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### SYSTEMSIdentification

Ali Karimpour

Assistant Professor

Ferdowsi University of Mashhad

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

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

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

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.

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

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:

Choose the regression vector φ(t)

Regression vector φ(t)

ARX, ARMAX, OE, …

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

2. Choose the mapping g(φ,θ)

?????

Black-box models

Functions for a scalar regressor

There are two main problems:

Choose the regression vector φ(t)

2. Choose the mapping g(φ,θ)

Scale or dilation

Coordinates

Location parameter

Basis functions

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

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

Several Regressors

Expansion into multiple regressors

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

Some non-linear model

Examples of “named” structures

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:

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.

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

Physical 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.

Physical modeling

Semi physical modeling

First of all consider a linear model for system

Solar heated house

The model can not fit the system so:

Let x(t): Storage temperature

So we have:

And also

So we have

Exercise1: Derive (I)

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

Linear regression

Linear regression means that the prediction is linear in parameters

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

For Wiener model we may choose

Exercise2:

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

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:

Local linear models

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:

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.

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

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