Lecture 10 - Model Identification

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1 Lecure - odel Idenificaion Wha is ssem idenificaion? Direc impulse response idenificaion Linear regression Regularizaion Parameric model ID nonlinear LS Conrol Engineering -

2 Wha is Ssem Idenificaion? Experimen Plan Daa Ssem Idenificaion odel Rarel used in real-life conrol Whie-box idenificaion esimae parameers of a phsical model from daa Example: aircraf fligh model Gra-box idenificaion given generic model srucure esimae parameers from daa Example: neural nework model of an engine Black-box idenificaion deermine model srucure and esimae parameers from daa Example: securi pricing models for sock marke Conrol Engineering -

3 Indusrial Use of Ssem ID Process conrol - mos developed ID approaches all plans and processes are differen need o do idenificaion canno spend oo much ime on each indusrial idenificaion ools Aerospace whie-box idenificaion speciall designed programs of ess Auomoive whie-box significan effor on model developmen and calibraion Disk drives used o do horough idenificaion shorer ccle ime Embedded ssems simplified models shor ccle ime Conrol Engineering -3

4 Impulse response idenificaion Simples approach: appl conrol impulse and collec he IPULSE RESPOSE daa IE Difficul o appl a shor impulse big enough such ha he response is much larger han he noise OISY IPULSE RESPOSE IE FIR modeling can be used for building simplified conrol design models from complex sims Conrol Engineering -4

5 Sep response idenificaion Sep bump conrol inpu and collec he daa used in process conrol.5 SEP RESPOSE OF PAPER WEIGH Acuaor bumped IE SEC Impulse esimae: impulse sep-sep- Sill nois.3.. IPULSE RESPOSE OF PAPER WEIGH IE SEC Conrol Engineering -5

6 oise reducion oise can be reduced b saisical averaging: Collec daa for muliple sep inpus and perform more averaging o esimae he sep/pulse response Use a parameric model of he ssem and esimae a few model parameers describing he response: dead ime rise ime gain Do boh in a sequence done in real process conrol ID packages Pre-filer daa Conrol Engineering -6

7 Linear Regression - univariae Simple fiing problem: Given model sep response And process sep response Find he gain facor + e.5.5 SEP RESPOSE OF PAPER WEIGH IE SEC Φ + e Φ e e e Soluion assuming uncorrelaed noise: Φ Φ Φ Conrol Engineering -7

8 Conrol Engineering -8 Linear Regression Linear regression is one of he main Ssem ID ools e j j j + Daa Regression weighs Regressor Error of he fi + e Φ Φ e e e O

9 Linear regression - leas squares akes sense onl when marix Φ is all > more daa available han he number of unknown parameers. Saisical averaging Leas square soluion: e min ˆ L L Φ Φ min Φ Φ Φ Φ Φ Can be compued using alab pinv or lef marix division \ Conrol Engineering -9

10 Conrol Engineering - Linear regression - leas squares Correlaion inerpreaion of he leas squares soluion Φ Φ Φ ˆ c R O R c ˆ Informaion marix Correlaion vecor Φ Φ R c Φ

11 Example: Firs-order ARA model a + gu + e Linear regression represenaion a u g ˆ Φ Φ Φ his pe of approach is considered in mos of he echnical lieraure on idenificaion alab Idenificaion oolbox Limied indusrial use Fundamenal issue: + + e Lennar Ljung Ssem Idenificaion: heor for he User nd Ed 999 Small error in a migh mean large change in he ssem response Conrol Engineering -

12 Regularizaion Linear regression where Φ Φ is ill-condiioned Insead of e min solve a regularized problem e + r min Φ + e where r is a small regularizaion parameer A..ikhonov 963 see hp://solon.cma.univie.ac.a/~neum/ms/reguorial.pdf Regularized soluion ˆ ri Φ Φ + Φ Cu off he singular values of Φ ha are smaller han r Conrol Engineering -

13 Regularizaion Analsis hrough SVD singular value decomposiion R U R ; S diag{ s Φ USV U U VV I Regularized soluion s j ri V ˆ Φ Φ + Φ diag U s j r + j Cu off he singular values of Φ ha are smaller han r V ; j} j Inverse singular values /s Regularized inverse s values s +. IVERSE SIGULAR VALUE Conrol Engineering -3 s

14 Linear regression for FIR model Idenifing impulse response b.5 appling muliple seps PRBS exciaion signal FIR impulse response model h k u Linear regression represenaion k u u Φ u u k Regularized LS soluion: u + e u ˆ O u ri Φ Φ + Φ PRBS EXCIAIO SIGAL PRBS Pseudo-Random Binar Sequence See IDIPU in alab u h + h Conrol Engineering -4

15 Example: FIR model ID PRBS exciaion inpu PRBS exciaion Simulaed ssem oupu: 4 samples random noise of he ampliude SYSE RESPOSE IE Conrol Engineering -5

16 Example: FIR model ID Linear regression esimae of he FIR model Impulse response for he simulaed ssem: FIR esimae Impulse Response ime sec H f[.5][. ]; P cdh.5; Conrol Engineering -6

17 onlinear parameric model ID Predicion model depending on he unknown parameer vecor u ODEL ˆ Opimizer Loss Index L onlinear regression: loss index L ˆ min Ieraive numerical opimizaion. Compuaion of L as a subrouine L ˆ u odel including he parameers sim Lennar Ljung Idenificaion for Conrol: Simple Process odels IEEE Conf. on Decision and Conrol Las Vegas V Conrol Engineering -7

18 Parameric SsID of sep response Firs order process wih deadime os common indusrial process model Response o a conrol sep applied a B τ g γ + g e B D / τ for for > B B D D D γ g τ D Example: Paper machine process Conrol Engineering -8

19 Conrol Engineering -9 Sep: Gain and Offse Esimaion wo-sep approach: linear regression + nonlinear regression For given he modeled sep response can be presened in he form his is a linear regression Parameer esimae and predicion for given ; D D g τ γ τ + ; k k k D τ τ D D Φ Φ Φ ˆ ˆ τ ˆ ˆ ˆ D D g τ γ τ + g D τ γ τ D

20 Sep : Rise ime & Dead ime Esimaion For an given τ D he loss index is L ˆ τ D Grid τ D and find he minimum of L L τ D Conrol Engineering -

21 Examples: Sep Response ID Idenificaion resuls for real indusrial process daa his algorihm works in an indusrial ool used in 5+ indusrial plans man processes each.6 Process parameers: Gain.34; del.; rise onlinear Regression ID Linear Regression ID of he firs-order model onlinear Regression ID ime in sec.; D response - solid; esimaed response - dashed Conrol Engineering -

22 Linear Filering in SsID A rick ha helps: pre-filer daa Consider daa model h * u + e u Plan SsID ĥ F is a linear filering operaor usuall LPF { F F h * u + Fe { f F h * u e Fh * u f h * Fu Plan Can esimae h from filered and filered u Or can esimae filered h from filered and raw u Pre-filer bandwidh limis he esimaion bandwidh u Conrol Engineering - F SsID F ĥ

23 ulivariable Idenificaion Sep/impulse response idenificaion is a ke par of he indusrial mulivariable odel Predicive Conrol packages Appl SISO ID o various inpu/oupu pairs eed n ess: excie each inpu in urn and collec all oupus a ha Conrol Engineering -3