f-domain expression for the limit model Combine: 5.12 Approximate Modelling What can be said about H(q, θ) G(q, θ ) H(q, θ ) with

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1 5.2 Approximate Modelling What can be said about if S / M, and even G / G? G(q, ) H(q, ) f-domain expression for the limit model Combine: with ε(t, ) =H(q, ) [y(t) G(q, )u(t)] y(t) =G (q)u(t) v(t) We know (convergence result) that: then follows: and = arg min V () = Ēε(t, ) 2 = 2π π π V () Φ ε (ω)dω (Parsseval; both expressions are equal to R ε ()) ε(t, ) =H(q, ) [[G (q) G(q, )]u(t) v(t)] Consequence of the identification criterion: V () = π Φ ε (ω)dω = 2π π V () = π G (e iω ) G(e iω,) 2 Φ u (ω)φ v (ω) dω 2π π H(e iω,) 2 This criterion plays the role of approximation criterion. Lec5a- Lec5a-2 Alternative form Write ε(t, ) =e(t) G (q) G(q, ) H(q, ) then u(t) H (q) H(q, ) e(t) H(q, ) = arg min [ π G (e iω ) G(e iω, ) 2 Φ 2π π H(e iω, ) 2 u (ω) ] H (e iω ) H(e iω, ) 2 σ 2 H(e iω, ) 2 e dω Expression shows how is obtained. Two mechanisms: Minimization of G (e iω ) G(e iω, ) 2 Φ u (ω) H(e iω, ) 2 Minimization of H (e iω ) H(e iω, ) 2 σ 2 e H(e iω, ) 2 Problems are coupled if H(q, ) is parametrized. Observation: Type of approximation of G by G(q, ˆ N ) is dependent on noise model H(q, ). Lec5a-3 Lec5a-4

2 Special case: fixed noise model H(q, ) =H (q) (e.g. OE). Then = arg min 2π π π G (e iω ) G(e iω,) 2 Φ u (ω) H (e iω ) 2 dω (second term in integrand is -independent). is determined by minimizing the integrated quadratic error G G() with weighting function Φ u (ω) H (e iω ) 2 At those frequencies where the weighting function is large, the model error will be small. Prefiltering of data Prefiltering of the data with filter L(q), leads to ε F (t, ) =L(q)ε(t, ) and Φ εf (ω) = L(e iω ) 2 Φ ε (ω) For a fixed noise model the weighting function becomes: Φ u (ω) L(e iω ) 2 H (e iω ) 2 Lec5a-5 Lec5a-6 Example Frequency response y(t) =G (q)u(t) G 5th order; Φ u (ω) =(white noise). To illustrate the effect of approximation 2nd order models will be estimated: Amplitude 2 G OE model ARX model 2 OE-model, 2nd order y(t) = ARX-model, 2nd order b q b 2 q 2 u(t) e(t) f q f 2 q 2 Phase (deg) 2 3 ( a q a 2 q 2 )y(t) = (b q b 2 q 2 )u(t) e(t) 4 Frequency (rad/s) Lec5a-7 Lec5a-8

3 Comparison of situations: OE-model ARX-model min π G G() 2 Φ u dω 2π π - Amplitude Bode plot min π G G() 2 Φ u A(e iω,) 2 dω 2π π Additional weighting with (a priori unknown) function frequency (rad/sec) Weighting function A(e iω, ˆ N ) in ARX-case. Lec5a-9 Lec5a- General situation: 2π π π G (e iω ) G(e iω,) 2 Φ u (ω) Φ v (ω) dω H(e iω,) 2 In case of independent parametrization or fixed noise model H = H : G(q, ˆρ N ) is obtained through ˆρ N = arg min π G G(ρ) 2 Φ u L 2 2π π H dω 2 This holds for OE, BJ, FIR. Influencing the approximation criterion (and so the resulting model) by Choice of input spectrum Φ u Choice of prefilter L Choice of noise model H In other cases: compromise in which a.o.φ v is playing a role in the approximation of G. Lec5a- Lec5a-2

4 Example S : y(t) =G (q)u(t) e(t) Since Z N is given, the only degree of freedom we have is to use a pre-filter L to shape the bias error with G 4th order with three delays. We have to use a given set of data Z N (N = 5) for the identification where u is the sum of a white noise of variance 5 and three high-frequencies sinus of amplitude Objective: Using the given data, identify a good model G(q, ˆ N ) for G in the frequency range [.7] in the reduced order model structure: We want a small bias error in the frequency range [.7] choose L such that L(e iω ) 2 Φ u (ω) is relatively (much) larger in the frequency range [.7] than in [.7 π] L Butterworth low pass filter of order 7 and cut-off frequency.7rad/s We filter u and y collected from S by this L and we obtain filtered data with which we perform the identification in M M = OE(n b =2,n f =2,n k =3) Lec5a-3 Lec5a-4 What if we do not use a pre-filter L? G (red) and G(ˆN ) (blue) identified with the filtered data. From u to y G (red) and G(ˆN ) (blue) identified with the data in Z N Amplitude Amplitude 2 From u to y Phase (degrees) Frequency (rad/s) Phase (degrees) G(ˆ N ) is OK. G(ˆ N ) is KO 4 2 Frequency (rad/s) Lec5a-5 Lec5a-6

5 Identification criteria - from PE to ML Prediction error characteristics: Extensions on identification methods and model structures Model structure represented by G(q, ),H(q, ) One step ahead predictor: Contents Identification criteria - from PE to ML The stochastic/deterministic identification paradigm Attractive model structures: OBF s ŷ(t t ; ) =H(q, ) G(q, )u(t)[ H(q, ) ]y(t) Prediction error: ε F (t, ) =L(q)[y(t) ŷ(t t ; )] Identification criterion: ˆ N = arg min ε F (t, ) 2 t= Characterization of results in terms of bias and variance of parameter ˆ N and related transfer functions G(q, ˆ N ) and H(q, ˆ N ). Lec5- Lec5-2 Minimization of quadratic cost function is not the unambiguous universal choice Alternative for quadratic prediction error criterion is a correlation approach: Most simple choice is ζ(t) =[u(t ) u(t d)] T Instrumental Variable (IV) estimator: leading to a parameter estimate that generates a prediction error signal ε F (t, ) that is uncorrelated to past inputs. ˆ N = arg { ζ(t)ε F (t, ) =} t= IV estimators are generally directed towards estimating G only. with ζ(t) R d, a signal vector with a dimension at least as large as dim(). Consistency properties are attractive and in line with general PE methods. ζ(t) = instrumental vector / signals. ˆ N is now obtained as the solution of a set of equations, rather How to compare different estimators? Here we will take a look at (asymptotic) estimator variance. than as minimizing argument of a cost function. Lec5-3 Lec5-4

6 PE methods: From covariance matrix to uncertainty interval Asymptotic result (under appropriate conditions): For N : N(ˆN ) N(,P ) where P can be specified for = (no bias). The sum of quadratic variables follows a χ 2 distribution: N(ˆN ) N(,P ) N(ˆ N ) T P (ˆ N ) χ 2 (n). Let P be decomposed as P = R T R then NR (ˆ N ) N(,I n ) i.e. a vector of n standard (independent) normally distributed random variables. Lec5-5 Lec5-6 Note that contour lines: while the level of probability related to the event N(ˆ N ) T P (ˆ N )=c ( ) T P ( ) <c/n are ellipsoidal contour lines of the multivariable Gaussian distribution: 2sqrt{cλ } is determined by the χ 2 (n)-distribution. Example χ 2 -distribution (2) w 2 w.6 ().4.2 (P has eigenvalues λ i and eigenvectors w i ) 5 5 Lec5-7 Lec5-8

7 Denote D = { ( ) T P ( ) < c χ(α, n) N } ellipsoid centered at then ˆ N D with probability α. Alternatively, denote: Dˆ N = { ( ˆ N ) T P ( ˆ N ) < c χ(α, n) N } ellipsoid centered at ˆ N then it can simply be verified that Central question in estimation theory: Does there exist - in a specified situation - a lower bound for the variance of a parameter estimator? ˆ N D DˆN so that DˆN with probability α. Lec5-9 Lec5- Cramé-Rao lower bound (CRLB) Consider observations from a random variable y with pdf f y (y, ), where is the unknown parameter. Then for any unbiased estimator ˆ of the parameter, its covariance matrix satisfies the inequality Remarks on CRLB CRLB requires knowledge of pdf f y (y; ) CRLB generally requires exact knowledge of Exception: Gaussian pdf s with linear regression model It provides a lower bound for unbiased estimators cov(ˆ) J with the Fisher Information Matrix: { J = E 2 log f y(y; ) 2 = } Independent of the particular estimation method Useful for issues as analysis, and experiment design Estimator that reaches CRLB does not necessarily exist! Lec5- Lec5-2

8 Example Measurement of a physical variable with 5 different sensors y i = e i i =, 5 and e i are independent zero-mean Gaussian random variables with variance σ 2 i σ σ 5. f y (y,)= (2π) 5/2 detσ exp[ 2 (y )T Σ (y )] with y:=[y y 2 y 5 ] T, =[ ] T and covariance matrix Σ=diag(σ 2,,σ2 5 ). log f y (y,)=c 5 (y i ) 2 ; 2 i= σ 2 i log f y (y; ) = 5 (y i ) i= σ 2 i Lec5-3 2 log f y (y; ) 2 = 5, var(ˆ) /σ 2. /σ2 5 σ 2 i= i CRLB decreases with every additional measurement (determined by essential information content in the data) Note that a LS estimator will provide and leads to ˆ = 5 For σi 2 =(.,, 5,, 2) this equals.444 while CRLB =.88 5 i= var(ˆ) = 25 y i 5 i= σ 2 i Lec5-4 Maximum Likelihood Estimator General estimator principle for situations where the probability density function (pdf) of the observed/measured variables is known. Goal: Estimate an unknown parameter in the pdf of a random variable y on the basis of observations of y. Example: random variable y has a Gaussian (unit variance) pdf with unknown μ y f y (y; ) = e (y )2 2 2π Maximum likelihood principle: For given observations y, determine such that L() is maximum. (Choose that pdf that - a posteriori - makes the observed value of the considered variable most likely) For observation y: L() = e (y )2 2 2π Maximizing L() leads to: ˆ = y For N independent observations y i : For given this is a pdf For given y and unknown this is a deterministic function of likelihood function L() L() =f(y,,y n ; ) = N f(y i ; ) i= Lec5-5 Lec5-6

9 Example: observation with model y= u e.4.2 pdf f (y;) y 5 Model: y(t) =ŷ(t t ; ) ε(t), withε =efor =. If {e(t)} independent r.v. s for different t with equal pdf f e, then: N f y (x N )= f e (x(t) ˆx(t t ; )) t= (probability density function of the observations) and for the particular observation x = y (a posteriori): likelihood L(;y=) 5 5 y When observing y =this leads to ˆ = arg max L(; y =). L y (; y N )= N f e (y(t) ŷ(t t ; )) t= the likelihood function, a deterministic function of. L y (; y N )= N f e (ε(t, )). t= Lec5-7 Lec5-8 If f e Gaussian: L y (; y N )= N e ε(t,) 2σe 2 2πσe t= ML-estimator: Maximizing L y is equivalent to minimizing log L y : log L y (; y N )= N 2 log2π Nlogσ e 2 2σe 2 t= ε(t, ) 2 If the noise e on the data is a sequence of independent observations (white noise) from a Gaussian pdf with zero mean and equal variance for all observations, then the ML estimator is equal to the least squares (LS) prediction error estimator. If σ e is either fixed or parametrized independent from, then min ε(t, ) 2 t= =LS Lec5-9 Lec5-2

10 Properties of the ML estimator for N : N(ˆN ) N(,NJ ) Or differently phrased: with J the Fisher Information Matrix, so that asymptotically in N If in the PE setting, the noise disturbance e is Gaussian and S M,then PE = ML cov(ˆ N )=J (Cramér-Rao lower bound). This implies that the ML estimator is consistent asymptotically reaches the smallest possible variance (CRLB) over all unbiased estimators no guarantees for properties in case of finite N Lec5-2 Lec5-22 Alternative interpretation of stochastic estimation in PE The stochastic/deterministic identification paradigm e H u G H e v y Probabilistic estimation framework results from assumptions on noise e Modelling the characteristics of e or v is a user s choice Probabilistic framework useful for random type of disturbances u G G(q,) v -- /H(q,) ε(t,) y system model ˆN (2) The (probabilistic) ellipsoidal uncertainty set is a selection of s for which the resulting ε(t, ) is likely to be a realization of a (Gaussian) white noise process ˆN () Assumptions on e determine the parameter uncertainty set Lec5-23 Lec5-24

11 Generalization Set-membership identification u G G(q,) v -- ε(t,) y system model Identification of model uncertainty bounds can be reformulated: Choose a noise set V of which the disturbance signal v is assumed to be a member For a given identification experiment (y N,u N ), determine the parameter uncertainty set as Driven by the desire to have a hard-bounded uncertainty set Θ unc,thenoisesetv can be chosen hard-bounded (deterministic) also: amplitude-bounded: v(t) c energy-bounded: v 2 (t) c N t=... In combination with a l-i-p model structure, the resulting identification algorithm becomes a linear programming problem (2) Θ unc = { ˆv(t, ) :=ε(t, ) V(with probability α)} If V non-probabilistic = Θ unc also () Lec5-25 Lec5-26 Set-membership identification (cont d) Characteristic properties Alternative noise sets allow for nonlinear, time-varying disturbances Close to the parameter uncertainty bounds, the related (worst-case) v becomes strongly correlated to u (Nature is playing against you) With the indicated noise sets, uncertainty bounds become hard-bounded but very large Conservatism results from a choice of V that is bigger than the realistically occurring set of noise signals Improvements can be made in V by including a bound on the cross-correlation v(t)u(t τ ) N c t= (Hakvoort et al., 995) This limits the knowledge that the (worst-case) noise signal has about the (past and future) input. Consistency is generally lost (set does not shrink to a single point for increasing N) Lec5-27 Lec5-28

12 Finite impulse response models: Attractive model structures: OBF s In PE results (Chapter 5) attractive properties of model structures: Linearity-in-the-parameters: LS leads to convex optimization G(z, ) = n b k= b k z k Output-Error structure: Consistent identification of G irrespective of the noise model Both properties are combined by the FIR-model: G(z, ) = n b k= b k z k H(z, ) = I (fixed) Disadvantage For systems with relatively high sampling rates and small damping (long tails), the necessary number of parameters n b can become very high General aim towards parsimonious parametrizations: Use parametrizations with few parameters, but in strategic locations Lec5-29 Lec5-3 Generalized basis functions Particular properties of the sequence {z,z,z 2, }: From linear combinations of shifted pulse functions towards linear combinations of smartly chosen dynamic functions They constitute a complete basis for the systems space H 2, Systems with a pulse response g(k) l 2 that satisfies k= g(k) 2 < The inner product in this space is <G,G 2 >:= π G 2π (eiω )G 2 (e iω )dω = g (k)g 2(k) π All basis functions are orthonormal, i.e. <z i,z j > =, i j k= n G(z, ) = c k F k (z) k= =, i = j When using a white noise input signal u, then E[q i u(t) q j u(t)] = i j Lec5-3 Lec5-32

13 Generalized orthonormal basis functions Generalization of this idea: Consider the sequence of functions Laguerre functions Consider the sequence of functions {, z ξ z ξ 2, z ξ 3, }, ( ξ i < ) { z a, (z a),, }, ( a < ) 2 (z a) 3 These functions constitute a complete basis for H 2. After Gram-Schmidt orthogonalization: [ ] a 2 az k F k (z) = k =, 2, z a z a with a < real-valued pole. Note: If a =then the FIR structure results. Classical result (Walsh, 935): basis functions are complete in H 2 if i= ( ξ i ) = (I.e. the poles should not tend to the unit circle). After Gram-Schmidt orthogonalization: ξk 2 k [ ξ ] F k (z) = i z z ξ k z ξ i= i }{{} all pass function Again: For ξ i =, i, the FIR structure results. Lec5-33 Lec5-34 Use in system identification The considered model structure leads to n ŷ(t t ; ) = c k F k (q)u(t) k= Then ε(t, ) =y(t) ϕ T (t) with ϕ(t) :=[F (q)u(t) F 2 (q)u(t) F n (q)u(t)] T. The LS-estimate is simply obtained by [ ˆ N = N ] [ ϕ(t)ϕ T (t) N t= ] ϕ(t)y(t) t= and all relevant properties of linear regression estimators are maintained. Lec5-35 Degrees of freedom: selection of basis poles Pole selection usually done periodically: {ξ,ξ 2, ξ n } = {ξ,ξ 2, ξ nb,ξ, ξ nb, }. Convergence result: If G (z) has poles {p i } i=,..,n, and the basis is induced by a set of poles {ξ j } j=,..,nb, then the slowest eigenvalue that determines the convergence rate of G (z) = c k F k (z) is given by max i n b j= k= p i ξ j ξ j p i = coefficients c k decay quickly if pole sets are close Lec5-36

14 Example: 3rd order G : p,2 =.985 ±.6i; p 3 =.75 ρ N. FIR Laguerre with ξ = Kautz with ξ,2 =.97 ±.i GOBF: ξ,2 =.97 ±.i, ξ 3 = GOBF: ξ,2 =.98 ±.5i, ξ 3 = N. : number of coefficients before magnitude is decayed below %. Lec5-37 Features in identification with GOBF s Use of uncertain prior knowledge in identification: The more accurate the priors (pole information) the more simple the ID-problem (less parameters) but no loss of generality (for increasing n still all systems can be represented) Additional properties: Variance analysis is available Noise model can be added (iteratively through GLS) Easily extendable to MIMO models Finite-time variance analysis Linear constraints can be added (static gain,...) Lec5-38 Summary Alternative interpretation of basis construction u(t) G b G b G b Interpretation of the PE identification criterion in a statistical ML framework Justification of a stochastic approach () (2) (n) x b (t) x b (t) x b (t) Balanced state readout c c 2 c n y(t) Discussion of deterministic alternatives Extension of the classical model structures towards attractive GOBF s (Generalized Orthonormal Basis Functions) Further reading in Heuberger, Van den Hof and Wahlberg (Springer 25) Through balanced-state representations of all-pass functions G b generated by the poles {ξ, ξ nb }. Generalization of so-called tapped delay line (G b = z ). What we could not discuss yet MIMO models and subspace identification (Lect. 7) Nonlinear dynamics Estimation of physical parameters (on-line/recursive) Lec5-39 Lec5-4