Computacion Inteligente Least-Square Methods for System Identification.

Computacion Inteligente

Least-Square Methods for System Identification

2Contents

System Identification: an Introduction Least-Squares EstimatorsStatistical Properties of least-squares estimatorsMaximum likelihood (ML) estimatorMaximum likelihood estimator for linear model LSE for Nonlinear ModelsDeveloping Dinamic models from DataExample: Tank level modeling

3System Identification: Introduction

Goal– Determine a mathematical model for an unknown system

(or target system) by observing its input-output data pairs


Purposes

– To predict a system’s behavior,

– As in time series prediction & weather forecasting

– To explain the interactions & relationships between inputs & outputs of a system


Context example

– To design a controller based on the model of a system,

– as an aircraft or ship control

– Simulate the system under control once the model is known

6Why cover System Identification

System Identification

It is a well established and easy to use technique for modeling a real life system.

It will be needed for the section on fuzzy-neural networks.

7Spring Example

Experiment Force(newtons) Length(inches)

1 1.1 1.5

2 1.9 2.1

3 3.2 2.5

4 4.4 3.3

5 5.9 4.1

6 7.4 4.6

7 9.2 5.0

What will the length be when the force is 5.0 newtons?

Experimental data

8Components of System Identification

There are 2 main steps that are involved

– Structure identification

– Parameter identification

9Structure identification

Structure identification

Apply a-priori knowledge about the target system to determine a class of models within which the search for the most suitable model is to be conducted

This class of model is denoted by a function y = f(u,) where:

• y is the model output

• u is the input vector is the parameter vector

10Structure identification

Structure identification

f(u,) depends on

– the problem at hand– the designer’s experience– the laws of nature governing the target system

11Parameter identification

– Training data is used for both system and model.

– Difference between Target System output, yi, and Mathematical Model output, yi, is used to update parameter vector, θ.

^

12Parameter identification

Parameter identification

– The structure of the model is known, however we need to apply optimization techniques

– In order to determine the parameter vector such that the resulting model

describes the system appropriately:

iii u to assignedy with 0yy

13System Identification Process

The data set composed of m desired input-output pairs

– (ui, yi) (i = 1,…,m) is called the training data

System identification needs to do both structure & parameter identification repeatedly until satisfactory model is found

14System Identification: Steps

– Specify & parameterize a class of mathematical models representing the system to be identified

– Perform parameter identification to choose the parameters that best fit the training data set

– Conduct validation set to see if the model identified responds correctly to an unseen data set

– Terminate the procedure once the results of the validation test are satisfactory. Otherwise, another class of model is selected & repeat step 2 to 4

15System Identification Process

Structure and parameter identification may need to be done repeatedly

16

Least-Squares Estimators

17Objective of Linear Least Squares fitting

Given a training data set {(ui, yi), i = 1, …, m} and the general form function:

Find the parameters 1, …, n , such thatestimate

18The linear model

The linear model:

y = 1 f 1(u) + 2 f2(u) + … + nfn(u)

= fT(u, )

where:

– u = (u1, …, up)T is the model input vector

– f1, …, fn are known functions of u

1, …, n are unknown parameters to be estimated

19Least-Squares Estimators

The task of fitting data using a linear model is referred to as linear regression

where:

– u = (u1, …, up)T is the input vector

– f1(u), …, fn(u) regressors

1, …, n parameter vector


We collect training data set {(ui, yi), i = 1, …, m}

System’s equations becomes:

mnmn2m21m1

2n2n222121

1n1n212111

y)u(f...)u(f)u(f

y)u(f...)u(f)u(f

y)u(f...)u(f)u(f

Which is equivalent to: A = y


Which is equivalent to: A = y

– where

)u(f)u(f

)u(f)u(f

A

mnm1

1n11

n

1

A = y = A-1y (solution)

m*n matrix n*1 vector m*1 vectorunknown


We have

– m outputs, and

– n fitting parameters to find

Or – m equations, and

– n unknown variables

Usually m is greater than n


Since

the model is just an approximation of the target system & the data observed might be corrupted,

Therefore

– an exact solution is not always possible!

To overcome this inherent conceptual problem, an error vector e is added to compensate

A + e = y


Our goal consists now of finding that reduces the errors between and

The problem: Find,

estimate


If e = y - A then:

We need to compute:

mi

1i

TT2Tii )Ay()Ay(ee)ay()(E


Theorem [least-squares estimator]

The squared error is minimized when satisfies the normal equation

if is nonsingular, is unique & is given by

is called the least-squares estimators, LSE

27Spring Example

– Structure Identification can be done using domain knowledge.

– The change in length of a spring is proportional to the force applied.

Hooke’s law

length = k0 + k1*force

28Spring Example

29

Statistical Properties of least-squares estimators

30Statistical qualities of LSE

Definition [unbiased estimator]

An estimator of the parameter is unbiased if

where E[.] is the statistical expectation


Definition [minimal variance]

– An estimator is a minimum variance estimator if for any other estimator *:

where Cov() is the covariance matrix of the random vector


Theorem [Gauss-Markov]:

– Gauss-Markov conditions:

• The error vector e is a vector of m uncorrelated random variables, each with zero mean & the same variance 2.

• This means that:


Theorem [Gauss-Markov]

LSE is unbiased & has minimum variance.

Proof:

34

Maximum likelihood (ML) estimator

35Maximum likelihood (ML) estimator

The problem

– Suppose we observe m independent samples

x1, x2, …, xm,

– coming from a probability density function with parameters 1, …, r


The criterion for choosing is:

– Choose parameters that maximize data probability

Which one do you prefer? Why?


Likelihood function definition:

– For a sample of n observations x1, x2, …, xm

– with independent probability density function f,– the likelihood function L is defined by

L is the joint probability density


ML estimator is defined as the value of which maximizes L:

or equivalently:


Example: ML estimation for normal distribution

– Suppose we have m indipendent samples x1, x2, …, xm, coming from a Gaussian distribution with parameters μ and σ2.

Which is the MLE for μ and σ2?

2x

2

1exp

2

1),;x(f



– For m observations x1, x2, …, xm, we have:

2

22 2

1

1,

2

ixm

i

L e



– For m observations x1, x2, …, xm, we have:

42

Maximum likelihood estimator for linear model

43Maximum likelihood estimator for linear model

– Let a linear model be given as

– Then

– here e has PDF pe(u,θ) (independent). The likelihood function is given by


– Asume a regression model where errors are distributed normally with zero mean.

– The likelihood function is given by


The maximum likelihood model

– Any algorithm that maximizes

– gives de Maximum likelihood model with respect to a

given family of possible models


– Same as maximizing

– Same as minimizing

47Connection to Least Squares

Conclusion

– The least-squares fitting criterion can be understood as emerging from the use of the maximum likelihood principle for estimating a regression model where errors are distributed normally.

– The applicability of the least-squares method is, however, not limited to the normality assumption.

48

LSE for Nonlinear Models

49LSE for Nonlinear Models

Nonlinear models are divided into 2 families

– Intrinsically linear– Intrinsically nonlinear

• Through appropriate transformations of the input-output variables & fitting parameters, an intrinsically linear model can become a linear model

• By this transformation into linear models, LSE can be used to optimize the unknown parameters

50LSE for Nonlinear Models

Examples of intrinsically linear systems

51

Developing Dinamic models from Data

52Dynamical System?

Input u(t) Output y(t)

System

))(),...,2(),1(),(),...,1(),(()(ˆ mtututuntytytySty

53The ARX model

In dynamic systems analysis, the independent variable is often time (k)

– A ARX model (AutoRegressive with eXogenous input model) is often used where

54The ARX model

Or equivalently

– writing

55The ARX model as a linear regressor

Input-output relationship can take the form

– where

Regression vector

Parameter vector to estimate

56Prediction error model estimation

The problem– Assume input-output data

– Build the predictor

– Such that minimizes Prediction ErrorPrediction Error


– The model is fitted to the data by minimizing the criterion function

2

1

1 N

Nk

V kN

Which gives the least squares criterion


Solution

– Normal equation

– Estimates

1 1

1 1N NT

Nk k

k k k y kN N

1 1

1 1N NT

Nk k

k k k y kN N

1

1 1

ˆ arg min ( ) ( ) ( ) ( ) ( )N N

LS TN N

k n

V k k k y k

1

1 1

ˆ arg min ( ) ( ) ( ) ( ) ( )N N

LS TN N

k n

V k k k y k


In matrix form, the solution is the standard linear least squares formula

1ˆ TN y

1ˆ T

N y

60

Example: Tank level modeling

61Example: Tank level modeling

62Example Tank level modeling

The identification goal– To explain how the voltage u(t) (the input) afects the

water level h(t) (the output) of the tank

Experimetal data

63Simple ARX modeling

A plausible first identification attempt is to try a simple linear regression model

– The parameters can easily be estimated using linear least squares, resulting in

64ARX model results

– Simulated water level follows the true level but at levels close to zero the linear model produces negative levels.

65Semiphysical modeling

Model equation is based on dynamic conservation of mass

– Accumulation of mass in the tank is equal to:

the mass flow rate into the tank

the mass flow rate out.

i o

dhA q q

dt minus

66Semiphysical modeling

While the inflow is roughly proportional to u(t) the outflow can be approximated using Bernoulli’s law

– The parameters can easily be estimated using linear least squares, resulting in

67Semiphysical model results

The RMS error of this model is lower and more importantly no simulated output is negative which indicates that the model is physically sound

68Sources

J-Shing Roger Jang, Chuen-Tsai Sun and Eiji Mizutani, Slides for Ch. 5 of “Neuro-Fuzzy and Soft Computing: A Computational Approach to Learning and Machine Intelligence”, First Edition, Prentice Hall, 1997.

Djamel Bouchaffra. Soft Computing. Course materials. Oakland University. Fall 2005

Henrik Melgaard, Identication of Physical Models. Institute of Mathematical Modelling, Technical University of Denmark. Ph.D. THESIS. 1994

Lucidi delle lezioni, Soft Computing. Materiale Didattico. Dipartimento di Elettronica e Informazione. Politecnico di Milano. 2004

Peter Lindskog, Fuzzy Identification from a Grey Box Modeling Point of View. Department of Electrical Engineering, Linkoping University. 1997

Jacob Roll, Local and Piecewise Afinne Approaches to System Identification. Department of Electrical Engineering, Linkoping University, Linkoping, Sweden. 2003

Computacion Inteligente Least-Square Methods for System Identification.

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