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
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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
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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
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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:
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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%
38
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!
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