6
A Comparison Study between Static and Dynamic Recurrent Neural Networks Based Adaptive Control Of Nonlinear Multivariable Systems T. A. Al-Zohairy Community collage in ALRiyadh King Saud University ALRiyadh, Kingdom of Saudi Arabia [email protected] Abstract – This paper considers the problem of real time adaptive control of nonlinear multivariable systems. Two neural networks techniques are presented to solve the problem mentioned above. The first technique combines the ability of a single-layer feedforward neural network for modeling purposes and a linear control law to design the controller. The second technique combines the ability of dynamic recurrent neural network for modeling purposes and a linear control law to design the controller. In this paper, we consider that the state of the system is accessible. A comparison between the simulation results for the above two techniques are presented to complete the study. I. INTRODUCTION There has been considerable interest in the past few years in exploring the applications of artificial neural networks for identification and adaptive control of dynamical systems [1]-[7]. It has been realized by systems theorists that most real dynamical systems are nonlinear [6]. However, linearizations of such systems around the equilibrium states yield linear models which are mathematically tractable. From [9], it was found that the discrete-time linear multivariable system with m inputs and m outputs described by the state equations ) ( ) ( ) ( ) ( ) 1 ( k Cx k y k Bu k Ax k x = + = + (1) where n k x ) ( is the state vector at instant k, m ) k ( u is the input vector, and m k y ) ( is the output vector, can be represented by ) k ( u E ) k ( x A C ) d k ( y ) k ( u E ) k ( x A C ) d k ( y ) k ( u E ) k ( x A C ) d k ( y m d m m m 2 d 2 2 2 1 d 1 1 1 m 2 1 + = + + = + + = + L L L (2) where B A C E i d i i 1 - = and i d is the relative degree of the ith output i y . In the following we will see that the form (2) is important for the control developments given in sections II and III. The adaptive linear controller for the linear multivariable system of the form (2) is given by ). ( ) ( ) ( ) ( ) ( 1 2 1 1 2 1 k Fr k Gx k r E k x A C A C A C E k u m d m d d + = + = - - L (3) In the following we will show how (3) can be used as an adaptive linear controller for a nonlinear multivariable system. Consider a discrete-time nonlinear multivariable system S be described by the state equations S: x(k) C y(k) k)] f[x(k), u( ) 1 x(k = = + (4) where , ) k ( y ), k ( u m and n ) k ( x are the input, output, and state, respectively, at time t and C f . Consider the reference input ) k ( r is defined as m T m 2 1 )] k ( r , ), k ( r ), k ( r [ ) k ( r = L is specified and the state variables are accessible. The control problem in this paper is to determine the input ) k ( u so that ) k ( y i follows the reference input ) k ( r i . Note that we consider that f is unknown and only input, output state variables are known. In this paper, two control strategies are proposed, the first combines a single-layer feedforward network model with self tuning indirect adaptive control and the second combines a dynamic recurrent neural network model with self tuning indirect adaptive control. The proposed control structures are based on linearization of both single-layer feedforward and dynamic recurrent network models at every operating point. A standard linear state space model of the form (2) is derived in both strategies and a state space feedback decoupling controller of the form (3) is applied. With simultaneous online training of both single layer feedforward neural network and recurrent neural network with control synthesis the resulting two algorithms are indirect adaptive control laws. Some prior information concerning the system is required to determine the controller which are the order of the system and relative degree i d of each output i y . For known system some additional conditions are required to determine the controller as given in [8] which are: 1) the linearized system S L is controllable observable and can be decoubled using state feedback. 2) the plant satisfies the minimum phase condition. The paper is organized as follows: In section II two neural network architectures used for the identification process is given. Section III study the linearization of the nonlinear process obtained from the neural network. In section IV the algorithms that used to solve the control problem is stated in details. Section V gives the simulation

[IEEE 2008 3rd International Design and Test Workshop (IDT) - Monastir, Tunisia (2008.12.20-2008.12.22)] 2008 3rd International Design and Test Workshop - A comparison study between

  • Upload
    t-a

  • View
    215

  • Download
    3

Embed Size (px)

Citation preview

Page 1: [IEEE 2008 3rd International Design and Test Workshop (IDT) - Monastir, Tunisia (2008.12.20-2008.12.22)] 2008 3rd International Design and Test Workshop - A comparison study between

A Comparison Study between Static and Dynamic

Recurrent Neural Networks Based Adaptive

Control Of Nonlinear Multivariable Systems T. A. Al-Zohairy

Community collage in ALRiyadh

King Saud University

ALRiyadh, Kingdom of Saudi Arabia

[email protected]

Abstract – This paper considers the problem of real time adaptive control of nonlinear multivariable systems. Two neural networks techniques are presented to solve the

problem mentioned above. The first technique combines the

ability of a single-layer feedforward neural network for

modeling purposes and a linear control law to design the controller. The second technique combines the ability of dynamic recurrent neural network for modeling purposes

and a linear control law to design the controller. In this paper, we consider that the state of the system is accessible. A

comparison between the simulation results for the above two techniques are presented to complete the study.

I. INTRODUCTION

There has been considerable interest in the past few

years in exploring the applications of artificial neural

networks for identification and adaptive control of

dynamical systems [1]-[7].

It has been realized by systems theorists that most real

dynamical systems are nonlinear [6]. However,

linearizations of such systems around the equilibrium

states yield linear models which are mathematically

tractable.

From [9], it was found that the discrete-time linear multivariable system with m inputs and m outputs

described by the state equations

)()(

)()()1(

kCxky

kBukAxkx

=

+=+ (1)

where nkx ℜ∈)( is the state vector at instant k,

m)k(u ℜ∈ is the input vector, and mky ℜ∈)( is the

output vector, can be represented by

)k(uE)k(xAC)dk(y

)k(uE)k(xAC)dk(y

)k(uE)k(xAC)dk(y

md

mmm

2d

222

1d

111

m

2

1

+=+

+=+

+=+

LLL

(2)

where BACE idii

1−= and id is the relative degree of the

ith output iy . In the following we will see that the form

(2) is important for the control developments given in

sections II and III. The adaptive linear controller for the linear

multivariable system of the form (2) is given by

).()()()(

)(12

1

12

1

kFrkGxkrEkx

AC

AC

AC

Eku

mdm

d

d

+=+

= −−

L

(3)

In the following we will show how (3) can be used as an

adaptive linear controller for a nonlinear multivariable

system.

Consider a discrete-time nonlinear multivariable system

S be described by the state equations

S:x(k) Cy(k)

k)]f[x(k), u()1x(k

=

=+ (4)

where ,)k(y ),k(u mℜ∈ and n)k(x ℜ∈ are the input,

output, and state, respectively, at time t and ∞∈Cf .

Consider the reference input )k(r is defined as

mTm21 )]k(r , ),k(r ),k(r[)k(r ℜ∈= L is specified and

the state variables are accessible. The control problem in

this paper is to determine the input )k(u so that

)k(yi follows the reference input )k(ri . Note that we

consider that f is unknown and only input, output state

variables are known.

In this paper, two control strategies are proposed, the

first combines a single-layer feedforward network model

with self tuning indirect adaptive control and the second

combines a dynamic recurrent neural network model with

self tuning indirect adaptive control. The proposed control

structures are based on linearization of both single-layer

feedforward and dynamic recurrent network models at every operating point. A standard linear state space model

of the form (2) is derived in both strategies and a state

space feedback decoupling controller of the form (3) is

applied.

With simultaneous online training of both single layer

feedforward neural network and recurrent neural network

with control synthesis the resulting two algorithms are

indirect adaptive control laws.

Some prior information concerning the system is

required to determine the controller which are the order of

the system and relative degree id of each output iy . For

known system some additional conditions are required to

determine the controller as given in [8] which are:

1) the linearized system SL is controllable observable and

can be decoubled using state feedback.

2) the plant satisfies the minimum phase condition.

The paper is organized as follows: In section II two

neural network architectures used for the identification

process is given. Section III study the linearization of the

nonlinear process obtained from the neural network. In

section IV the algorithms that used to solve the control

problem is stated in details. Section V gives the simulation

Page 2: [IEEE 2008 3rd International Design and Test Workshop (IDT) - Monastir, Tunisia (2008.12.20-2008.12.22)] 2008 3rd International Design and Test Workshop - A comparison study between

study on a real example for the two methods mentioned

above. A comparison between simulation results for the

two presented methods is given in section VI and the

conclusion of our study is shown in section VII.

II. NETWORK ARCHITECTURES

The network architectures used in this paper depends on

the concept of relative degree of a nonlinear multivariable

system S given in (4).

It can be shown that if the vector relative degree exists for the multivariable system S in some neighborhood Ω of

the equilibrium point ( 0u ,0 ==x ), the nonlinear system

S can be expressed as

)]( ),([)(

)]( ),([)(

]u(k) ),([)(

222

111

kukxdky

kukxdky

kxdky

mmm Ψ=+

Ψ=+

Ψ=+

LLL (5)

Assume that the unknown nonlinear multivariable

system to be considered is expressed by (5) where

,)( ),( mdkyku ℜ∈+ and nkx ℜ∈)( . The two architectures

are described as follows:

A. Single-layer Feedforward Network and its Dynamics

The single-layer feedforward network architecture used

in this case is shown in Fig. 1, in which the input layer is

composed of )1mn( ++ nodes for n states, m inputs all at

time k and an additional node for input bias its value is always 1 and the output layer of m nodes for the m outputs

of the system at times m21 dk , ,dk ,dk +++ L . The

interconnection matrices are n,myxW ℜ∈ and 1m,myuW +ℜ∈ , respectively among the output-state nodes

and output-input nodes.

Fig. 1. The neural network structure.

The dynamics of the network is described by the following

equations:

][ˆ)(ˆ jjj Sdky Ψ=+ (6)

∑∑+

==

+=1m

1

yuj

n

1i

iyxjij )k(uw)k(xwS

l

ll (7)

The activation functions Ψ are hyperbolic tangent given

by (8)

+

ℜ∈+

−=Ψ b a, )(ˆ

asas

asas

ee

eebs (8)

Then the neuro model for the system (5) can be expressed

as

]u(k) ),([ˆ)(ˆ kxdky jj Ψ=+ (9)

B. Recurrent Network Architecture and its Dynamics

The recurrent network architecture taken in this paper is

that given in [10] in which a recurrent network is

composed of n neuron units, and m external inputs.

The dynamics of each neuron unit is described by

∑ ∑∈ ∈

+=+U X

xk

ukk twtwts

l l

llll )(x )(u )1( (10)

)1(ˆ)1(ˆ +=+ tsftx kk (11)

In the above equation xkwl

represents the weight of

connection from the thl neuron to the kth neuron, ukwl

represents the weight of the connection from the thl input

neuron to the kth neuron, )(tsk is an intermediate variable

and f is a nonlinear function, an hyperbolic tangent,

given by (12):

+−

ℜ∈+

−= b a, )ˆ(ˆ

ˆˆ

ˆˆ

xaxa

xaxa

ee

eebxf (12)

This equation can be rewritten, similarly to a nonlinear

state space model

Wu(t), ),(ˆˆ)1(ˆ txftx =+ (13)

Fig. 2 shows a recurrent network with three neuron units

and two external inputs. For this particular example the

neuron outputs are given as follows with 2 ,1 ,0=l .

)( )( )(

)( )( )( ˆ)1(

352413

322110

tuwtuwtuw

txwtxwtxwftx

lll

llll

++

+++=+ (14)

where )(3 tu is an additional input for bias its value is

always 1.

Fig. 2. Example of a recurrent network with three neuron units and two inputs.

The epochwise backpropagation through time algorithm

[10] is used to train a recurrent network.

III. LINEARIZATION OF THE NEURAL MODELS

The linearized models for the two nonlinear neural

models (9) and (13) can be obtained by computing the

derivatives from the outputs with respect to the inputs of

the network (x(k),u(k)).

A. Linearization of the Single-Layer Feedforward Network

The linearized system for the single layer network is

given by

Page 3: [IEEE 2008 3rd International Design and Test Workshop (IDT) - Monastir, Tunisia (2008.12.20-2008.12.22)] 2008 3rd International Design and Test Workshop - A comparison study between

u(k) K)k(x H)dk(y jj +=+ (15)

where the two matrices H and K are of orders )nm( × and

)1mm( +× respectively which are defined by

ˆ)(

ˆΨ′=

Ψ∂= yxW

kxH (16)

ˆW)k(u

ˆK yuΨ

Ψ′=

∂= (17)

B. Linearization of the Recurrent Network

The linearized system for the recurrent network is given

by

u(t)K )(ˆ )1(ˆ +≈+ txHtx (18)

where the two matrices H and K are of orders )( nn× and

)1( +× mn respectively which are defined by

)(

fW

kx

fH x ′=

∂= (19)

)(

fW

ku

fK

u ′=∂

∂= (20)

Experimentally, the linearized systems (15) and (18) are

similar to the linear system given in (2).

IV. ALGORITHMS DESCRIPTION

Fig. 3 shows the adaptive control structure when we use a single-layer and recurrent networks which consists of

1) the system (5).

2) a single-layer feedforward network or a recurrent

network which estimate ] , , ,[ m21 ΨΨΨΨ L= or

] , , ,[ 21 mffff L= respectively.

3) a controller realized by the linearization of the neural

model.

Fig. 3. Adaptive control structure when we use single layer network and

recurrent network.

A. Algorithm When single Layer Network Is Used

The neural model for the unknown system (5) can be

expressed as (9)

]u(k) ),k(x[ˆ)dk(y jj Ψ=+

where )dk(y jj + is the jth output of the neural network

and Ψ is the estimate of the functons m21 , , , ΨΨΨ L . The

training algorithm used guarantees that

.min)]dk(y)dk(y[

m

1j

2jjjj =+−+∑

=

(21)

The objective function J is defined as follows:

∑=

=m

1j

2j )k(e

2

1J (22)

where

)dk(y)dk(y)k(e jjjjj +−+= . (23)

The elements of the gradient of J with respect to both yxW

and yuW is given by the following two equations

ijjyxji

x].S[.eW

JΨ ′−=

∂ (24)

ijjyuji

u].S[.eW

JΨ ′−=

∂ (25)

The weight changes may be made along the negative

gradient of of J by means of the equations

yxji

yxji

W

JW

∂−= η∆ (26)

yuji

yuji

W

JW

∂−= η∆ (27)

Once the weights are changed, the linearization process

start using the process stated in section III. Then the

linearized system of the form (18)

u(k) K)k(x H)dk(y jj +=+

is obtained and the state feedback control low can be

defined as (3).

Then equation (3) obtained can now be used in computer

program for real time control.

A pesudocode outline of the algorithm when a single layer

is used:

Select initial weights

Select the state

Select the values of the desired set points r(k)

u(k) = any random value (very small) or equal to zero.

Repeat up to approximately no change occur in the control signal

produce )dk(y jj + using (9);

find J using (22);

update the weights using (24), (25), (26) and (27);

find the linearized model (15);

compute new control signals u(k+1) using (3);

)1k(u)k(u += ;

B. Algorithm When Recurrent Network Is Used

The neural model for the unknown system (4) can be

expressed as (14)

Wu(t), ),(ˆˆ)1(ˆ txftx =+

Where W is the weight matrix obtained by training the

network using BTT algorithm [10].

Using the linearization process given in section III the

linearized system of the form (18)

u(t)K )(ˆ )1(ˆ +≈+ txHtx

can be obtained.

Using the matrices H, K, and C consider that the relative

degree of the multivariable nonlinear system under study is

known, the linear decoupling system of the form (2) can be

obtained. Using the linear decoupling system (2) if the

desired output vector is denoted by )(tr , it follows that the

desired )(tu can be expressed as a linear combination of

Page 4: [IEEE 2008 3rd International Design and Test Workshop (IDT) - Monastir, Tunisia (2008.12.20-2008.12.22)] 2008 3rd International Design and Test Workshop - A comparison study between

)(tr and )(ˆ tx as given in (3). Now equation (3) can now be

used in computer program for control implementation.

A pesudocode outline of the algorithm when a recurrent

network is used:

Select initial weights

Select the state

Select the values of the desired set points r(t)

u(t) = any random value (very small) or equal to zero.

Repeat up to approximately no change occur in the control

signal

produce )1(ˆ +tx using (14);

update the weights using BTT algorithm [10];

find the linearized model (18);

compute new control signals u(t+1) using (3);

)1k(u)k(u += ;

V. SIMULATION RESULTS

Note that the learning rate used for the two networks used

in this section takes the value 0.001. The values of both a

and b in the activation function takes the values 1 and 8

respectively and random initial weights in the

range 0.1) ,1.0( − .

The plant considered here is a third-order system given

in [8] which is described by the state equations

)k(x)k(y );k(x)k(y

)k(u)]k(x2sin[3)1k(x

)k(x1

)k(x)]k(x4sin[1)k(x)1k(x

)k(u])k(x1

)k(x2)k(x

)k(u])k(x1

)k(u)k(x5.12[)]k(xsin[)k(x9.0)1k(x

2211

213

23

3332

221

11

121

11211

==

+=+

+++=+

++

++

++=+

where T

321 )]k(x ),k(x ),k(x[)k(x = represents the state,

T21 )]k(u ),k(u[)k(u = the input, and

T21 )]k(y ),k(y[)k(y = the output, at instant k.

The linearized system is controllable, observable, and is

of minimum phase. Hence, a nonlinear decoupling

controller exists for the nonlinear plant in a neighborhood of the origin. Our objective is to realize this nonlinear

controller using neural networks to make the two outputs

)k(y1 and )k(y2 follow two independent reference

signals )k(y*1 and )k(y*

2 , respectively. We assume that

)k(r)dk(y 11*1 =+ and )k(r)dk(y 22

*2 =+ are

specified at instant k. Hence, the control problem is to

determine a control input u(k) so that

)2,1i( 0)k(r)dk(ylim iiik ==−+∞→ .

A. Simulation Results When A Single-Layer Network Used

Using the network architecture and Algorithm given in

sections II and IV respectively it was found that:

The state input output neural model obtained in this case

is given by:

]. u w u w

u w x w x w x w[ˆ)2k(y

]. u w u w

u w x w x w x w[ˆ)1k(y

2yu121

yu11

0yu102

yx121

yx110

yx102

2yu021

yu01

0yu002

yx021

yx010

yx001

+

++++=+

+

++++=+

Ψ

Ψ

where 1u2 = is the additional input used for bias.

After a weight change obtained using (26) and (27) the linearization process started using the method given in

section II which in our example is given by

+

=

+

+

2

1

0

2

1

1

1

0

1

2

0

1

0

0

0

2

1

0

2

1

1

1

0

1

2

0

1

0

0

0

2

1

u

u

u

u

ˆ

u

ˆ

u

ˆ

u

ˆ

u

ˆ

u

ˆ

x

x

x

x

ˆ

x

ˆ

x

ˆ

x

ˆ

x

ˆ

x

ˆ

)2k(y

)1k(y

ΨΨΨ

ΨΨΨ

ΨΨΨ

ΨΨΨ

(28)

Now to find a control low that match our purpose, we

will use a control low similar to that given in (3). Then we

replace )1k(y1 + and )2k(y2 + by the desiresd

prespecified setpoints )k(r1 and )k(r2 . Then we have to

find a control signals 1u and 2u in terms of the states and

desiresd prespecified setpoints. The process is shown in the

following equations:

+

=

2

1

0

2

1

1

1

0

1

2

0

1

0

0

0

2

1

0

2

1

1

1

0

1

2

0

1

0

0

0

2

1

u

u

u

u

ˆ

u

ˆ

u

ˆ

u

ˆ

u

ˆ

u

ˆ

x

x

x

x

ˆ

x

ˆ

x

ˆ

x

ˆ

x

ˆ

x

ˆ

)k(r

)k(r

ΨΨΨ

ΨΨΨ

ΨΨΨ

ΨΨΨ

(29)

. ˆ

ˆ

u

ˆ

ˆ

ˆ

ˆ

x

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

)(

)(

2

2

1

2

0

1

0

1

1

0

1

1

0

0

0

2

1

0

2

1

1

1

0

1

2

0

1

0

0

0

2

1

u

u

u

u

uu

uu

x

x

xxx

xxx

kr

kr

Ψ∂

Ψ∂

+

Ψ∂

Ψ∂

Ψ∂

Ψ∂

+

Ψ∂

Ψ∂

Ψ∂

Ψ∂

Ψ∂

Ψ∂

=

because 1u 2 = then the controller can be given as

.

x

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ)(

ˆ)(

ˆ

ˆ

ˆ

ˆ

u

2

1

0

2

1

1

1

0

1

2

0

1

0

0

0

2

12

2

01

1

1

1

0

1

1

0

0

0

1

0

Ψ∂

Ψ∂

Ψ∂

Ψ∂

Ψ∂

Ψ∂

Ψ∂−

Ψ∂−

Ψ∂

Ψ∂

Ψ∂

Ψ∂

=

x

x

xxx

xxx

ukr

ukr

uu

uu

u (30)

Page 5: [IEEE 2008 3rd International Design and Test Workshop (IDT) - Monastir, Tunisia (2008.12.20-2008.12.22)] 2008 3rd International Design and Test Workshop - A comparison study between

equation (30) represent the state feedback control low used

which is similar to that given in (3).

Once the control signals are obtained using (30) it can

be used instead of the old value.

The processes of the training, linearization, and

obtaining the control signals are continued until no change

occur in the control signals.

The performance of the proposed method for controlling

the system given in this section was tested with the sinusoidal reference inputs given by

+

=

+

=

20

k2sin75.0

30

k2sin75.0)k(r

10

k2sin75.0

50

k2sin75.0)k(r

1

1

ππ

ππ

(31)

is shown in figures 4, 5, and the control signals are shown

in Fig. 6.

-2

-1

0

1

2

0 30 60 90 120 150 180 210 240 270 300

k

r1 &

y1

-2

-1

0

1

2

0 30 60 90 120 150 180 210 240 270 300

k

r2 &

y2

Fig. 4. Performance of linear controller when the plant equations are unknown; solid line is the desired output, and dotted line is the actual

output of the plant.

-2

-1

0

1

2

0 30 60 90 120 150 180 210 240 270 300

k

r1 &

yL

1

-2

-1

0

1

2

0 30 60 90 120 150 180 210 240 270 300

k

r2 &

yL

2

Fig. 5. Performance of linear controller on a linearized model; solid line is

the desired output, and dotted line is the output of the linear model.

-1

-0.5

0

0.5

1

0 30 60 90 120 150 180 210 240 270 300

k

u1

& u

2

Fig. 6. The required control signals; solid line represent 1u and dotted

line represent 2u .

B. Simulation Results When A Recurrent Network Used

Using the network architecture and the algorithm, given

in sections II and IV respectively, with an epoch size

3=T it was found that after a weight change obtained

using the BTT training algorithm the linearization process

started using the method given in section III which in our

example at 1=t is given by

+

=

02

01

00

02

12

01

12

00

12

02

11

01

11

00

11

02

10

01

10

00

10

02

01

00

02

12

01

12

00

12

02

11

01

11

00

11

02

10

01

10

00

10

12

11

10

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

u

u

u

u

f

u

f

u

f

u

f

u

f

u

f

u

f

u

f

u

f

x

x

x

x

f

x

f

x

f

x

f

x

f

x

f

x

f

x

f

x

f

x

x

x

(32)

The linearization process for t = 2, 3 can be obtained easily

by the same method.

In case of t = 1, 2, 3, 1221202 === uuu is the additional

input used for bias. Then the corresponding output at t =1

is given by

=

12

11

10

11

10

ˆ

ˆ

ˆ

0 1 0

0 0 1

ˆ

ˆ

x

x

x

y

y

(33)

The output at t = 2, 3 can be obtained.

The linear decoupling system in this case at time 1=t is

given by

[ ] [ ]

+

=

02

01

00

020100

02

01

00

02

12

01

12

00

12

02

11

01

11

00

11

02

10

01

10

00

10

10 E1 E1 1

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

0 0 1ˆ

u

u

u

E

x

x

x

x

f

x

f

x

f

x

f

x

f

x

f

x

f

x

f

x

f

y

[ ] [ ]

+

=

02

01

00

121110

02

01

00

02

12

01

12

00

12

02

11

01

11

00

11

02

10

01

10

00

10

11 E1 E1 1

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

0 1 0ˆ

u

u

u

E

x

x

x

x

f

x

f

x

f

x

f

x

f

x

f

x

f

x

f

x

f

y

(34)

Where

00

1000

ˆ1

u

fE

∂= .

01

1001

ˆ1

u

fE

∂= .

02

1002

ˆ1

u

fE

∂= .

00

12

02

11

00

11

10

11

00

10

00

1110

ˆ

ˆ

ˆˆ

ˆ

ˆˆ

ˆ

ˆ1

u

f

x

f

u

f

x

f

u

f

x

fE

∂+

∂+

∂= .

01

12

02

11

01

11

10

11

01

10

00

1111

ˆ

ˆ

ˆˆ

ˆ

ˆˆ

ˆ

ˆ1

u

f

x

f

u

f

x

f

u

f

x

fE

∂+

∂+

∂= .

02

12

02

11

02

11

10

11

02

10

00

1112

ˆ

ˆ

ˆˆ

ˆ

ˆˆ

ˆ

ˆ1

u

f

x

f

u

f

x

f

u

f

x

fE

∂+

∂+

∂= .

The linear decoupling system (34) can be represented in

the following form

+

=

02

01

00

121110

020100

02

01

00

121110

02

10

01

10

00

10

11

10

E1 E1 1

E1 E1 1

ˆ

ˆ

ˆ

A A

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

u

u

u

E

E

x

x

x

A

x

f

x

f

x

f

y

y

(35)

Page 6: [IEEE 2008 3rd International Design and Test Workshop (IDT) - Monastir, Tunisia (2008.12.20-2008.12.22)] 2008 3rd International Design and Test Workshop - A comparison study between

Where

00

12

02

11

00

11

01

11

00

10

00

1110

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

x

f

x

f

x

f

x

f

x

f

x

fA

∂+

∂+

∂= .

01

12

02

11

01

11

01

11

01

10

00

1111

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

x

f

x

f

x

f

x

f

x

f

x

fA

∂+

∂+

∂= .

02

12

02

11

02

11

01

11

02

10

00

1112

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

x

f

x

f

x

f

x

f

x

f

x

fA

∂+

∂+

∂= .

From the system (35) the controller can be represented as

=

02

01

00

121110

02

10

01

10

00

10

1201

0200

1

1110

0100

01

00

ˆ

ˆ

ˆ

A A

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

1

1

E1 1

E1 1

x

x

x

A

x

f

x

f

x

f

Er

Er

E

E

u

u

(36)

The linear decoupling systems at times 2=t and 3=t can

be obtained in a similar fashion.

Once the control signals are obtained using (36) it can be

used instead of the old value.

The process of the training, linearization, and obtaining the

control signals are continued until no change occur in the

control signal.

The performance of this method for controlling the

system was tested with the sinusoidal reference inputs

given by (31) is shown in figures 7 and 8.

VI. COMPARISON BETWEEN SINGLE-LAYER AND

RECURRENT CONTROLLERS In this section, we will make a comparison between the

two models given in section V to solve the control problem

mentioned in section I.

From the results obtained during the simulation, it is

clear that, recurrent controller needs intensive

computations compared with the single-layer controller for

computation of u(k).

In comparison between the two models presented in

this paper it was found that:

1) the single-layer model use a simple training algorithm

while the recurrent model is computationally intensive. 2) the single-layer model is more suitable for real time

problems than that of recurrent model.

3) the performance given by the single-layer model is more

accurate than that for the recurrent model the root mean

square error for the single-layer network equal 0.3 610−×

and for the recurrent network equal 0.000006 which give

an advantage for a single-layer controller .

VII. CONCLUSION

In this paper we study the ability of both a single layer

and recurrent models to find a real time adaptive

controller. From our study it was found that the two

models used are very important in the field of adaptive control of nonlinear dynamical systems and they supply us

with very acceptable results. Also it was found that the

effectiveness of the single-layer strategy was validated in

real time tracking of a multivariable system than that of the

recurrent strategy because of its simple architecture and

lowest of computationally used in the learning process.

Future work will include a study the adaptive control

problem using radial-Basis function networks. The

comparison between the two mentioned models with the

radial-Basis function model will also be made.

-2

-1

0

1

2

0 30 60 90 120 150 180 210 240 270 300

t

r1 &

y1

-2

-1

0

1

2

0 30 60 90 120 150 180 210 240 270 300

t

r2 &

y2

Fig. 7. Performance of the recurrent neural network controller when the state variables are accessible, but the plant equations are unknown; solid line is the desired output, and dotted line is the actual output of the plant.

-1

-0.5

0

0.5

1

0 30 60 90 120 150 180 210 240 270 300

t

u1

& u

2

Fig. 8. The required control signals; solid line represent 1u and dotted

line represent 2u .

VIII. REFERENCES

[1] O. Patrick and N. Sadegh, "control of discrete-time systems via online learning and estimation," International conference on advanced intelligent mechatronics, California, USA, 24-28 July, 2005. [2] M. I. Jordan and R. A. Jacobs, “Learning to control an unstable system with forward modeling,” Advances in Neural Inform. Processing Syst. Vol. 2, pp. 324-331, 1990.

[3] K. S. Narendra and K. Parthasarathy, “Identification and control of dynamical systems using neural networks,” IEEE Tran. Neural Networks, vol. 1, pp. 4-27, 1990.

[4] A. U. Levin and K. S. Narendra, “control of nonlinar dynamical systems using neural networks: Controllability and stabilization,” IEEE Tran. Neural Networks, vol. 4, no. 2. pp. 192-205, 1993.

[5] A. U. Levin and K. S. Narendra, “Control of nonlinear dynamical

systems using neural networks Pare II: Observability, Identification,

and control,” IEEE Tran. Neural Networks, vol. 7, no. 1, pp. 30-42, 1996. [6] K. S. Narendra and S. Mukhopadhyay, “Adaptive control of dynamical using neural networks and approximate models,” IEEE Tran.

Neural Networks, vol. 8, no. 3, pp. 475-485, 1997. [7] T. A. Al-zohary and A. M. Wahdan, "Adaptive Control for nonlinear multivariable systems using a single layer neural network and conventional linear controller," the 11th Mediterranean conference on control & automation, Greek, 2003. [8] K. S. Narendra and S. Mukhopadhyay, “Adaptive control of nonlinear multivariable systems using neural networks,” Neural Networks, vol. 7, no. 5, pp. 737-752, 1994. [9] P. L. Falb and W. A. Wolovich, “Decoupling in the design and synthesis of multivariable control systems,” IEEE Tran. On Automatoc control, vol. 12, pp. 651-659, 1967. [10] R. J. Williams and D. Ziposer, "A learning algorithm for continually

running fully recurrent neural networks," Neural Computation, vol. 1, no. 2, pp. 270 – 280, 1989.