SYSTEMS Identification

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SYSTEMSIdentification

• Ali Karimpour
• Assistant Professor
• Ferdowsi University of Mashhad

Reference System Identification Theory For The
User Lennart Ljung
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Lecture 1
Introduction

• Topics to be covered include
• Dynamic systems.
• Models.
• An Archetypical problem-ARX models and LSM.
• The system identification procedure.

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Dynamic systems
System An object in which variables of different
kinds interact and produce observable signals.
Stimuli External signals that affects system.
Dynamic System A system that the current output
value depends not only on the current external
stimuli but also on their earlier value.
Time series A dynamic system whose external
stimuli are not observed.
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Dynamic systems
Stimuli
Input
Disturbance
It can be manipulated by the observer.
It can not be manipulated by the observer.
Measured
Unmeasured
Dynamic system
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A solar heated house
Dynamic system
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Speech generation
Dynamic system
Time series A dynamic system whose external
stimuli are not observed.
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Models
Model Relationship among observed signals.
1- Mental models
2- Graphical models
3- Mathematical (analytical) models
4- Software models

• Split up system into subsystems,

• Joined subsystems mathematically,

1- Modeling

• Does not necessarily involve any
  experimentation on the actual system.

• It is directly based on experimentation.

2- System identification

• Input and output signals from the system are
  recorded.

3- Combined
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The fiction of a true model
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A Basic problem and the linear least squares
method
Perhaps the most basic relationship between the
input and output is the linear difference
equation
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Least Square Method
Suppose that we dont know the value of
parameters ?, but the recorded input and output
over a time interval 1 t N is
Now define
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First order difference equation
Consider the simple model
Then
Now we have
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First order difference equation
Consider the simple model
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An Archetypical problem of system identification
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Linear Regression
That are linear in ? are known in statistics as
linear regressions.
The vector f(t) is called the regression vector
and its components are the regressors.
In this models y(t) explained with the regression
vector f(t) which contains older value of y(t)-
so it is called Auto-Regression with eXtera
inputs. ARX-models
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Model quality and experimental design
Consider a finite impulse response (FIR)
e(t) is a white noise sequence with variance ?
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Model quality and experimental design
The estimate is consequently unbiased.
The covariance matrix of the parameter error is
Exercise1 Derive PN .
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Model quality and experimental design
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The system identification procedure
The construction of a model from data involves
three basic identities
1- A data set, like ZN .

• The data must be maximally informative, subject
  to constraints.

2- The set of candidate models a model structure.

• The most important and at the same time, the
  most difficult choice.

• Model sets with adjustable parameters with
  physical interpretation may be called gray boxes.

• Models whose parameters do not reflect physical
  consideration may be called black boxes.

3- A rule by which candidate models can be
assessed using the data, like the least square
selection rule.
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Model validation
We arrived at a particular model.

• Whether the model is good enough?

• Whether it is valid for its purpose?

Note A model can never be accepted as a final
and true description of the system.
Rather, it can at best be regarded as a good
enough description of certain aspects that are
particular interest to us.
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The system identification loop
The model may be deficient for varietyof reasons

• The numerical procedure failed to find the
  best model according to our criterion.

• The criterion was not well chosen.

• The model set was not appropriate.

• The data set was not informative enough.