SYSTEMS Identification Ali Karimpour Assistant Professor Ferdowsi University of Mashhad Reference: “System Identification Theory For The User” Lennart Ljung

lecture 1 Ali Karimpour Nov Lecture 8 Convergence & Consistency Topics to be covered include: v Conditions on the Data Set v Prediction-Error Approach v Consistency and Identifiability v LTI Models: A Frequency-Domain Description of the Limit Model v The Correlation Approach lecture 8

lecture 1 Ali Karimpour Nov Introduction lecture 8 Previously different methods to determine models from data are described. To use these methods in practice, we need insight into their properties: How well will the identified model describe the actual system? Are some identification methods better than others? How should the design variables be chosen? The questions depend to mapping: 1- Simulation. 2- Analysis.

lecture 1 Ali Karimpour Nov Lecture 8 Convergence & Consistency Topics to be covered include: v Conditions on the Data Set v Prediction-Error Approach v Consistency and Identifiability v LTI Models: A Frequency-Domain Description of the Limit Model v The Correlation Approach lecture 8

lecture 1 Ali Karimpour Nov lecture 8 Conditions on the Data Set Data generation configuration Let Analysis means:

lecture 1 Ali Karimpour Nov Conditions on the Data Set lecture 8 D1 A Technical Condition D1

lecture 1 Ali Karimpour Nov Conditions on the Data Set lecture 8 Remind:

lecture 1 Ali Karimpour Nov Conditions on the Data Set lecture 8 S A True System S

lecture 1 Ali Karimpour Nov Conditions on the Data Set lecture 8 Exercise1: Prove the lemma. When S1 holds, a more explicit version of conditions D1 can be given

lecture 1 Ali Karimpour Nov Conditions on the Data Set lecture 8 Information Content in the Data Set. Whether the data set Z  allows us to distinguish between different models in the set? The data set is informative if it is capable of distinguishing between different models. Definition: A quasi-stationary data set Z  is informative enough with respect to the model set M * if, for any two models W 1 (q) and W 1 (q) in the set, Definition: A quasi-stationary data set Z  is informative if it is informative enough with respect to the model set L *, consisting of all linear, time invariant models. Recall:

lecture 1 Ali Karimpour Nov Conditions on the Data Set lecture 8 Remember Theorem: A quasi-stationary data set Z  is informative if the spectrum matrix for z(t)=[u(t) y(t)] T is strictly positive definite for almost all . Proof: We need to show 2-65

lecture 1 Ali Karimpour Nov Lecture 8 Convergence & Consistency Topics to be covered include: v Conditions on the Data Set v Prediction-Error Approach v Consistency and Identifiability v LTI Models: A Frequency-Domain Description of the Limit Model v The Correlation Approach lecture 8

lecture 1 Ali Karimpour Nov Prediction-Error Approach lecture 8 In PEM Clearly it depends to For quadratic criterion and a linear, uniformly stable model structure M, we have Using D1 Condition: Uniformly Stable since M is stable

lecture 1 Ali Karimpour Nov Prediction-Error Approach lecture 8

lecture 1 Ali Karimpour Nov Prediction-Error Approach lecture 8 Since D M is compact It may happen that does not have a unique/global minimum.

lecture 1 Ali Karimpour Nov Prediction-Error Approach lecture 8

lecture 1 Ali Karimpour Nov Prediction-Error Approach lecture 8 Example 8.1 Bias in ARX Structures Suppose that the system is given by Where {u(t)} and {e 0 (t)} are independent white noises with unit variances. Let the model structure be given by The prediction-error variance is Exercise2 : Proof I

lecture 1 Ali Karimpour Nov Prediction-Error Approach lecture 8 Example 8.1 Bias in ARX Structures The prediction-error variance is But for true value of parameters When we apply PEM the estimates will converge, according to pervious theorem. But one of the parameters is biased. It is clear that bias is beneficial for the prediction performance of the model. So it gives a strictly better predictor. Exercise3 : Proof and explain the example by suitable simulation.

lecture 1 Ali Karimpour Nov Prediction-Error Approach lecture 8 Example 8.2 Wrong Time Delay Consider the system {e(t)} and {w(t)} are independent white-noise sequence with unit variances. Let the model structure be given by The associated prediction-error variance is: Exercise4 : Proof I

lecture 1 Ali Karimpour Nov Prediction-Error Approach lecture 8 Example 8.2 Wrong Time Delay Consider the system Hence Now the predictor Is a fairly reasonable one for

lecture 1 Ali Karimpour Nov Lecture 8 Convergence & Consistency Topics to be covered include: v Conditions on the Data Set v Prediction-Error Approach v Consistency and Identifiability v LTI Models: A Frequency-Domain Description of the Limit Model v The Correlation Approach lecture 8 Ali Karimpour Nov 2009

lecture 1 Ali Karimpour Nov The first condition: lecture 8 Ali Karimpour Nov 2009 Consistency and Identifiability Exercise5: Prove the above-mentioned theorem. Suppose that assumption S1 holds so that we have a true system. Will PE identification recover the true plant? So

lecture 1 Ali Karimpour Nov lecture 8 Ali Karimpour Nov 2009 Consistency and Identifiability

lecture 1 Ali Karimpour Nov Consistency and Identifiability lecture 8 Ali Karimpour Nov 2009

lecture 1 Ali Karimpour Nov Consistency and Identifiability lecture 8 Ali Karimpour Nov 2009 Example 8.3 First Order Output Error Model Suppose that the true system is given by Let we define a first-order output error model as: By pervious theorem the estimates of and will converges to the true value of and.

lecture 1 Ali Karimpour Nov Lecture 8 Convergence & Consistency Topics to be covered include: v Conditions on the Data Set v Prediction-Error Approach v Consistency and Identifiability v LTI Models: A Frequency-Domain Description of the Limit Model v The Correlation Approach lecture 8 Ali Karimpour Nov 2009

lecture 1 Ali Karimpour Nov LTI Models: A Frequency-Domain Description of the Limit Model lecture 8 Ali Karimpour Nov 2009

lecture 1 Ali Karimpour Nov LTI Models: A Frequency-Domain Description of the Limit Model lecture 8 Ali Karimpour Nov 2009 An Expression for Under assumption S1 we have

lecture 1 Ali Karimpour Nov LTI Models: A Frequency-Domain Description of the Limit Model lecture 8 Ali Karimpour Nov 2009 An Expression for Monic Independent So Overbar is complex conjugate.

lecture 1 Ali Karimpour Nov LTI Models: A Frequency-Domain Description of the Limit Model lecture 8 Ali Karimpour Nov 2009 An Expression for Let So

lecture 1 Ali Karimpour Nov LTI Models: A Frequency-Domain Description of the Limit Model lecture 8 Ali Karimpour Nov 2009 We now have a characterization of in the frequency domain. with indicated weightings.

lecture 1 Ali Karimpour Nov LTI Models: A Frequency-Domain Description of the Limit Model lecture 8 Ali Karimpour Nov 2009 Open Loop Case

lecture 1 Ali Karimpour Nov LTI Models: A Frequency-Domain Description of the Limit Model lecture 8 Ali Karimpour Nov 2009 Open Loop Case Consider an independently noise model with Spectral factorization

lecture 1 Ali Karimpour Nov LTI Models: A Frequency-Domain Description of the Limit Model Open Loop Case =

lecture 1 Ali Karimpour Nov LTI Models: A Frequency-Domain Description of the Limit Model Open Loop Case

lecture 1 Ali Karimpour Nov LTI Models: A Frequency-Domain Description of the Limit Model lecture 8 Ali Karimpour Nov 2009 Closed Loop Case

lecture 1 Ali Karimpour Nov LTI Models: A Frequency-Domain Description of the Limit Model Closed Loop Case =

lecture 1 Ali Karimpour Nov LTI Models: A Frequency-Domain Description of the Limit Model lecture 8 Ali Karimpour Nov 2009 Example 8.5 Approximation in the Frequency Domain

lecture 1 Ali Karimpour Nov LTI Models: A Frequency-Domain Description of the Limit Model Example 8.5 Approximation in the Frequency Domain

lecture 1 Ali Karimpour Nov LTI Models: A Frequency-Domain Description of the Limit Model Example 8.5 Approximation in the Frequency Domain

lecture 1 Ali Karimpour Nov LTI Models: A Frequency-Domain Description of the Limit Model Example 8.5 Approximation in the Frequency Domain

lecture 1 Ali Karimpour Nov Lecture 8 Convergence & Consistency Topics to be covered include: v Conditions on the Data Set v Prediction-Error Approach v Consistency and Identifiability v LTI Models: A Frequency-Domain Description of the Limit Model v The Correlation Approach lecture 8 Ali Karimpour Nov 2009

lecture 1 Ali Karimpour Nov LTI Models: A Frequency-Domain Description of the Limit Model lecture 8 Ali Karimpour Nov 2009 In section 7.5 we defined the correlation approach to identification,with the special cases of PLR and IV methods.

lecture 1 Ali Karimpour Nov LTI Models: A Frequency-Domain Description of the Limit Model lecture 8 Ali Karimpour Nov 2009 Basic Convergence Result Consider the function And correlation vector ξ(t,θ) is obtained by linear filtering of past data: where

lecture 1 Ali Karimpour Nov LTI Models: A Frequency-Domain Description of the Limit Model lecture 8 Ali Karimpour Nov 2009

lecture 1 Ali Karimpour Nov LTI Models: A Frequency-Domain Description of the Limit Model lecture 8 Ali Karimpour Nov 2009

lecture 1 Ali Karimpour Nov LTI Models: A Frequency-Domain Description of the Limit Model lecture 8 Ali Karimpour Nov 2009 Instrumental-variable Methods

lecture 1 Ali Karimpour Nov The Correlation Approach lecture 8 Ali Karimpour Nov 2009

lecture 1 Ali Karimpour Nov The Correlation Approach lecture 8 Ali Karimpour Nov 2009

lecture 1 Ali Karimpour Nov The Correlation Approach lecture 8 Ali Karimpour Nov 2009

lecture 1 Ali Karimpour Nov LTI Models: A Frequency-Domain Description of the Limit Model lecture 8 Ali Karimpour Nov 2009