Computacion Inteligente Least-Square Methods for System Identification

Content  System Identification: an Introduction  Least-Squares Estimators  Statistical Properties & the Maximum Likelihood Estimator  LSE for Nonlinear Models

3 System Identification: Introduction  Goal –Determine a mathematical model for an unknown system (or target system) by observing its input-output data pairs

4 System Identification: Introduction  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

5 System Identification: Introduction  Context –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

6 System Identification: Introduction  There are 2 main steps that are involved –Structure identification –Parameter identification

7 System Identification: Introduction  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

8 System Identification: Introduction  Structure identification  f(u,  ) depends on –the problem at hand –the designer’s experience –the laws of nature governing the target system

9 System Identification: Introduction  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:

10 Block diagram for parameter identification System Identification: Introduction

11 System Identification: Introduction  The data set composed of m desired input-output pairs –(u i, y i ) (i = 1,…,m) is called the training data  System identification needs to do both structure & parameter identification repeatedly until satisfactory model is found

12 System 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

13  Least-Squares Estimators

14 Least-Squares Estimators  General form: y =  1 f 1 (u) +  2 f 2 (u) + … +  n f n (u) (14) where: –u = (u 1, …, u p ) T is the model input vector –f 1, …, f n are known functions of u –  1, …,  n are unknown parameters to be estimated

15 Least-Squares Estimators  The task of fitting data using a linear model is referred to as linear regression where: –u = (u 1, …, u p ) T is the input vector –f 1 (u), …, f n (u)regressors –  1, …,  n parameter vector

16 Least-Squares Estimators  We collect a training data set {(u i, y i ), i = 1, …, m} Equation (14) becomes: Which is equivalent to: A  = y

17 Least-Squares Estimators  Which is equivalent to: A  = y –where A  = y   = A -1 y (solution) m*n matrixn*1 vectorm*1 vector unknown

18 Least-Squares Estimators  We have –m outputs & –n fitting parameters to find  Or m equations & n unknown variables  Usually m is greater than n

19 Least-Squares Estimators  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

20 Least-Squares Estimators  Our goal consists now of finding that reduces the errors between and  The problem: find, estimate

21 Least-Squares Estimators  If e = y - A  then: We need to compute:

22 Least-Squares Estimators  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

23  Statistical Properties of least-squares estimators

24 Statistical qualities of LSE  Definition [unbiased estimator] An estimator of the parameter  is unbiased if where E[.] is the statistical expectation

25 Statistical qualities of LSE  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 

26 Statistical qualities of LSE  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:

27 Statistical qualities of LSE  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:

28 Statistical qualities of LSE  Theorem [Gauss-Markov] LSE is unbiased & has minimum variance. Proof:

29  Maximum likelihood (ML) estimator

30 Maximum likelihood (ML) estimator –ML is one of the most widely used technique for parameter estimation of a statistical distribution  ML definition: –For a sample of n observations (of a probability density function ) x 1, x 2, …, x n, the likelihood function L is defined by:

31 Maximum likelihood (ML) estimator  The criterion for choosing  is: –“pick a value of  that provides a high probability of obtaining the actual observed data x 1, x 2, …, x n ” –Therefore, ML estimator is defined as the value of  which maximizes L: or equivalently:

32 Maximum likelihood (ML) estimator  Example: ML estimation for normal distribution –For m observations x 1, x 2, …, x m, we have:

33 Maximum likelihood (ML) estimator  Example: ML estimation for normal distribution –For m observations x 1, x 2, …, x m, we have:

34 Statistical Properties & the ML Estimator  Equivalence between LSE & MLE  Theorem –Under the Gauss conditions and if each component of the vector e follows a normal distribution then: LSE of  = MLE of 

35 Statistical Properties & the Maximum Likelihood Estimator (5.7) (cont.)  Maximum likelihood (ML) estimator (cont.) –Equivalence between LSE & MLE –Theorem: Under the Gauss conditions & if each component of the vector e follows a normal distribution then the LSE of  = MLE of 

36  LSE for Nonlinear Models

37 LSE 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

38 LSE for Nonlinear Models  Examples of intrinsically linear systems