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Cerebellar Model Articulation Controller Simple Adaptive

Control

Shiqi An

Automation and Electrical Engineering Institute

Qingdao University of Science & Technology

Qingdao, Shandong Province, China

anshiqi@126.com

Abstract - Combined Cerebellar Model Articulation

Controller neural network with Simple Adaptive Control, a kind

of new control method, Cerebellar Model Articulation Controller

Simple Adaptive Control is proposed, structures and learning

algorithms of this control method are derived in this paper. In

the design, fast learning of Cerebellar Model Articulation

Controller neural network and simple structure of Simple

Adaptive Control are combined. The simulation results show that

the proposed method has fine accuracy, dynamic performance

and robustness, and it is feasible and effective to be used to

control high-order linear systems and nonlinear systems.

Index Terms - Cerebellar Model Articulation Controller;

neural network; learning algorithm; Simple Adaptive Control;

controller.

I. INTRODUCTION

Most self-tuning and adaptive control algorithms usually

use reference models, controllers, or identifiers of the same

order as the controlled plant. Since the dimension of the plants

in the real word may be very large or unknown,

implementation of adaptive control may be difficult, or

sometimes impossible. Simple Adaptive Control (SAC)

techniques that have been developed for over 20 years that

can use low-order model reference and controllers, no

observers or identifiers are used in the adaptation process.

But the traditional SAC is limited to the linear controlled

plants, adopting PI control algorithms, its function and scope

is limited. In fact, the form of the SAC algorithms are various,

the controlled plants may be linear ones, or nonlinear ones.

This paper adopts cerebella model neural network and

learning algorithms to single input and single output system,

proposes the Cerebellar Model Articulation Controller

(CMAC) SAC.

II. CEREBELLAR MODEL ARTICULATION CONTROLLER

The CMAC is a kind of neural network proposed by

Albus in 1970’s [1]. CMAC has the characteristic of the local

association and learning ability by Lookup Table. It has the

advantages of fast learning, so it is suitable for real-time

control.

Learning in the network of the CMAC uses the ? learning

rate. Its formula is

??

/ cdwsFF

i

))((

0

??

(1)

? is learning rate, C is the constant of generalization, F0 is

teacher's signal, F(si) is for the actual output of the network,

F(si)=wx0? x0 is the output vector, chooses C unit

components from unit components of x0, equals 1, and the

others are zero.

If ? =1, then the modification of the weight

F(w

???

0

F0 means the expectation output, F(si) means the actual

output, and the required accuracy can be satisfied by

regulating weight. Rewrite the formula (2) to be each weight

modification formula

tw

ij

??

) 1(

c/ ))s(F

i

(2)

C

tw

i

ij

?

?

?

?

)(

(3)

Where,

The characteristics of the CMAC are as follows:

(1) Learning structure based on the local area.

(2) Adopting simple ? learning rule.

(3) Because the CMAC is local network, the weight

adjusted each time is C, the learning speed is quick, and the

local smallest value does not exist.

(4) The CMAC generalization ability is related to

generalization constant C, as C enlarges, the generalization

ability strengthens.

(5) The main parameters

performance are generalization constant C, overlap extent

between close network input and input quantification, which

influence approximation accuracy, generalization ability and

learning speed.

(6) When the input dimensions increase, the storage also

increases.

III. SAMPLE ADAPTIVE CONTROL

SAC is a new direct adaptive control algorithm proposed

by K.Sobel and H.Kaufman, SAC has the characteristics of

simple control structure and less regulated parameters. The

system performance relates to the reference model selected. It

is irrespective to the controlled plant in control system design

and applicable to the single variable, multi-variable and

nonlinear system. The SAC has the extensive development

foreground in the industry process control.

The structure of SAC is showed in figure 1.

) s (FF

iii

??

0

?

?

iji

w)s(F

determined network

1-4244-0828-8/07/$20.00 © 2007 IEEE.

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Proceedings of the 2007 IEEE

International Conference on Mechatronics and Automation

August 5 - 8, 2007, Harbin, China

Page 2

Consider the following controllable and observable but

parameter plant model of order n with single input and single

output

??

?

)(

k

pp

xCy

The output of the controlled plant is required to follow the

output of a low-order model

?

?

)(

k

mm

xCy

The low-order model incorporates within it the desire

input-output behaviour of the plant, but is otherwise free. For

example, plant of order (say) 100 may be required to follow

the input-output behaviour of well-designed first-or second-

order models.

The objective is to design an adaptive controller such that

the plant output yp(t) approximates the ym(t) of the reference

model by using the available information of the plant.

Furthermore, it is required to achieve this aim by using

simple adaptive controllers of the form

)()()()(

kkkk

mxyep

xKeKu

??

Where

)(

k

y

e

?

)([)(

kk

e

KK

?

)([)(

kk

y

er

?

The control objective is to achieve the following relation

lim

k

As a result, the basic adaptive algorithm is given as

following

)(

k

KK

?

()()(

?

yP

kkk

reK

( ) 1()(

???

yII

kkk

eKK

??

?

???

)(

)()( 1)(

k

kkk

p

ppppp

uBxAx

(4)

?

?

???

)(

)()( 1)(

k

kkk

m

mmmmm

uBxAx

(5)

)()()()()(

kk

~

K

kk

~

K

k

~~

mu

ru

??

(6)

)

K

()(

kk

~

K

x

pm

y

)

)

k

y

?

(7)

~

u

)](

k

m

u

)]((

(

k

T

k

~~

x

T

m

(8)

(9)

TT

0)(

?

??

k

y

e

(10)

)

T

()(

k

~

K

k

~~

IP

T

)

r

?

(11)

0

??

PP

T

??

II

TT

)

T

P

T

T

,

k

~

(12)

(13)

0)(

T

I

,

~~

T

IV. CEREBELLAR MODEL ARTICULATION CONTROLLER

SAMPLE ADAPTIVE CONTROL

The adaptive controller in the traditional SAC is replaced

by the CMAC, the CMACSAC is composed by controlled

plant, reference model and the CMAC, etc (figure 2).

The controlled plant is single input and single output

linear or nonlinear system. The nonlinear plant can be

described as follows

(, ),1(()(

ppp

nkykyfky

???

?

(14)

e

desc

?

)()

kk

ppp

pppp

xC

(15)

Where, xp is the n-order plant state vector, up

or, yp is the output vector, and Ap, Bp, Cp are matrices with

the appropriate dimensions.

In Figure 2, the input of the CMAC is u , x , e , the

output of the CMAC is uc, and the total input of the controlled

plant is u.

The CMAC is placed parallelly on PID controller, its

ructu stre is similar to the control structure of Miller

changes the excitation signal of the CMAC, that is to say, take

reference model um, reference model state xm and the

output tracking error of the system ey as input, regulate the

weight by the difference between the CMAC output uc and the

total input u, so using the information of reference model of

the SAC, joining the fast learning in the CMAC, increase the

performance of the control system.

V. CEREBELLAR MODEL ARTICU

SAMPLE ADAPTIVE CONTROL LEARNING ALGORITHM

The CMACSAC has two processes: control and learn

principle is all weights of the CMAC are zero in the

initial state. While control, serve the quantization input um, xm,

ey of the CMAC as address, input the CMAC, find out

corresponding C addresses in memory of internal mapping of

CMAC, and add these weights in the C addresses, get the

output of the CMAC, namely?

?

?

vect

mmy

[8]

LATION CONTROLLER

ing.

The

j

jk

(

w

1

)

(16)

Then add with PID controller out

cont

rol input of the controlled plant u(k), namely

)(

ku

At the end of each control perio

add

ress, input the CMAC, compute the corresponding output

uc(k) of the CMAC, and compare with total control input u(k),

modify weight, enter into learning process. The purpose of

learning is to make the difference between total control input

and the output of the CMAC is minimum; That is to say, by

learning, the total control input of the system is mainly

produced by the CMAC. Its weight regulating rule is

kwkw

ll

)( ) 1(

???

p

))(,),1( );

bppa

nkuku

??

?

When the plant is linear system, the plant can b

ribed as follows

??

(

y

?

???

)()( ) 1(

kukk

p

BxAx

is the control

, but it

?

c

ck

(

u

)

put u (k), get the total

)()(

kuku

pc

??

(17)

d, use um, xm, ey as

c

kuku

c

)()(

?

?

(18)

?

???

??

??

??

Ke

Kx

Ku

ey

ym

xm

um

yp

Gm(s)

Gp(s)

Fig 1. Simple Adaptive Control system

??

??

?

?

PID

CMAC

C ontroller

ey

ym

xm

u

yp

Model

Plant

uc

up

um

Fig 2. Cerebellar Model Aroller Simple Adaptive Control ticulation Contr

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Use linear function in quantization process

round?

??

?

??

?

??

mini maxi

minii iq

xx

xxx

)(

i N

(19)

Where, xi is the input member of the CMA

quantization value of the input member, ximax is the maximum

valu

e of the input member, ximin is the minimum value, Ni is

corresponding value of quantization xi , round(?) is integer

function.

The control algorithm of the system is

C, xiq is the

c

?

?

i

)

k

Where, ai is the binary select vector, C is the

generalization constant of the CMA

corr

esponding output of CMAC, up(k) is the output of the

conventional controller PID.

The adjustment of the CMAC is

kukE

()(

?

(

2

uku

kw

)(

???

?

iic

wku

1

?

)(

a (20)

) (21)

(()(

kuuku

pc

?

C, uc(k) is the

c

u

ku

a

1

i

2

))(

(22)

c

)

?

i

p

i

w

c

a

k)(

c

w

(23)

a

c

(

k

)()(

??

?

))1()(() ) 1

?

()(

??????

kkkwkwkw

?

(24)

Where, ? is learning rate of the network, ??(0,1)?? is a

momentum???(0,1).

When the system begins to work, place w=0, at this time,

uc=0, u=up, the system

troller.?By the learning of the CMAC, the output up(k) of

the PID is to be zero gradually, and the output uc(k) of the

CMAC approaches to total output u(k) of controller gradually.

VI. SIMULATION RESULTS

Choose the linear high-order controlled plant and the

led plant respectively to make veri

ctiveness and feasibility of the CMACSAC, use above-

mentioned control algorithms to carry on the simulation

research. Compared conveniently, the following simulation

process uses the same ideal reference model.

The ideal reference model

?

??

. 04 . 0

??

1

is controlled by conventional

con

nonlinear control

effe

fication of

?

??

?

?? ??

k

)( 7 . 01

?

)(

)(

0

1

0

xky

ku

mm

mm

(25)

Its zero-poles respectively are: Z=-0.7; P=-0.8,

the good dynamic and static characteristic.

The high- order controlled plant

zG

zz

?

?

?

)(

32

kx

?

??

??

??

) 1(kxm

0.4, has

0625 . 0

?

05. 0 02. 0 26 . 0

234

??

8 . 0

?

z

)(

?

?

z

z

(26)

In fact, this is a four-order plant with three ste

delay, its zero point is Z=-0.8; its extremity is P=0.3 ±0

0.4

±0.3i. Figure 3 shows the unit square wave response; there

are the greater overshoot and static error.

p

ps pure

.4i and

0 50100 150200250300350 400

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

Time/Step

Up

Yp

Fig 3. The unit square wave response of high-order controlled plant

model is ±1, the period is 50 steps square signals. Choose the

C=1

sponse and the output tracing error, such as figure 4 show. It re

is thus clear that, the CMACSAC is more successful to the

control of the high-order plant, the plant output is can tracking

to given signal, the average error of steady state close to in

zero.

The input value of reference that establishes the reference

put 0, the controlled plant that simulation get the out

0 50100150 200 250 300350 400

-2

-1

0

1

2

0 50100150200250 300 350 400

-2

-1

0

1

2

050100150200250300350400

-4

-2

0

2

4

time/step

Um

U

Ym

Yp

Ey

Fig 4. Simulation results to high-order controlled plant (26)

To investigate the adaptive situations of the control

algorithms to controlled plant parameter, start to change a

couple

ant from the 76 steps of simulation, namely change

p=0

.3±0.4i into p=0.3±0.6i, the mathematics model becomes?

zGp

trolled

of extremities in mathematics model of the con

pl

1125 . 0

?

21. 022. 02 . 0

?

8 . 0

2

z

)(

34

??

?

?

zzz

z

(27)

has stronger adaptation ability.

The nonlinear controlled plant

) 1( 5 . 0 ) (

???

kyky

pp

Figure 6 shows the unit square wave response of the

nonl

inear controlled plant (formula

Figure 5 gives simulation results; this control algorithm

) 3

?

k

(8 . 0

?

) 2

?

k

( ) 3

?

k

( ) 2

?

k

( 3 . 0

?

) 2

?

k

(

?

uuyyy

ppppp

(28)

28), the characteristics of

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nonlinear and dissymmetry of the controlled plan

v ob ious.

t are

0 50 100150200250300350 400

-2

-1

0

1

2

0 50100150 200 250300 350400

-4

-2

0

2

0 50 100 150 200 250300 350400

-4

-2

0

2

4

time/step

Um

U

Ym

Yp

Ey

Fig 5. The simulation results of nonlinear controlled plant while parameter

changed

0 50 100150200 250 300350 400

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

Time/Step

Up

Yp

Fig 6. The unit square wave response of nonlinear controlled plant

0 50100 150200 250300 350400

-1

-0.5

0

0.5

1

0 50 100 150 200250300350 400

-2

-1

0

1

0 50100 150200250300 350400

-2

-1

0

1

2

time/step

Um

U

Ym

Yp

Ey

Fig 7. The simulation results of nonlinear controlled plant (28)

Choose C=10, get the simulation output responses of the

controlled plant and output tracing error, see figure 7. The

result shows, the CMACSAC is also more successful to the

control of nonlinear plants and the dissymmetry plants, the

plant output can track to the given signal, and the average

error of steady state closes to zero.

Although the control algorithm of the CMACSAC is

trained by the output of the controller of PID, it is not simple

replication of the controller of the PID.?Joining the controller

of PID is for training the CMACSAC and judging the

performance, strengthen the stability of the system, and

restrain disturbances.?When PID controls alone, the gain kP

value determines the control result to a large extent, but use

the PID plus the CMACSAC, the result does not depend on

the kP value, the kP value only needs within a reasonable

scope.

From the simulation results, in the beginning, the

conventional PID controller mainly acts, by the learning to the

output of the conventional controller continuously, gradually,

the ou tput of the CMACSAC starts from control action.

Join the CMACSAC, the control result is better than that

of simple PID control, while input square wave, reduce

overshoot, speed up control response, embody the

characteristics of combining the CMAC with the SAC fully?

that is small output error, good real time, and strong

robustness, etc.

In condition of input variety, discover that the control

system enters the stable output state quickly because of

joining the CMACSAC, especially to the nonlinear controlled

plant, have more obvious control result. The CMACSAC

overcomes unavoidable disadvantage of the conventional

controller to the certain extent, the control result gets raised.?

The simulation results prove that it is effective and feasible for

the control algorithms to the high-order controlled plants and

nonlinear controlled plants.

VII. CONCLUSIONS

The paper proposes the structure and learning algorithms

of the CMACSAC, which inherits the succinct structure of the

SAC, and have the advantage of the CMAC, can u

rol the high-order controlled plants and nonlinear

controlled plants. Because of the variety of parameter learning

algorithm composed controller, have greater flexibility in

actual design. By the simulation to linear high-order

controlled plants and nonlinear controlled plants, the results

show proposed

application, also have stronger learning ability and adaptable

ability to the change of controlled plants. Pay attention in the

simulation, the selection of the generalization constant C,

learning rate of the network and a momentum influence

control result, the selection of the parameter of the PID also

influences the beginning control of the system greatly. In

actual control, the selection of the parameter of the PID, we

can pass the simulation to make sure the choice scope of the

PID parameter, generalizati

an be determined by si

se to

cont

control algorithm has broad arrange of

on constant C, network learning

mulation. rate and momentum c

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