NN-based model structures for identification and control

of nonlinear systems.Non-fully connected neural

networksEduard Petlenkov,TTÜ automaatikainstituudi dotsenteduard.petlenkov@dcc.ttu.ee

Model Based Control

Model structure independent Model structure dependent

Neural Networks (NN) based Models

•Choosing proper structure of the model may increase accuracy of the model•Adaptivity by using network’s ability to learn

Model may become•Nonanalytical or hardly analytical•Overparameterized

•Easy realization of the model’s structure relevant to the control algorithm •Adaptivity by using network’s ability to learn•Analytical models (for example, state-space representation of nonlinear models)

•Restricted class of systems to which the control technique can be applied

Linear dynamic system with static actuator nonlinearity

H(s), H(z)

Neural Network Structure for dynamic systems with static actuator nonlinearity

DC Servo Motor with Nonlinear Driver

-0.4 -0.2 0 0.2 0.4 0.6-10

-8

-6

-4

-2

0

2

4

6

8

10

Vi - input of the driver (control signal)

Vo

- ou

tput

of

the

driv

er (

inpu

t vo

ltage

of

the

mot

or)

aa

a

aaa

a

t

a

v

a

a

a

a

Vi

y

VLi

J

B

J

kL

k

L

Ri

dt

d

010

0

1

Vi(t)Vo(t) y(t)

DC Servo Motor

Identification of the DC Servo Motor

-1.5 -1 -0.5 0 0.5 1 1.5-60

-40

-20

0

20

40

60

Driver input

velo

city

, ra

d/se

cmotor velocitymodel output

sects 001.0

Nonlinear Discrete-Time Input-Output Models

))1(,),1(),(),1(,),1(),(()( ntututuntytytyfnty

NARX (Nonlinear Autoregressive Exogenous) model:

))1(),1(())1(),1(())(),(()( 21 ntuntyftutyftutyfnty n

ANARX (Additive Nonlinear Autoregressive Exogenous) model:

or

n

ii ituityfnty

1

))1(),1(()(

Advantages of ANARX Structure over NARX

• Easy to change the order of the model

• Always realizable in the classical state-space form

n

ii ituityfnty

1

))1(),1(()(

)()(

))(),(()(

))(),(()()(

))(),(()()(

))(),(()()(

1

1

111

1232

1121

txty

tutxftx

tutxftxtx

tutxftxtx

tutxftxtx

nn

nnn

• Linearizable by dynamic feedback

Neural Network based ANARX (NN-ANARX) Model

n-th sub-layer

1st sub-layer

n

i

Tiii ituityWCnty

1

)1(),1()(

i i

is a sigmoid functioni

ANARX Model based Dynamic Output Feedback Linearization Algorithm

)(),()()1(

)(),()()1(

)(),()()1(

)()(),(

1

112

221

11

tutyftvt

tutyftt

tutyftt

ttutyfF

nn

nnn

NN

Tnnnn

Tnnnnn

T

T

tutyWCtvt

tutyWCtt

tutyWCtt

ttutyWCF

)(),()()1(

)(),()()1(

)(),()()1(

)()(),(

1

11112

22221

1111

n

ii ituityfnty

1

))1(),1(()(

ANARX model NN-ANARX model

n

i

Tiii ituityWCnty

1

)1(),1()(

))(),(()( 11 ttyFtu

NN-ANARX Model based Control of Nonlinear Systems

y(t)u(t)

Reference signal v(t)

Nonlinear SystemDynamic Output

Feedback Linearization

NN-based ANARX model

Parameters (W1, …, Wn, C1, …, Cn)

* HSA=History-Stack Adaptation

Controller

NN-ANARX Model based Control of Nonlinear MIMO Systems:Problem Statement

)()(),( 1111 ttUtYWCF T )(tU

SISO systems: Numerical calculation of u(t) by Newton’s Method, which needs several iterations to converge

MIMO systems: Numerical algorithms do not converge or calculation takes too much time

NN-based Simplified ANARX Model (NN-SANARX)

n-th sub-layer(nonlinear)

n

iiii itZWCtZWCty

211 )()1()(

1tZ

ntZ

NN-SANARX Model based Control (1) )(,),(,,,,, 11111111 tttututytyWC m

Trm

TrT

m tutuWtytyW ,,,, 112111

=

1111 WCT 1212 WCT Let’s define:

)(,),(,,,,, 1111111 tttututytyWC mT

rm (*)

mmT 1rmT 2

)(,),(,,,, 1111211 tttutuTtytyT mT

rT

m

where

If m=r and T2 is not singular or close to singular then

Tm

Tm

Tr tytyTttTtutu ,,,,,, 11,11,1

121

NN-SANARX Model based Control (2)

Tm

Tm

Tr tytyTttTtutu ,,,,,, 11,11,1

121

Trmnnn

mT

mnn

TTrmnnn

Tmnn

Tmnn

Trm

Tm

Tm

tututytyWC

tvtvtt

tututytyWC

tttt

tututytyWC

tttt

,,,,,

,,1,,1

,,,,,

,,1,,1

,,,,,

,,1,,1

11

1,11,1

11111

,11,1,21,2

11222

,21,2,11,1

Numerical Example

)(2.0)(4.0)(1

)()(2.0)1(

)(5.0)(2.0)(1

)()(4.0)1(

1322

2

222

2312

1

111

tututu

tutyty

tututu

tutyty

Identification by MIMO NN-SANARX model

T

TT

tututytyWC

tututytyWCtyty

2,2,2,2

1,1,1,1,

2121222

21211121

21 l 52 l

I

I

e

eI

1

12

u1

u2

y1

y2

Numerical Example: NN-based SANARX model

6805.07675.02253.02669.1

7308.03075.02909.19239.01W

5827.07948.0

8987.01414.0 1C

1096.0 0513.0 1615.0 2699.0

1054.0 0501.0 1549.0 2637.0

0373.00094.00167.0 0259.0

0473.00553.02258.0 1837.0

0424.00046.00214.0 0358.0

2W

9052.258546.271403.226542.19867.1

8476.63793.79394.1071466.06868.842C

1.02.0sin4.01 ttuv

ttuv 1.0sin8.02

41036.1 MSE

0 10 20 30 40 50 60 70 80 90 100-1

-0.5

0

0.5

1

1.5

Time Steps

Outp

ut

first output of the systemsecond output of the systemfirst output of the modelsecond output of the model

y1

y2

Numerical Example: Control (1/2)

6805.07675.02253.02669.1

7308.03075.02909.19239.01W

1573.10039.0

0161.02720.11111 CWT

9773.02028.0

5061.07341.01122 CWT

1940.13299.0

8231.05896.112T

2253.02669.1

2909.19239.011W

6805.07675.0

730.03075.012W

6.0det 2 T 22 Trank

TTT

TTT

tututytyWCtvtvtt

tytyTttTtutu

2121222212,11,1

2112,11,11

221

,,,,1,1

,,,

Numerical Example: Control (2/2)

ty

tyT

tu

tu

ty

ty

WCtv

tvT

tu

tu

2

11

2

1

2

1

2222

112

2

1

1

1

1

1

1

1

TTT tytyTttTtutu 2112,11,11

221 ,,,

TTT tututytyWCtvtvtt 2121222212,11,1 ,,,,1,1

0 10 20 30 40 50 60 70 80 90 100-1

-0.5

0

0.5

1

1.5

Time Steps

Out

puts

reference signal for the first outputreference signal for the second outputfirst output of the systemsecond output of the systemy1

y2

0 10 20 30 40 50 60 70 80 90 100-1

-0.5

0

0.5

1

Time Steps

Con

trol

Sig

nals

first control signalsecond control signalu1

u2

0 10 20 30 40 50 60 70 80 90 100

-1

-0.5

0

0.5

Time stepsO

utpu

ts

reference signal for the first outputreference signal for the second outputfirst output of the systemsecond output of the system

y1

y2

Model Reference Control

u(t)

y(t)

v(t)

Parameters(W1, …, Wn, C1, …, Cn)

Nonlinear system

NN-based ANARX model

Dynamic Output Feedback Linearization

Algorithm

Parameters(a1, …, an, b1, …, bn)

reference model

Reference model zVzHzY m

ntvbtvbntyatyaty nn 11 11

nn

nnnnn

nn

mazazazaz

bzbzbzbzH

12

21

1

12

21

1

Dynamic output feedback linearization of ANARX model

tyatutyftvbt

tya

tutyftvbtt

tyatutyftvbtt

ttvbtyatutyfF

nnnn

n

nnnn

)(),()1(

)(),()()1(

)(),()()1(

)()(),(

1

1

1112

22221

1111

NN-ANARX Model based Reference Model Control

NN-ANARX model

n

i

Tiii ituityWCnty

1

)1(),1()(

Dynamic output feedback linearization of NN-ANARX model

)()(),( 111111 ttvbtyatutyWCF T

tyaWCtvbt

WC

tyatvbtt

WC

tyatvbtt

nT

nnnnn

Tnnn

nnnn

T

tuty

tuty

tuty

)(),(

)(),(

)(),(

)1(

)()1(

)()1(

1

111

1112

222

2221

Numerical Example: JCSTR model

Input-Output equation:

)1()(5611.0)1()1(592.0

)1(3921.0)1()(014.1

)1(6407.0)1(4801.0

)(231.0)1(7653.0)2(

2

2

tutytuty

tytyty

tytu

tytyty

Example: Control Task

72.01.0150

1975

19

8.09.0

5.0

75

192

zz

z

zz

zzH

Linear second order discrete time reference model with poles p1=-0.9, p2=0.8, zero n1=-0.5, steady-state gain Kss=1

It defines the desired behavior of the closed loop system (=control system)

Parameters of the reference model for control algorithm are as follows:

72.02 a1.01 a150

192 b75

191 b

Example: NN-based modeland control

First sub-layer is linear

11 WCD

TT tutyWCtutyDty 2,21,1)( 222

Here

)(111)(),( ttvbtyaD Ttuty We know that

TtutyWCtyatvbt )(),(222221 )1(

Thus,

and ][ 21 ddD

tyatydtvbWCtyatvbd

tu Ttuty 11122222

2

)1(),1(1 11

Example: NN-based modeland control

0 5 10 15 20 25 300

0.2

0.4

0.6

0.8

1

Time Steps

Outp

uts

reference model output

system output

reference signal