SYSTEMS Identification Ali Karimpour Assistant Professor Ferdowsi University of Mashhad Reference: “System Identification Theory For The User” Lennart Ljung(1999)

lecture 4 Ali Karimpour Nov Lecture 4 Models of Linear Systems Topics to be covered include: v Frequency Response Function (FRF). v Data in Time and Frequency Domain. v Linear Model Structures (Without Noise). v Linear Model Structures (With Noise). v Fitting Parameterized Linear Models to Data. * Time Domain Data * Frequency Domain Data v The Asymptotic Properties of the Estimate. v Model Sets, Model Structures and Identifiability.

lecture 4 Ali Karimpour Nov Identification of Linear Systems Topics to be covered include: v Frequency Response Function (FRF). v Data in Time and Frequency Domain. v Linear Model Structures (Without Noise). v Linear Model Structures (With Noise). v Fitting Parameterized Linear Models to Data. * Time Domain Data * Frequency Domain Data v The Asymptotic Properties of the Estimate. v Model Sets, Model Structures and Identifiability.

lecture 4 Ali Karimpour Nov Frequency Response Function (FRF) A linear system is characterized by its transfer function G(s) This leads to FRF G(jω) This could be a way of determining G. All frequency at the same time

lecture 4 Ali Karimpour Nov Identification of Linear Systems Topics to be covered include: v Frequency Response Function (FRF). v Data in Time and Frequency Domain. v Linear Model Structures (Without Noise). v Linear Model Structures (With Noise). v Fitting Parameterized Linear Models to Data. * Time Domain Data * Frequency Domain Data v The Asymptotic Properties of the Estimate. v Model Sets, Model Structures and Identifiability.

lecture 4 Ali Karimpour Nov Data in Time and Frequency Domain Spectral Analysis.

lecture 4 Ali Karimpour Nov Data in Time and Frequency Domain Spectral Analysis.

lecture 4 Ali Karimpour Nov Data in Time and Frequency Domain Spectral Analysis. Exercise 1: Derive the figure by an experimental data.

lecture 4 Ali Karimpour Nov Identification of Linear Systems Topics to be covered include: v Frequency Response Function (FRF). v Data in Time and Frequency Domain. v Linear Model Structures (Without Noise). v Linear Model Structures (With Noise). v Fitting Parameterized Linear Models to Data. * Time Domain Data * Frequency Domain Data v The Asymptotic Properties of the Estimate. v Model Sets, Model Structures and Identifiability.

lecture 4 Ali Karimpour Nov Linear Model Structures (Without Noise) A complete model is given by A particular model thus corresponds to specification of the function G. We try to parameterize coefficients so: Where θ is a vector in R d space. Sets of models We thus have:

lecture 4 Ali Karimpour Nov Linear Model Structures (Without Noise) Finite Impulse Response Model u y The output error model structure If we suppose that the relation between input and input can be written as: With So FIR model

lecture 4 Ali Karimpour Nov Linear Model Structures (Without Noise) Output error model structure u y The output error model structure If we suppose that the relation between input and undisturbed output w can be written as: With So OE model

lecture 4 Ali Karimpour Nov Linear Model Structures (Without Noise) For most physical systems it is easier to construct models with physical insight in continuous time: We can derive the transfer operator from u to y State Space model Another important model that is used in process system is: Static gain, time constant, delay Process model

lecture 4 Ali Karimpour Nov For most physical systems it is easier to construct models with physical insight in continuous time: θ is a vector of parameters that typically correspond to unknown values of physical coefficients, material constants, and the like. Let η(t) be the measurements that would be obtained with ideal, noise free sensors We can derive the transfer operator from u to η Linear Model Structures (With Noise)

lecture 4 Ali Karimpour Nov Sampling the transfer function Let Then x(kT+t) is So x(kT+T) is We can derive the transfer operator from u to η Linear Model Structures (With Noise)

lecture 4 Ali Karimpour Nov Identification of Linear Systems Topics to be covered include: v Frequency Response Function (FRF). v Data in Time and Frequency Domain. v Linear Model Structures (Without Noise). v Linear Model Structures (With Noise). v Fitting Parameterized Linear Models to Data. * Time Domain Data * Frequency Domain Data v The Asymptotic Properties of the Estimate. v Model Sets, Model Structures and Identifiability..

lecture 4 Ali Karimpour Nov A complete model is given by A particular model thus corresponds to specification of the function G, H. Linear Model Structures (With Noise) Same as We try to parameterize coefficients so: Where θ is a vector in R d space. Sets of models We thus have:

lecture 4 Ali Karimpour Nov u e y The ARX model structure Equation error model structure AR part X part Define So we have: where Adjustable parameters in this case are ARX model Linear Model Structures (With Noise)

lecture 4 Ali Karimpour Nov ARMAX model structure AR part X part MA part with So we have: now where Let Linear Model Structures (With Noise) + u e y The ARMAX model structure

lecture 4 Ali Karimpour Nov Other equation error type model structures + u e y The equation error model family With AR part X part AR part ARARX model X part AR part We could use an ARMA description for error ARMA part ARARMAX model Linear Model Structures (With Noise)

lecture 4 Ali Karimpour Nov Output error model structure + u e y The output error model structure If we suppose that the relation between input and undisturbed output w can be written as: Then With So OE model Linear Model Structures (With Noise)

lecture 4 Ali Karimpour Nov Box-Jenkins model structure + u e y The BJ model structure A natural development of the output error model is to further model the properties of the output error. Let output error with ARMA model then BJ model This is Box and Jenkins model (1970) Linear Model Structures (With Noise)

lecture 4 Ali Karimpour Nov A general family of model structure + u e y General model structure The structure we have discussed in this section may give rise to 32 different model sets, depending on which of the five polynomials A, B, C, D, F are used. For convenience, we shall therefore use a generalized model structure: General model structure Linear Model Structures (With Noise)

lecture 4 Ali Karimpour Nov u e y General model structure Sometimes the dynamics from u to y contains a delay of n k samples, so So But for simplicity Linear Model Structures (With Noise)

lecture 4 Ali Karimpour Nov The structure we have discussed in this section may give rise to 32 different model sets, depending on which of the five polynomials A, B, C, D, F are used. General model structure Some common black-box SISO models as special cases of generalized model structure Polynomial used Name of model Structure B FIR (finite impulse response) AB ARX ABC ARMAX AC ARMA ABD ARARX ABCD ARARMAX BF OE (output error) BFCD BJ (Box-Jenkins) Linear Model Structures (With Noise)

lecture 4 Ali Karimpour Nov State Space models Noise Representation and the time-invariant Kalman filter A straightforward but entirly valid approach would be: with {e(t)} being white noise with variance λ. Note: The θ-parameter in H(q, θ) could be partly in common with those in G(q, θ) or be extra. {w(t)} and {v(t)} are assumed to be sequences of independent random variables with zero mean and process noise measurement noise

lecture 4 Ali Karimpour Nov State Space models Noise Representation and the time-invariant Kalman filter {w(t)} and {v(t)} are assumed to be sequences of independent random variables with zero mean and process noise measurement noise {w(t)} and {v(t)} may often be signals whose physical origins are known. The load variation T l (t) was a “process noise”. The inaccuracy in the potentiometer angular sensor is the “measurement noise”. In such cases it may of course not always be realistic to assume that the signals are white noises.

lecture 4 Ali Karimpour Nov State Space models Exercise 2: (4G.2) Colored measurement noise: (I)

lecture 4 Ali Karimpour Nov State Space models For state space descriptions, The conditional expectation of y(t), given data y(s), u(s), s≤t-1, is: The conditional expectation of x(t), by Kalman filter is: where is obtained as the psd solution of the stationary Riccati equation: Here K(θ) is given by

lecture 4 Ali Karimpour Nov State Space models For state space descriptions, The conditional expectation of y(t), given data y(s), u(s), s≤t-1, is: The conditional expectation of x(t), by Kalman filter is: The conditional expectation of x(t) is: The predictor filter can thus be written as: Exercise 3: Show that covariance matrix of state estimator error is

lecture 4 Ali Karimpour Nov State Space models Innovation representation Innovation=Amounts of y(t) that cannot be predicted from past data Innovation Let it e(t) The innovation form of state space description Exercise 4: Show that the covariance of e(t) is:

lecture 4 Ali Karimpour Nov State Space models Innovation representation The innovation form of state space description Directly Parameterized Innovations form Let suppose Which one involve with lower parameters? Both according to situation.

lecture 4 Ali Karimpour Nov State Space models Innovation representation It is ARMAX model

lecture 4 Ali Karimpour Nov State Space models Example 4.2 Companion form parameterization Let and So we have an ARMAX model with

lecture 4 Ali Karimpour Nov Identification of Linear Systems Topics to be covered include: v Frequency Response Function (FRF). v Data in Time and Frequency Domain. v Linear Model Structures (Without Noise). v Linear Model Structures (With Noise). v Fitting Parameterized Linear Models to Data. * Time Domain Data * Frequency Domain Data v The Asymptotic Properties of the Estimate. v Model Sets, Model Structures and Identifiability.

lecture 4 Ali Karimpour Nov Fitting Models to Data Curve fitting Time Domain Frequency Domain

lecture 4 Ali Karimpour Nov 2010 Fitting Models to Data Pre filtering: A Further Degree of Freedom The fit is focused to the frequency ranges where L is large.

lecture 4 Ali Karimpour Nov 2010 Linear Regression Model structures such as That are linear in θ are known in statistics as linear regressions. Now define:

lecture 4 Ali Karimpour Nov 2010 First order difference equation Consider the simple model Now we have Then

lecture 4 Ali Karimpour Nov 2010 First order difference equation Consider the simple model Exercise 5: Suppose for t=1 to 6 the value of u and y are: Derive

lecture 4 Ali Karimpour Nov u e y The ARX model structure Equation error model structure So we have ARX model 41 Now if we introduce Linear regression Linear Regression for ARX model

lecture 4 Ali Karimpour Nov Exercise 6: (4E.1) Consider the ARX model structure Linear regression in statistic where b 1 is known to be 0.5. Write the corresponding predictor in the following linear regression form. Linear Regression for ARX model

lecture 4 Ali Karimpour Nov ARMAX model structure Now Let Pseudo Linear Regression for ARMAX model 43 Pseudo linear regressions Now if we introduce

lecture 4 Ali Karimpour Nov Output error model structure + u e y The output error model structure w(t) is never observed instead it is constructed from u So Let Pseudo Linear Regression for ARMAX model

lecture 4 Ali Karimpour Nov General model structure Predictor A pseudolinear form for general model structure Predictor error is: Pseudo Linear Regression for general model structure

lecture 4 Ali Karimpour Nov So we have: Pseudo Linear Regression for general model structure

lecture 4 Ali Karimpour Nov Pseudo Linear Regression for general model structure

lecture 4 Ali Karimpour Nov Identification of Linear Systems Topics to be covered include: v Frequency Response Function (FRF). v Data in Time and Frequency Domain. v Linear Model Structures (Without Noise). v Linear Model Structures (With Noise). v Fitting Parameterized Linear Models to Data. * Time Domain Data * Frequency Domain Data v The Asymptotic Properties of the Estimate. v Model Sets, Model Structures and Identifiability.

lecture 4 Ali Karimpour Nov The Asymptotic Properties of the Estimate  Assume that data have been generated by  Properties of the estimate θ N  A parameter free assessment of model quality for linear systems and models  How to affect the model quality?

lecture 4 Ali Karimpour Nov Identification of Linear Systems Topics to be covered include: v Frequency Response Function (FRF). v Data in Time and Frequency Domain. v Linear Model Structures (Without Noise). v Linear Model Structures (With Noise). v Fitting Parameterized Linear Models to Data. * Time Domain Data * Frequency Domain Data v The Asymptotic Properties of the Estimate. v Model Sets, Model Structures and Identifiability.

lecture 4 Ali Karimpour Nov General Notation Some notation It is convenient to introduce some more compact notation One step ahead predictor is:

lecture 4 Ali Karimpour Nov Definition 4.1. A predictor model of a linear, time-invariant system is a stable filter W(q). Definition 4.2. A complete probabilistic model of a linear, time-invariant system is a pair (W(q),f e (x)) of a predictor model W(q) and the PDF f e (x) of the associated errors. Clearly, we can also have models where the PDFs are only partially specified (e.g., by the variance of e) We shall say that two models W 1 (q) and W 2 (q) are equal if General Notation

lecture 4 Ali Karimpour Nov 2010 General Notation Example: Unstable system. If A(q) has zeros outside the unit disc, then the map from u → y is unstable. But Is always stable.

lecture 4 Ali Karimpour Nov 2010 Model Sets Definition: A model set is a collection of models Examples:

lecture 4 Ali Karimpour Nov 2010 Parameterization of Model Sets Let a model be index by a parameter θ, W(q, θ) We require W(q, θ) to be differentiable with respect to θ So

lecture 4 Ali Karimpour Nov 2010 Model Structure Definition: A model structure m is a differentiable mapping from a connected subset D m of R d to a model set m *, such that the gradients of the predictor functions are stable.

lecture 4 Ali Karimpour Nov 2010 Parameterization of Model Sets Example: An ARX model.

lecture 4 Ali Karimpour Nov 2010 Definition: A model structure m is a differentiable mapping from a connected subset D m of R d to a model set m *, such that the gradients of the predictor functions are stable. Differentiability of T ( G and H )Differentiability of W Model Structure

lecture 4 Ali Karimpour Nov 2010 Definition: A model structure m is a differentiable mapping from a connected subset D m of R d to a model set m *, such that the gradients of the predictor functions are stable. Model Structure Lemma: The predictor For θ confined to D m ={θ | F(q)C(q) has no zeros outside the unit disc} is a model structure. Proof: We need only verify that the gradients of With respect to θ are analytical in |z| ≥ 1 for θ D m.

lecture 4 Ali Karimpour Nov 2010 Definition: A model structure m is a differentiable mapping from a connected subset D m of R d to a model set m *, such that the gradients of the predictor functions are stable. Model Structure Lemma: For the Kalman filter predictor Assume that the entries of the matrices A(θ), B(θ), C(θ) and K(θ) are differentiable with D m ={θ | all eigenvalues of A(θ)-K(θ)C(θ) are inside the unit circle} Then the parameterization is a model structure. Exercise 7: Proof the Lemma. (4D.1)

lecture 4 Ali Karimpour Nov 2010 Definition: A model structure m is said to have an independently parameterized transfer function and noise model if Independent Parameterization We can define a model set as the range of a model structure: We can define union of different model structures:

lecture 4 Ali Karimpour Nov Identifiability Identifiability properties The problem is whether the identification procedure will yield a unique value of the parameter θ, and/or whether the resulting model is equal to the true system. Definition 4.6. A model structure M is globally identifiable at θ* if Definition 4.7. A model structure M is strictly globally identifiable if it is globally identifiable at all at

lecture 4 Ali Karimpour Nov Identifiability This definition is quite demanding. A weaker and more realistic property is: Definition A model structure M is globally identifiable at θ* if Definition A model structure M is strictly globally identifiable if it is globally identifiable at all at Definition A model structure M is globally identifiable if it is globally identifiable at almost all at For corresponding local property, the most natural definition of local identifiability of M at θ* would be to require that there exist an ε such that

lecture 4 Ali Karimpour Nov Identifiability Use of the Identifiability concept The identifiability concept concerns the unique representation of a given system description in a model structure. Let such a description as: Let M be a model structure based on one-step-ahead predictors for Then define the set D T (S,M) as those θ-values in D M for which S=M (θ) The set is empty in case Now suppose that so that S=M(θ 0 ) Very important property

lecture 4 Ali Karimpour Nov Identifiability Consider the model structure: SISO Transfer function models Idenifiability of ARX Model? Idenifiability of OE Model? Idenifiability of other Models?

lecture 4 Ali Karimpour Nov Identifiability ARX Model So ARX Model is strictly identifiable

lecture 4 Ali Karimpour Nov Identifiability OE Model So OE Model is not generally strictly identifiable

lecture 4 Ali Karimpour Nov 2010 Identifiability Theorem: Consider the general model structure The model structure is globally identifiable at if and only if all of following condition hold. Exercise 8: (4E.6)