systems identification

SYSTEMSSYSTEMSIdentificationIdentification

Ali Karimpour

Assistant Professor

Ferdowsi University of Mashhad

Reference: “System Identification Theory For The User” Lennart Ljung(1999)

lecture 3

Ali Karimpour Oct 2010

2

Lecture 3

Simulation and predictionSimulation and prediction

Topics to be covered include:

Simulation.

Prediction.

Observers.

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3

Simulation

Suppose the system description is given by

)()()()()( teqHtuqGty

Let

.....,3,2,1,)( ttu

Undisturbed output is

)()()( tuqGty

Let (by a computer)

.....,3,2,1,)( tteDisturbance is

)()()( teqHtv

Output is

)()()()()( teqHtuqGty

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Prediction (Invertibility of noise model)

Disturbance is

0

)()()()()(k

ktekhteqHtv

H must be stable so:

0

)(k

kh

Invertibility of noise model

?)(known is )( if)()()( tetvteqHtv

)()(~

)( tvqHte

With

0

)(~

k

kh

)(~

)(?

qHqH

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5

Lemma 1 Consider v(t) defined by

0

)()()()()(k

ktekhteqHtv

Assume that filter H is stable and let:

0

)()(k

kzkhzH

Assume that 1/H(z) is stable and:

0

)(~

)(

1

k

kzkhzH

Define H-1(q) by

0

1 )(~

)(k

kqkhqH

Then H-1(q) is inverse of H(q) and

)()(~

)()()( 1 tvqHtvqHte


Exercise1: Proof Lemma1

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Example 1 A moving average process

)1()()( tcetetvLet

That is11)( cqqH

This is a moving average of order 1, MA(1). Then

z

czczzH

11)(

If |c| < 1 then the inverse filter is determined as

0

1 )()(k

kk zccz

zzH

So e(t) is:

0

)()()(k

k ktvcte

Exercise3: Exercise 3T.1

Exercise2: Show the validity of example 1 by an example for c=0.2 and c=0.9 c=1.2.

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Prediction (One-step-ahead prediction of v)

Now we want to predict v(t) based on the pervious observation

10

)()()()()()(kk

ktekhtektekhtv

Now the knowledge of v(s), s ≤t-1 implies the knowledge of e(s), s ≤t-1 according to inevitability. Also we have

)1()()()()()(1

tmtektekhtetvk

Suppose that the PDF of e(t) be denoted by fe(x) so:

xxfxxtexP e )())((

Now we want to know fv(x) so:

))(()( 1 t

v vxxtvxPxxf ))1()(( 1 tvxxtmtexP

))1()()1(( 1 tvxtmxtetmxP xtmxfe ))1((

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xtmxfxxf ev ))1(()(So the (posterior) probability density function of v(t), given observation up to time t-1, is ))1(()( tmxfxf ev

1- Maximum a posteriori prediction (MAP): Use the value for which PDF has its maximum.

)(max)(ˆLet xftv v

2- Conditional exceptation: Use the mean value of the distribution in question.

)1|(ˆ)(ˆLet ttvtv

We use mostly the blocked one.

?)(ˆ tv

Exercise4: Exercise 3D.3

Exercise5: Exercise 3E.4

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)1()()( tmtetv?)1|(ˆ ttv

)1()1|(ˆ tmttv )()(1

teqkhk

k

)(1)( teqH

)()(

1)(tv

qH

qH )()(1 1 tvqH

1

)()(~

k

ktvkh

Conditional exceptation

1

)()(~

)1|(ˆk

ktvkhttv

Alternative formula

1

)()()1|(ˆ)(k

ktvkhttvqH

1

)()(k

ktekh

Exercise6: Exercise 3T.1

Suppose that H(q) is inversely stable and monic, what about H-1(q) ?

?

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Example 3.3

1)()(0

akteatvk

kLet

That is

011

1)(

k

kk

azzazH


1

)()(~

)1|(ˆk

ktvkhttv

Alternative formula

1

)()()1|(ˆ)(k

ktvkhttvqH

1

)()(~

)1|(ˆk

ktvkhttv )1()1|(ˆ tavttv

11 1)( azzH

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Prediction (One-step-ahead prediction of y)

)()()()()()()()( teqHtuqGtvtuqGty Let

Suppose v(s) is known for s ≤ t-1 and u(s) are known for s ≤ t . Since

)()()()( tvtuqGty )1|(ˆ)()()1|(ˆ ttvtuqGtty

)1|(ˆ)()()1|(ˆ ttvtuqGtty )()(1)()( 1 tvqHtuqG

)()()()(1)()( 1 tuqGtyqHtuqG

)()(1)()()()1|(ˆ 11 tyqHtuqGqHtty

)(1)()()()1|(ˆ)( tyqHtuqGttyqH

)()(~

)()()1|(ˆ11

ktykhktuklttykk

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Prediction (One-step-ahead prediction of y)

Unknown initial condition

)()(~

)()()1|(ˆ11

ktykhktuklttykk

Since only data over the interval [0 , t-1] exist so

)()(~

)()()1|(ˆ11

ktykhktuklttyt

k

t

k

The exact prediction involves time-varying filter coefficients and can be computed using the Kalman filter.

The prediction error

)()()()()()1|(ˆ)( 1 tetytuqGqHttyty

So the variable e(t) is the part of y(t) that can not be predicted from past data. It is also called innovation at time t.

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Prediction (k-step-ahead prediction of v)

k-step-ahead predictor of v

kl

lktelhtktv )()(0)|(ˆ

First of all we need k-step-ahead prediction of v

0

)()()(l

ltelhtv

0

)()()(l

lktelhktv

kl

k

l

lktelhlktelhktv )()()()()(1

0

)()(~

)()()( teqHkteqHktv kk

Known at t

Unknown at t

kl

klk

k

l

lk qlhqHqlhqH )()(

~)()( where

1

0

)()(~

teqH k )()()(~ 1 tvqHqH k

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Prediction (k-step-ahead prediction of y)

)()()(~

)()(~

)()()|(ˆ 1 tvqHqHteqHlktvlhtktv kkl

k

k-step-ahead prediction of v is:

Suppose we have measured y(s) for s≤ t and u(s) is known for s≤ t+k-1. So let

)()()()( ktvktuqGkty

)|(ˆ)()()|(ˆ),|(ˆ 1 tktvktuqGtktyuykty ktt

)()()(~

)()( 1 tvqHqHktuqG k

)()()()()(~

)()( 1 tuqGtyqHqHktuqG k

)()()(~

)()()()(~

1)|(ˆ 11 tyqHqHktuqGqHqHqtkty kkk

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Prediction (k-step-ahead prediction of y)

)()()(~

)()()()(~

1)|(ˆ 11 tyqHqHktuqGqHqHqtkty kkk

)()(1)()()()|(ˆ tyqWtuqGqWktty kk k-step-ahead prediction of y is:

)(qWk )(qWqq kkk

Define prediction error of k-step-ahead prediction as:

)|(ˆ)()( tktyktyktek

Exercise8: Show that prediction error of k-step-ahead prediction is a moving average of e(t+k) , … ,e(t+1)

Exercise9: Exercise 3E.2

Exercise7: Show that the k-step-ahead prediction of can)()()()( tvtuqGty

also viewed as a one-step-ahead predictor associated with the model:

)()()()()( 1 tvqWtuqGty k

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Observer

)()()( tuqGty In many cases we ignore noises, so deterministic model is used

This description used for “computing,” “guessing,” or “predicting”. So we need the concept of observer.

As an example let:1

1

1

1

1)()(

az

bzzabzG

k

kk

)(1

)(1

1

tuaq

bqty

This means that

1

1 )()()(k

k ktuabty

)()(1 11 tubqtyaq )1()1()( tbutayty

)1()1()1|(ˆ tbutaytty

1

1 )()()1|(ˆk

k ktuabtty

So we have

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Observer

)()1()1()1|(ˆ IItbutaytty )()()()1|(ˆ1

1 Iktuabttyk

k

So we have

If input output data are lacking prior to time t = 0 , first one suffers from an error, but second one still is correct for t > 0.In the other hand first one is un affected by measurement errors in the output, but second one affected.

So the choice of predictor could be seen as a trade-off between sensitivity with respect to output measurement errors and rapidly

decaying effects of erroneous initial conditions.

Exercise (3E.3): Show that if )()()(

1)(

1

1

teqHtuaq

bqty

Then for the noise model H(q)=1, (I) is the natural predictor, whereas the noise model

0

)()(k

kk qaqH

Leads to the predictor (II)

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Observer

kl

lqwqW 11)(

)()()()()( tuqGqWtyqW

)()()()()(1)( tuqGqWtyqWty

)()()( tuqGty A family of predictor for

So the choice of predictor could be seen as a trade-off between sensitivity with respect to output measurement errors and rapidly decaying effects of erroneous initial conditions.

To introduce design variables for this trade-off, choose a filter W(q) such that

Applying it to both sides we have

Which means that

The right hand side of this expression depends only on y(s), s≤t-k, and u(s). s ≤t-1. So

)()()()()(1)1|(ˆ tuqGqWtyqWtty

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Observer




)()()1|(ˆ)()( tvqWttytyte

The trade-off considerations for the choice of W could then be

1. Select W(q) so that both W and WG have rapidly decaying filter coefficients in order to minimize the influence of erroneous initial conditions.

2. Select W(q) so that measurement imperfections in y(t) are maximally attenuated.

The later issue can be shown in frequency domain. Suppose that

)()()( tvtyty M The prediction error is:

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Observer




)()()1|(ˆ)()( tvqWttytyte

The prediction error is:

)()()(2

v

ieW

The spectrum of this error is, according to Theorem 2.2:

The problem is thus to select W, such that the error spectrum has an acceptable size and suitable shape.

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Observer

Fundamental role of the predictor filterLet

)()()()()( teqHtuqGty )()()( tuqGty or

Then y predicted as:

)()(1)()()()1|(ˆ 11 tyqHtuqGqHtty

)()()()()(1)1|(ˆ tuqGqWtyqWtty or

They are linear filters since:

Linear Filter

)(ty

)(tu)1|( tty

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