system iden7 recursive estimation methods.ppt - Ali …karimpor.profcms.um.ac.ir/imagesm/354/...recursive_estimation_meth… · Lecture 7 Ali Karimpour Dec 2014 4 Introduction In

SYSTEMSYSTEMIDENTIFICATIONIDENTIFICATION

Ali KarimpourAssociate Professor

Ferdowsi University of Mashhad

References are appeared in the last slide.

Last update: 09.12.2014 (1393/09/18)

Lecture 7

Ali Karimpour Dec 2014

2

Lecture 7

Recursive estimation methodsRecursive estimation methodsTopics to be covered include:

Introduction.

The Recursive Least-Squares Algorithm.

The Recursive IV Method.

Recursive Prediction-Error Methods.

Recursive Pseudolinear Regressions.

Lecture 7


3

Topics to be covered include:

Introduction.





Recursive estimation methods

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4

Introduction

In many cases it is necessary, or useful, to have a model of the system available in real time.

The reason for parameter varying system:

• Aging.

• Changing the dynamic of process.

• Changing the ambient of process.

Type of data in system identification:

• Batch data.

• Sequential data.

Offline identification.

Online identification.

Both Online and offline identification can be used in Real-time identification.

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5

Introduction

In many cases it is necessary, or useful, to have a model of the system available in real time.The need for such real-time model is required in order to:

• Which input should be applied at the next sampling instant?

• How should the parameters of a matched filter be tuned?

• What are the best predictions of the next few output?

• Has a failure occurred and, if so, of what type?

Adaptive

Adaptive control, adaptive filtering, adaptive signal processing, adaptive prediction.

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6

Introduction

The real-time computation of the model must completedduring one sampling interval.

Identification techniques that comply with this requirement will be called:

• Recursive identification methods.

• On-line identification. • Real-time identification.

• Adaptive parameter estimation. • Sequential parameter estimation.

• Recursive identification methods. Used in this Course.

Apart from the use of recursive methods in adaptive schemes, they are important since:

• They will carry their own estimate of the parameter variance.

Data collected from the system and processed until a sufficient degree of accuracy reached.

• This algorithms will also quite competitive with off-line situations.

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7

Introduction

Algorithm format

General identification method: ),(ˆ tt ZtF

))(),(),1((ˆˆ))(),(),1(()1()(

1 tutytXQ

tutytXQtXtX

ttt

Xt

This form cannot be used in a recursive algorithm, since it cannot be completed in one sampling instant.

Instead following recursive algorithm must comply:

Minimizing argument of some function or…

Information state

Since the information in the latest pair of measurement { y(t) , u(t) } normally is small compared to the pervious information so there is a more suitable form

Small numbers reflecting the relative information value in the latest measurement.

))1((ˆ))(),(),1(,()(

tXh

tutytXtHtX

t

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8

Introduction

Mean derive from a recursive procedure

?1 NN

N

NN xNN

N 111

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9


Introduction.






Lecture 7


10

The Recursive Least-Squares Algorithm

LS Criterion: The estimate for the weighted least squares is:

Where

Let us try to utilize the relationship of parameters at time t and t+1

2

1)()(minarg)(ˆ

t

k

T kkyt

Yt TT 1)(̂

)()()(ˆ tbtPt

1

1

1 )()()(

t

k

TT kktP

t

k

T kyktYttb1

)()()()()(

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11



Let us try to utilize the relationship of parameters at time t and t+1

)()()(ˆ tbtPt 1

1

1 )()()(

t

k

TT kktP

t

k

T kyktYttb1

)()()()()(

1

1

11

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

t

k

TTt

k

T ttkkkktP

11 )1()1()()1( tttPtP T

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

1

1

tytkykkyktb

t

k

t

k

)1()1()()1( tyttbtb

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12



Recursive algorithm:

)()()(ˆ tbtPt 1

1

1 )()()(

t

k

TT kktP

t

k

T kyktYttb1

)()()()()(

11 )1()1()()1( tttPtP T

)1()1()()1( tyttbtb ................

)(ˆ)1()1()1()1()(ˆ)1(ˆ tttyttPtt T

)1(

)|1(ˆ)1()1()1()(ˆ)1(ˆ

t

ttytyttPtt

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13


Ordinary LS Criterion:

Recursive algorithm: Yt TT 1)(̂

11 )1()1()()1( tttPtP T

)1(

)|1(ˆ)1()1()1()(ˆ)1(ˆ

t

ttytyttPtt

Benefits of recursive algorithm?

Order of complexity.Ordinary LS: O(d2t)

Recursive LS: O(d3)

Problem of recursive LS?

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14


11 )1()1()()1( tttPtP T

Version with Efficient Matrix Inversion

To avoid inverting at each step, let remember matrix inversion lemma:

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

ttPt

tPtttPtPtPT

T

)1(

)|1(ˆ)1()1()1()(ˆ)1(ˆ

t

ttytyttPtt

Mor

e ef

ficie

nt

recu

rsiv

e LS

111111 DABDACBAABCDA

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15


Ordinary LS Criterion:

Recursive algorithm: Yt TT 1)(̂

11 )1()1()()1( tttPtP T

)1()1()1()(ˆ)1(ˆ tttPtt

Order of complexity?

Ordinary LS: O(d2t) Recursive LS: O(d3)

More efficient recursive algorithm:

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

ttPt

tPtttPtPtPT

T

)1()1()1()(ˆ)1(ˆ tttPtt

More efficient recursive LS: O(d2)

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Initial values?

More efficient recursive algorithm:

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

ttPt

tPtttPtPtPT

T

)1()1()1()(ˆ)1(ˆ tttPtt

1- Try estimation on a set of batch data,

?)0(?)0(ˆ P

2- Start with arbitrary initials,

1000100)0(0)0(ˆ IP

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17


Exercise 7-1: Try to find 6000 suitable data from following system,

y(t)+ay(t-1)=bu(t-1)+e(t) e(t) is WGN (variance is 0.1)

a) Suppose a=0.7 and b=2 try to find parameters through ordinary LS and recursive LS.

b) Draw parameters variation versus iteration and explain the convergence behavior.

c) Suppose a=0.7 and b=2 in first 3000 samples and suddenly they change to a=1 and b=3 in the next 3000 samples. Draw parameters variation versus iteration and explain the convergence behavior in first and in second part.

d) What happened in part ‘c’ and why?

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18


Draw back of recursive LS:

2

1)()(minarg)(ˆ

t

k

T kkyt

Using forgetting factor:

2

1)()(minarg)(ˆ

t

k

Tkt kkyt

999.095.0

QYQt TT 1)(̂

New estimation is:

021 ,...,, ttdiagQ

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

ttPt

tPtttPtPtPT

T

0)( tP

Modification for time-varying parameters

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Forgetting factor

11

The number of data kept in the memory can roughly be calculated as:

noise. toeSusceptibl

.capability trackingParameter

λ is chosen as a trade-off between noise sensitivity and parameter tracking capability.

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Exercise 7-2: Show that recursive least square with forgetting factor is derived by:

2

1)()(minarg)(ˆ

t

k

T kkyt

Using forgetting factor:

2

1)()(minarg)(ˆ

t

k

Tkt kkyt

999.095.0

QYQt TT 1)(̂

New estimation is:

021 ,...,, ttdiagQ

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

ttPttPttItPtP

T

T

)1()1()1()(ˆ)1(ˆ tttPtt

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Multivariable case

Now for MIMO

Remember SISO

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

ttPttPttItPtP

T

T

)1()1()1()(ˆ)1(ˆ tttPtt

2

1)()(minarg)(ˆ

t

k

Tkt kkyt

Exercise 7-3: Show that recursive least square with forgetting factor is derived by:

)()()()()(minarg)(ˆ 1

1kkykkkyt T

Tt

k

Tkt

)()1()1()()1()1()1()()(1)1( 1 tPtttPttttPtPtP TT

)1()1()1()1()(ˆ)1(ˆ 1 ttttPtt

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Recursive LS.

Draw back?

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

ttPttPttItPtP

T

T

)1()1()1()(ˆ)1(ˆ tttPtt

2

1)()(minarg)(ˆ

t

k

Tkt kkyt

1- A single forgetting factor can be chosen for the complete model.

2- A “blow-up” can also be happen for constant λ and low excitation.

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Use of state estimation to use in feedback loop

x x

M

G

u y

F

H

r

States are not available!

x Estate Estimator

x̂

Condition for xx ˆ

Kalman filterA common application is for state-feedback control.

Lecture 7


Discrete Kalman filter:

If the pair {A, C} is observable . Then the eigenvalues of A-kC can arbitrarily be assigned.

24

( 1) ( ) ( )( )

x t Ax t Bu ty Cx t

ˆ( ) ( ) ( )

ˆ ˆ ˆ( 1) ( 1) ( ) ( ) ( ( ) ( ))

( 1) ( ) ( ) ( ) ( ) (0)e t x t x t t

x t x t A x t x t kC x t x t

e t A kC e t e t A kC e

Kalman filter

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Kalman filter in presence of noise

{w(t)} and {v(t)} are assumed to be sequences of white noise independent random variables with zero mean and

1

2

12

( ) ( )

( ) ( )

( ) ( )

T

T

T

E w t w t R

E v t v t R

E w t v t R

(0 ) (0 ) ( ) 0

(0 ) (0 ) ( ) 0

T

T

E x x w t

E x x v t

( 1) ( ) ( ) ( ) (( ) ( ) ( )x t Ax t Bu t w ty t Cx t v t

ˆ ˆ ˆ( 1) ( ) ( ) ( ) ( ) (x t Ax t Bu t K y t Cx t

( 1) ( ) ( ) (e t Le t t

Kalman filter:

Lecture 7


We show that

26


( 1) ( ) ( ) (e t Le t t L A KC

( ) ( ) ( )t K v t w t Where

The gain matrix K must be chosen such that:

First part of e(t) goes to zero by suitable K but second part?

)()()( minimize and0 teteEtPeE T

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( 1) ( ) ( ) ( )E e t E Le t t LE e t

( 1) ( ) ( ) (e t Le t t L A KC

( ) ( ) ( ) (TP t E e t e t

2 12 12 1( 1) ( ) (T T T TP t A KC P t A KC KRK R K KR R

( ) ( ) ( )t K v t w t

)()()( minimize and0 teteEtPeE T

P(t+1) can be rewritten as:

Lecture 7


28


P(t+1) can be rewritten as

1

1

12 2

1

2 12 2

1

12 2 12

( 1) ( )

( ) ( )

( ) ( ) ( )

( ) ( ) ( ) (22)

T

T T

TT T T

TT T T

P t AP t A R

K R AP t C R CP t C

R CP t C K R AP t C R CP t C

R AP t C R CP t C R AP t C

Minimizing error covariance matrix leads to:1

12 2( ) ( ) ( )T TK t AP t C R CP t C R

1

1

12 2 12

( 1) ( )

( ) ( ) ( )

T

TT T T

P t AP t A R


And finally P(t+1) can be rewritten as

Lecture 7


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{w(t)} and {v(t)} are assumed to be sequences of white noise independent random variables with zero mean and

1

2

12

( ) ( )

( ) ( )

( ) ( )

T

T

T

E w t w t R

E v t v t R

E w t v t R

(0 ) (0 ) ( ) 0

(0 ) (0 ) ( ) 0

T

T

E x x w t

E x x v t


ˆ ˆ ˆ( 1) ( ) ( ) ( ) ( ) (x t Ax t Bu t K y t Cx t

Kalman filter:

1

12 2( ) ( ) ( )T TK t AP t C R CP t C R

1

1

12 2 12

( 1) ( )

( ) ( ) ( )

T

TT T T

P t AP t A R


Lecture 7


30


Exercise 7-6: Consider following system.

a) Let r1=0.5 and r2=0.25 and x(0)=[10 10]T draw x(t).

)()(]01[)(

)()(995.00599.0

)1(

tvtxty

twtxtx

t.independen areand)}()({)}()({

2

1

vwrtvtvEIrtwtwE

T

T

b) Design a Kalman filter and let [0 0]T as initial value for filter.

c) Draw estimated states and true states in the same figure.

d) Repeat parts a, b, and c for r1=0 and r2=0.

Lecture 7



31


Recursive least square by Kalman Filter Interpretation The Kalman Filter for estimating the state of system

Kalman Filter

The linear regression model can be cast to above form as:

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

tetttytt

T

)(ˆ tx

IA )(tC T

0B 01 R

)}()({2 teteER T

Exercise 7-5: Show that recursive least square for MIMO system can be derived from Kalman filter by suitable assumption.

Hint: let λ=1 and Λ=R2

Lecture 7



32


The Kalman Filter for estimating the state of system

Kalman Filter

The linear regression model can be cast to above form as:

)()()()()()()1(tetttytwtt

T

)(ˆ tx

IA )(tC T

0B1R

)}()({2 teteER T

Coping with Time-varying Systems

Exercise 7-6: Derive the Kalman filter for above mention system and compare it with recursive least square with forgetting factor.

)](ˆ)()()[()(ˆ)1(ˆ tttytKtt T 1

2 ])()()()[()( RttPttPtK T 1

1

2 )()()]()()()[()()()1( RttPttPtRttPtPtP T

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Kalman Filter Interpretation

Kalman filter interpretation gives important information, as well as some practical hints:

Lecture 7


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Recursive LS.

Draw back?1- A single forgetting factor can be chosen for the complete model. 2- A “blow-up” can also be happen for constant λ and low excitation.Kalman filter

)](ˆ)()()[()(ˆ)1(ˆ tttytKtt T 1

2 ])()()()[()( RttPttPtK T

1

1

2 )()()]()()()[()()()1( RttPttPtRttPtPtP T 1- Different weight on different parameter.

2- No “blow-up” can be happen in the case of low excitation.

)()1()1()()1()1()1()()(1)1( 1 tPtttPttttPtPtP TT

)1()1()1()1()(ˆ)1(ˆ 1 ttttPtt

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Introduction.






Lecture 7


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The Recursive IV Method

The recursive instrumental variable method is:

Remember SISO recursive LS

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

ttPttPttItPtP

T

T

)1()1()1()(ˆ)1(ˆ tttPtt

2

1)()(minarg)(ˆ

t

k

Tkt kkyt

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

ttPttPttItPtP

T

T

)1()1()1()(ˆ)1(ˆ tttPtt

Lecture 7


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Introduction.






Lecture 7


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Recursive Prediction-Error Methods Analogous to the weighted LS case, let us consider a weighted quadratic prediction-error criterion

Where

so we have

the gradient with respect to θ is

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Remember the general search algorithm developed for PEM as:

For each iteration i, we collect one more data point, so

now define

As an approximation let:

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As an approximation let:

With above approximation and taking μ(t)=1, we thus arrive at the algorithm:

But this terms are not recursive.

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Recursive Prediction-Error Methods

Lecture 7


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Recursive Prediction-Error Methods

Family of recursive prediction error methods

• According to the model structure

• According to the choice of R

Wide family of methods

We shall call “RPEM”

For example, the linear regression

If we consider R(t)=I

Where the gain could be a given sequence or normalized as

This is recursive least square method

This scheme has been widely used, under the name least mean squares (LMS)

Lecture 7


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Recursive Prediction-Error Methods Example 7-1 Recursive Maximum Likelihood

)()()()()()( teqCtuqBtyqA

Tcba nttntutuntytyt ),(...),1()(...)1()(...)1(),(

Tnnn cbacccbbbaaa ......... 212121

Consider ARMAX model

where

and

Remember chapter 10

By rule 11.41

This scheme is known as recursive maximum likelihood (RML)

Lecture 7


44

Recursive Prediction-Error Methods Projection into DM

In off-line minimization this must be kept in mind as a constraint.

The model structure is well defined only for giving stable predictors. MD

The same is true for the recursive minimization.

m

m

DiftDiftttRttt

~)1(ˆ

~~)()()()()1(ˆ)(ˆ1

Experience shows that the test typically take place in the few samples in the beginning.

Lecture 7


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Introduction.






Lecture 7


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Recursive Pseudolinear Regressions

Consider the pseudo linear representation of the prediction

And recall that this model structure contains, among other models, the general linear SISO model:

A bootstrap method for estimating θ was given by (Chapter 10, 10.64)

By Newton - Raphson method

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Recursive Pseudolinear Regressions By Newton - Raphson method

By rule 11.41

Lecture 7


48

Recursive Pseudolinear Regressions Family of RPLRsThe RPLR scheme represents a family of well-known algorithms when applied to different special cases of

The ARMAX case is perhaps the best known of this. If we choose

This scheme is known as extended least squares (ELS).Other special cases

Lecture 7


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References

2- “System Identification Theory For The User” Lennart Ljung(1999)

3- “Nonlinear System Identification” Oliver Nelles (2001)

1- “System Modeling and Identification” Rolf Johansson (2010)

Documents

system iden7 recursive estimation methods.ppt - Ali …karimpor.profcms.um.ac.ir/imagesm/354/...recursive_estimation_meth… · Lecture 7 Ali Karimpour Dec 2014 4 Introduction In