. A discrete-time T-S fuzzy model is employedas a dynamical model of an unknown continuous-

time plant with unmeasurable states. By propertraining, the T-S fuzzy model is able to achieveinput-output behaviour close to the plant and provideuseful information about the states of the plant. A

feedback linearization control is calculated entirelybased on the model, using the model structure,

parameters and states, while the parameters of themodel are updated on-line by RLSE method.Sufficient conditions stability and tracking error

convergence are derived for stable nonlinear plant. Itshould be noted here that although the errorboundedness or the closed-loop global stability are

proven under the assumption that the premiseparameters of the model are the same as in the plant,a local stability or the error boundedness are

achieved if the initial parameter values are closeenough to the plant parameter values. Simulationexamples verify the effectiveness, adaptation and

tracking performance of the proposed control scheme.

ACKNOWLEDGEMENT

This work has been partly supported by UniversitiesUK on behalf of the Department for Education and

Skills (DfES) through the UK funding councils andby the Polish State Committee for ScientificResearch under grant No. 4 T11A 008 25.

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