SYSTEMS Identification

SYSTEMSSYSTEMSIdentificationIdentification

Ali Karimpour

Assistant Professor

Ferdowsi University of Mashhad

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

lecture 5

Ali Karimpour Nov 2010

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Lecture 5

Models for Non-Linear SystemsModels for Non-Linear SystemsTopics to be covered include: General Aspects Black-box models

• Choice of regressors and nonlinear function

• Functions for a scalar regressor

• Expansion into multiple regressors

• Examples of “named” structures

Grey-box Models• Physical modeling

• Semi-physical modeling

• Block oriented models

• Local linear models

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Models for Non-Linear SystemsModels for Non-Linear Systems

Topics to be covered include: General Aspects Black-box models









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General Aspects

A mathematical model for the system is a function from these data to the output at time t, y(t), in general

),()1|(ˆ 1 tzgtty t

Let Zt as input-output data.

A parametric model structure is a parameterized family of such models:

),()|(ˆ 1 tzgty

The difficulty is the enormous richness in possibilities of parameterizations.

There are two main cases:

• Black-box models: General models of great flexibility• Grey-box models: Some knowledge of the character of the actual system.

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Black-box models

Let the output is scalar so:),()|(ˆ 1 tzgty

There are two main problems:

1. Choose the regression vector φ(t)

A parametric model structure is a parameterized family of such models:

),()( 1 tZt )),((ˆ tgy dt Rz 1: RRg d :

Rzg t 1:

2. Choose the mapping g(φ,θ)

Regression vector φ(t) ARX, ARMAX, OE, …

For non-linear model it is common to use only measured (not predicted)

?????

Choice of regressors and nonlinear function

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Black-box models

There are two main problems:1. Choose the regression vector φ(t) 2. Choose the mapping g(φ,θ)

N

kkkkg

1

)(),(

Basis functions

Coordinates Scale or dilation

Location parameter

ansform Fourier tr )cos()( xx

0

101)(

else

xx e

2

1)( 2

x-

2

x

00

01)(

x

xx

1

1)(

xex

Global Basis Functions: Significant variation over the whole real axis.

Local Basis Functions: Significant variation take place in local environment.

Functions for a scalar regressor

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Several Regressors

N

kkkkg

1

)(),(

In the multi dimensional case (d>1), gk is a function of several variables:

Expansion into multiple regressors

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Some non-linear model

Examples of “named” structures

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Simulation and prediction

)1()1()( tutytLet

The (one-step-ahead) predicted output is:

,)1()1()|(ˆ Tp tutygty

A tougher test is to check how the model would behave in simulation i.e. only the input sequence u is used. The simulated output is:

,)1(),1(ˆ),(ˆ Tss tutygty

There are some important notations:

Outputs)( kty inputs)( ktu

Outputs model Simulated),(ˆ ktys Outputs model Predicted)|(ˆ kty p

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Choose of regressors

There are some important notations: Outputs)( kty inputs)( ktu

Outputs model Simulated),(ˆ ktys Outputs model Predicted)|(ˆ kty p

)()(...)1()( 1 tentubtubty bnb

)()(...)1()(...)1()( 11 tentubtubntyatyaty bnan ba

)(...)1()(

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

1

11

cn

bnan

ntectecte

ntubtubntyatyaty

c

ba

)(...)1()(...)1()( 11 bnfn ntubtubntwftwftwbf

)()()( tetwty

Regressors in NFIR-models use past inputs

Regressors in NARX-models use past inputs and outputs

Regressors in NOE-models use past inputs and simulated outputs

Regressors in NARMAX-models use past inputs and predicted outputs

Regressors in NBJ-models use all four types.

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Network of non-linear systems

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Recurrent networks

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Grey-box ModelsPhysical modeling

Perform physical modeling and denote unknown physical parameters by θ

So simulated (predicted) output is:

The approach is conceptually simple, but could be very demanding in practice.

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Grey-box ModelsPhysical modeling

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Grey-box ModelsSemi physical modeling

First of all consider a linear model for system

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

)2()1()(

2121

21

tIctIctubtub

tyatyaty

The model can not fit the system so:

)()1( tyty

So we have:

And also

So we have

)(...)2(

)1()2()1()1()1()( 31 I

tu

tutydtydty

Let x(t): Storage temperature

Exercise1: Derive (I)

Solar heated house

)1()( txtx )(2 tId )(3 txd )()(0 tutxd

)(1 tyd)()(0 tutxd

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Grey-box ModelsBlock oriented models

It is common situation that while the dynamics itself can be well described by a linear system, there are static nonlinearities at the input and/or output.

Hammerstein Model:

Wiener Model :

Hammerstein Wiener Model :

Other combination

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Grey-box ModelsLinear regression

Linear regression means that the prediction is linear in parameters

)(),(),()|(ˆ 1111 tyuyuty Ttt

ddtt

p

The key is how to choose the function φi(ut,yt-1)

GMDH-approach considers the regressors as typical polynomial combination of past inputs and outputs.

For Hammerstein model we may choose

mmuuuuf ...)( 2

21

For Wiener model we may choose

mm yyyyf ...)( 2

21

Exercise2: Derive a linear regression form for equation (I) in solar heated house.

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Grey-box ModelsLocal linear models

Non-linear systems are often handled by linearization around a working point.

Local linear models is to deal with the nonlinearities by selecting or averaging over some linearized model.

Example: Tank with inflow u and outflow y and level h:

Operating point at h* is:

Linearized model around h* is:

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Sampled data around level h* leads to:

Total model

Let the measured working point variable be denoted by . If working point partitioned into d values , the predicted output will be:

)(t

k

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To built the model, we need:

If the predicted corresponding to is linear in the parameters, the whole model will be a linear regression.

)(ˆ ty kk

It is also an example of a hybrid model.

Sometimes the partition is to be estimated too, so the problem is considerably more difficult.Linear parameter varying (LPV) are also closely related.

SYSTEMS Identification

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