Lecture 8 - Model Identification

1 EE392m - Winter 2003 control Engineering8-1 Lecture 8 - Model Identification What is system Identification ? Direct pulse response Identification Linear regression Regularization Parametric Model ID, nonlinear LSEE392m - Winter 2003 control Engineering8-2 What is System Identification ? White-box Identification estimate parameters of a physical Model from data Example: aircraft flight Model Gray-box Identification 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 marketDataIdentificationModelExperimentP lantRarely used inreal-life controlEE392m - Winter 2003 control Engineering8-3 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.

2 Short cycle timeEE392m - Winter 2003 control Engineering8-4 Impulse response Identification Simplest approach: apply control impulse and collect thedata Difficult to apply a short impulse big enough such that theresponse is much larger than the IMPULSE RESPONSETIME Can be used for building simplifiedcontrol design models from complex simsEE392m - Winter 2003 control Engineering8-5 Step response Identification Step (bump) control input and collect the data used in process control Impulse estimate still noisy: impulse(t) = step(t)-step(t-1) RESPONSE OF PAPER WEIGHTTIME (S E C ) RESPONSE OF PAPER WEIGHTTIME (S E C )Actuator bumped EE392m - Winter 2003 control Engineering8-6 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.

3 Dead time, risetime, gain Do both in a sequence done in real process control ID packages Pre-filter dataEE392m - Winter 2003 control Engineering8-7 Linear regression Mathematical aside linear regression is one of the main System ID tools)()()(1tettyjNjj+= = DataRegression weightsRegressorErrorey+ = = = = =)()1(,,)()()1()1(,)()1(111 NeeeNNNyyyKKKMMKMOMKM EE392m - Winter 2003 control Engineering8-8 Linear regression Makes sense only when matrix istall, N > K, more data available thanthe number of unknown parameters. Statistical averaging Least square solution: ||e||2 min Matlab pinvor left matrix division \ Correlation interpretation:()yTT = 1 ey+ = = = ======NtKNtNtKNtKNtKNttyttytNcttttttNR11 112111112)()()()(1,)()()()()()(11 MKMOMKcR1 = EE392m - Winter 2003 control Engineering8-9 Example: linear first-order Model Linear regression representation)()1()1()(tetgutayty+ + = = = =gatuttyt )1()()1()(21 This approach is considered in most of the technicalliterature on Identification Matlab Identification Toolbox Industrial use in aerospace mostly Not really used much in industrial process control Main issue.

4 Small error in a might mean large change in responseLennart Ljung, System Identification : Theory for the User, 2nd Ed, 1999()yTT = 1 EE392m - Winter 2003 control Engineering8-10 Regularization Linear regression, where is ill-conditioned Instead of ||e||2 min solve a regularized problemr is a small regularization parameter Regularized solution Cut off the singular values of that are smaller than rey+ = Tmin22 + re()yrITT + = 1 EE392m - Winter 2003 control Engineering8-11 Regularization Analysis through SVD (singular value decomposition) Regularized solution Cut off the singular values of that are smaller than rnjjmmnnTsSRURVUSV1,,}diag{.

5 == = ()yUrssVyrITnjjjTT += + == 121diag +ssEE392m - Winter 2003 control Engineering8-12 Linear regression for FIR Model Linear regression representation)()()()(1tektukhtyKk+ = = = = =)()1()()()1()(1 KhhKtuttutKMM ()yrITT + = 1 Identifying impulse response byapplying multiple steps PRBS excitation signal FIR (impulse response) EXCITATION SIGNALPRBS =Pseudo-Random Binary Sequence,see IDINPUT in MatlabEE392m - Winter 2003 control Engineering8-13 Example: FIR Model ID PRBS excitationinput Simulated systemoutput: 4000samples, randomnoise of theamplitude RESPONSETIMEEE392m - Winter 2003 control Engineering8-14 Example: FIR Model ID Linear regressionestimate of the FIRmodel Impulse responsefor the simulatedsystem:T=tf([1.)]

6 5],[1 1]);P=c2d(T, ); e s tima teImpulse ResponseTime (s e c ) - Winter 2003 control Engineering8-15 Nonlinear parametric Model ID Prediction Model depending onthe unknown parameter vector Loss index Iterative numerical of V as a subroutinesimModel including theparametersOptimizer Loss Index V)(tu =2)|( )( tytyV =2)|( )( tytyJ)|( ) Model ()( tytu )(ty Lennart Ljung, Identification for control : Simple Process Models, IEEE Conf. on Decision and control , Las Vegas, NV, 2002EE392m - Winter 2003 control Engineering8-16 Parametric ID of step response First order process with deadtime Most common industrial process Model Response to a control step applied at tBExample.

7 () > += DBDBTttTttTttegtyDBfor ,0for ,1)|(/)( PapermachineprocessDT g =DTg EE392m - Winter 2003 control Engineering8-17 Gain estimation For given , the modeled step response can bepresented in the form This is a linear regression Parameter estimate and prediction for given),|()|(1 DTtygty += ==21)()|(kkktwty DT, ()yTwTTD = 1),( ),|( ),|( 1 DDTtygTty +=1)(),|()(21121====tTtytwgwD DT, EE392m - Winter 2003 control Engineering8-18 Rise time/dead time estimation For given , the loss index is Grid and find the minimum ofDT, = =NtDTtytyV12),|( )( ),(DTVV =DT, EE392m - Winter 2003 control Engineering8-19 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 each01020304050607080-0.

8 4-0 . parameters: Gain = ; Tdel = ; Trise = in s e c .; MD re spons e - s olid; e s tima te d re sponse - dashedLinear Regression IDof the first-ordermodelNonlinear Regression IDNonlinear Regression IDEE392m - Winter 2003 control Engineering8-20 Linear filtering L is a linear filtering operator, usually LPF A trick that helps: pre-filter data Consider data modeleuhy+=*{{)(**)()*()*(LuhuLhuhLLeuhL Lyffey==+= 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 2003 control Engineering8-21 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.}}