Lecture 8 - Model Identification

Lecture 8 - Model Identification What is system identification?Direct pulse response identificationLinear regressionRegularizationParametric model ID, nonlinear LSEE392m - Winter 2003Control Engineering8-1

What is System del White-box identification– estimate parameters of a physical model from data– Example: aircraft flight model Gray-box identificationRarely used inreal-life control– given generic model structure estimate parameters from data– Example: neural network model of an engine Black-box identification– determine model structure and estimate parameters from data– Example: security pricing models for stock marketEE392m - Winter 2003Control Engineering8-2

Industrial Use of System ID Process control - most developed ID approaches– all plants and processes are different– need to do identification, cannot spend too much time on each– industrial identification tools Aerospace– white-box identification, specially designed programs of tests Automotive– white-box, significant effort on model development and calibration Disk drives– used to do thorough identification, shorter cycle time Embedded systems– simplified models, short cycle timeEE392m - Winter 2003Control Engineering8-3

Impulse response identification Simplest approach: apply control impulse and collect thedataIMP ULS E RES P ONS E10.80.60.40.2001234567TIME Difficult to apply a short impulse big enough such that theresponse is much larger than the noiseNOIS Y IMP ULS E RES P ONS E10.50-0.501 Can be used for building simplifiedcontrol design models from complex simsEE392m - Winter 2003Control Engineering23456TIME8-47

Step response identification Step (bump) control input and collect the data– used in process controlS TEP RES P ONS E OF P AP ER WEIGHT1.51Actuator bumped0.500200400600TIME (S EC)8001000 Impulse estimate still noisy: impulse(t) step(t)-step(t-1)IMP ULS E RES P ONS E OF P AP ER WEIGHT0.30.20.100EE392m - Winter 2003100Control Engineering200300400TIME (S EC)5006008-5

Noise reductionNoise can be reduced by statistical averaging: Collect data for mutiple steps and do more averaging toestimate the step/pulse response Use a parametric model of the system and estimate a fewmodel parameters describing the response: dead time, risetime, gain Do both in a sequence– done in real process control ID packages Pre-filter dataEE392m - Winter 2003Control Engineering8-6

Linear regression Mathematical aside– linear regression is one of the main System ID toolsDataRegression weightsRegressorNy (t ) θ jϕ j (t ) e(t )Errory Φθ ej 1 y (1) ϕ1 (1) K ϕ K (1) θ1 e(1) OM , θ M , e M y M , Φ M y ( N ) ϕ1 ( N ) K ϕ K ( N ) θ K e( N ) EE392m - Winter 2003Control Engineering8-7

Linear regression Makes sense only when matrix Φ istall, N K, more data available thanthe number of unknown parameters.– Statistical averaging Least square solution: e 2 miny Φθ eθˆ (Φ T Φ ) Φ T y 1– Matlab pinv or left matrix division \ Correlation interpretation:N N 2 K ϕ K (t )ϕ1 (t ) ϕ 1 (t )t 11 t 1 ,MOMR NN N2 ϕ1 (t )ϕ K (t ) K ϕ K (t ) t 1 t 1EE392m - Winter 2003Control Engineeringθˆ R 1c N ϕ1 ( t ) y ( t ) 1 t 1 Mc N N ϕ K (t ) y (t ) t 1 8-8

Example: linear first-order modely (t ) ay (t 1) gu (t 1) e(t ) Linear regression representationϕ1 (t ) y (t 1) a θ ϕ 2 (t ) u(t 1) g 1 TTˆθ (Φ Φ ) Φ y This approach is considered in most of the technicalliterature on identificationLennart Ljung, System Identification: Theory for the User, 2nd Ed, 1999 Matlab Identification Toolbox– Industrial use in aerospace mostly– Not really used much in industrial process control Main issue:– small error in a might mean large change in responseEE392m - Winter 2003Control Engineering8-9

Regularization Linear regression, where Φ T Φ is ill-conditioned Instead of e 2 min solve a regularized problem22e r θ miny Φθ er is a small regularization parameter Regularized solution 1 TTˆθ (Φ Φ rI ) Φ y Cut off the singular values of Φ that are smaller than rEE392m - Winter 2003Control Engineering8-10

Regularization Analysis through SVD (singular value decomposition)Φ USV T ;V R n ,n ;U R m ,m ; S diag{s j }nj 1 Regularized solutionn s 1 θˆ (Φ T Φ rI ) Φ T y V diag 2 j U T y s j r j 1 Cut off the singular values of Φ that are smaller than rREGULARIZED INVERS E21.5ss 2 0.110.50EE392m - Winter 2003012Control Engineering3s458-11

Linear regression for FIR modelP RBS EXCITATION S IGNAL Identifying impulse response by 10.5applying multiple steps0-0.5 PRBS excitation signal-1 FIR (impulse response) model 0Ky ( t ) h ( k )u ( t k ) e( t )k 110203040PRBS Pseudo-Random Binary Sequence,see IDINPUT in Matlab Linear regression representationϕ1 (t ) u(t 1)Mϕ K (t ) u (t K )EE392m - Winter 2003 h (1) θ M h ( K ) Control Engineering 1 TTˆθ (Φ Φ rI ) Φ y8-1250

Example: FIR model IDP RBS e xcita tion PRBS S YS TEM RES P ONS E Simulated system 1output: 40000.5samples, random 0-0.5noise of the-1amplitude 0.50200400600TIMEEE392m - Winter 2003Control Engineering8-13

Example: FIR model IDFIR es tima te Linear regression 0.2estimate of the FIR0.150.1model0.050-0.0501234567567Impuls e Re s pons e Impulse responsefor the simulatedsystem:0.20.150.10.050-0.05T tf([1 .5],[1 1.1 1]);0P c2d(T,0.25);EE392m - Winter 20031234Time (s ec)Control Engineering8-14

Nonlinear parametric model ID Prediction model depending onthe unknown parameter vector θu (t ) MODEL(θ ) yˆ (t θ )Loss Index VV y (t ) yˆ (t θ ) Loss indexJ y (t ) yˆ (t θ )θOptimizer22 Iterative numerical optimization.Computation of V as a subroutiney (t )u (t )Model including theparameters θsimLennart Ljung, “Identification for Control: Simple Process Models,”IEEE Conf. on Decision and Control, Las Vegas, NV, 2002EE392m - Winter 2003Control Engineering8-15

Parametric ID of step responseτ First order process with deadtime Most common industrial process model Response to a control step applied at tB g (1 e( t tB TD ) /τ ), for t tB TDy (t θ ) γ 0,for t tB TD γ g θ τ TD gTDExample:PapermachineprocessEE392m - Winter 2003Control Engineering8-16

Gain estimation For given τ , TD , the modeled step response can bepresented in the formy (t θ ) γ g y1 (t τ , TD ) This is a linear regression2w1 gk 1w2 γy (t θ ) wkϕ k (t )ϕ1 (t ) y1 (t τ , TD )ϕ 2 (t ) 1 Parameter estimate and prediction for given τ , TDwˆ (τ , TD ) (Φ Φ ) Φ T yTEE392m - Winter 2003 1yˆ (t τ , TD ) γˆ gˆ y1 (t τ , TD )Control Engineering8-17

Rise time/dead time estimation For given τ , TD , the loss index isNV y (t ) yˆ (t τ , TD )2t 1 Grid τ , TD and find the minimum of V V (τ , TD )EE392m - Winter 2003Control Engineering8-18

Examples: Step response ID Identification results for real industrial process data This algorithm works in an industrial tool used in 500 industrial plants, many processes eachP roce s s pa ra me te rs : Ga in 0.134; Tde l 0.00; Tris e 119.896911.61.4NonlinearRegression ID0.80.61.210.40.8LinearRegression IDof the first-ordermodel0.20-0.2-0.4010203040EE392m - Winter 20035060700.6NonlinearRegression ID0.40.2080-0.20100200300400500600700time in s e c.; MD re s pons e - s olid; e s tima te d re s pons e - da s he dControl Engineering8-19800

Linear filtering A trick that helps: pre-filter data Consider data modely h *u e L is a linear filtering operator, usually LPFLy L( h * u ) {Le{eyffL( h * u ) ( Lh ) * u h * ( Lu ) Can estimate h from filtered y and filtered u Or can estimate filtered h from filtered y and ‘raw’ u Pre-filter bandwidth will limit the estimation bandwidthEE392m - Winter 2003Control Engineering8-20

Multivariable ID Apply SISO ID to various input/output pairs Need n tests - excite each input in turn Step/pulse response identification is a key part of theindustrial Multivariable Predictive Control packages.EE392m - Winter 2003Control Engineering8-21

Industrial Use of System ID Process control - most developed ID approaches – all plants and processes are different – need to do identification, cannot spend too much time on each – industrial identification tools Aerospace – white-box identification, specially designed programs of tests Automotive