1

Use of Multiple Integration and Laguerre Models for System Identification:

Methods Concerning Practical Operating Conditions

Yu-Chang Huang (黃宇璋 )Department of Chemical and Materials EngineeringNational Kaohsiung University of Applied Sciences

2010/09/24

2

Outline

Introduction and available identification methods Identification of continuous-time SISO systems

Use of multiple integration Two-stage algorithms

Identification of discrete-time MIMO systems Use of a single Laguerre model Augmented order to deal with unknown disturbances

Identification of discrete-time MIMO systems Use of double Laguerre models Suited to obtaining a process model of reduced order

Conclusions and future work

3

Introduction (I)

System identification finding process and disturbance models based on input-output testing is often faced with practical operating conditions as follows: unsteady and unknown initial states load disturbances of unknown dynamics and

unpredicted nature stochastic disturbances unknown model structure (order and delay) and

parameters constraints on the input signal to a test experiment continuous-time or discrete-time

4

Introduction (II)

Available identification methods for linear systems Diamessis (1965) assumed implicitly that all initial conditions

were zero – multiple integration Lecchini and Gevers (2004) delivered a Laguerre analysis

under zero initial conditions and no disturbances Hang et al. (1993), Shen et al. (1996) and Park et al. (1997)

resolved static load disturbances but not slow and periodic disturbances – relay tests

Hwang and Wang (2003) developed a time- and frequency-weighted method to deal with non-static disturbances

Hwang and Lai (2004) and Liu and Gao (2008) presented methods based on specified test signals

5

Identification Method for Continuous-Time SISO Systems

Use of multiple integration to avoid time derivatives of the input-output signals

A sequential least-squares method that identifies a parametric model using a two-segment test signal (first complicated and then simple) in face of the practical difficulties

A convenient technique to determine the model structure based on the same test data

The method is robust with respect to unsteady initial states, unknown load disturbances, noise, and model structure mismatch

6

Nth-Order Continuous System

( ) ( 1) ( )1 0

( 1)1 0

( ) ( ) ( ) ( )

( ) ( ) ( )

n n mn m

mm

y t a y t a y t b u t d

b u t d b u t d t

y(t) and u(t): output and input signals

n and m: system orders

d: time delay

ai, bi ： model parameters

(t): unknown disturbance

7

Multiple Integration

To avoid time derivatives of a signal x(t), we define a multiple integral filter as

nj

dddxttX j

t

t t tbajb

a

j

a a

,,2,1

,)(),( 2

211

8

Underlying Identification Model

L

n

j

jjn

m

jjmnjm

n

jjjn

dtfor

tfdtdUbtYaty

001

),(),0()(

– unsteady initial states are unknown

Ldtct ,)(– static disturbance (an offset)

– fi accounts for the effects of the nonzero initial states and the unknown offset c

)0()(iy

– the number of parameters to be estimated is high

9

Sequential Algorithms Based on Two-Segment Testing

LttuLttu

tu),(,)(

)(2

1 1 2( ) ( ) foru t u t t L

First segment gives the estimation of d and bi

Second segment gives the estimation of n and ai

Treat the identification problem as two sub-problems sequentially

Two-segment testing signal

10

The Input Signal for Plant Tests

arbitrary)(1 tu

))(sin()( 222 LtActu

u2 = 0 → a pulse test with arbitrary shape

u2 → a simple combination of a step and a sinusoid

11

First-Stage of Estimation

Ld

)cos()sin(),0()( 2101

ththtgtYatyn

j

jjn

n

jjjn

The intermediate parameters gi consist of fi

and those resulting from the two input functions A goodness-of-fit criterion En can be developed to

determine the best value of n

t

Applying the ordinary least-squares gives rise to estimates of the model parameters ai

12

Second-Stage of Estimation (I)

Ldt

m

jjmnjm

n

j

jjn

n

jjjn

dtLUb

thth

tgtYatyt

0

21

01

),(

)cos()sin(

),0()()(

j

dt

L L Lj ddduudtLU j

2

211211 )]()([),(

dL

where

Applying the ordinary least-squares gives rise to estimates of the model parameters bi

13

Second Stage of Estimation (II)

A goodness-of-fit criterion Ed can be developed to determine the best value of d

Order m can be set to n – 1 or specified by users

The number of parameters to be estimated at each stage is minimized

A complicated input can be employed at this stage to enhance estimation accuracy

14

Rejection of Slow Disturbances

Modifying the regression equation for the first-stage estimation as

)cos()sin(

),0()(

21

01

thth

tgtYatyp

j

jjn

n

jjjn

In practice, p = n+1 or n+2

15

Rejection of Periodic Disturbances

22 )( ctu Employ pulse testing, i.e. Modifying the regression equation for the

first-stage estimation as

)cos()sin(

),0()(

21

01

thth

tgtYaty

dddd

p

j

jjn

n

jjjn

: frequency of the periodic disturbanced

16

Cancellation of Measurement Noise

The use of the integral filter could eliminate the effect of measurement noise to a certain extent

In the presence of severe noise, it is better to employ the wavelets de-noising procedure based on multi-level decomposition and reconstruction of the output signal (Mallat, 1989)

17

Fitting Model Predictions to Output Measurements

)()()( tytyty iMp

:)(tyi

:)(tyM response of the model assuming the zero conditions

Model verification Once fi are calculated, the model predictions

can be obtained as

effects of the nonzero initial states and disturbance

18

Simulation Study

(3) (2) (1) (2)

(1)

( ) 2 ( ) 2 ( ) ( ) 0.5 ( 1)

1.5 ( 1) ( 1) ( )

y t y t y t y t u t

u t u t t

2)0(,1)0(,1)0( )2()1( yyysubjected to

Case I: a static disturbance (offset) & NSR = 10%

Case II: a slowly changing load (drift) & NSR = 5%

Case III: a periodic disturbance & NSR = 5%

19

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

-5 0 5 10 15 20 25

u

Time

Input test signal Model predictions for Cases I and II

-1

-0.5

0

0.5

1

1.5

2

2.5

0 5 10 15 20 25

Measured Output (Case I)Model predictions (Case I)Measured output (Case II)Model predictions (Case II)

y

Time

20

Identification input-output data for Case III

-1

0

1

2

3

4

-5 0 5 10 15 20 25

Time

u

y

21

The goodness-of-fit function En versus order n

Finding the Model Order

0.001

0.01

0.1

1

10

1 2 3 4

Case ICase IICase III

En

Order

22

Case I ___________

Case II __________________________________

Case III ___________

2a

1a

0a

2b

1b

0b d

1.9691 1.9719 0.9924 0.5112 -1.4970 0.9951 1.00

3p 0.6267 0.6487 -0.0457 -0.8618 0.5481 0.0824 1.50

5p 2.0370 2.0280 1.0377 0.2967 -1.5565 0.9051

1.10

6p 2.0247 2.0250 1.0095 0.5583 -1.5323 1.0538 1.00

1.9800 1.9764 0.9911 0.5176 -1.4861 0.9948

1.00

Estimated parameters under different test conditions

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Identification Method Based on a Single Laguerre Model for Discrete-Time MIMO Systems

Use of a Laguerre ARX model with a time-scaling factor in face of unpredicted load and unknown stochastic disturbances

The idea of augmented order is introduced to account for the MISO process and distinct load dynamics

Three error criteria are developed to find the best values for the time-scaling factor, load entering time, and process delays

Not suited to finding a process model of reduced order Persistent excitation for the input is required

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Discrete Laguerre expansions (I)

1

,i ii

G z g L z

2 11 1,

i

i

T zL zz z

: time-scaling factor

gi: Laguerre coefficients

T: sample time

Laguerre IIR (infinite impulse response) model (Wahlberg, 1991)

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Discrete Laguerre expansions (II)

1

( ) ( , )r

i ii

G z g L z

Laguerre FIR (finite impulse response) model

1

1

( , )( )

1 ( , )

n

i ii

n

i ii

b L zG z

a L z

Laguerre ARX (autoregressive with an exogenous input) model

ia, ib : Laguerre

coefficients

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Conversion Relationships between , and ,

min( 1, )2 1 2

1 max(0, )

( ) (1 ) ( )p n in

n i n p p i p qi i n i q q p

p q p i

a C T C C a

min( 1, )2 1 2

1 max(0, )

(1 ) ( )p n in

n p p i p qi n i q q p

p q p i

b T C C b

( )( )( )

B zG zA z

1

( )n

n n ii

i

A z z a z

1

( )n

n ii

i

B z b z

ia ibia

ib

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SISO Identification Model (I)

L

P D I L1

L2 V

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

( ) ( ) ( ) ( )

Y z G z U z G z G z S z

G z S z z G z E z

Y(z), UD(z): z–transforms of the output y(k) and delayed input uD(k) = u(k - )GI(z): initial statesGL1(z), GL2(z), L: first and second load disturbances and load entering time GV(z): stochastic disturbance

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SISO Identification Model (II)

L

P I L1D

P P P L1

- 1L2

P L2

1

z1

B z zB z zB zY z U z

A z A z z A z A z

B zz A z A z

AP(z): denominator polynomial for process of order nP

AL1(z), AL2(z): distinct load dynamics

AL (z): monic polynomial of degree nL, the least common multiple of AL1(z), AL2(z)

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SISO Identification Model (IV)

L

L

D1 1 1

110 0

1

, , ,

, in z-domain1 1

n n n

i i i i i ii i i

n

i ii

Y z a L z Y z b L z U z c zL z

c z d zd z L zz z

Applying the Laguerre ARX model gives

D,1 1 1

0 L 0 L1

( ) ( ) ( )

( ) ( )

n n n

i i i i i ii i i

n

i ii

y k a y k bu k c k

c d k d I k

Regression equation in time domain

30

Recovery of Augmented Model

Simulated outputs,

(a ) (a )P P D

ˆˆ ( ) ( ) ( )Y z G z U z

P P(a ) (a )

P P D1 1

ˆ ˆ( ) ( , ) ( ) ( , ) ( )n n

i i i ii i

Y z e L z Y z h L z U z

.

( t )P

ˆ ( )G z

aP̂Y z

Applying the least–squares estimation leads to the construction of

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MIMO Identification Model (I)

L,

P, D, I, L1,1

L2, V,

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

( ) ( ) ( ) ( )l

m

l lj lj l lj

l l

Y z G z U z G z G z S z

G z S z z G z E z

For a w outputs, m inputs system, the lth MISO subsystem can be expressed by

Augmented order

P, L,l l ln n n

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MIMO Identification Model (II)

L,

L,

D,1 1 1

0,

1

11 0,

1

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

( , )1

( , )1

l l

l

lll

n nm

l li i l l lji i l lji j i

nl

li i li

nl

li i li

Y z a L z Y z b L z U z

c zc zL z

z

d zd z L z

z

33

21

aOE , L, 1

0

1, , ,N

l l l lm lk

J OE kN

Two Error Criteria for Identification under Deterministic Disturbances

21

a

0

1 ˆN

l lk

y k y kN

Find the best values for l, L,l, and lj The first is the output error criterion:

34

The second is the relative error criterion

1

2RE L, 1

0

1( , , , , ) ( )N

l l l lm lk

J RE kN

( t )P,ˆ ( ) :ly k the process–only outputs predicted by

the Laguerre ARX models of true order

FIR,ˆ ( ) :ly k outputs predicted by the Laguerre FIR models

1 2(t )P, FIR,

0

1 ˆ ˆ( ) ( )N

l lk

y k y kN

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Error Criterion for Identification under Stochastic Disturbances

The filtered output error criterion1 2

(t )FOE

0

1 ˆ( ) ( ) ( )N

l lk

J F q OE kN

11( ) 1 F

F

nnF q f q f q

( t ) ( t )ˆ( ) ( ) ( )l l lOE k y k y k

where

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MIMO study: Example 2

12.8 18.916.7 1 21 1( )

6.6 19.410.9 1 14.4 1

s ss

s s

PG

,

1 37 3

δ

, T = 1

Wood and Berry (1973)

0.744 0.8790.942 0.954( )

0.579 1.3020.912 0.933

z zz

z z

PG The exact discrete model

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Load Disturbances: Cases A and B

2 80

L,1 4 5

5 15( 2 1)( )(16.7 1)(21 1) ( 1) (5 1)

ss s eG ss s s s

Case A: 2

L,1 4 5

10 15( 2 1)( )(16.7 1)(21 1) ( 1) (5 1)

s sG ss s s s

L,2 2 4

10 15(15 1)( )(10.9 1)(14.4 1) (5 1) (30 1)

sG ss s s s

Case B:

80

L,2 2 4

5 15(15 1)( )(10.9 1)(14.4 1) (5 1) (30 1)

ss eG ss s s s

― both subjected to measurement noise of NSR = 5%

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ln Subsystem 1 Subsystem 2 3 ( t )

P,110.731ˆ

0.944G

z

( t )P,12

0.881ˆ0.953

Gz

11 12 L,1

1 OE

ˆ ˆ ˆ1, 3, 30,

0.93, 0.0338J

( t )P,21

0.589ˆ0.916

Gz

( t )P,22

1.314ˆ0.930

Gz

21 22 L,2

2 OE

ˆ ˆ ˆ7, 3, 36,

0.93, 0.0378J

2 ( t )

P,110.760ˆ

0.937G

z

( t )P,12

0.914ˆ0.952

Gz

11 12 L,1

1 OE

ˆ ˆ ˆ1, 3, 22,

0.96, 0.0494J

( t )P,21

0.556ˆ0.928

Gz

( t )P,22

1.330ˆ0.928

Gz

21 22 L,2

2 OE

ˆ ˆ ˆ7, 3, 41

0.88, 0.0581J

Identification results for Case A

39

Identification results for Case B

ln Subsystem 1 Subsystem 2 3 ( t )

P,110.745ˆ

0.942G

z

( t )P,12

0.877ˆ0.953

Gz

11 12 L,1

1 OE

ˆ ˆ ˆ1, 3, 95,

0.93, 0.0341J

( t )P,21

0.567ˆ0.921

Gz

( t )P,22

1.328ˆ0.927

Gz

21 22 L,2

2 OE

ˆ ˆ ˆ7, 3, 112,

0.94, 0.0391J

2 ( t )

P,110.784ˆ

0.941G

z

( t )P,12

0.875ˆ0.945

Gz

11 12 L,1

1 OE

ˆ ˆ ˆ1, 3, 99

0.89, 0.0671J

( t )P,21

0.568ˆ0.933

Gz

( t )P,22

1.291ˆ0.928

Gz

21 22 L,2

2 OE

ˆ ˆ ˆ7, 3, 113

0.87, 0.0767J

40

Comparison of the actual outputs and load disturbances with thosepredicted by the identified models for Case A

41

Comparison of the actual outputs and load disturbances with thosepredicted by the identified models for Case B

42

Type I: V,1( ) 1G s , V,2 ( ) 1G s

Type II: V,11( )

16.7 1G s

s

, V,2

1( )10.9 1

G ss

Type III: V,11( )

(16.7 1)(21 1)G s

s s

V,21( )

(10.9 1)(14.4 1)G s

s s

Three types of noise characteristics:

,

Stochastic Disturbances

43

Mean of time-scaling factor versus NSR

44

Effect of NSR on identification reliability

45

Identification Method Based on Double Laguerre Models for Discrete-Time MIMO Systems

A good reduced-order model for the process is sometimes desired for controller design

Use of double Laguerre ARX models to account for the process and distinct load dynamics separately

Two different time-scaling factors and need to be sought for each MISO subsystem

46

SISO Identification Model (I)

L

P ID

P P

- 1L1 L2

P L1 P L2

z1 1

B z zB zY z U z

A z A z

zB z B zz A z A z z A z A z

L

P P D I

- 1L1 L2

L1 L2

z1 1

A z Y z B z U z zB z

zB z B zz A z z A z

Assume n = nP and multiply the above equation by A(z) = AP(z) yields

47

SISO Identification Model (II)

L L

L L

D1 1 1

11 10 0

1 1

, , ,

, ,1 1

n n n

i i i i i ii i i

nn

i i i ii i

Y z a L z Y z b L z U z c zL z

c z d zd z L z e z L zz z

Applying double Laguerre ARX models gives

L

D, 01 1 1

L 0 L L1 1

( ) ( ) ( )

( ) ( ) ( )

n n n

i i i i i ii i i

nn

i i ii ii i

y k a y k bu k c k c

d k d I k e k

Regression equation in time domain

48

OE , , L, 1, , ,l l l l lmJ

Two Error Criteria for MIMO Systems

RE , L, 1( , , , , )l l l l lmJ

49

MIMO Example under Load Disturbances

L,1

2 70

12 1 3 1 4 1 5 1

10 2 110 1 12 1 14 1 16 1

s

G ss s s s

s s es s s s

5 92 1 3 1 4 1 5 1

10 131 9 1 6 1 7 1

PGs s s s

s

s s s s

1 23 1

δ

, ,

1T

L,2

70

11 6 1 7 1 9 1

10 3 12 1 8 1 11 1 12 1

s

G ss s s s

s es s s s

― subjected to measurement noise of NSR = 5%

50-10

-5

0

5

10

15

0 50 100 150 200 250 300

(a)

Output measurementsActual disturbancesOutput predictionsPredicted disturbances

Out

put d

tat o

f sub

syst

em 1

Time

Comparison of the actual outputs and load disturbances with thosepredicted by the identified models for the two subsystems

-15

-10

-5

0

5

10

15

0 50 100 150 200 250 300

(b)

Output measurementsActual disturbancesOutput predictionsPredicted disturbances

Out

put d

ata

of s

ubsy

stem

2

Time

51

Comparison of the actual and identified disturbances by virtueof Nyquist plots

-8

-6

-4

-2

0

2

-4 -2 0 2 4 6 8 10 12

(a)

Actual disturbancesPredicted disturbances

Imag

inar

y

Real

-8

-6

-4

-2

0

2

-4 -2 0 2 4 6 8 10

(b)

Actual disturbancesPredicted disturbances

Imag

inar

y

Real

52

Conclusions (I)

We have developed three effective methods to deal with system identification based on plant tests under practical operating conditions

The first method using multiple integration and a sequential algorithm can identify a continuous-time SISO process from a relatively simple test experiment

53

Conclusions (II)

The second method based on a single Laguerre model with an adjustable time-scaling factor can identify a discrete-time MIMO process if the process order is not too high

The third method based on double Laguerre models with different time-scaling factors can identify a good reduced-order model for a discrete-time MIMO process

54

Future Work

Extend the first method to the Identification of continuous-time MIMO systems

Consider the use of other orthogonal functions for system identification

Extend the second and third methods to the identification of nonlinear processes

55

Thanks for your attention!