SYSTEMS Identification

SYSTEMSSYSTEMSIdentificationIdentification

Ali Karimpour

Assistant Professor

Ferdowsi University of Mashhad

[email protected]

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

lecture 6

Ali Karimpour Nov 2010

2

Lecture 6

Nonparametric Time and Frequency domain methods

Topics to be covered include:

Transient-Response Analysis and Correlation Analysis

Frequency Response Analysis

Fourier Analysis

Spectral Analysis

Estimating the Disturbance Spectrum

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Fourier Analysis

Spectral Analysis


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4


Impulse Response Analysis

Suppose

Let the input as

Then the output will be

So we have

The error is:

This simple idea is impulse Response Analysis.

Its basic weakness is that many physical processes do not allow pulse input so the the error be small, moreover that input make the system exhibit nonlinear effects.

)()()()( 0 tvtuqgty

00

0)(

t

ttu

)()()( 0 tvtgty

)(

)(ˆty

tg

)(tv

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5


Step Response Analysis

Suppose

Let the input as


So we have

The error is:

)()()()( 0 tvtuqgty

00

0)(

t

ttu

)()()(1

0 tvkgtyt

k

)1()(

)(ˆ

tyty

tg

)1()( tvtv

It would suffer from large error in most practical applications.

It is acceptable for delay time, static gain, dominating time constant.

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6


Correlation Analysis

Suppose

If the input is a quasi stationary sequence with

Then we have

• If the input is white noise

Then if the input is white noise

An estimate of the impulse response is thus obtained from

)()()()( 0 tvtuqgty

)()()( uRtutuE

0)( uR

)(

)(0yuR

g

)()()()(1

0 tvktukgtyk

operation loop-open 0)()( tvtuE

1

0 )()()()()(k

uyu kRkgRtutyE

)(

)(0yuR

g

N

t

Nyu tuky

NR

)()(

1)(

• If the input is not white noise

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Correlation Analysis

Suppose

If the input is a quasi stationary sequence with

Then we have

• If the input is white noise

Then if the input is not white noise

Choose the input so that (I) and (II) become easy to solve.

)()()()( 0 tvtuqgty

)()()( uRtutuE

)()()(1

)(ˆ ItutuN

RN

t

Nu

)()()()(1

0 tvktukgtyk

operation loop-open 0)()( tvtuE

1

0 )()()()()(k

uyu kRkgRtutyE

• If the input is not white noise

)()(ˆ)(ˆ)(ˆ1

IIkRkgRM

k

Nu

Nyu

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Exercise 1:

Suppose)()4(1.0)3(25.0)2(5.0)1()( tvtututututy

and v(t) is normal noise such that

)2,0()( Ntv

a) Derive with Impulse Response Analysis let u(t)<3)(ˆ tg

)(ˆ tgb) Derive with Step Response Analysis let u(t)<3

)(ˆ tgc) Derive with correlation analysis with white noise input let u(t)<3

)(ˆ tgd) Derive with correlation analysis with non white noise input u(t)<3

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Fourier Analysis

Spectral Analysis


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Frequency-Response Analysis

Sine Wave testing

Suppose

Let the input as


This is known as frequency analysis and is a simple method for obtaining detailed information about a linear system.

Bode plot of the system can be obtained easily.

One may concentrate the effort to the interesting frequency ranges.

Many industrial processes do not admit sinusoidal inputs in normal operation.

Long experimentation periods.

)()()()( 0 tvtuqgty

...,2,1,0cos)( tttu

parttransient )(cos)()( 0 tvtegty i ??

)()( 0 iegt

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Frequency-Response Analysis

Sine Wave testing

...,2,1,0cos)( tttu parttransient )(cos)()( 0 tvtegty i ??

)()( 0 iegt

Frequency analysis by the correlation method.

If v(t) does not contain a pure periodic component of frequency ω .

0

Define ,sin)(1

)(,cos)(1

)(11

ttyN

NIttyN

NIN

ts

N

tc

0

0 0

2/

)()()(ˆ

22

NINI

eg csiN

)(

)()(ˆ

NI

NIarctgeg

c

siN

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Exercise 2:

Suppose)()4(1.0)1(25.0)2(5.0)1()( tvtututututy

and v(t) is normal noise such that

)2,0()( Ntv

a) Derive and discuss about the value of N.)(ˆ iN eg

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Fourier Analysis

Spectral Analysis


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Fourier Analysis

In a linear system different frequencies pass through the system independently.

So extend the frequency analysis estimate to the case of multifrequency inputs.

Empirical Transfer-Function Estimate

Properties of ETFE

Claim:

Remember:

)(

)()(ˆ

N

NiN U

Yeg

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Fourier Analysis

Properties of ETFE

Claim:

Remember:

Now suppose

Let

By above claim

)()()()( 0 tvtuqgty

N

t

tiN etv

NV

1

)(1

)(

)(

)(

)(

)()()(ˆ 0

N

N

N

NiiN U

V

U

Regeg

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Fourier Analysis

Properties of ETFE

Since v(t) is assumed zero mean

So

)()()()( 0 tvtuqgty

N

t

tiN etv

NV

1

)(1

)(

)(

)(

)(

)()()(ˆ 0

N

N

N

NiiN U

V

U

Regeg

0)(NEV

)(

)()()(ˆ 0

N

NiiN U

RegegE

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Fourier AnalysisProperties of ETFE

So

N

t

tiN etv

NV

1

)(1

)(

)(

)(

)(

)()()(ˆ 0

N

N

N

NiiN U

V

U

Regeg

)(

)()()(ˆ 0

N

NiiN U

RegegE

)()(ˆ)()(ˆ 00 ii

Nii

N egegegegE )()()()( 0 tvtuqgty

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Fourier Analysis

Spectral Analysis


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Spectral Analysis

Smoothing the ETFEThe true transfer function is a smooth function.

If the frequrncy distance 2π/N is small compared to how quickly

Changes then

)(0jeG

Are uncorrelated and if we assume

To be constant over the interval

)(0jeG

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For large N we have

If transfer function is not constant

Spectral Analysis

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If noise spectrum is not known and don’t change very much over frequency intervals

Then

Spectral Analysis

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Connection with the Blackman-Tukey Proceture

If N

If

Spectral Analysis

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If noise spectrum don’t change much over frequency intervals

dUW NNu

2

00 )()()(ˆ

similarly

Spectral Analysis

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The fourier cofficients for periodogram 2

)(NU

The fourier cofficients of the function W2

Spectral Analysis

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smooth function is chosen so that its fourier coefficients vanish

for

Weighting Function: The Frequency Window

Spectral Analysis

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Spectral Analysis

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Asymptotic Properties of the Smoothed Estimate

1-Bias

2-Variance

Here

Spectral Analysis

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5/1

5/122

.)(

)()(4N

W

RM

v

u

opt

Value of the width parameter that minimizes the MSE is

The optimal choice of width parameter leads to a MSE error that decays like

5/4. NCMSE

Spectral Analysis

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Spectral Analysis

Smoothing the ETFEExercise 3:

)()2(5.0)1()2(7.0)1(5.1)( tetututytyty

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Another Way of Smoothing the ETFE

Spectral Analysis

The ETFEs obtained over different data sets will also provide uncorrelated estimates, and another approach would be to form averages over these. Split the data set into M batches, each containing R data (N=R . M). Then form the ETFE corresponding to the kth batch:

The estimate can then be formed as a direct average

ikR eG )(ˆ Mk ,...2,1

M

k

ikR

iN eG

MeG

1

)(ˆ1ˆ

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Or one that is weighted according to the inverse variances:

with

M

k

kR

M

k

ikR

kR

iN

eGeG

1

)(

1

)()(

)(

ˆ)(ˆ

2)()( )( kR

kR U

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Estimating The Disturbance Spectrum

)()()()( 0 tvtuqgty Suppose

dUW NNv

2

00 )()()(ˆ 1-Bias

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

1)()(ˆ

1 NOCOME Nvv

Nv

2-Variance)1()(

)()(ˆ 2

NON

WVar Nv

Nv ,0

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The Residual Spectrum

)()(ˆ)()(ˆ tuqGtytv N

dUeGYW Ni

NNNv

2

0 )()(ˆ)()()(ˆ

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We have

)(ˆ

)(ˆ)(ˆ)(ˆ

2

Nu

NyuN

yNv

Coherency Spectrum:

Denote

)(ˆ)(ˆ

)(ˆ)(ˆ

2

Nu

Ny

NyuN

yuk

Then

2)(ˆ1)(ˆ)(ˆ N

yuNy

Nv k

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Fourier Analysis

Spectral Analysis


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Properties of ETFE

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SYSTEMS Identification