Direct continuous-time approaches to system identification. Overview and benefits for practical applications

Transcription

1 Direct continuous-time approaches to system identification. Overview and benefits for practical applications Hugues GARNIER University of Lorraine, CNRS Research Centre for Automatic Control (CRAN Nancy, France Aim of this lecture ü To provide an introduction theory of direct time-domain methods for continuous-time parametric black-box model identification ü The key computational method, we refer to, is Optimal Instrumental variable (IV method 2

2 Outline. Performance Evaluation of Discrete and Continuous-time Methods The Rao-Garnier benchmark 2. Direct Approaches for Linear Model Identification Traditional SVF-based estimators SRIVC: a reliable optimal instrumental variable estimator Practical side of continuous-time model identification 3. Software Aspects and Benefits in Practical Applications The CONTSID toolbox: a guided tour A selection of successful practical applications 4. Refined Instrumental Variables for More Advanced Situations Identification of Box-Jenkins models, multiple input, LTV systems, 3 System identification ü Aims at building mathematical models that describe the I/O behavior of a dynamic system based on observed data ü Common use of the identified model for better understanding, simulation, prediction, control design or fault detection ü Models used in different areas from mechanical to biomedical engineering and many successful applications are reported 4

3 Applications of system identification ü Mature field and successful applications can be found in the special issue of IEEE Control Systems Magazine, Oct In-Flight Vibration Monitoring of Aeronautical Structures Electric Load Forecasting Dynamic Model Identification for Industrial Robots High-Purity Distillation column Identification of Stellar Interferometer models from empirical data Space Weather Forecasting 5 The system identification procedure System Identification: an iterative procedure ü often perceived more as an art than a science Data not OK Should the data be filtered? Choice of the model structure and estimation methods Model structure not OK Green rectangles: computer s main responsability Blue ovals: practionner s main responsability Adapted from Ljung 999 no Construct the experiment and collect data Data Polish and present data Data Fit the model to the data Model Validate the model Can the model be accepted? yes The practionner has to make many choices: ü well-planned data acquisition ü Sampling period, type of input, ü data-preprocessing ü Filtering, detrending ü type of models to be estimated: ü linear or non linear ü continuous or discrete-time ü estimation methods ü PEM or IV These choices will impact the SYSID procedure and require active participation of a specifically trained practitioner 6

4 Types of parameterized models ü Grey-box models parameters have a direct physical interpretation Models constructed in continuous-time from physical principles main shortcomings: cannot be built in the case of complex systems where the physical principles are not well established or too involved some parameters might not be identifiable ü Black-box models parameters have no direct physical interpretation are families of flexible models of general applicability can be continuous-time or discrete-time Ø Black-box continuous-time linear model identification is mainly considered in this lecture 7 Discrete-time (DT models of linear systems ü A model that directly expresses the relationship between the values of the I/O signals at the time-instants is called a discrete-time model Difference equation / polynomial / transfer function model y + a y ++ a na y na = b u ++ b nb u nb A(q y = B(q u G( z = Y( z B( z = U( z A( z q u = u shift operator z: Z transform variable A(q =+ a q ++ a na q n a B(q = b q ++ b nb q n b State-space model # x+ = Ax + Bu " $# y = Cx + Du G( z = C ( zi A B + D 8

5 ü A model that describes the relationship between time continuous I/O signals is called a continuous-time model Differential equation / polynomial / transfer function model d n y(t dt n Continuous-time (CT models of linear systems d n y(t + a dt n d m u(t ++ a n y(t = b 0 dt m + b d m u(t dt m ++ b m u(t A( py(t = B( pu(t pu(t = du(t dt differentiation operator A( p = p n + a p n ++ a n B( p = b 0 p m + b p m ++ b m G(s = Y(s U(s = B(s A(s s: Laplace variable State-space model x(t = Ax(t + Bu(t y(t = Cx(t + Du(t G(s = C ( si A B + D 9 System identification A brief historical review ü Early research: focussed on identification of CT models from CT data ü From 970, with the developments in digital data acquisition and computers DT model identification from sampled data took over the field Matlab SID toolbox has greatly helped to disseminate the methods Book ref.: L. Ljung, System Identification. Theory for the user, 999 ü Many practitioners appear unaware CT model identification from sampled data not only exists but may be better suited to their data-based modelling problems Book ref.: H. Garnier, L. Wang (Eds. Identification of CT models from sampled data, 2008 Ø The last decade has witnessed a growing interest in direct CT methods to system identification 0

6 Main approaches to identify a black-box CT linear models from time-domain sampled data? u u(t y(t t k hold t CT system t Sampled data T s y u, y t k Indirect approach G(z SID toolbox, Delicate conversion? CT black-box linear model G(s Direct approach ü less known ü quite straightforward ü presents many advantages ü 4th order simulated system The Rao-Garnier benchmark G o ( p = " $ $ # K( Tp + 2ζ %" p +' p 2 + 2ζ % $ 2 p +' 2 ω n, ' $ 2 & ω ω # n,2 n,2 ' & p 2 ω n, 6400p +600 = p 4 + 5p p p +600 p=d/dt ü unstable zero ü 2 pseudo-oscillatory modes K = ω n, = 2 rad / s ; ω n,2 = 20 rad / s ζ = 0.; ζ 2 =

7 Simulation setup ü Use of a Monte Carlo simulation (00 runs to get an average performance evaluation of the direct CT and indirect DT methods # " $# x(t = G o ( pu(t y = x + e u hold u(t G o (p x(t ü Simulations conditions u(t: PRBS (kept the same during the MCS respects the ZOH assumption T s =0 ms fast but not irrealistic in view of the resonance frequency of 20 rad/s f s 0 times the system bandwidth, an often given rule + T s e(tk + y e : DT white noise, SNR=0dB G.P. Rao, H. Garnier, Numerical illustrations of the relevance of direct continuous-time model identification, 5th IFAC World Congress, Barcelona, Spain, Example of the I/O data used to identify both DT and CT models for one realization of the noise 4

8 CT and DT models and related methods/toolboxes tested ü DT model and tested estimation methods from the SID toolbox G(q,θ = b q + b 2 q 2 + b 3 q 3 + b 4 q 4 + a q + a 2 q 2 + a 3 q 3 + a 4 q 4 4 parameters in the numerator artificial zeros due to ZOH discretisation ARX (basic LS IV4 (four step IV iterative IV N4SID (subspace OE (pem ü CT model and tested estimation methods from the CONTSID toolbox b G( p,θ = 0 p + b p 4 + a p 3 + a 2 p 2 + a 3 p + a 4 A priori knowledge about the number of parameters to be estimated for the numerator easy to accommodate LSSVF (basic LS IVSVF (two step IV SRIVC (iterative IV COE (pem All routines from the SID and CONTSID toolboxes used in their default mode with R205a Matlab release 5 Frequency responses of the stable discrete-time ARX models (SID toolbox 6

9 Frequency responses of the stable discrete-time IV4 models (SID toolbox 7 Frequency responses of the stable discrete-time N4SID models (SID toolbox Magnitude (db true N4SID Phase (deg Rate of stable estimated models: 67 % Frequency (rad/s 8

10 Frequency responses of the stable discrete-time OE models (SID toolbox Magnitude (db true DT OE models Phase (deg Rate of stable estimated models: 89% Frequency (rad/s 9 Assessment of DT model identification methods First comments ü Intriguing MCS results Why does IV4 perform so badly? Is-it a general result for IV estimators? Why does OE/PEM perform so badly? ü Let us review the theory Theory tells us that in this noisy (white output measurement situation IV/OE methods should not be biased ü This proves again that theory has its own limitations N, implementation aspects, robustness issues ü Any explanation for these bad results obtained by DT estimators? Problems in the simulations? ü L. Ljung confirmed that the Rao-Garnier benchmark gives severe problems for the traditional DT model identification methods L. Ljung, XXXXXXXX aspects for subspace and output error identification methods, ECC 03, Cambridge, Sept

11 What can be concluded from the benchmark results? ü Standard DT model techniques encounter pbs due to bad initialisation PEM/IV4 are both initiated from ARX model estimates Pb : initial ARX model estimates are very biased ARX fit focusses in the high frequencies unless we prefilter ü Data prefiltering: often a crucial stage to get success in SYSID Data filtering is non-inherent in standard DT model identification ü Remedy: move the focus in the model fit to lower frequencies by pre-filtering The experienced practitionner should use one of the two solutions: use of a data prefiltering strategy use of a data filtering-decimation strategy 2 Frequency responses of the DT models estimated from prefiltered data (SID toolbox u(t G o (p x(t N4SID T s T s + + e u y L(q u f L(q: low-pass filter y f L(q G(q,θ = b q + b 2 q 2 + b 3 q 3 + b 4 q 4 + a q + a 2 q 2 + a 3 q 3 + a 4 q 4 IV4 OE 22

12 Frequency responses of the DT models estimated from prefiltered-decimated data (SID toolbox u(t G o (p x(t N4SID T s T s + + e u y L(q u f L(q: low-pass filter y f L(q r Decimation u f (rt k T s =rt s R=0 y f (rt k G(q,θ = b q + b 2 q 2 + b 3 q 3 + b 4 q 4 + a q + a 2 q 2 + a 3 q 3 + a 4 q 4 r IV4 OE 23 CT and DT models and related methods/toolboxes tested ü DT model and tested estimation methods from the SID toolbox G(q,θ = b q + b 2 q 2 + b 3 q 3 + b 4 q 4 + a q + a 2 q 2 + a 3 q 3 + a 4 q 4 4 parameters in the numerator artificial zeros due to ZOH discretisation ARX (basic LS IV4 (four step IV iterative IV N4SID (subspace OE (pem ü CT model and tested estimation methods from the CONTSID toolbox b G( p,θ = 0 p + b p 4 + a p 3 + a 2 p 2 + a 3 p + a 4 A priori knowledge about the number of parameters to be estimated for the numerator easy to accommodate LSSVF (basic LS IVSVF (two step IV SRIVC (iterative IV COE (pem All routines from the SID and CONTSID toolboxes used in their default mode with R205a Matlab release 24

13 Frequency responses of the stable continuous-time LSSVF models (CONTSID toolbox Magnitude (db true Direct CT LSSVF models Phase (deg Rate of stable estimated models: 00% Frequency (rad/s 25 Frequency responses of the stable continuous-time IVSVF models (CONTSID toolbox Magnitude (db true Direct CT IVSVF models Phase (deg Rate of stable estimated models: 00% Frequency (rad/s 26

14 Frequency responses of the stable continuous-time SRIVC models (CONTSID toolbox Magnitude (db true Direct CT SRIVC models Phase (deg Rate of stable estimated models: 00% Frequency (rad/s 27 Frequency responses of the stable continuous-time COE models (CONTSID toolbox Magnitude (db true Direct CT COE models Phase (deg Rate of stable estimated models: 00% Frequency (rad/s 28

15 What can be concluded from the benchmark results? ü Traditional DT techniques encounter pbs PEM/IV4 are both initiated from highly biased ARX model estimates Numerical issues for fast sampling Require special care from the user: data prefiltering data (prefiltering decimation Difficult for the user to choose a priori: prefilter cut-off frequency new sampling rate Should the data be filtered? Choice of the model structure Construct the experiment and collect data Data Polish and present data Data Fit the model to the data Model ü Direct CT approaches Free of these difficulties Data not OK Model structure not OK Validate the model Include inherent data prefiltering no Can the model be accepted? It seems worth to know more about these direct CT approaches Green rectangles: computer s main responsability Blue ovals: practionner s main responsability From Ljung 999 yes 29 Outline. Performance Evaluation of Discrete and Continuous-time Methods The Rao-Garnier benchmark 2. Direct Approaches for Linear Model Identification Traditional SVF-based estimators SRIVC: a reliable optimal instrumental variable estimator Practical side of continuous-time model identification 3. Software Aspects and Benefits in Practical Applications The CONTSID toolbox: a guided tour A selection of successful practical applications 4. Refined Instrumental Variables for More Advanced Situations Identification of Box-Jenkins models, multiple input, LTV systems, 30

16 Aim of this part ü Provides an introduction theory of some reliable direct time-domain methods for continuous-time parametric linear model estimation ü The key computational methods, we refer to, are State-Variable Filtering (SVF approach for estimating the I/O time-derivatives Instrumental variable (IV method for estimating the parameters of the continuous-time model 3 True system ü Assumptions on the true system: S = {G o ( p;h o ( p} e H o ( p y = G o ( pu + H o ( pe v p = d dt differentiation operator u G o ( p x y ü The measured output y is assumed to be made up of two distinct contributions: G o (pu : dependent of the choice of the input signal u(t the disturbance v =H o (pe : independent of the input signal u(t 32

17 General system identification objective ü Objective: Find the best parametric models G(p,θ and H(p,θ for the unknown transfer functions G o (p and H o (p using a set of measured data u and y M = ( G( p,θ ; H( p,θ, θ R n θ { } ü In the beginning, we will make the following assumption: θ o such that G(p,θ o = G o (p and H(p,θ o = H o (p i.e. ü The objective can therefore be restated as follows: Find an estimate of the unknown parameter vector θ o using a set of N input and output data: Z N = { u, y k =...N } generated by the true system, i.e. S M y = G o ( pu + H o ( pe 33 ü Model structure: Black-box continuous-time model structures ü General parametrization G( p,θ = M = ( G( p,θ ; H( p,θ, θ R n θ { } B( p,θ C( p,θ A( p,θ e τp H( p,θ = D( p,θ θ T = a a n b 0 d A( p,θ = p n + a p n + + a n B( p,θ = b 0 p m + b p m + + b m C( p,θ = p n c + c p n c + + c nc D( p,θ = p n d + d p n d + + d nd 34

18 Main black-box CT model structures ü Main model structures used in practice: M = ( G( p,θ ; H( p,θ, θ R n θ { } CARX G( p,θ = COE G( p,θ = CBJ G( p,θ = hybrid CBJ G( p,θ = B( p,θ A( p,θ e τp H( p,θ = B( p,θ A( p,θ e τp H( p,θ = A( p,θ B( p,θ C( p,θ A( p,θ e τp H( p,θ = D( p,θ B( p,θ A( p,θ e τp H(q,θ = C(q,θ D(q,θ 35 Distinction between model structures ü CARX model can be written in linear regression form A( p,θ y = B( p,θ u + e y ( n = ϕ T θ + e L The model is not very realistic in practice There are common denominators in G and H J The model is a linear function in θ Important computational advantages ü COE and CBJ models have an independent parametrization of G(p,θ and H(p,θ y = B( p,θ A( p,θ u + e y = B( p,θ A( p,θ u(t C( p,θ k + D( p,θ e L Models are no longer linear-in-the-parameters J There are no common parameters in G and H Advantages for independent identification of G and H and models more realistic in practice 36

19 A priori knowledge to use/apply system identification ü User/practitioner expected to have an extensive background in statistics estimation theory signal processing optimization to understand various model parameter estimation methods associated decision variables in terms of bias-variance trade-offs 37 Principal sources of error in estimation theory ü Consider the estimate ˆθ N of the true parameter vector θ o based on N data Error = Bias + Variance ü Bias = systematic error Unbiased estimate if E ( ˆθ N = θ o ü Variance = fluctuations of the estimates around its expected value P θ = E " ( ˆθ N θ o ( ˆθ N θ o T % $ ' # & Affected by number of data N, noise, SNR ü Consistency When N, if the bias and variance 0, then the estimator is said to be consistent 38

20 Illustration of bias-variance trade-off for estimators Estimator A Estimator B Estimator C Center of the target (in red represents θ o Estimator A: biased (average value of the estimates are not in the center of the target Estimator B: unbiased but quite large fluctuations around the mean value large variance Estimator C: unbiased and small variance 39 Objectives when developing an estimator ü First focus: construct estimators to be unbiased ü Second focus: construct estimators with minimum variance for the estimates, and in any case as small as possible Ø Construct optimal estimator: unbiased and minimum variance ü Further aspects: strive to get a robust estimator robustness to assumptions (noise model, robustness to experiment design (measurement setup, T s, robustness to algorithmic aspects (prefiltering, initialization, 40

21 Issue in CT model identification: time-derivative measurement problem ü DT model identification - difference equation model y(k + a y(k + + a na y(k n a = b u(k + + b nb u(k n b ü CT model identification - differential equation model Unlike the DT model, where only sampled input and output data appear, the CT differential equation model contains I/O time-derivatives d n y(t dt n n d y(t + a dt n m d u(t ++ a n y(t = b 0 dt m + b d m u(t dt m ++ b m u(t Well-known approach to handle the time-derivative problem: apply a linear transform Not measured to both I/O in most data can be seen as a data prefiltering practical cases strategy 4 Two-stage approach for direct CT model identification u ( t B ( p A ( p v ( t + + y ( t LT : Linear Transforms Inherent data prefiltering LT LT primary stage Arises out of the nonmeasurable I/O time- derivatives problem * u ( t y * ( t ( * + B ˆ p z ( t * A ˆ ε ( t ( p Estimation algorithm criterion J secondary stage Most of the standard DT parametric model estimation methods can be used 42

22 Main linear transforms developed for the primary stage Linear Transforms Integral methods Linear Filters Num. Int. Methods: BPF, TPF, Simpson function methods: Laguerre, Legendre, Hermite, Chebychev polynomials, Walsh, Fourier functions LIF (Sagara RPM (Trigeassou SVF (Young GPMF (Rao Modulating Functions FMF (Pearson HMF (Unbehauen H. Garnier, M. Mensler, A. Richard, Continuous-time model identification from sampled data: implementation issues and performance evaluation. IJC, 76(3, Traditional SVF method y ( n (t + a y ( n (t ++ a n y(t = b 0 u ( m (t ++ b m u(t Apply a stable filter L(p=/E(p on both sides, the prefiltered DE model obeys exactly (except for a possible transient y f ( n (t + a y f ( n (t ++ a n y f (t = b 0 u f ( m (t ++ b m u f (t x(t E( p p i E( p p n E( p x f (t x f ( n (t How to choose L(p=/E(p? E( p = ( p + λ n Bank of SVF filters The filtered time-derivatives can then be exploited to estimate the parameters of the differential equation model 44

23 Simple least squares-based SVF estimator ü At t=t k, the prefiltered DE model can be rewritten in linear regression form ( n y f = ϕ T f θ +ε ϕ T ( f = y n ( m f y f u f u f T θ = a a n b 0 b m ü From N samples observed at t, t N, the LS-based SVF parameter estimates are computed as # ˆθ lssvf = arg min% θ % $ N " ˆθ lssvf = $ # $ N N k= N k= % ϕ f ϕ T f ' & ' ( n ( y f ϕ T f θ 2 & ( ( ' " $ # $ N N k= % ( n ϕ f y f ' & ' 45 Simple LS-based SVF estimator u(t B( p A( p N samples u v E( p = ( p + λ n y u E( p p i E( p p n E( p E( p p i E( p p m E( p SVF filter Bank y f y f (n u f u f (m LS algorithm ˆθ lssvf " ˆθ lssvf = N % " ϕ N f ϕ T N % $ f ' $ ( n ϕ # $ k= & ' N f y f ' # $ k= & ' This simple LS-based SVF estimator represents the simplest archetype of CT model identification from sampled data 46

24 ü Consider a second-order system LSSVF method - Example y (2 (t + a y ( (t + a 2 y(t = b 0 u(t + e(t ü Apply a second-order SVF filter L(p=/(p+λ 2 p 2 ( p + λ + a 2 ( p 2 + a p + a 2 y(t = b 0 u(t + e(t p ( p + λ + a 2 2 ( p + λ 2 y(t = b 0 p + λ ( 2 u(t + y f (2 (t + a y f ( (t + a 2 y f (t = b 0 u f (t + e f (t ( p + λ e(t 2 ü At t=t k (2 y f = y ( f y f u f a a 2 b 0 + e f 47 LSSVF method - Example ü At t=t k ü For t=t, t N, we have (2 y f = y ( f y f u f y f (2 (t y f (2 (t 2 y f (2 (t N = y f ( (t y f (t u f (t y ( f (t 2 y f (t 2 u f (t 2 y ( f (t N y f (t N u f (t N a a 2 b 0 a a 2 b 0 + e f + e f (t e f (t 2 e f (t N Y N = Φ N θ + E N ˆθ lssvf = Φ T N Φ N ΦN T Y N 48

25 LS-based SVF estimator Implementation aspects ˆθ lssvf = Φ T N Φ N T N ΦN Y = ϕ (t ϕ T (t N f k f k k= ü As for DT ARX model identification by simple LS, use numerically stable and computationally efficient algorithms for computing the LS-based SVF estimates N k= ( n ϕ f y f ü Do not compute the normal equation solution above, but use instead: SVD Singular Value Decomposition (pinv in Matlab Θ=PINV(Φ*Y computes the solution to Y=Φ Θ QR factorization (matrix division \ in Matlab Θ = Φ \Y computes also the solution to Y=Φ Θ ü Recommended implementation of the LSSVF solution in Matlab ˆθ lssvf = Φ N \Y N 49 SVF-based estimators Implementation aspects ü Roles of the SVF filters Reconstruct the time-derivatives in the bandwidth of interest Improve the statistical efficiency of the estimates (filter out the high-frequency noise u(t ü User parameters of the SVF filter Filter order: should be chosen larger or equal than the system order n Simplest choice: minimal-order SVF, L(s=/(s+ λ n ( p + λ n p i ( p + λ n p n ( p + λ n u f (t u f ( i (t u f ( n (t Note that so called minimal-order GPMF where L(s=/(s+ λ n+ is often more robust against the noise than basic SVF (see lsgpmf in CONTSID Cut-off frequency λ of the SVF filter L(s=/(s+ λ n, chosen in order to emphasize the frequency band of interest 50

26 Bode plot of SVF filters ü The outputs of the SVF filter bank will provide a smoothed estimate of the I/O timederivatives in the frequency band of interest s i L i (s = si E(s = ( s + λ n L 0 (s = ( s + λ 3 s L (s = s + λ ( 3 s 2 L 2 (s = ( s + λ 3 s 3 L 3 (s = ( s + λ 3 5 SVF-based estimators Implementation aspects ü Digital implementation of the CT SVF filtering operations The computation of the LSSVF parameter estimates requires the value of prefiltered signals at the time-instants t k, k =,, N y f (2 (t y f (2 (t 2 y f (2 (t N = y f ( (t y f (t u f (t y ( f (t 2 y f (t 2 u f (t 2 y ( f (t N y f (t N u f (t N a a 2 b 0 + e f (t e f (t 2 e f (t N Y N = Φ N θ + E N ˆθ lssvf = Φ N \Y N The digital implementation method has to be selected carefully according to the assumption about the filter input intersample behavior: choice of the hold block u u t k t k hold? u(t u(t t t ( p + λ n p i ( p + λ n p n ( p + λ n u f (t u f ( i (t u f ( n (t u f u f ( i u f ( n 52

27 Digital implementation of the CT SVF filtering operations If the filter input intersample behavior is known (e.g. piecewise constant or piecewise linear or if the input takes a particular form (e.g. a sine or sum of sines: an exact solution to the filtering operation at specified time-instants can be obtained If the filter input intersample behavior is not known: approximate solution to the filtering operation can be obtained only approximation errors depend on T s and fast sampling is often preferred in CT model identification Fast sampling is however not required for all CT methods, e.g. SRIVC (see later on One efficient approach is implemented in the Matlab lsim routine where the state-space representation of the SVF filter bank is discretized assuming the best zoh or foh assumption for the input intersample behavior x(t = A c x(t + B c u(t y(t = Cx(t hold T s x+ = F d x +G d u y = Cx 53 LSSVF implementation in Matlab 2 nd -order example ü Simple second-order COE model x(t = G o ( pu(t y = x + e G o ( p = 2 ( p + 3 ( p + = 2 p 2 + 4p + 3 ü Simulations conditions u(t: PRBS T s =0 ms N=500 2 output measurement situations Noise-free e : DT white Gaussian noise, σ e =0.2 u ZOH u(t G o (p x(t + T s e(tk + y 54

28 LSSVF estimator - Matlab implementation Noise-free case B=2; % B(p=2 A=[ 4 3]; % A(p=p 2 +4p+3 True system Ts=0.0; u=prbs(4,00; % PRBS input from the CONTSID N=500; t=(0:n- *Ts; x=lsim(b,a,u,t; % simulation of the noise-free output data0=iddata(x,u,ts;idplot(data0; % Primary stage - SVF filtering lambda=3; % λ: SVF filter cut-off frequency den_l=[ 2*lambda lambda^2]; % denominator of the SVF filter num_l0=; % numerator of L0(p=/(p+λ 2 num_l=[ 0]; % numerator of L(p =p/(p+λ 2 num_l2=[ 0 0]; % numerator of L2(p =p 2 /(p+λ 2 xf0=lsim(num_l0,den_l,x,t; % Computation of the SVF filter bank outputs xf=lsim(num_l,den_l,x,t; xf2=lsim(num_l2,den_l,x,t; uf0=lsim(num_l0,den_l,u,t; % Secondary stage - LS estimates Phi_N=[-xf -xf0 uf0]; % Regression matrix Y_N=xf2; % Output vector theta_lssvf=phi_n\y_n % LSSVF estimates theta_lssvf % see also the LSSVF routine in the CONTSID toolbox % Mlssvf=lssvf(data0,[2 0],lambda 55 LSSVF estimator - Matlab implementation Noisy case B=2; % B(p=2 A=[ 4 3]; % A(p=p 2 +4p+3 True system u=prbs(4,00; % PRBS input from the CONTSID N=500; Ts=0.0; t=(0:n- *Ts; x=lsim(b,a,u,t; % simulation of the noise-free output y=x+0.2*randn(n,; % white noise added to the noise free output data=iddata(y,u,ts;idplot(data; % Primary stage - SVF filtering lambda=3; % λ: SVF filter cut-off frequency den_l=[ 2*lambda lambda^2]; % denominator of the SVF filter num_l0=; % numerator of L0(p=/(p+λ 2 num_l=[ 0]; % numerator of L(p =p/(p+λ 2 num_l2=[ 0 0]; % numerator of L2(p =p 2 /(p+λ 2 yf0=lsim(num_l0,den_l,y,t; % Computation of the SVF filter bank outputs yf=lsim(num_l,den_l,y,t; yf2=lsim(num_l2,den_l,y,t; uf0=lsim(num_l0,den_l,u,t; % Secondary stage - LS estimates Phi_N=[-yf -yf0 uf0]; % Regression matrix Y_N=yf2; % Output vector theta_lssvf=phi_n\y_n % LSSVF estimates theta_lssvf % see also the LSSVF routine in the CONTSID toolbox % Mlssvf=lssvf(data,[2 0],lambda 56

29 Frequency responses of the LSSVF models (recap of the Rao-Garnier benchmark results Magnitude (db A small bias can be observed L( p = ( p + λ 4 λ =3 rad / sec true Direct CT LSSVF models Phase (deg Frequency (rad/s 57 Basic LSSVF estimator Statistical analysis ü Assume the data-generating system is described as S: y ( n = ϕ T θ o +v where θ o is the true parameter vector ü Assume that v is a stationary stochastic process independent of u. After the SVF filtering, the data-generating system can be rewritten as y f ( n = ϕ f T θ o +v f ˆθ lssvf = N N k= ˆθ lssvf = θ o + N T ϕ f ϕ f (tk N k= ϕ f ϕ T f N N N k= ( n ϕ f y f (tk N k= ϕ f v f 58

30 Basic LSSVF estimator Statistical analysis ˆθ lssvf = θ o + N ü Under weak conditions, the normalized sums tend to the corresponding expected values as N tends to infinity. Hence ˆθ lssvf N k= θ o if N ϕ f ϕ T f { } E ϕ f ϕ f T { } = 0 E ϕ f v f is nonsingular The first condition is satisfied in most cases The second condition is never satisfied ü LSSVF estimates are always biased because of the correlation between the regression vector ϕ f and the noise v f even if v is white noise, v f becomes colored due to the SVF filtering N N k= ϕ f v f 59 Simple LSSVF estimator Conclusions ü Simple LSSVF method has some attractive properties Simple, analytical solution easy to compute, low computational complexity ü Main shortcomings always biased in noisy output measurement situations E { ˆθ lssvf } θ o since E { ϕ f v f } 0 quite sensitive to the SVF filter cut-off frequency Ø Motivation for studying more advanced methods L( p = ( p + λ n We can do better 60

31 Traditional solutions to get optimal estimates ü Maximum Likelihood Method (ML If the disturbances on the system are Gaussian, the ML method coincides with the Prediction Error Method (PEM ü Instrumental Variable Method (IV 6 Prediction Error Method (PEM ü Main idea: model the noise ü General approach applicable to a wide range of model structures: OE, BJ, ü Conditions to obtain optimal PEM estimates are well-established N ˆθ pem = arg min ε 2,θ θ = arg min y ŷ,θ 2 θ k= ü If assumptions about the noise valid: delivers optimal estimates ü Involves often solving a non-convex optimization problem relies on iterative nonlinear optimization (computationally quite demanding special care required for the initialization of the iterative search may be trapped in false solutions that correspond to local minima N k= 62

32 Frequency responses of the CONTSID COE models (recap of the Rao-Garnier benchmark results The iterative COE(PEM-based search is initialized from IVSVF parameter estimates (see later. Being well initiated, it delivers optimal estimates here Magnitude (db true Direct CT COE models Phase (deg Rate of stable estimated models: 00% Frequency (rad/s 63 Frequency responses of the TFEST models (SID toolbox (tested on the Rao-Garnier benchmark The recent PEM-based TFEST routine is initialized from SRIVC parameter estimates (see later and explains why such good results are obtained here 64

33 Instrumental Variable (IV method ü Main idea: model the noise ü General approach applicable to a wide range of model structures: OE, BJ, ü Conditions to obtain optimal IV estimates are well-established N ˆθ opt iv = arg min z opt f L opt ( p y ϕ T θ θ k= ( Q 2 Need to specify the instrument z f and the prefilter L(p ü If the assumptions about the noise are valid: delivers optimal estimates ü If the assumptions about the noise are not valid: delivers unbiased estimates ü Based on (pseudo linear regression (do not rely on nonlinear optimization low computational complexity (comparable to the LS method Surprisingly IV methods appear to be under-appreciated 65 Two main references for IV methods See also the following recent papers T. Söderström, How accurate can instrumental variable models become? Chapter in Garnier & Wang (Eds, 202 P.C. Young, Refined instrumental variable estimation: ML optimization of a unified BJ model, Automatica,

34 Main idea: Instrumental Variable (IV method ü Recap: LSSVF estimates always biased because of the correlation between the regression vector ϕ f and the noise v f ü Main idea of IV: introduce a vector z f called instrument or instrumental variable which components are uncorrelated with v f E { z f v f } = 0 N N ( z f v f = 0 with v f = y n f ϕ T f θ k= ˆθ iv = sol θ N " ˆθ iv = $ # $ N N k= N ( n z f y f ϕ T f θ = 0 k= % z f ϕ T f ' & ' ( How should the instrument z f be chosen? " $ # $ N N k= % ( n z f y f ' & ' 67 Basic two step IV-based SVF estimator ü The instrument must be chosen so that it is: not correlated with the measurement noise E { z f v f } = 0 sufficiently correlated with the filtered regression vector ϕ T " f = L( p y ( n y u ( m u #$ % &' L( p = ü In the basic two-step IVSVF estimator, the instrument is built as z T f = L( p ˆx ( n ˆx u ( m u N z f ϕ T f k= ( p + λ n ˆx = G( p, ˆθ lssvf u is the estimated noise-free output calculated from an a priori LSSVF estimate ü The basic IV-based SVF estimator can then be computed from " ˆθ ivsvf = $ # $ N N k= % z f ϕ T f ' & ' " $ # $ N N k= % ( n z f y f ' & ' 68

35 Two-step IV-based SVF estimator - Summary u(t B( p A( p x(t First step N samples Second step N samples u v E( p = ( p + λ n y u E( p p i E( p p n E( p E( p p i E( p p n E( p y f y f (n u f u f (m LS algorithm ˆθ lssvf y f y (n f u f u (m f IV algorithm ˆθ ivsvf Bank of SVF filter u ˆB lssvf ( p  lssvf ( p Auxiliary model ˆx E( p p i E( p p n E( p ˆx f ˆx f ( n (tk 69 IVSVF estimator - Matlab implementation Noisy case B=2; % B(p=2 A=[ 4 3]; % A(p=p 2 +4p+3 True system u=prbs(4,00; % PRBS input from the CONTSID N=500; Ts=0.0; t=(0:n- *Ts; x=lsim(b,a,u,t; % simulation of the noise-free output y=x+0.2*randn(n,; % white Gaussian noise added data=iddata(y,u,ts;idplot(data; % First step LSSVF estimation lambda=3; % λ: SVF filter cut-off frequency den_l=[ 2*lambda lambda^2]; % denominator of the SVF filter num_l0=; % numerator of L0(p=/(p+λ 2 num_l=[ 0]; % numerator of L(p =p/(p+λ 2 num_l2=[ 0 0]; % numerator of L2(p =p 2 /(p+λ 2 yf0=lsim(num_l0,den_l,y,t; % Computation of the SVF filter bank outputs yf=lsim(num_l,den_l,y,t; yf2=lsim(num_l2,den_l,y,t; uf0=lsim(num_l0,den_l,u,t; Phi_N=[-yf -yf0 uf0]; % Regression matrix Y_N=yf2; % Output vector theta_lssvf=phi_n\y_n % LSSVF estimates theta_lssvf % see also the LSSVF routine in the CONTSID toolbox % Mlssvf=lssvf(data,[2 0],lambda 70

36 % Second step IVSVF estimation % Construction of the auxiliary model Blssvf=theta_lssvf(3 ; % Auxiliary model Alssvf=[ theta_lssvf(:2 ]; % Simulation of the auxiliary model output xest=lsim(blssvf,alssvf,u,t; % Computation of the SVF filter bank outputs for the auxiliary model xestf0=lsim(num_l0,den_l,xest,t; % filtered auxiliary model output xestf=lsim(num_l,den_l,xest,t; % st-order time-derivative of the filtered auxiliary model output % Construction of the IV matrix Z_N=[-xestf -xestf0 uf0]; % Instrumental variable matrix % IVSVF estimates IVSVF estimator - Matlab implementation Noisy case theta_ivsvf=(z_n *Phi_N\Z_N *Y_N; theta_ivsvf % IVSVF solution % see also the ivsvf routine in the CONTSID toolbox % Mivsvf=ivsvf(data,[2 0],lambda % The bias has clearly by reduced % Run several times your program to get a feel for the bias reduction (which can vary depending on the noise realization or even better run a Monte Carlo simulation 7 Frequency responses of the IVSVF models (recap of the Rao-Garnier benchmark results Magnitude (db The bias has clearly been removed L( p = ( p + λ 4 λ =3 rad / sec true Direct CT IVSVF models Phase (deg Frequency (rad/s 72

37 Basic IVSVF estimator Conclusions ü Some attractive properties simple analytical solution low computational complexity unbiased estimates in output measurement noise situations S M,G G o ü But IVSVF estimates quite sensitive to the choice of the SVF filter not minimum variance ü Motivation for studying more advanced IV methods We can still do better How to choose the instrument to get optimal estimates? 73 Extended Instrumental Variable ü To improve the basic IV estimate accuracy, some extensions are introduced operate a prefiltering by L(ρ on both I/O data ρ = q or ρ = p enlarge the instrument vector z such that n z n θ ü The so-called extended IV estimate is then given (Söderström & Stoica 983 N ˆθ xiv = arg min L( ρ z(t θ N k L( ρ ϕ T θ N L( ρ z(t k= N k L( ρ y k= ##### "##### $ ##### "##### $ R N L(ρ is a stable prefilter, Q a positive definite weighting matrix ü It is the weighted LS solution of an overdetermined system of linear equations ˆθ xiv = ( R T N QR N ( R T N Qr N This solution is then well suited for the consistency analysis of the IV estimators r N 2 Q x Q 2 = x T Q x 74

38 Optimal IV General results ü Data-generating system (ρ=p or ρ=q e y = B o ( ρ A o ( ρ u + H o ( ρ e y =ϕ T θ o +v ü IV estimates are consistent if u G o ( ρ x H o ( ρ y E { L( ρ z L( ρ v } = 0 { } is nonsingular E L( ρ z L( ρ ϕ T ü IV estimates are optimal if (Söderström and Stoica 983 Q = I n z = n θo L opt ( ρ = H o ( ρ A o ( ρ z opt = ϕ o : noise-free version of ϕ ü IV estimates are asymptotically Gaussian distributed dist ( ( ( L( ρ z T N ˆθ iv θ o N(0,P N iv P iv = σ 2 e E L( ρ z 75 Optimal IV General results ü Data-generating system (ρ=p or ρ=q e y = B o ( ρ A o ( ρ u + H o ( ρ e y =ϕ T θ o +v u G o ( ρ x H o ( ρ y ü Optimal accuracy if (Söderström & Stoica 983. See also Young 976. Optimal IV derives from the ML equations. See recent paper P.C. Young, Refined instrumental variable estimation: ML optimization of a unified BJ model, Automatica, 205 L opt ( ρ = H o ( ρ A o ( ρ z f opt = L opt ( ρ ϕ o ϕ o : noise-free version of ϕ ü Inherent filtering: a distinguishing feature of optimal IV Interesting for CT model identification, the filtering ensures minimum variance estimates provides a convenient way for generating the time-derivatives Ø can be automatically (and optimally chosen 76

39 Implementation of the optimal IV solution ü Usual dilemma met with accuracy optimization requires the knowledge of the true plant and noise models ϕ o : noise-free version of ϕ requires the knowledge of the noise-free output x z opt f =L opt ( ρ ϕ o L opt ( ρ = H o ( ρ A o ( ρ ü Two different main implementations have been suggested Multistep procedure (Söderström & Stoica 983 Example: IV4 (4 steps routine in the SID toolbox assumes a (rather peculiar ARARX model structure may be quite unreliable in practice (see later on Iterative (or refined procedure (Young 976, 984 Example: SRIVC routine in the CONTSID toolbox assumes a COE model structure is particularly reliable in practice 77 Iterative (refined implementation of optimal IV: SRIVC for COE models ü Data-generating system: a CT output error (COE model x(t = B o # ( p " A o ( p u(t # $ y = x + e u G o ( p x e y H o ( p = ü Optimal choice for the instrument and filter # z opt f =L opt ( pϕ o " L opt # ( p = $ A o ( p ü Requires the knowledge of the true plant model and noise-free output ü Solution (P.C. Young use of an iterative procedure where the instrument and prefilter are iteratively adapted until they converge on their optimal value i+ N = z f, ˆθ i ϕ T f, ˆθ i k= ˆθ srivc ϕ T o = x ( n x u ( m u N k= z f, ˆθ i ( n y f, ˆθ i 78

40 Optimal SRIVC method for COE models u B x o ( p A o ( p e y u N samples Any algo. ˆθ 0 y u Â( p p i Â( p p n Â( p Â( p p i Â( p p m Â( p y f y (n f u f u f (m N samples CT plant Model estimation by IV algorithm i ˆθ srivc u ˆB( p Â( p Auxiliary model ˆx Â( p p i Â( p p m Â( p ˆx f ˆx f ( n Convergence occurs in about 4 to 5 iterations 79 SRIVC parametric error covariance matrix estimate ü Good empirical estimates of the uncertainty in the SRIVC parameter estimates Provided by the parametric error covariance matrix estimate = E ˆθ Pˆθ srivc srivc θ o ( ( ˆθ srivc θ o T J J : Fischer Inf. Matrix ˆP srivc = σ 2 N ê z N N f, ˆθ srivc z T f, ˆθ srivc k= where ê = y ŷ, ˆθ srivc even for small sample size N can be used to in the procedure to select the best model structure (see YIC criterion later 80

41 Frequency responses of the estimated SRIVC models (recap of the Rao-Garnier benchmark results The SRIVC algorithm provides a quick and reliable approach to CT model identification 8 To sump up LSSVF IVSVF SRIVC Simple LSSVF: always biased 2-step IVSVF: unbiased but not minimum variance Iterative SRIVC: optimal (unbiased & minimum variance for COE models unbiased with low (but not minimum variance when the additive noise is colored The SRIVC algorithm provides a reliable and robust approach to CT model identification It is recommended for day-to-day use 82

42 Instrumental variable: take-home messages ü Include inherent (possibly optimal data prefiltering ü Conditions to obtain optimal IV estimates are well-established ü Provide consistent estimates even for an imperfect noise structure Choice of the instrument and prefilters influences the variance only, while the consistency properties are secured S M,G G o ü Implementation of the optimal IV solution Iterative algorithms: much more preferable than multistep algorithms ü Offer similar good performance as PEM methods in general ü Iterative IV implementations present one major advantage over PEM are much less sensitive to the initialization stage 83 Outline. Performance Evaluation of Discrete and Continuous-time Methods The Rao-Garnier benchmark 2. Direct Approaches for Linear Model Identification Traditional SVF-based estimators SRIVC: a reliable optimal instrumental variable estimator Practical side of continuous-time model identification 3. Software Aspects and Benefits in Practical Applications The CONTSID toolbox: a guided tour A selection of successful practical applications 4. Refined Instrumental Variables for More Advanced Situations Identification of Box-Jenkins models, multiple input, LTV systems, 84

43 The practical side of CT model identification Construct the experiment and collect data Data Data not OK Should the data be filtered? Choice of the model structure and estimation methods Model structure not OK Polish and present data Data Fit the model to the data Model Validate the model Green rectangles: computer s main responsability Blue ovals: practionner s main responsability Adapted from Ljung 999 no Can the model be accepted? yes 85 Model structure/order determination ü One of the most difficult stages of the system identification procedure ü Recommended procedure. Start with non-parametric estimates (impulse, step or frequency-responses Can provide useful information about the time-delay, system order and bandwidth 2. Begin with COE model structure x(t = G( pu(t τ y = x + e 3. Select polynomial degrees and time-delay according to the flexibility/parsimony trade-off m b i s i i=0 G(s = e n s i i=0 a i n k T s s 4. Try more complex model structures if necessary (Box-Jenkins model for example 86

44 Flexibility-parsimony trade-off Measure of model goodness the Mean Squared Error (MSE ( 2 V( ˆθ np,z N = N ε 2 (t N k = N y(t N k ŷ, ˆθ np k= k= MSE consists of a (squared bias contribution and a variance contribution A more flexible model => reduces the bias but increases the variance A more parsimonious model => reduces the variance but increases the bias Trade-off between flexibility and parsimony there is no ideal solution/method 87 Model order selection Practical considerations In practice, it is easy to identify a model for a range of possible model orders Select the best order for which a criterion J(n p,z N is minimal or does not decrease significantly anymore m best ˆn p = arg min J(n p,z N b n p ( i s i G(s = i=0 n e k T s s n s i J(n p,z N i=0 a i ˆn p best n p = n + m :number of parameters to be estimated 88

45 Traditional criteria for model order selection ü The criterion J(n p,z N is formulated from two functions one term measuring the model fit based on the loss function one term penalizing the model complexity J(n p,z N = logv( ˆθ np,z N + β(n p,z N β(n p,z N is a function which should increase with the model order but decrease to zero when N ü Basic approach: pick the model that minimizes AIC (Akaike s Information Criteria FPE (Final Prediction Error AIC(n p,z N = logv( ˆθ np,z N + 2n p N + n p FPE(n p,z N = N n V( ˆθ np,z N p N ü Both criteria tend to over-estimate the model order If too many parameters are used, the fit to the estimation data will be good but the fit to another validation data can be very poor 89 ü Data-generating system Statistical properties of SRIVC estimates y = G o ( pu + e or u G o ( p x e y y =ϕ T θ o +v S M ü When the system is in the model set, SRIVC delivers optimal estimates ü Expected value E( ˆθ srivc = θ o ü Covariance matrix estimates Pˆθ = E ˆθ srivc srivc θ o ( ( ˆθ srivc θ o T = σ 2 N ε z f, ˆθ srivc z T f, ˆθ srivc k= 90

46 Recommended SRIVC-based model order selection approach ü Computation of two criteria Coefficient of determination (measure of the model fit R T 2 = σ ε 2 σ y 2 YIC (Young s Information Criteria (parsimonious principle YIC = log σ 2 ε 2 σ + log n y p optimisation of R 2 T only: over-parametrization of the model optimisation of YIC only: under-parametrization of the model ü Recommended approach ε = y ŷ, ˆθ np n p j= σ 2 ε ˆp jj ˆθ j 2 ˆp jj : j thdiagonal element of ˆPˆθ ˆθ j 2 : square of the j th SRIVC parameter n p : number of parametersto be estimated The covariance matrix of the SRIVC parameter estimates is taken into account Select the model order that has the most negative YIC with the highest associated R 2 T σ 2 ε = N ε(t N k ε k= σ 2 y = N y(t N k y k= ( 2 ( 2 9 Example: Rao-Garnier benchmark Different model structures in the range [nb nf nk] = [ 3 0] to [2 5 0] have been computed for a data set 5 best models sorted according to YIC np nb nf nk RT2 YIC Niter FPE AIC The second model with [nb nf nk]= [2 4 0] seems to be quite clear cut It has the second most negative YIC= 8.9, with the highest associated R 2 T = 0.92 See srivcstruc and selcstruc in the CONTSID toolbox 92

47 Principles of model validation ü Try to verify that observations behave according to modeling assumptions ü Recommended tests. Compare simulated model output with estimation data Compute a model fit criterion: e.g. the coefficient of determination R T 2 2. Compare step or frequency responses of the identified model with measured responses if available 3. Interpret the main features of the identified model in a physical sense 4. Perform statistical tests on prediction errors Plot the autocorrelation of the residuals and the cross-correlation between the input and the residuals 5. Compare simulated model output with validation data 93 Compare simulated model output with estimation data Measured and simulated outputs Coefficient of determination R T 2 = measured output simulated output Time 94

48 Compare step or frequency responses of the identified model with measured responses Amplitude (db Frequency response estimate 5/6 Msrivc model response Frequency (Hz 95 Interpret the main features of the identified model in a physical sense 96

49 Compare simulated model output with validation data Measured and simulated outputs Coefficient of determination R T 2 = measured output simulated output Time 97 Statistical tests on prediction errors Correlation function of residuals. Output # lag Cross corr. function between input and residuals from output lag 98

50 Outline. Performance Evaluation of Discrete and Continuous-time Methods The Rao-Garnier benchmark 2. Direct Approaches for Linear Model Identification Traditional SVF-based estimators SRIVC: a reliable optimal instrumental variable estimator Practical side of continuous-time model identification 3. Software Aspects and Benefits in Practical Applications The CONTSID toolbox: a guided tour A selection of successful practical applications 4. Refined Instrumental Variables for More Advanced Situations Identification of Box-Jenkins models, multiple input, LTV systems, 99 Software aspects ü Several actively maintained toolboxes are available Comprehensive Mathworks SID toolbox (L. Ljung FDIDENT toolbox (I. Kollar, J. Schoukens UNIT toolbox (B. Ninness CAPTAIN toolbox (P. Young ü No software entirely dedicated to direct CT approaches first released in

51 CONTSID toolbox - Key features CONtinuous-Time System IDentification ü Supports direct CT identification approaches Basic linear black-box models Transfer function and state-space models regularly and irregularly sampled data Time-domain or frequency domain data More advanced black-box models errors-in-variables and closed-loop situations Nonlinear systems: block-structured, LPV or LTV models ü May be seen as an add-on to the Matlab SID toolbox Uses the same syntax, data and model objects M=srivc(data,nn ü P-coded version freely available from: 0 Main features of the latest version 7. ü Core of the routines mainly based on iterative optimal IV: SRIVC CONTSID includes also a few PEM and subspace-based methods ü SRIVC-based parameter estimation schemes for more advanced identification closed-loop identification: CLSRIVC simple process models: PROCSRIVC time-delay models: TDSRIVC TVP models: recursive RSRIVC LPV models: LPVSRIVC Hammerstein models: HSRIVC, ü Includes a flexible GUI and many demos to illustrate its use and the recent developments 02

52 CONTSID graphical user interface Allows the user to easily apply the iterative process of system identification 03 CONTSID toolbox Demonstration programs >>idcdemo 04

53 What have CT models to offer? ü CT models have certain advantages in relation to their equivalent DT models Are more intuitive to control engineers in their every-day practice Many practical control design are still based on CT models CT models are often preferred for fault detection reveal faults more directly than their DT counterparts Parameter values are independent of T s G o ( p = p 2 + p + ZOH T s = 0.s; G Ts (q = q q q q 2 T s =s; G Ts (q = q q q q 2 05 What have direct CT methods to offer? ü CT methods present many advantages in relation to their equivalent DT methods include inherent data prefiltering are well-suited to fast sampling situations are well-adapted to identify stiff systems can cope easily with irregularly sampled data H. Garnier, P.C. Young, The advantages of directly identifying continuous-time transfer function models in practical applications, IJC, 87(7, ,

54 Irregularly sampled data ü Appears in many situations: in biomedical or environmental systems, in transport and traffic systems, losses in data transmission manual measurements event-based sampling signals are sampled only when they pass certain limits 07 CT identification works with irregularly sampled data ü A constant T s is assumed for DT models y(t 3 y(t 4 y(t N = y(t 2 y(t u(t 2 u(t y(t 3 y(t 2 u(t 3 u(t 2 y(t N y(t N 2 u(t N u(t N 2 when T s varies, DT model parameters become time-varying Estimation still possible but much more difficult a a 2 + b b 2 e(t 3 e(t 4 e(t N ü CT models represent the system at every t k # # # # "# y f (2 (t y f (2 (t 2 y f (2 (t N $ & # & # & = # & # %& "# y f ( (t y f (t u f (t y ( f (t 2 y f (t 2 u f (t 2 y ( f (t N y f (t N u f (t N $ & &# &# &# " %& a a 2 b 0 $ # & # & + # & % # "# e f (t e f (t 2 e f (t N $ & & & & %& The time-instants do not need to be equidistantly spaced Appropriate ODE solver for the digital implementation of the CT filtering operations is only required 08

55 Example: tracer experiment identification from irregularly sampled data ü Data from a real-life tracer experiment in a wetland located in Florida Tracer experiment: traditional experiment in environmental science to model transport and dispersion of pollutants in rivers and wetlands Conservative potassium bromide (KBr: used as the tracer material Irregularly sampled data arise because the sampling is rapid to start with, so that the peak of the event is captured well the samples are then taken more slowly over the subsequent slow recession 09 Dispersion modelling of a pollutant in a wetland Irregular sampling of the data is clear, with T s varying from 2 to 40 h 0

56 SRIVC model identification from irregularly sampled data R T 2 = 0.99 M srivc 8 apple < 0.69 x(t = + 5.4p p u(t : y =x +e Ø SRIVC model explains the data very well Ø 99 % of the downstream concentration is explained by the SRIVC model output Identification of stiff systems LED temperature( C Current applied to the LED (A Time (seconds Time (seconds Stiff systems are challenging: combine both slow & fast dynamics ü makes the selection of T s difficult requires rapid sampling to capture the fast dynamics requires slow sampling to identify the slow dynamics these requirements are incompatible need for a compromise: normally, the selection of fast sampling 2

57 CT identification more easily identifies stiff systems In case of rapidly sampled data for stiff systems, fitting an accurate model is not straightforward both algorithmic and numerical aspects play a role ü DT methods poles of the estimated DT model will be very close to the unit circle possible numerical issues might affect the estimated DT model ü CT methods are particularly well-suited to fast sampling situations work well with rapidly sampled data from stiff systems ü Typical example of stiff systems: Light-emitting diodes (LED 3 Light-emitting diodes (LED are everywhere ü Semi-conductors: more and more used in applications as diverse as automotive lighting general lighting traffic signals, LED lamps have a lifespan and electrical efficiency that is several times better than incandescent lamps 4

58 Example: thermal stiff response of LEDs ü Typical example of stiff response thermal transient response of High-powered light-emitting-diodes (LEDs combine both slow and fast thermal phenomena ü Transient LED response can be used to detect possible thermal defects LED driven by a current step source and the thermal stiff response is measured Goal: to fit a model from the data that captures the stiff behavior 5 Thermal step response of the LED LED temperature( C LED temperature( C Fast sampling: f s =50 khz ; T s =0.02 ms Large number of data: N= Time (seconds Time (seconds 6

59 Step responses of the different estimated models G(s = 6 i= K i + τ i s LED temperature ( C Measured output CT SRIVC model output CT COE model output CT TFEST model output DT OE model output Time (seconds 7 Applications of CT model identification ü More examples listed in the special issue of Int. Journal of Control, July 204 Identification for a class of hybrid systems with application to power electronics Identification of aircraft structural modes from short-duration flight test Identification of the steering dynamics of a ship on a river Identification of a smoking cessation intervention Identification of blood glucose dynamics for type diabetes 8

60 Yesterday s message LSSVF IVSVF SRIVC Simple LSSVF: always biased 2-step IVSVF: unbiased but not minimum variance Iterative SRIVC: optimal (unbiased & minimum variance for COE models unbiased with low (but not minimum variance when the additive noise is colored The SRIVC algorithm provides a reliable and robust approach to CT model identification It is recommended for day-to-day use 9 Outline. Performance Evaluation of Discrete and Continuous-time Methods The Rao-Garnier benchmark 2. Direct Approaches for Linear Model Identification Traditional SVF-based estimators SRIVC: a reliable optimal instrumental variable estimator Practical side of continuous-time model identification 3. Software Aspects and Benefits in Practical Applications The CONTSID toolbox: a guided tour A selection of successful practical applications 4. Refined Instrumental Variables for More Advanced Situations Identification of Box-Jenkins models, multiple input, LTV systems, 20

61 SRIVC - Extensions to more advanced situations ü SRIVC: proven to be reliable in the day-to-day use. It has been extended to Box-Jenkins models Multiple input (MISO systems Fractional time-delay systems Simple process models Frequency domain Systems operating in closed loop Errors-in-Variables (EIV models Linear time-varying (LTV systems via recursive algorithms Linear parameter varying (LPV systems Block-structured nonlinear systems Fractional order systems Partial differential equations 2 SRIVC extension Colored noise situation ü In practice L Measurement noise is colored: does not have the nice white properties SRIVC still yields asymptotically unbiased estimates with low but not minimum variance prefilters are not designed to account for the color of the noise ü Solution J Model for the noise H( p e u G( p x v y The optimal filter depends on the noise model structure So, what model parametrization of the noise dynamics should be chosen? 22

62 Optimal IV for CT Box-Jenkins models ü Box-Jenkins (BJ models can represent a very large practical situations present the clear advantage that the dynamics from the input and from the noise are described with independent parameters ü (Fully continuous-time Box-Jenkins data-generating system e y = G o ( pu + H o ( pe y = B o ( p A o ( p u + C o ( p D o ( p e A o ( p y u B o ( p x C o ( p D o ( p 23 Optimal IV for CT Box-Jenkins models ü Fully CT Box-Jenkins data-generating system y = G o ( pu + H o ( pe G o ( p = B o ( p A o ( p ; H o ( p = C o ( p D o ( p ü Optimal IV parameter estimates e L opt ( p = H o ( pa o ( p = D o ( p C o ( pa o ( p z f opt = L opt ( p ϕ o u B o ( p A o ( p x C o ( p D o ( p y ˆθ opt iv = N N k= z opt f ϕ T f N N k= z opt ( n f y f

63 Optimal IV for hybrid CT BJ models ü Key idea: to avoid the difficult problem of treating CT random processes Assume that the disturbance noise can be written at the sampling instances as filtered discrete-time white noise (Young 80, Johansson 94, Pintelon et al ü Hybrid CT BJ modeling: CT model for the process + DT model for the noise y = G o ( pu + H o (q e y = B o ( p A o ( p u + C o (q D o (q e L opt ( ρ = f d ( ρ f c ( ρ = z opt = ϕ o H o ( ρ A o ( ρ ρ = q or ρ = p Optimal choice for the instrument and filter requires the knowledge of the true CT plant and DT noise models ü Practical implementation use of an iterative procedure (RIVC that refines an initial estimate of the parameter vector P.C. Young, H. Garnier, M. Gilson, Refined IV identification of hybrid CT Box-Jenkins models. In Identification of CT models from sampled data, H. Garnier & L. Wang (Eds., Springer-Verlag, Optimal RIVC method for CT hybrid BJ models - Summary H o ( ρ e N samples N samples N samples u u G o ( ρ x y y u Any algo. ˆθ 0 u ˆB( p Â( p Auxiliary model y u ˆx Â( ρ Ĥ( ρ p i Â( ρ Ĥ( ρ p n Â( ρ Ĥ( ρ Â( ρ Ĥ( ρ p i Â( ρ Ĥ( ρ p m Â( ρ Ĥ( ρ Â( ρ Ĥ( ρ p i Â( ρ Ĥ( ρ p m Â( ρ Ĥ( ρ y f y (n f u f u f (m ˆx f ˆx f ( n CT Plant model estimation by IV algorithm i ˆθ rivc DT Noise model estimation by any ARMA estimation algorithm Ĥ i 26

64 Robustness of the SRIVC/RIVC methods ü Recap on one of the objectives when constructing an estimator: strive to get robust estimators ü MCS analysis, again with the Rao-Garnier benchmark, but in colored noise situations to test the SRIVC/RIVC robustness to the noise model ü Simulation setup T s =50 ms, u(t: PRBS Noise: ARMA DT noise, SNR=0dB H o (q e y = G o ( pu + H o (q e v G o ( p H o (q = C o (q D o (q = +0.98q u(t y q q 2 k x RIVC with the correct orders RIVC with long AR (n d =23 SRIVC and COE (H(q= S M S M,G G o S M,G G o 27 MCS for the Rao-Garnier benchmark ARMA noise case S M,G G o S M,G G o S M S M,G G o SRIVC/RIVC: good robustness to violation of the assumption about the noise model structure 28

65 Multistep implementation of optimal IV: IV4 for ARARX model (SID toolbox ü Data-generating system: ARARX model rather unrealistic A o (q y = B o (q u + D o (q e y = ϕ T θ + D o (q e ü Optimal choice for the instrument and filter z f opt =L opt ( pϕ o L opt (q = D o (q No prefiltering by /A o (q L opt (q = ü Multistep implementation (Söderström & Stoica 983. See also Ljung 999, p step implementation of the optimal IV in IV4 (SID toolbox H o (q A o (q ; zopt = ϕ o ϕ T o = x x na u u nb - Initialization. D(q=. Estimate θ 0 of the resulting ARX model by simple LS 2 - Set L(q=D(q=. Generate the instrument and estimate θ by IV 3 - Compute the residuals and estimate an AR(na+nb noise model H(q=/D(q by simple LS 4 Set L(q=D(q. Prefilter both instrument and data by L(q and estimate θ 2 by IV 29 Frequency responses of the DT IV4 models (recap of the Rao-Garnier benchmark results IV4 is not very robust and has quite poor performance in general 30

66 Iterative implementation of optimal IV: SRIV for DT OE models ü Data-generating system: a DT output error (OE model x = B o (q u(t A o (q k y = x + e ü Optimal choice for the instrument and filter z opt f =L opt (q ϕ o L opt (q = A o (q u G o (q ü Requires the knowledge of the true plant model and noise-free output ü Iterative implementation (P.C. Young use of an iterative procedure where the instrument and prefilter are iteratively adapted i+ N = z f, ˆθ i ϕ T f, ˆθ i k= ˆθ sriv N k= x e z f, ˆθ i y f, ˆθ i y H o ( q = ϕ T o = x x na u u nb 3 Iterative SRIV method of the CONTSID toolbox for DT OE models - Summary u B o (q u A o (q x e y y u N samples ARX model estima tion by LS ˆθ 0 y u q Â(q q i Â(q q na Â(q q Â(q q i Â(q q nb Â(q y f - y f -na u f - u f -nb N samples ARX model estimation by IV algorithm i ˆθ sriv ˆD(q q ˆx f u ˆB(q Â(q Auxiliary model ˆx ˆD(q q i ˆD(q q na ˆx f na 32

67 Frequency responses of the DT SRIV models Rao-Garnier benchmark results DT SRIV provides overall good results, only outlier estimate, larger uncertainty in high frequency band than CT SRIVC models (due to artificial zeros introduced by the appropriate ZOH assumption in the CT to DT transformation 33 SRIVC extension Multiple input (MISO systems ü In practice L Many systems include more than one input ü Solution J SRIVC for multiple input linear TF models with common denominators x(t = B ( p A( p u (t ++ B n ( p u A( p u n (t u y = x + e SRIVC for multiple input linear TF models with different denominators x(t = B ( p A ( p u (t ++ B n ( p u A nu ( p u n (t u y = x + e H. Garnier, M. Gilson, P.C. Young, E. Huselstein. An optimal IV technique for identifying continuous-time transfer function model of multiple input systems. Control Engineering Practice,

68 SRIVC for MISO TF models with common denominators ü Data-generating MISO system: x(t = B o, ( p A o ( p u (t ++ B o,n ( p u A o ( p u n (t u y = x + e u (t B o, ( p x (t u nu (t A o ( p B o,nu ( p A o ( p x nu (t e(t y(t y ( n = ϕ T o θ + e ϕ T o = x ( n ( m x u ( m (tk u u u (tk u nu nu ü The model is still linear-in-the-parameters J The iterative SRIVC procedure similar to the SISO case where the instrument and prefilter are iteratively adapted can be used " i+ N % = $ z f, ˆθ i ϕ T f, ˆθ i ' # $ k= & ' ˆθ srivc z f, ˆθ i y f ( n, ˆθ i # " # $ z opt f =L opt ( pϕ o L opt ( p = A o ( p 35 SRIVC for MISO TF models with common denominators ü Example: 2 inputs - output case y = B ( p A( p u + B 2 ( p A( p u 2 + e A( py = B ( pu + B 2 ( pu 2 + e y ( n = ϕ T θ + e ϕ T = y ( n ( m y u ( m (tk u u 2 2 (tk u 2 ü SISO version of SRIVC can be modified in a straightforward way to handle MISO TF models with common denominators J ü Application of the common denominator assumption is rather limited in practice L 36

69 SRIVC for MISO TF models with different denominators ü Data-generating MISO system: x(t = B o, ( p A o, ( p u (t ++ B o,n ( p u A o,nu ( p u n (t u y = x + e u (t B o, ( p x (t u nu (t A o, ( p B o,nu ( p A o,nu ( p x nu (t e(t y(t ü The model is no longer linear-in-the-parameters L ü Proposed SRIVC-based approach The MISO model is converted into n u SISO models, e.g. for i= y ˆB 2 ( p ˆB  2 ( p u 2 nu ( p  nu ( p u n (t u k = B ( p A ( p u + e ######## "######## $ ξ ξ = B ( p A ( p u + e θ is partitioned into classes θ,, θ nu such that the error is affine with respect to the parameters of any of these classes when the parameters of all others are fixed It is then possible to search for θ i by applying successively the SISO version of the SRIVC algorithm to estimate the parameters of each class in turn, with a cyclic exploration of all classes 37 SRIVC for MISO TF models with different denominators ü Example: 2 inputs - output case y(t = B ( p A ( p u (t + B 2 ( p A 2 ( p u 2 (t + e(t convert the MISO model into 2 SISO models use the SISO SRIVC method to estimate in a bootstrap way the parameters of each transfer function ( i ˆθ 2 y(t ˆB 2 ( p  2 ( p u 2 (t ### "### $ ξ (t = B ( p A ( p u (t + e(t ξ (t = B ( p A ( p u (t + e(t cyclic exploration y(t ˆB ( p  ( p u (t ### "### $ ξ 2 (t = B 2 ( p A 2 ( p u 2 (t + e(t ξ 2 (t = B 2 ( p A 2 ( p u 2 (t + e(t Use of the SISO version of SRIVC ˆθ ( i Use of the SISO version of SRIVC 38

70 Properties and implementation aspects ü Properties provided that the inputs are independent of the noise, the SRIVC-based parameter estimates of the multiple input TF models: Asymptotically unbiased (even if the noise assumptions are incorrect ü Implementation aspects initial estimates are crucial estimates for a particular transfer function i can be obtained using SISO SRIVC modeling of the output y against each input u i in turn Some issues are still remaining for MISO TF model identification See for example: V. Pascu, H. Garnier, L. Ljung, A. Janot, Developments towards formalizing a benchmark for MISO continuous-time model identification. CDC, Las Vegas, Numerical example ü Simulated system : 2 inputs - output 0.5p + x(t = u p 2 (t p + y = x + e 3p + 2 ( p + p + 3 ( u 2 (t ü Simulation conditions Input signals: 2 uncorrelated multisines of 25 frequencies T s : 50 ms Monte Carlo simulation of 200 runs e : white Gaussian SNR=0 db TF model orders assumed known u (t B o, ( p x (t u 2 (t A o, ( p B o,2 ( p A o,2 ( p x 2 (t ü Tested methods SRIVC-based method for MISO TF models with common denominators SRIVC-based method for MISO TF models with different denominators e(t y(t 40

71 MCS results SRIVC for MISO models with common denominators y(t = B ( p A( p u (t + B 2 ( p A( p u 2 (t + e(t Fails, as expected, to give good estimates 4 MCS results SRIVC for MISO models with different denominators y(t = B ( p A ( p u (t + B 2 ( p A 2 ( p u 2 (t + e(t Delivers much better parameter estimates 42

72 Winding process modeling ü is a prototype of an industrial 3 inputs / 3 outputs winding process 43 Cross-validation results for the winding process SRIVC for MISO models with different denominators 44

73 SRIVC extension Fractional time-delay ü In industrial practice L Many systems include a pure time delay G(s = B(s A(s e τs Furthermore, the delay can be a non-integer number of T s Difficult to be handled by DT model identification (assumes τ=n k T s ü Solution J SRIVC recently coupled to a gradient-based search for the time-delay TF parameters and fractional time-delay are simultaneously estimated in a cyclic way " $ # $ % B( p x(t = A( p u(t τ y = x + e F. Chen, H. Garnier, M. Gilson, Robust identification of continuous-time models with arbitrary time-delay, Journal of Process Control, 25, 9-27, SRIVC extension Simple process models ü In industrial practice L identification for control: often reduced to the determination of simple process models of low order with input delay such as G(s = K e T d s +Ts G(s = K ( +T z s e T d s ( +T p s ( +T p2 s ü Solutions J Many ad hoc methods exist SRIVC naturally adapted to identify simple process models from step responses and similar F. Chen, H. Garnier, M. Gilson, Robust identification of CT models with arbitrary time-delay, Journal of Process Control, 25,

74 SRIVC extension Frequency domain ü CT model identification can also be done in the frequency domain Similar iterative IV procedure can be used N ( 2 ˆθ = arg min G(ω k,θ Ĝ(ω k θ k= N " ˆθ = arg min G(ω k,θ Y(ω k % θ $ # U(ω k ' & Ø SRIVC: naturally extended to work with frequency-domain data k= 2 M. Gilson, J. Welsh, H. Garnier, Frequency-domain IV based method for wide band system identification. ACC SRIVC extension Closed-loop identification ü In industrial practice L Many systems have feedback that cannot be interrupted for an identification experiment y c + - ε C r G v y " $ y(t = G( pu(t +v(t # %$ u(t = r(t +C( p y c (t y(t ( To get reliable estimates, IV methods have to be adapted: input is contaminated by noise ü Solutions J Optimal SRIVC/RIVC method for closed-loop identification M. Gilson, H. Garnier, P.C. Young, P. Van den Hof, IV methods for continuous-time closed-loop model identification, in H. Garnier & L. Wang (Eds., Springer, 2008 Two stage-based SRIVC approach P.C. Young, H. Garnier, M. Gilson. Simple refined IV methods of closed-loop system identification, SYSID

75 Extension Errors-in-Variables (EIV identification ü In practice L The measured input signal can also be contaminated by a stochastic noise Errors-in-variables (EIV model u(t G y(t v T s T s w u m y m To get reliable estimates, direct CT methods have to be adapted ü Solutions J Second-order statistic SVF-based approaches K. Mahata, H. Garnier, Identification of continuous-time errors-in-variables models, Automatica, March 2006 Higher-order statistic SVF-based approaches S. Thil, H. Garnier, M. Gilson, Third-order cumulants based method for continuous time errors in variables model identification, Automatica, Sept SRIVC extension Linear time-varying (LTV systems ü In practice L The nature of the system can be time-varying A time-invariant model built on initial operating data would obviously be unable to capture the changing system characteristics ü First solution J Use LTI model with time-varying parameters Ø Recursive SRIVC version for tracking the varying parameters A. Padilla, H. Garnier, P.C. Young, J. Yuz, Real-time identification of continuous-time linear time-varying systems. Submitted to CDC, Las Vegas,

76 SRIVC extension Linear parameter varying (LPV systems ü In industrial practice L The nature of the system can be time-varying ü Second solution J Use a LPV model to parametrize the time-varying nature of the parameters (" n % * $ a i ( ρ $ tk pi ' ' x(t = # i= & * + * y = x +v " $ $ # m i=0 ( pi b i ρ tk % ' ' u(t & ρ: scheduling variable p=d/dt Ø SRIVC/RIVC methods for LPV input/output models V. Laurain, M. Gilson, R. Toth, H. Garnier, Refined Instrumental Variable Methods for Identification of LPV Box- Jenkins Models, Automatica, June 200. V. Laurain, R. Toth, M. Gilson, H. Garnier, Direct identification of continuous-time linear parameter-varying input/ output models, IET Control Theory & Applications, May 20 5 SRIVC extension Block-structured system identification ü Further models of nonlinear systems SRIVC/RIVC methods for block-structured models SRIVC/RIVC for Hammerstein model identification SRIVC/RIVC for Hammerstein-Wiener model identification e H u u y f(. G f(. G g(. H e y Hammerstein system Hammerstein-Wiener system V. Laurain, M. Gilson, H. Garnier, Refined Instrumental Variable Methods for Hammerstein Box-Jenkins Models. In System Identification, Environmental Modelling, and Control System Design, L. Wang and H. Garnier (Eds., Springer, pp , 202. B. Ni, M. Gilson, H. Garnier, A Refined Instrumental Variable Method for Hammerstein-Wiener Continuous-time Model Identification, IET Control Theory & Applications, 7(9, ,

77 SRIVC extension Fractional systems ü In practice L Many systems obey to diffusive phenomena diffusion of charges in acid batteries, thermal diffusion, stress in viscoelastic materials, ü Solution J Use a general fractional order or simpler commensurate order model to parametrize the diffusive nature of the system n a i p α m i x(t = b i p β i u(t i=0 i=0 y = x +v α i, β i : fractional orders n a i p iγ m x(t = b i p iγ u(t i=0 i=0 y = x +v γ: single commensurate order Ø SRIVC method for fractional order transfer function model identification S. Victor, R. Malti, H. Garnier, A. Oustaloup, Parameter and differentiation order estimation in fractional models, Automatica, 49(4, April SRIVC extension PDE models ü In practice L Many engineering and environmental systems involve spatio-temporal behavior heat transmission/exchange in mechanical structures transport/dispersion of contaminants in rivers and the atmosphere can be described partial differential equation (PDE models ü Solution J PDE model identification from I/O data sampled in space x and time t z t ( x,t a 2 z ( x,t = b t 2 ou( x,t y( x l,t k = z( x l,t k + e( x l,t k Ø SRIVC method for PDE model identification J. Schorsch, H. Garnier, M. Gilson, P.C. Young, Instrumental variable methods for identifying partial differential equation models, IJC,

78 Final take-home messages ü For more than 40 years, the general mainstream identification approach has been to identify DT models from sampled data ü To obtain satisfying results, DT model identification requires the active participation of an experienced practitioner to pre-process the data (decimation or prefiltering accentuates the feeling that system identification is more an art than a science ü We argue that direct CT model identification has been under-appreciated although it includes many advantages: well adapted to irregularly and rapidly sampling situations requires less participation from the user (inherent pre-filtering makes the application of the SYSID procedure much easier 55 Final take-home messages ü Amongst the different CT methods, one is particularly recommended SRIVC: iterative instrumental Variable (IV method robust to assumptions and algorithmic aspects extended to handle wider practical applications: colored noise, closed loop, frequency-domain, MISO, TVP, LPV systems, ü Direct CT methods now available as user-friendly algorithms in: CONTSID toolbox Matlab SID toolbox, It is hoped that these toolboxes will lead to a better appreciation of direct continuous-time system identification in the general systems and control community 56

79 To know more ü ü See for this lecture, in particular, H. Garnier, Direct continuous-time approaches to system identification. Overview and benefits for practical applications, European Journal of Control, 24, 50 62, July 205 Chapters, 4 and 9 in H. Garnier and L. Wang (Eds., Identification of continuous-time models from sampled data, Springer-Verlag, 2008 But also H. Garnier, P.C. Young (Guest editors, Special issue on «Applications of Continuous-time Model Identification and Estimation», International Journal of Control, July 204 H. Garnier, T. Söderström and J. Yuz (Guest editors, Special issue on «Continuous-time model identification», IET Control Theory & Applications, May 20 57