SYSTEMS%20Identification

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SYSTEMSIdentification

• Ali Karimpour
• Assistant Professor
• Ferdowsi University of Mashhad

Reference System Identification Theory For The
User Lennart Ljung(1999)
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Lecture 5
Models for Non-Linear Systems

• Topics 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 Systems

• Topics 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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General Aspects
Let Zt as input-output data.

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

A parametric model structure is a parameterized
family of such models
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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Models for Non-Linear Systems

• Topics 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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Black-box models
Choice of regressors and nonlinear function
A parametric model structure is a parameterized
family of such models
Let the output is scalar so
There are two main problems

1. Choose the regression vector f(t)

Regression vector f(t)
ARX, ARMAX, OE,
For non-linear model it is common to use only
measured (not predicted)
2. Choose the mapping g(f,?)
?????
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Black-box models
Functions for a scalar regressor
There are two main problems

1. Choose the regression vector f(t)

2. Choose the mapping g(f,?)
Global Basis Functions Significant variation
over the whole real axis.
Local Basis Functions Significant variation take
place in local environment.
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Several Regressors
Expansion into multiple regressors
In the multi dimensional case (dgt1), gk is a
function of several variables
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Some non-linear model
Examples of named structures
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Simulation and prediction
Let
The (one-step-ahead) predicted output is
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
There are some important notations
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Choose of regressors
There are some important notations
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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Models for Non-Linear Systems

• Topics 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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Grey-box Models
Physical modeling
Perform physical modeling and denote unknown
physical parameters by ?
So simulated (predicted) output is
The approach is conceptually simple, but could be
very demanding in practice.
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Grey-box Models
Physical modeling
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Grey-box Models
Semi physical modeling
First of all consider a linear model for system
The model can not fit the system so
So we have
And also
So we have
Exercise1 Derive (I)
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Grey-box Models
Block 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 Models
Linear regression
Linear regression means that the prediction is
linear in parameters
The key is how to choose the function fi(ut,yt-1)
GMDH-approach considers the regressors as typical
polynomial combination of past inputs and outputs.
For Hammerstein model we may choose
For Wiener model we may choose
Exercise2 Derive a linear regression form for
equation (I) in solar heated house.
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Grey-box Models
Local 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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Grey-box Models
Local linear models
Sampled data around level h leads to
Total model
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Grey-box Models
Local linear models
To built the model, we need
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.