SYSTEMS Identification Ali Karimpour Assistant Professor Ferdowsi University of Mashhad Reference: “System Identification Theory For The User” Lennart

SYSTEMSSYSTEMSIdentificationIdentification

Ali Karimpour

Assistant Professor

Ferdowsi University of Mashhad

Reference: “System Identification Theory For The User”

Lennart Ljung

2

lecture 7

Ali Karimpour Nov 2009

Lecture 7

Parameter Estimation MethodParameter Estimation Method

Topics to be covered include: Guiding Principles Behind Parameter Estimation Method.

Minimizing Prediction Error.

Linear Regressions and the Least-Squares Method.

A Statistical Framework for Parameter Estimation and the Maximum Likelihood Method.

Correlation Prediction Errors with Past Data.

Instrumental Variable Methods.

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Models of linear time invariant systemModels of linear time invariant system

Topics to be covered include:

Guiding Principles Behind Parameter Estimation Method.






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Guiding Principles Behind Parameter Estimation Method

Parameter Estimation Method

MDMM |)(

)(),()(),()( teqHtuqGty

)(),()(),()|(ˆ:)( tuqWtyqWtyM uy

Suppose that we have selected a certain model structure M. The set of models defined as:

For each θ , model represents a way of predicting future outputs. The predictor is a linear filter as:

Suppose the system is:

Where:

),(),(),(,),(1),( 11 qGqHqWqHqW uy

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Suppose that we collect a set of data from system as:

)(,)(,...,)2(,)2(,)1(,)1( NuNyuyuyZ N

Formally we are going to find a map from the data ZN to the set DM

MNN DZ ̂

Such a mapping is a parameter estimation method.

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Evaluating the candidate model

Let us define the prediction error as:

)|(ˆ)(),( tytyt

When the data set ZN is known, these errors can be computed for t=1, 2 , … , N

A guiding principle for parameter estimation is:

Based on Zt we can compute the prediction error ε(t,θ). Select so that the

prediction error t=1, 2, … , N, becomes as small as possible.

N̂

,)ˆ,( Nt

?

We describe two approaches

• Form a scalar-valued criterion function that measure the size of ε.7_2 till 7_4

• Make uncorrelated with a given data sequence. 7_5 and 7_6 )ˆ,( Nt

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Minimizing Prediction Error

Clearly the size of prediction error

)|(ˆ)(),( tytyt

is the same as ZN

Let to filter the prediction error by a stable linear filter L(q)

),()(),( tqLtF Then use the following norm

N

tF

NN tl

NZV

1

),(1

),(

Where l(.) is a scalar-valued positive function.

The estimate is then defined by:N̂

),(minarg)(ˆˆ NN

D

NNN ZVZ

M

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),()(),( tqLtF

N

tF

NN tl

NZV

1

),(1

),(

),(minarg)(ˆˆ NN

D

NNN ZVZ

M

Generally the term prediction error identification methods (PEM) is used for the family of this approaches.

Particular methods with specific names are used according

to:

• Choice of l(.)

• Choice of L(.)

• Choice of model structure

• Method by which the minimization is realized

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),()(),( tqLtF

N

tF

NN tl

NZV

1

),(1

),(

),(minarg)(ˆˆ NN

D

NNN ZVZ

M

Choice of L

The effect of L is best understood in a frequency-domain interpretation. Thus L acts like frequency weighting.

See also >> 14.4 Prefiltering

Exercise: Consider following system

)(),()(),()( teqHtuqGty Show that the effect of prefiltering by L is identical to changing the noise model from

),()(),( 1 qHqLqH

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),()(),( tqLtF

N

tF

NN tl

NZV

1

),(1

),(

),(minarg)(ˆˆ NN

D

NNN ZVZ

M

Choice of l

A standard choice, which is convenient both for computation and analysis.

2

2

1)( l

See also >> 15.2 Choice of norms: Robustness (against bad data)

One can also parameterize the norm independent of the model parameterization.

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Linear Regressions and the Least-Squares Method

)()()|(ˆ ttty T We introduce linear regressions before as:

φ is the regression vector and for the ARX structure it is

Tba ntutuntytyt )(...)1()(...)1()( μ(t) is a known data dependent vector. For simplicity let it zero in the reminder of this section.

Least-squares criterion

)()(),( :iserror Prediction ttyt T

Now let L(q)=1 and l(ε)= ε2/2 then

211

)()(2

11),(

1),( tty

Ntl

NZV T

N

t

N

tF

NN

This is Least-squares criterion for the linear regression

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Least-squares criterion

211

)()(2

11),(

1),( tty

Ntl

NZV T

N

t

N

tF

NN

The least square estimate (LSE) is:

N

t

N

t

TNLSN tyt

Ntt

NZV

1

1

1

)()(1

)()(1

),(minargˆ

)(NR )(Nf

)()(ˆ 1 NfNRLSN

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Properties of LSE

The least square method is a special case of PEM (prediction error method)

So we have

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Properties of LSE

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Weighted Least Squares

Different measurement could be assigned different weights

2

1

)()(1

),(

N

t

Tt

NN tty

NZV

or

2

1

)()(),(),(

N

t

TNN ttytNZV

N

t

N

t

TLSN tyttNtttN

1

1

1

)()(),()()(),(ˆ

The resulting estimate is the same as previous.

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Colored Equation-error Noise

if the disturbance v(t) is not white noise, then the LSE will not converge to the true value ai and bi .

To deal with this problem, we may incorporate further modeling of the equation error v(t) as discussed in chapter 4, let us say

We show that in a difference equation

)()(...)1(

)(...)1()(

1

1

tvntubtub

ntyatyaty

bn

an

b

a

)()()( teqktv Now e(t) is white noise, but the new model take us out from LS environment, except in two cases:

• Known noise properties

• High-order models

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Colored Equation-error Noise • Known noise properties

)()(...)1(

)(...)1()(

1

1

tvntubtub

ntyatyaty

bn

an

b

a

Suppose the values of ai and bi are unknown, but k is a known filter (not too realistic a situation), so we have

)()()( teqktv

Filtering through k-1(q) gives

where

Since e(t) is white, the LS method can be applied without problems.

Notice that this is equivalent to applying the filter L(q)=k-1(q) .

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Colored Equation-error Noise

Now we can apply LS method. Note that nA=na+r, nB=nb+r

• High-order models

)()(...)1(

)(...)1()(

1

1

tvntubtub

ntyatyaty

bn

an

b

a

Suppose that the noise v can be well described by k(q)=1/D(q) where D(q) is a polynomial of order r. So we have

)()()( teqktv

)()(

1)()()()( te

qDtuqBtyqA

or

)()()()()()()( tetuqDqBtyqDqA

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Consider a state space model as

)()()()(

)()()()1(

tvtDutCxty

twtButAxtx

To derive the system

1- Parameterize A, B, C, D as in section 4.3

2- We have no insight into the particular structure and we would like to find any suitable matrices A, B, C, D.

Note: Since there are infinite number of such matrices that describe the same system (similarity transformation), we will have to fix the coordinate basis of the state space realization.

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Consider a state space model as)()()()(

)()()()1(

tvtDutCxty

twtButAxtx

Note: Since there are infinite number of such matrices that describe the same system (similarity transformation), we will have to fix the coordinate basis of the state space realization.Let us for a moment that not only y and u are measured the states are also measured. This would, by the way, fix the state-space realization coordinate basis.

Now with known y, u, x the model becomes a linear regression

)(

)()(,

)(

)()(,,

)(

)1()(

tv

twtE

tu

txt

DC

BA

ty

txtY

Then

)()()()(

)()()()1(

tvtDutCxty

twtButAxtx

)()()( tEttY

But there is some problem? States are not available to measure!

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Estimating State Space Models Using Least Squares Techniques(Subspace Methods)

)(

)()(,

)(

)()(,,

)(

)1()(

tv

twtE

tu

txt

DC

BA

ty

txtY

By subspace algorithm x(t+1) derived from observations.

Chapter 10

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A Statistical Framework for Parameter Estimation and the Maximum Likelihood Method

Estimation and the Principle of Maximum Likelihood

);(),...,,;( 21N

yN xfxxxf That is:

N

Ax

Ny

N dxxfAyPN

);()(

The area of statistical inference, deals with the problem of extracting information from observations that themselves could be unreliable.

Suppose that observation yN=(y(1), y(2),…,y(N)) has following probability density function (PDF)

θ is a d-dimensional parameter vector. The propose of the observation is in fact to estimate the vector θ using yN.

dNN RRy )(̂Suppose the observed value of yN is yN

*, then

)(ˆˆ**Ny

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Estimation and the Principle of Maximum Likelihood

dNN RRy )(̂

Many such estimator functions are possible.

),( *N

y yf

A particular one >>>>>>>>> maximum likelihood estimator (MLE) .

),(maxarg)(ˆ**N

yN

ML yfy

The probability that the realization(=observation) indeed should take the value yN* is

proportional to

This is a deterministic function of θ once the numerical value yN* is inserted and it is

called Likelihood function.

A reasonable estimator of θ could then be

where the maximization performed for fixed yN* . This function is known as the

maximum likelihood estimator (MLE).

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Example: Let Niiy ,...,1,)(

Be independent random variables with normal distribution with unknown means θ0

and known variances λi ),()( 0 iNiy A common estimator is the sample mean:

N

i

NSM iy

Ny

1

)(1

)(̂

To calculate MLE, we start to determine the joint PDF for the observations. The PDF for y(i) is:

i

i

i

x

2

)(exp

2

1 2

Joint PDF for the observations is: (since y(i) are independent)

N

i i

i

i

Ny

xxf

1

2

2

)(exp

2

1);(

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Example: Let Niiy ,...,1,)(

Be independent random variables with normal distribution with unknown means θ0

and known variances λi

A common estimator is the sample mean:

N

i

NSM iy

Ny

1

)(1

)(̂

Joint PDF for the observations is: (since y(i) are independent)

N

i i

i

i

Ny

xxf

1

2

2

)(exp

2

1);(

So the likelihood function is:);( N

y yf

Maximizing likelihood function is the same as maximizing its logarithm. So

);(logmaxarg)(ˆ Ny

NML yfy

N

i ii

N

i

iyN

1

2

1

)(

2

1

2

12log

2maxarg

N

i iN

ii

NML

iyy

1

1

)(

/1

1)(ˆ

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lecture 7


2 4 6 8 10-20

0

20

40

Different experimentsD

iffer

ent

estim

ator

s


Example: Let Niiy ,...,1,)( Be independent random variables with normal distribution with unknown means θ0

and known variances λi),()( 0 iNiy

Suppose N=15 and y(i) is derived from a random generation (normal distribution) such that the means is 10 but variances are:

10, 2, 3, 4, 61, 11, 0.1, 121, 10, 1, 6, 9, 11, 13, 15 The estimated means for 10 different experiments are shown in the figure:

N

i

NSM iy

Ny

1

)(1

)(̂

N

i iN

ii

NML

iyy

1

1

)(

/1

1)(ˆ

)(ˆ NSM y

)(ˆ NML y

Exercise:Do the same procedure for another experiments and draw the corresponding figure.

Exercise:Do the same procedure for another experiments and draw the corresponding figure. Suppose all variances as 10.

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Relationship to the Maximum A Posteriori (MAP) Estimate

Maximum likelihood estimator (MLE)

),(maxarg)(ˆ**N

yN

ML yfy

The Bayesian approach is used to derive another parameter estimation problem.

In the Bayesian approach the parameter itself is thought of as a random variable.

Let the prior PDF for θ is:

)()( zPzg

After some manipulation the Maximum A Posteriori (MAP) estimate is:

)().,(maxarg)(ˆ gyfy N

yN

MAP

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Cramer-Rao Inequality

TNN yyEP 00 )(ˆ)(ˆ

The quality of an estimator can be assessed by its mean-square error matrix:

True value of θ

We may be interested in selecting estimators that make P small. Cramer-Rao inequality give a lower bound for P

M is Fisher Information

matrix

32

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Asymptotic Properties of the MLE

TNN yyEP 00 )(ˆ)(ˆ Calculation of Is not an easy task.

Therefore, limiting properties as the sample size tends to infinity are calculated instead.

For the MLE in case of independent observations, Wald and Cramer obtain

Suppose that the random variable {y(i)} are independent and identically distributed, so that

N

iiiyNy xfxxxf

1)(21 );(),...,,;(

Suppose also that the distribution of yN is given by fy(θ0 ;xN) for some value θ0. Then tends to θ0 with probability 1 as N tends to infinity, and )(ˆ N

ML y

0)(ˆ NML yN

converges in distribution to the normal distribution with zero mean covariance matrix given by Cramer-Rao lower bound M-1.

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Likelihood function for Probabilistic Models of Dynamical Systems

)|(ˆ)(),(

);,()|(ˆ:)( 1

tytyt

ZtgtyM t

Suppose

);,(PDF

thehave andt independen is

txfe

Recall this kind of model a complete probabilistic model.

We note that, the output is:);,( PDF thehas ),( where),()|(ˆ)( txftttyty e

Now we must determine the likelihood function

);( Ny yf

Probabilistic Models of Dynamical Systems

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?);( Ny yf

Lemma: Suppose ut is given as a deterministic sequence, and assume that the generation of yt is described by the model

),( is )( of PDF lconditiona the where)(),()( 1 txfttZtgty et

Then the joint probability density function for yt , given ut is:

)(),,()()|,( 1

1 IkZkgkyfuytft

k

ke

ttm

Proof: CPDF of y(t), given Zt-1 , is tZtgxfZxp tte

tt ),,()|( 11

Using Bayes’s rule, the joint CPDF of y(t) and y(t-1), given Zt-2 can be expressed as:

1),,1(.),,( 21

1

tZtgxftZtgxf tte

tte

)|().,)1(|()|,( 21

21

21

t

tt

ttt

tt ZxpZxtyxpZxxp

Similarly we derive (I)

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lecture 7



)|(ˆ)(),(

);,()|(ˆ:)( 1

tytyt

ZtgtyM t

Suppose

);,(PDF

thehave andt independen is

txfe

Now we must determine the likelihood function

Probabilistic Models of Dynamical Systems

By previous lemma

N

t

te

Ny tZtgtyfyf

1

1 ;),;,()();(

N

te ttf

1

;),,(

Maximizing this function is the same as maximizing

N

te

Ny ttf

Nyf

1

;),,(log1

);(logN

1

If we define

);,(log),,( tftl e

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Probabilistic Models of Dynamical SystemsMaximizing this function is the same as maximizing

N

te

Ny ttf

Nyf

1

;),,(log1

);(logN

1

If we define

);,(log),,( tftl eWe may write

N

t

NML ttl

Ny

1

);),,((1

minarg)(ˆ

The ML method can thus be seen as a special case of the PEM.

Exercise: Find the Fisher information matrix for this system.

Exercise: Derive a lower bound for .ˆNCov

37

lecture 7










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Correlation Prediction Errors with Past Data

0),()(1

1

N

t

ttN

Ideally, the prediction error ε(t,θ) for good model should be independent of the past data Zt-1

If ε(t,θ) is correlated with Zt-1 then there was more information available in Zt-1 about y(t) than picked up by )|(ˆ ty

To test if ε(t,θ) is independent of the data set Zt-1we must check

This is of course not feasible in practice.

Uncorrelated with

All transformation of ε(t,θ)

All possible function of Zt-1

Instead, we may select a certain finite-dimensional vector sequence {ζ(t)} derived from Zt-1 and a certain transformation of {ε(t,θ)} to be uncorrelated with this sequence. This would give

Derived θ would be the best estimate based on the observed data.

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),()(),( tqLtF

Choose a linear filter L(q) and let

Choose a sequence of correlation vectors

),,(),( 1 tZtt

Choose a function α(ε) and define

N

tF

NN tt

NZf

1

),(),(1

),( Then calculate

0),(ˆ

NN

DN Zfsol

M

Instrumental variable method (next section) is the best known representative of this family.

40

lecture 7



0),(ˆ

NN

DN Zfsol

M

N

tF

NN tt

NZf

1

),(),(1

),(

Normally, the dimension of ξ would be chosen so that fN is a d-dimensional vector.

Then there is many equations as unknowns. Sometimes one use ξ with higher dimension than d so there is an over determined set of equations, typically without solution. so

),(minargˆ NN

DN Zf

M

Exercise: Show that the prediction-error estimate obtained from

),(minarg)(ˆˆ NN

D

NNN ZVZ

M

can be also seen as a correlation estimate for a particular choice of L, ζ and α.

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),()|(ˆ tty T

Pseudolinear Regressions

We saw in chapter 4 that a number of common prediction models could be written as:

Pseudo-regression vector φ(t,θ) contains relevant past data, it is reasonable to require the resulting prediction errors be uncorrelated with φ(t,θ) so:

)(

),(),( tt

N

t

TPLRN ttyt

Nsol

1

0),()(),(1ˆ

Which the term PLR estimate.

42

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Instrumental Variable Methods

)()|(ˆ tty T

Consider linear regression as:

The least-square estimate of θ is given by

N

t

TLSN ttyt

Nsol

1

0)()()(1ˆ

So it is a kind of PEM with L(q)=1 and ξ(t,θ)=φ(t)

Now suppose that the data actually described by

)()()( 00 tvtty T

We found in section 7.3 that LSE will not tend to θ0 in typical cases.N̂

44

lecture 7



N

t

TLSN ttyt

Nsol

1

0)()()(1ˆ )()()( 00 tvtty T


N

t

TIVN ttyt

Nsol

1

0)()()(1ˆ

Such an application to a linear regression is called instrumental-variable method.

The elements of ξ are then called instruments or instrumental variables.

Estimated θ is:

)()(1

)()(1ˆ

1

1

1

tytN

ttN

N

t

N

t

TIVN

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lecture 7



N

t

TLSN ttyt

Nsol

1

0)()()(1ˆ

?method IVin asˆ Does 0 NN

Exercise: Show that will be exist and tend to θ0 if following equations exists.

IVN̂

0)()(

rnonsingula be)()(

0 tvtξE

ttE T


N

t

TIVN ttyt

Nsol

1

0)()()(1ˆ )()(

1)()(

1ˆ1

1

1

tytN

ttN

N

t

N

t

TIVN

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Suppose an ARX model:

)()(...)1(

)(...)1()(

1

1

tentubtub

ntyatyaty

bn

an

b

a

Choices of instruments

A natural idea is to generate the instruments similarly to above model. But at the same time not let them be influenced by this leads to

0{ ( )}v t

( ) ( ) ( 1) ( 2)... ( ) ( 1)... ( )T

a bt K q x t x t x t n u t u t n

Where K is a linear filter and x(t) is generated from the input through a linear system

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Here

( ) ( ) ( ) ( )N q x t M q u t

11

10 1

( ) 1 ...

( ) ...

nnnn

nmnm

N q n q n q

M q m m q m q

( )tMost instruments used in practice are generated in this way. Obviously, is obtained from past inputs by linear filtering and can be written, consequently, as

1( ) ( , )tt t u

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If the input is generated in open loop so that is does not depend on the noise in the system. Then clearly the following property holds:

0 ( )v t

0( ) ( ) 0E t v t

Since both the -vector and -vector are generated form the same input sequence, it might be expected that the following property should hold in general.

( ) ( ) be nonsingularTE t t

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Model-dependent Instruments

It may be desirable to choose the filetrs N and M to those of the true system

0 0( ) ( ); ( ) ( )N q A q M q B q

They are clearly not known, but we may let the instruments depend on the parameters in the obvious way

( , ) ( )[ ( 1, )... ( , ) ( 1)... ( )]

( ) ( , ) ( ) ( )

Ta bt K q x t x t n u t u t n

A q x t B q u t

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The IV method could be summarized as follows

( , ) ( )[ ( ) ]TF t L q y t

( , ) ( , ) ( )ut K q u t

ˆ ( , ) 0IV NN N

DMsol f Z

N

tF

NN tt

NZf

1

),(),(1

),(

( , ) ( , ) ( )ut K q u t

In general, we could write the generation of ( , )t

Where is a d-dimentional column vector of linear filters( , )uK q

where

Documents

SYSTEMS Identification Ali Karimpour Assistant Professor Ferdowsi University of Mashhad Reference: “System Identification Theory For The User” Lennart