Lecure - odel Idenificaion Wha is ssem idenificaion? Direc impulse response idenificaion Linear regression Regularizaion Parameric model ID nonlinear LS Conrol Engineering -
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 -
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
Impulse response idenificaion Simples approach: appl conrol impulse and collec he IPULSE RESPOSE daa 3 4 5 6 7 IE Difficul o appl a shor impulse big enough such ha he response is much larger han he noise.8.6.4. OISY IPULSE RESPOSE.5 -.5 3 4 5 6 7 IE FIR modeling can be used for building simplified conrol design models from complex sims Conrol Engineering -4
Sep response idenificaion Sep bump conrol inpu and collec he daa used in process conrol.5 SEP RESPOSE OF PAPER WEIGH Acuaor bumped.5 4 6 8 IE SEC Impulse esimae: impulse sep-sep- Sill nois.3.. IPULSE RESPOSE OF PAPER WEIGH 3 4 5 6 IE SEC Conrol Engineering -5
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
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 4 6 8 IE SEC Φ + e Φ e e e Soluion assuming uncorrelaed noise: Φ Φ Φ Conrol Engineering -7
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
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
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
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 -
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 -
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
Linear regression for FIR model Idenifing impulse response b.5 appling muliple seps PRBS exciaion signal -.5 - FIR impulse response model 3 4 5 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
Example: FIR model ID PRBS exciaion inpu.5 -.5 PRBS exciaion Simulaed ssem oupu: 4 samples random noise of he ampliude.5-4 6 8.5 -.5 - SYSE RESPOSE 4 6 8 IE Conrol Engineering -5
Example: FIR model ID Linear regression esimae of he FIR model Impulse response for he simulaed ssem:..5..5 -.5 3 4 5 6 7..5..5 FIR esimae Impulse Response -.5 3 4 5 6 7 ime sec H f[.5][. ]; P cdh.5; Conrol Engineering -6
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
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
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
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 -
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 9.8969.8.6 onlinear Regression ID.4..4. -. Linear Regression ID of he firs-order model.8.6.4. onlinear Regression ID -.4 3 4 5 6 7 8 -. 3 4 5 6 7 8 ime in sec.; D response - solid; esimaed response - dashed Conrol Engineering -
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
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