2.160 System Identification, Estimation, and Learning Lecture Notes No. 17 April 24, 2006

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1 .6 System Idetfcato, Estmato, ad Learg Lectre Notes No. 7 Aprl 4, 6. Iformatve Expermets. Persstece of Exctato Iformatve data sets are closely related to Persstece of Exctato, a mportat cocept sed adaptve ad learg cotrols. See the block dagram of a drect adaptve cotrol system below. he cotrol system motors pt-otpt data order to detfy the plat model real, ad modfes the feedback cotrol as the plat dyamcs vary; hece the cotrol system s adaptve to varyg plat dyamcs. Adaptato Law Model + - Feedback Cotrol ( Plat y ( (Idrec Adaptve Cotrol Sccess of ths adaptve cotrol system hges o the data. he cetral qesto s whether the pt-otpt data obtaed real-tme are formatve eogh to detfy the plat model qely. hs s ofte qestoable, sce the cotrol system teds to drve the plat to a specfc set pot or to follow a specfc trajectory. he trajectory may ot be rch eogh to excte the system. he followg theory of persstet exctato ad formatve expermet are fdametal to these qestos. Defto 4 A qas-statoary sgal{ (}, wth spectrm Φ (ω ), s sad to be persstetly exctg of order, f the codto: mples where Φ ( ω) M () e M () e M ( s a arbtrary lear flter of form: M = mq + mq + + mq (3)

2 Remarks:. Note ( ω) e M Φ s the power spectrm of v ( = M ( t ). herefore, a sgal ( that s persstetly exctg of order caot be fltered to zero by ay (-)st order movg average flter (3), hece t s called persstetly exctg. ( M ( v (. Cosder fcto M ( z) M ( z ), assocated wth M. e M ( zm ) ( z ) = ( mz + mz + + mz )( mz+ mz + + mz) = ( m + m z+ + m z )( m + m z+ + m z ) If a+b s a zero of + M ( z) M ( z ) (4), a-b s also a zero, sce the fcto has all a b real coeffcets. Also, f a+b s a zero, the ts recprocal s also a zero a + b of the fcto sce the fcto s symmetrc wth respect to the t crcle, z z. See the fgre below. hs fcto ca have at most (-) zeros o the t crcle. herefore, M ( e ) = M ( e ) M ( e ) may be zero for at most (-) dfferet freqeces. I coseqece, f Φ ( ω) for at least dfferet freqeces; π < ω,, ω < π the ( s persstetly exctg. a + b a b a + b he followg lemma provdes a sefl method for checkg persstece of exctato: Lemma: Let ( be a qas-statoary sgal. Cosder the matrx gve by he ( R () R () R ( ) R R R R () () ( ) = R R ( ) R () s persstetly exctg of order f ad oly f R s o-sglar. (5)

3 Proof Pt the coeffcets of M ( to a -dmesoal vector: ( m m m ) R m =. Cosder a qadratc form: ( R s o-sglar ) ( m R m = m R m. It s kow that the followg two are eqvalet mples m = ) Compte m R m = ( ) mrm m m = ( m m ) R() R() m R () R () m R() m mr () + mr () + + mr mr () mr () + + mr ( ) + = mm R ( j ) = mm R ( j) j j j j j π M e ( ω) d π [ ] = E mt ( ) m t j = E M( qt ) j j = Φ π ω = mm je () t t ( j) = mm je ( t ) t j (6) herefore, m R m = M e ω ω ω meas, for almost all Φ. Matrx R s o- sglar, s eqvalet to M ( e ) Φ ( ω) M = herefore, ( s persstetly exctg of order f ad oly f R s o-sglar.. Codtos for Iformatve Expermets Based o the persstetly exctg codto, how ca we desg expermets so that ay two models of a model set ca be dstgshed,.e. formatve expermets? Cosder two models: =, of a model set M, 3

4 G = G( q, ), H = H ( q, ), ε ε ( t, ) ( = (7) ad ther dfferece G = G( G, H = H ( H( (8) Note ε ( = y yˆ t = y ( ) [ H G + ( H ) y] = H y H G Let s compte the dfferece of predcto error betwee the two models: ε = ε ε = H y H G ε [ ε ] = H y G H = H G G G+ y Hε G H ε = + H [ Gqt Hq ε ] ( q Usg the tre system model: y = G ( + H e () ( t (9) ε ca be wrtte as t ε () t = H y H G [ ] = H G + H e G = ( G G ) ( + H e( H () Combg (9) ad () yelds, G G H ε() t = G H () t H () H + + e t H H () Aq Bq Sppose that the expermet s carred ot ope-loop, so that ( ad e ( are correlated. he mea of ε ( s gve by (See Lectre Note No.5. eq.(9)) 4

5 π E ε () t = A ( e ) Φ ( ω) + B ( e ) λ π π H (3) where λ = E[ e ( ] ad ( e ) dω G e G e Ae ω ω = Ge + H e H( e ) (4) B e ( e ) H = H e H( e ) (5) herefore, E ( ε t ε t ) E ( yˆ t yˆ t ) [ ] = [ ] = mples Ae ω ad Φ ( ω) : both mst be detcally zero. (6) Be ω ( e ) Φ ( ω) G (7) If (7) mples G ( e ) formatve eogh w.r.t. He ( ω ) Eq. (5), the the two models are eqal, ad the expermet s M. hs last codto s bascally eqvalet to the persstetly exctg codto gve by Defto 3 ad Lemma. heorem 3 Cosderg a cocrete model for G ( q, ) leads to the followg theorem. Cosder a model set { (, ), (, ) M } M G q H q D M of SISO systems: = (8) where G( q, ) s a ratoal fcto: ( b ) ( b + b q + + b q ) k B( q, ) q G( q, ) = = (9) F( q, ) + f q + + f b f q f 5

6 ad H ( q, ) s versely stable. he a ope-loop expermet wth a pt that s persstetly exctg of order b + f Proof For two dfferet models, G ad G ; herefore (7) becomes Relatg BF BF q s formatve eogh w.r.t. M. B B B F B F G( = = () F F F F B e F e B e F e Φ( ω) () M ( e ω) to ( e ) k M e ω (5), we fd that - Factorg ot does ot chage B F B F b + f - he remag part s a polyomal of order. Sce the pt( s persstetly exctg of order +, () mples BF BF,.e. G QED. he pot s: (he order of persstet exctato) ( he mber of parameters to be estmated) b f.3 Sgal-to-Nose Rato ad Covergece Speed From heorem we kow that, f a model set cldes the tre system, ad the data set s formatve eogh wth respect to the model set, the estmated model coverges to the tre system,.e. cosstet. Frthermore, from heorem 3 we kow that, as log as the pt seqece has the order of persstet exctato greater tha the mber of parameters volved the model set, the model coverges to the tre model. hs covergece s garateed regardless the magtde of ose. However, the covergece speed may deped o the ose magtde or, more specfcally, the sgal-to-ose rato. he followg s to exame the covergece characterstcs. Usg the tre system dyamcs gve by (), the predcto error of a model M ( ) = { G, H; D M } s gve by ε ( t, ) = H = H [ y( G ( ] = H [( G G ) ( + H e( ] [( G G ) ( + ( H H ) e ( ] + e ( () 6

7 Note that the secod term the last expresso, ( ) H H e() t, does ot cota e () t bt s a fcto of e (, becase both H ad H are moc. herefore, the three terms are correlated to each other. As a reslt, the power spectrm of ε(, t ) s gve by Φ G G H H ε ( ω, ) = Φ ( ω) + λ + λ (3) H H where Φ ( ω) s the power spectrm of the pt, ad λ s the varace of the Whte ose. Applyg eq.(9) Lectre Note No 5 to the above power spectrm Φ ε ( ω, ), we obta the followg reslt. heorem 4 Let ε(, t ) be the predcto error of a model a lear tme-varat model set M ( ) = { G, H; D M }. Assme that the tre system gve by y ( = G ( + H( e ( ad the model process are qas-statoary processes, the the optmal parameter that mmzes the mea sqared predcto error V( ) = E[ ε( t, ) ] s gve by π H( e ) H ( e ) Φ ω π H( e ) H( e ) ˆ = arg m { G ( e ) G ( e ) + λ } dω (4) DM where Φ ( ω) s the power spectrm of the pt (, ad λ s the varace of the Whte ose e () t. he proof s obvos. he covergece process of the above system s complcated. If we assme that the ose model H s kow or fxed: H q = H q (5) the (4) redces to ˆ Φ ( ω) = arg m G( e ) G( e, ) dω (6) H ( e ) he followg observato ca be made for ths smplfed expresso. Remarks: 7

8 he model G s pshed towards the tre system Go( sch a way that the weghted mea sqared dfferece the freqecy doma be mmzed. Φ ( ω) he weght,, s the rato of the pt power spectrm to the ose H ( e ) power spectrm (f the varace of e o ( s ty). I other words, t s a sgal-toose rato. At freqeces where the sgal-to-ose rato s hgher, the model coverges to the tre system more rapdly. See the fgre below. G ( e ) Good agreemet for hgh S/N weght Model Poor agreemet as the S/N becomes small Hgh freqecy ose Ipt magtde re system 8