S t M d li d Id tifi tiSystem Modeling and Identification

L t N t #3Lecture Note #3(Chap 6)(Chap.6)

CHBE 702Korea University

Prof Dae Ryook YangProf. Dae Ryook Yang

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Ch 6 Id tifi ti f TiChap.6 Identification of Time-Series ModelSeries Model

• Identification of time series modelIdentification of time series model– Model structure

Parametric estimation– Parametric estimation• Least square method

ll i h di b l d– Excellent properties when disturbances are uncorrelated– Otherwise, there may be systematic errors and bias

• More sophisticated methods are needed– Handling correlated disturbancesg– Extension of linear regression

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Model structure• Time series

T– Multivariable time series– Multidimensional time series (temporal+spatial)

1 2[ ]Tk k k kmx x x x

• Model classification– SISO/MIMO– Linear/Nonlinear– Deterministic/StochasticDeterministic/Stochastic– For linear discrete-time systems

• Difference equation and ARMAX modelsDifference equation and ARMAX models• Transfer function model• State-space model

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ARMAX M d l d DiffARMAX Models and Difference EquationsEquations

• ARMAX (AutoRegressive Moving Average with eXogenous ( g g g ginput) model 1 1 1( ) ( ) ( )d

k k kA z y z B z u C z w

where d is time delay and 1 1 21 2

1 1 20 1 2

( ) 1

( )

A

A

B

B

nn

nn

A z a z a z a z

B z b b z b z b z

with unknown parameters

1 1 21 2( ) 1

B

C

C

nnC z c z c z c z

21 1 1[ ]Ta a b b c c with unknown parameters 1 1 1[ ]

A B Cn n n wa a b b c c

1( ) (AR)k kA z y w 1( ) (MA or FIR)k ky C z w1 1( ) ( ) (ARX)d

k k kA z y z B z u w 1 1( ) ( ) (ARMA)k kA z y C z w

1 1 1( ) ( ) ( ) (ARIMAX or CARIMA)k k kA z y B z u C z w

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( ) ( ) ( ) ( )k k ky

Integrated Controlled

• ARMAX: General model• ARX: Controlled autoregressive model and Linear

regression model when disturbance is measuredregression model when disturbance is measured• AR: Model harmonics compounded with noise, and

Truncated impulse response (FIR) modelTruncated impulse response (FIR) model• ARMA: Model-based spectrum analysis• ARIMAX: For nonzero means and drift or nonstationaryARIMAX: For nonzero means and drift or nonstationary

disturbance cases1 1 1( ) ( ) ( )k k kA z y B z u C z w ( ) ( ) ( )k k ky

1 1 1 1 1( ) ( ) ( )(1 )A y B u C w ( ) ( ) ( )(1 )k k kA z y B z u C z z w

Integrated white noise: Nonstationary (random walk)

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Prediction Error Method (PEM)– Methods to predict y based on previous data and the identified model– PEM

N N

It i i i th i f th di ti t h d f th t t

2 2

1 1min ( ( ) ) min ( )

N N

kk k k kk k

E y y E

• It minimizes the variance of the prediction steps ahead of the output ywhere the prediction is based on the present data.

(1 )k k k ky A y Bu Cw

1

( )(1 ) (1 )

(1 ) (1 )( ( ))

k k k k

k k k k

k k k k k

y yA y Bu C w w

A y Bu C C Ay Bu w

( ) ( )( ( ))k k k k k

k k

y yz w

2 2{( ) } {( ) }E y y E w y

2 2 2

{( ) } {( ) }

{( ) } { }k k kk k k k

k k wk k

E y y E z w y

E z y E w

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Minimum attainable variance

Transfer Function Models• Transfer function models

*( ) ( ) ( { } )k u k v k i j v ijy H z u H z v E v v k u k v k i j v ij

1 11

1 1

( ) ( )( )( ) ( )k k k

B z C zA z y u wF z D z

1

1

( ) (Output error (OE) model)( )k k k

B zy u vF z

1 1

1 1

( ) ( ) (Box-Jenkins (BJ) model)( ) ( )k k k

B z C zy u wF z D z

– OE model: No assumption on disturbance sequence {vk} – BJ model: Filtered white noise {wk} sequence by C/D

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• Difference between output error and prediction error1 1 (Output error method)k k ky ay bu

( di i h d)b

– Output error relies more on the accuracy of future output modeling.Prediction error uses actual output

1 1 (Prediction error method)k k ky ay bu

– Prediction error uses actual output.– Output error identification is a nonlinear estimation problem.– Prediction error identification is a linear estimation problem.p

• Algorithm for OE identification1. Least square identification to find initial estimate of F and B.

2. Filter the data according to

1 1: ( ) ( )k k kM F z y B z u v

1 1/ ( ) / ( )F FF F

3. Subsequence estimation of F and B from the model.1 1: ( ) ( )F FM F z y B z u v

1 1/ ( ), / ( )F Fk k k ky y F z u u F z Prewhitening filter

4. Repeat 2-3 until the estimate converges.: ( ) ( )k k kM F z y B z u v

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• Comparison of error models for identification

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Maximum-Likelihood Method• Select estimate so that the observation Y is most probable.

max ( ) ( )p Y p Y

Likelihood function• Example 6.4

Likelihood function

( { } 0 and { } 0)TY v E v E vv

– Assume that– Likelihood function:

1/ 2 1( ) ((2 ) det ) exp( 0.5 )N Tv vp v v v

1/ 2 1( ) ((2 ) det ) exp( 0.5( ) ( ))N Tv vp v Y Y

– If the model is linear in parameters with normally distributed white noise, the maximum likelihood estimate is same as Markov estimate.

1log ( ) log ( ) 0.5log((2 ) det ) 0.5N Tv vL p

the maximum likelihood estimate is same as Markov estimate.• Cramer-Rao lower bound

1 12log log log( )

TL L LC E E

g g g( ) TCov E E

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Fisher information matrix

F ARMAX d l• For ARMAX model1 1 1( ) ( ) ( )d

k k kA z y z B z u C z w 1log ( ) 0 5log((2 ) det ) 0 5 ( ) ( )N TL

wherelog ( ) 0.5log((2 ) det ) 0.5 ( ) ( )v vL

Tk k ky

1 1 1 1 1A A Bk k n k n k d k d ny a y a y b u b u Unknown

1 1 A A B

C C

Tk k n k n k kv c v c v v

T1 1 1[ ]

A B C

Tk k k n k d k dn k k ny y u u v v

– The empirical likelihood function when v=v2I (v

2 unknown)1 1 1[ ]

A B C

Tn n na a b b c c

2 2 2 2l ( ) ( / ) l ( ) ( / ) l ( ) ( / ) ( )N

2 2 2 2

12 2

log ( , ) ( / 2) log(2 ) ( / 2) log( ) (1/ 2 ) ( )

( / 2) log(2 ) ( / 2) log( ) (1/ ) ( )

v v v kk

v v N

L N N

N N V

2 2log ( , ) (1/ ) ( ) 0v v NL V

( ) 0NV

2 2 4 2

2 log ( , ) (1/ ) ( ) ( / 2 ) 0v v N vv

L V N

2 (2 / ) ( )v NN V

( 1) ( ) ( ) 1 ( )( ( )) ( ) (Newton-Raphson method)i i i ii N NV V

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( ( )) ( ) (Newton Raphson method)i N NV V

E l 6 5 LS d ML id tifi ti• Example 6.5 LS and ML identification

1 1 1: k k k k kS y ay bu w cw

Local minimum

– For colored noise, ML identification performsbetter than LS.

– LS can estimates only a and b.

• Example 6.6 Pseudolinear regression1 1 1: ( ) ( ) ( )k k kS A z y B z u C z w

– Estimate high order polynomials A and B by least squares– The computed residual sequence {k} yields a good approximation of

white noise sequence {w }white noise sequence {wk}.– Extend the regressor with {k} and then estimate the polynomials of A, B

and C using least squares identification.– It is also called two-step linear regression.

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Kalman Filter• State-space model

1k k k kx x u v { } 0, { } 0k kE v E e

• Optimal estimate of xk based on the input-output datak k k ky Cx Du e

1 2 0 0 0{ } , { } , (0) { }T T TE vv R E ee R P E x x R

k

• Kalman filter (Kalman-Bucy filter)

211min ( ) {( ) }k kk kJ x E x x

– Kalman filter will minimize the above minimization when vk and ek are independent and normally distributed.

( )K C 1 1 1

12

1

( )

( )

k k kk k k k k k

T Tk k k

x x u K y Cx

K P C R CP C

11 1 2( )T T T T

k k k k kP P R P C R CP C CP

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D i ti• Derivation– The prediction error– The prediction error dynamics ( )x K C x v K e

1 11k kk kx x x The prediction error dynamics

– The mean prediction error– The mean square prediction error

1 ( )k k k k k kx K C x v K e

1{ } ( ) { }k k kE x K C E x

1 1{ } {[( ) ][( ) ] }

( ) { }( )

T Tk k k k k k k k k k k k

T T Tk k k k v k e k

E x x E K C x v K e K C x v K e

K C E x x K C K K

T T– Let { } and T Tk k k k e kP E x x Q CP C

1T T T T T

k k k k k k v k k kP P K CP P C K K Q K 1T T T1

11 1( ) ( )

T T Tk k v k k k

T T Tk k k k k k k

P P P C Q CP

K P C Q Q K P C Q

– Minimization of Pk+1 gives1( )T T

k k e kK P C CP C k k e k

11 ( )T T T T

k k v k e k kP P P C CP C CP (Riccati equation)

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• Cases for time-varying parameters

1k k kv { } 0, { } 0k kE v E e

1 ( )Tk k k k k kK y

Tk k k ky e

11

21

1 1 2

( )

( )

k k k k k kT

k k k k k kT T

k k k k k k k k k

K P R P

P P R P R P P

– Excellent for time-varying systems– R1 and R2 are important design parameter that should match the temporal

k k k k k k k k k

R1 and R2 are important design parameter that should match the temporal variations of k and the observation noise, respectively.

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Instrumental Variable Method

• Correlation between the regressors and the prediction error g pleads to bias of the parameter estimates obtained from least-square solutions to the linear regression problem

• Replace regressor by some other variable Z: IV method– In order to make the estimator consistent { } 0TE Z v ( )Trank Z p

1( )z T TZ Z Y

– The instrumental variable should be chosen so that they are

1( )z T TZ Z Y 1 1( ) {( )( ) } ( ) ( )z z z T T T T

vCov E Z Z Z Z

The instrumental variable should be chosen so that they are simultaneously uncorrelated with v and highly correlated with .

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E l 6 8• Example 6.8

Biased least square estimate of parameters1 1 1: 0.9 0.1 0.7k k k k kS y y u w w 2 2( { } 0, { } )k k wE w E w

– Biased least-square estimate of parameters

– Instrumental variable

1[ ] ( ) [0.957 0.047]T T T Ta b Y 1 1

1 1N N

y u

y u

1 1k k kz az bu

1 1

1 1N N

z uZ

z u

1( ) [0.918 0.075]z T T TZ Z Y

• Shows reduced bias

• Example 6.9For a choice of IV

1 1N Nz u

0 1u uZ

0

– For a choice of IV2 1N N

Zu u

1( ) [0 413 0 047]z T T TZ Z Y

– Gives very poor estimate.– It might be difficult to choose appropriate instrumental variables.

( ) [0.413 0.047]Z Z Y

– Thus, an iterative procedure are usually used.

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E l 6 10 (Th Y l W lk ti )• Example 6.10 (The Yule-Walker equations)– Consider the AR process

1 2 2: ( ) ; ( { } 0 { } )k k k kS A z y w E w E w : ( ) ; ( { } 0, { } )k k k k wS A z y w E w E w *

1 1( ) { } {( ) } { } { }

A An nT T T

yy k k i k i k k i k i k k ki i

C E y y E a y w y a E y y E w y

2

1

1

( ) , 0( )

( ), 0

A

A

ni yy wi

yy ni yyi

a C iC

a C i

– Choosing numbers M> nA and p>nA and1[ ]

A

Tk k k ny y

[ ] / ( )Tz y y M k p M p 2 2[ ] / ( , , )k k k pz y y M k p M p

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

yy yy yy A yyaC i C i C i n C iaC i C i C i n C i

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

( ) ( 1) ( 1) ( )A

yy yy yy A yy

nyy yy yy A yy

aC i C i C i n C i

aC i p C i p C i p n C i p

( )TZ ( )TZ Y1( )z T TZ Z Y

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( )

Some Aspects of Application• Prefiltering, smoothing, prewhitening

1 1( ) ( ) ( ), ( ) ( ) ( )f fY z F z Y z U z F z U z

– For periodic variation, use F(z1)=1 zd when d is the period of trend.

1 10: ( ) ( ) ( )f f

k k kM A z y B z u v w

• Bias reduction– Trend eliminationTrend elimination

Diff i i f d

1 1( ) / , ( ) /N N

k kk ky y N u u N

1 1: ( )( ) ( )( )k k kM A z y y B z u u v – Differentiation of data

( )( ) ( )( )k k ky y1 1: ( ) ( )k k kM A z y B z u v 1 1: ( ) ( )M A z y B z u v

It introduces new noise correlation

It gives improved accuracy– Offset estimation via an extra parameter

: ( ) ( )k k kM A z y B z u v It gives improved accuracy

1 10: ( ) ( ) ( )k k kM A z y B z u v w

Extra parameter

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p

Convergence and Consistency

• Convergence in Lp, 0<p<∞g p, p

• Convergence almost surelylim { } 0p

kkE x x

g y

• Convergence in probabilitylim { , , 0} 1kn

P x x k n

• Central limit theorem{ , 0} 0kP x x

– Let {xk} be a sequence of independent random variables with common distribution function F with finite mean and variance 2.X has a limiting normal distribution with mean 0 and variance 1 as N→∞– XN has a limiting normal distribution with mean 0 and variance 1 as N→∞.

1If , thenN

N kkS x

(0,1)

distN

NS NX Normal

N

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Effi i t ti t

• Efficient estimate,

• Consistent estimate

2 2{( ) } {( ) } for any other estimate E E

• Consistent estimate2lim {( ) } 0NN

E

lim { 0} 0 plimP

Probability limit

• Unbiased and asymptotically unbiased estimateslim { , 0} 0 plimN NN

P

{ }E

(Unbiased estimate)

lim { }NNE

{ }NE (Unbiased estimate)

(Asymptotically unbiased estimate)

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