Lecture Note #7 (Chap.11)

Transcription

1 Sstem Modelng and Identfcaton Lecte ote #7 (Chap. CBE 7 Koea nvest of. Dae Roo Yang Recsve estmaton of a constant Consde the followng nos obsevaton of a constant paamete φθ + v, Ev { }, Evv { j} σδj ( φ, he least-saes estmate s fond as the sample aveage θ (/ Recsve fom θ θ θ+ (/ ( θ ( θ Vaance estmate of the least-saes estmate p ( ote that p as σ φφ E θ θ θ θ σ p + σ σ + p p p p {( ( } Chap. Real-tme Identfcaton Real-tme dentfcaton Spevson and tacng of tme vang paametes fo Adaptve contol, flteng, pedcton Sgnal pocessng Detecton, dagnoss atfcal neal netwos, etc. Identfcaton methods based on set of measements ae not stable Onl few data needed to be stoed Dawbacs Rees a po nowledge on model stcte Iteatve soltons based on lage data sets ma be dffclt to oganze Devaton of ecsve least-saes dentfcaton Consde as sal the egesso φ and the obsevaton Φ φ φ Y [ [ he least-saes cteon based on samples s V( θ (/( Y Φθ ( Y Φ θ (/ θ ( θ ( he odna least-saes estmate Φ Φ Φ θ ( Y ( φφ ( φ Intodce the matx ( ( Φ Φ φφ + φφ ( θ ( φ φ ( θ φ θ φ ( φθ Altenatve fom (avodng nveson of matces ( Φ Φ ( Φ Φ + φφ ( + φφ φ ( I + φ φ φ cf ( A BC A A BI ( CA B CA ( αi, wth α + + Matx nveson lemma

2 Recsve Least-Saes (RLS Identfcaton he ecsve least-saes (RLS dentfcaton algothm θ θ + φ θ : the paamete estmate φθ : the pedcton eo φφ, gven : the paamete covaance estmate except + φ φ σ Some popetes of RLS estmaton aamete accac and convegence Q( θ (/( θ θ ( θ θ (/ θ θ ( Q( θ Q( θ θ θ θ θ θ θ + φ φφ θ ( θ + θ φ + φ φ ( θ φ + + ( + φ φ ( θ φ + + φ φ + φ φφ φ + φ φ φ φ φ φ opetes of matx s a postve defnte and smmetc matx. ( > as he matx s asmptotcall popotonal to the paamete estmate covaance povded that a coect model stcte has been sed. It s often called the covaance matx. Compason between RLS and offlne LS dentfcatons If ntal vale and θ can be chosen to be compatble wth the eslts of the odna least-saes method, the eslt obtaned fom RLS s same as that of offlne least-saes dentfcaton. hs, calclate the ntal vales fo RLS fom the odna LS method sng some bloc of ntal data. nde the lnea model assmpton, φθ + v so that θ φ + v Q( θ Q( θ v φ φ + If v fo all, Q deceases n each ecson step If Q tends to zeo, t mples that θ tends to zeo as the seence of weghtng matx s an nceasng seence of postve defnte matx whee fo all >. heoem. he eos of estmated paametes and the pedcton eo fo leastsaes estmaton have a bond detemned b the nose magntde accodng to V( θ + Q( θ ( θ θ ( + θ θ vv It mples θ vv ( θ θ ( θ ΦΦ θ he paamete convegence can be obtaned fo a statona stochastc pocess {v } f ΦΦ >ci p p. (c s a constant hs, poo convegence s obtaned n cases of lage dstbance and a an-defcent ΦΦ matx. Modfcaton fo tme-vang paametes he RLS gves eal weghtng to old data and new data. If the paametes ae tme-vang, pa less attenton to old data. Fogettng facto (λ J( θ λ ( ( φθ < λ Modfed RLS θ θ + φ φθ ( φφ, gven λ λ + φ φ Dsadvantages he nose senstvt becomes moe pomnent as λ deceases. matx ma ncease as gows f the npt s sch that the magntde of - φ s small (-matx exploson o covaance matx exploson

3 Choce of fogettng facto ade-off between the eed ablt to tac a tme-vang paamete (.e., a small vale of λ and the nose senstvt allowed A vale of λ close to : less senstve to dstbance bt slow tacng of apd vaatons n paametes Defalt choce:.97 λ.995 Rogh estmate of data ponts n memo (tme constant: /( λ Example.3: Choce of fogettng facto φθ+ v, Ev Evv { }, { j} σδj ( φ, θ θ + φ φθ φφ (, λ λ+ φ φ θ λ.99,.98,.95 Kalman flte ntepetaton Assme that the tme-vang sstem paamete θ ma be descbed b the state-space eaton θ+ θ + v, Ev { }, Evv { j} Rδj,, j φθ + e, Ee { }, Eee { j} Rδj,, j Kalman flte fo estmaton of θ θ θ + K K φ /( R + φ φ φθ φφ + R R+ φ φ Dffeences fom RLS θ θ + φ φθ φφ (, gven λ λ + φ φ he dnamc of changes fom exponental gowth to lnea gowth ate fo φ de to R. of the Kalman flte does not appoach zeo as fo a nonzeo seence {φ }. (RLS Faste tacng Delta model he dsadvantage of z-tansfomaton s that the z-tansfomaton paametes do not convege to the Laplace tansfomaton contnos paametes, fom whch the wee deved, as samplng peod deceases. Ve small samplng peods eld the ve small nmbes fom the tansfe fncton nmeato. he poles of tansfe fncton appoach the nstable doman as the samplng peod deceases. hese dsadvantages can be avoded b ntodcng a moe stable dscete model. δ-opeato: δ ( z/ h x+ Φ x +Γ δ x Φ x +Γ (/ h( Φ I x + (/ h Γ Cx Cx hs fomlaton maes the state-space ealzaton and the coespondng sstem dentfcaton less eo-pone de to favoable nmecal scalng popetes of the Φ and Γ matces as compaed to the odna z- tansfom based algeba. Othe foms of RLS algothm Basc veson: - omalzed gan veson θ θ + K [ φθ K φ λ + φ φ ( φφ λ λ + φ φ φ Mltvaable case: R and R γ R θ θ + γ R φ [ φθ R R + γ ( φφ R γ λ + γ θ λ φθ φθ agmn j [ Λ [ θ j+ θ θ + K [ φθ ( K φ λ Λ + φ φ ( φ( λλ + φ φ φ λ Λ Λ + γ ( Λ f λ λ, j, λ λ j j j+ Otpt eo covaance (edcton gan pdate (aamete eo covaance pdate (Otpt eo covaance pdate

4 Recsve Instmental Vaable (RIV Method he odna IV solton RIV θ θ θ + K K z /( + φ z z Z ZY z φ z φθ z φ + φ z ( ( ( Φ Standad choce of nstmental vaable: z ( x xn A nb he vaable x ma be, fo nstance, the estmated otpt. RIV has some stablt poblem assocated wth the choce of IV and the pdatng of matx. Stochastc gadent methods A faml of REM Also, called stochastc appoxmaton o least mean sae (LMS ses steepest descent method to pdate the paametes ψ / θ ( φθ / θ φ fo lnea model φθ he algothm: (me-vang egesso-dependent gan veson θ θ + γφ γ φθ Qφ / ( Q Q > + φ Q φ Rapd comptaton as thee s no matx to evalate Good detecton of tme-vang paametes Slow convegence and nose senstvt Modfcaton fo tme vang paametes λ φq φ λ Keepng the facto at a lowe magntde REM Recsve edcton Eo Methods (REM Consde a weghted adatc pedcton eo cteon J( θ γ λ ( ( φθ γλ J ( θ γ λ ψ ( ψ / θ [( / J ( ( θ γ [ ψ J ( θ γ λ γ θ ψ J + Geneal REM seach algothm J ( θ J ( θ + γ ψ( θ ( θ J( θ ( θ: optmal θ θ R J ( θ θ + γ R ψ R R + γ ( φφ R REM fo mltvaable case θ θ + γ R ψ Λ ( θ ( R R + γ ψ Λ ψ R Λ Λ + γ ( Λ ojecton of paametes nto paamete doman D M θ θ + K [ φθ θ f θ DM θ θ f θ D M

5 Recsve sedolnea Regesson (RLR Recsve psedolnea egesson (RLR Also, called Recsve ML estmaton, Extended LS method he egesson model: φθ + v θ ( a an b b A n c c B nc he ecsve algothm θ θ + K K φ /( + φ φ φθ + φ φ φφ he egesson vecto: θ ( na nb nc he algothm ma be modfed to teate fo the best possble. Estmate of v REM to geneal npt-otpt models n a Sstem: B ( C ( A ( + a + + an a A ( + e nb F ( D ( B ( b + + bn b edcto nf F ( + f + + fn f D ( A ( D ( B ( nc ( θ + C ( + c + + cn c C ( C ( F ( nd D ( + d + + dn d Eo defntons ( ( ( ( ( D ( B D θ θ A v C ( F ( C ( B ( w( θ v ( θ A ( w( θ F ( aamete vecto θ a an b a bn f b fn c f cn d c d nd Regesso ϕ θ w w v v ( na nb nf nc nd Applcaton to Models REM to state-space nnovaton model edcto Algothm x ( ( ( ( ( + θ F θ x θ + G θ + K θ v H( θ x( θ Λ Λ + γ Λ R R + γ ψ Λ ψ R θ θ + γ R ψ Λ x Fx + G + K + H x + + W ( F KH W + M KD ψ + W H + D ( θ whee F F( θ G G( θ H H( θ K K( θ d ψ( θ ( θ dθ d W( θ x( θ dθ Innovaton v ( θ D( θ H( θ x( θ θ M F x G K θ [ ( θ + ( θ + ( θ θ,,, x Eo calclatons w ( θ b + + b fw ( θ f w ( θ nb nb nf nf v ( θ + a + + a w ( θ n a n a ( θ v ( θ + dv ( θ + d v ( θ c ( θ c ( θ nd nd nc nc Expesson fo pedcton eo ( θ v ( θ + dv ( θ + d v ( θ c ( θ c ( θ nd nd nc nc + a + + a b b + fw ( θ + + f w ( θ θ ϕ( θ Gadent expessons na na nb nb nf nf c ( θ c ( θ + dv ( θ + + d v ( θ nc nc nd nd ( θ C ( F ( ( θ F( C ( D ( A ( + D B ( ( ( θ ψ ( θ θ ( θ D ( a C ( ( θ D ( b C ( F ( ( θ ( D ( ψ θ w ( θ f C ( F ( ( θ ( θ c C ( ( θ v ( θ d C (

6 Algothm ϕ R R + γ ψψ R θ θ + γ R ψ w b + + b fw f w ψ n nb n n b f f v + a + + a w na na w w + na+ nb+ nf + v v nc+ nd+ v + dv + + dn v d n c d cn c nc θ ϕ d + + d c c nd nd nc nc + d + + d g g nd nd n n g g w w + dw + + d w gw g w nd nd n n g g c cn c nc v v cv c v nc nc w w + na+ nb+ nf + v v nc+ nd+ g C F coeffcents of ( ( Extended Kalman Flte Algothm Gven x, θ, (stat fom K [ FH + R [ HH + R X FX + G( θ + K [ H X + FF + R K [ HH + R K + o avod the calclaton of lage matces, patton the matces se of latest avalable measements to pdate the paamete estmates K [ FH + R [ HH + R X X + K [ H X X FX + G( θ + FF + R K [ HH + R K + df( X, dh( X, F( X H( X dx dx X X X X Sbspace Methods fo Estmatng State-Space Models Kalman flte fo nonlnea state-space model Sstem: x+ F ( x, θ + G( θ + w ( Eww { j} δjr H ( x, θ + e ( Eee { } δ R, Ewe { } δ R j j j j Wth extended state vecto: x X X F( X + G( θ + + w θ H( X + e whee F ( x, θ G w F( X G w H( X H ( x, θ θ Lneazaton df( X, dh( X, F H dx dx X X X X Estmaton of sstem matces, A, B, C, and D offlne n x+ Ax + B + w x R, R p Cx + D + v R Assmng mnmal ealzaton If the estmates of A and C ae nown, estmates of B and D can be obtaned sng lnea least-saes method. CI ( A B + D + v o CI ( A xδ + CI ( A B + D + v he estmates of B and D wll convege to te vales f A and C ae C CA O CA exactl nown o at least consstent. If the (extended obsevablt matx (O s nown, then A and C can be estmated. (>n m

7 Fo lnea tansfomaton, x x O O Fo now sstem ode (n * n G O p n C O p n * ( (:,: Fo nnown sstem ode (n * >n O ( p+ : p,: n O (: p (,: na * G O ( p n G SV (SV D atton the matces dependng on the sngla vales and neglect the poton fo smalle sngla vales G SV G SV O OV S O he estmate of the obsevablt matx can be O S o O, etc. O R ( R s nvetble Obtan estmates of A and C fom the estmated obsevablt matx os estmate of the extended obsevablt matx G SV SV + (othe tems O + E If O explans the sstem well, the E stems fom nose. If E s small, the estmate s consstent Defne vectos D CB D + + Y S CA B CA B CB D hen Y Ox + S + V Intodce Y [ Y Y Y X [ x x x hen [ V [ V x V Y O X+ S + V o emove -tem, se Π I ( YΠ XΠ + VΠ snce Π ( O Choose a matx Φ/Ν so that the effect of nose vanshes G YΠ Φ O XΠ Φ + VΠ Φ O + V lmv lm VΠ Φ lm lm XΠ Φ ( has fl an n (pojecton othogonal to sng weghtng matces n the SVD Fo flexblt, peteatment befoe SVD can be appled G WGW SV SV hen the estmate of extended obsevablt matx becomes W R O When the nose s pesent, W has mpotant nflence on space spanned b and hence on the alt of the estmate of A and C. Estmatng the extended obsevabltmatx he basc expesson Cx + D + v CAx + CB + Cw + D + v CA x + CA B + CA B + + CB + D CA w + CA w + + Cw + v Fndng good nstment s s s Let F [ ϕ ϕ ϕ s s VΠ Φ V ϕ V Fom the law of lage nmbes s s lm VΠ { ( } { ( } { ( } Φ EV ϕ EV R E ϕ whee R E ( ( ( ( ϕ { ( } (If V and ae ndependent hs, choose ϕ s so that the ae ncoelated wth V. pcal choce s s s ϕ s

8 Fndng the states and Estmatng the nose statstcs -step ahead pedcto Y Θ ϕ +Γ + E Y Θ F +Γ + E s Least-saes estmate of paametes [ ΦΦ Φ ( Θ Γ YΦ Y Θ YΠ Φ ΦΠ Φ Φ edcted otpt Y ( Y Y YΠ Φ ΦΠ Φ Φ SVD and deletng small sngla vales Y SV OR SV O X Altenatvel X LY x x whee L R [ ose chaactestcs can be calclated fom w x Ax B v Cx D + X R SV O R (, ( X R SV R Y onlnea Sstem Identfcaton Geneal nonlnea sstems xt ( f ( xt ( + gxt ( (, t ( + vt ( t ( hxt ( (, t ( Dscete-tme nonlnea models Hammesten models Bz ( F ( Az ( Wene models Bz ( F( Az ( x f ( x, + + v hx (, + w Sbspace dentfcaton algothm. Fom the npt-otpt data, fom G YΠ Φ. Select weghtng matx W and W and pefom SVD G WGW SV SV MOES: W I, W ((/ ΦΠ Φ ΦΠ 4SID: W I, W ((/ ΦΠ Φ Φ / / IVM: W ((/ YΠ Y, W ((/ ΦΦ CVA: / / W ((/ YΠ Y, W ((/ ΦΠ Φ 3. Select a fll an matx R and defne O W R. hen solve fo C and A. (pcal choces ae RI, RS, o RS / C O (: p,: n O( p+ : p,: n O(: p (,: na 4. Estmate B and D and x fom the lnea egesson poblem agmn CI ( A B D CI ( A xδ BDx,, 5. If a nose model s soght, calclate X and estmate the nose contbtons. Wene Models onlnea aspects ae appoxmated b Lagee and Hemte sees expansons Laee opeatos: sτ a z a L ( s L ( z + sτ + sτ az az Hemte polnomal: ( x d x H( x ( e e dx Dnamcs s appoxmated b Lagee flte and the statc nonlneat s appoxmated b the Hemte polnomal. x L( z n n n n c H ( x H ( x H ( x c H ( x H ( x H ( x n n n n

9 Voltea-Wene Models Voltea sees expanson L L L L L L h h h h Voltea Kenel (L s same as the model hozon n MC n-dmensonal weghtng fncton, Lmtatons of Voltea-Wene models Dffclt to extend to sstems wth feedbac Dffclt to elate the estmated Volteaenels to a po nfomaton h (mltdmensonal mplse esponse coeffcents Contnos-tme veson Sstem: xax bx+ b nnown paametes θ ( a b b Solvng ODE: t+ t+ t+ t+ xdt a xdt b xdt + b dt Collecton of data: t t t t t t t x x xdt xdt xdt t t t a b t t t x x b xdt xdt dt t t t Least-saes solton: θ Φ Φ Φ Y owe-sees expanson of geneal nonlnea sstem ( m xt ( ax ( t bx ( t ( t + vt ( j j j Smlal, he paametes can be obtaned fom the least-saes method owe Sees Expansons Example: me-doman dentfcaton of nonlnea sstem Sstem: x ax bx + + b nnown paametes: θ ( a b b Collecton of data: x x x a b x x x b Least-saes solton: θ ( Φ Φ Φ Y opetes of ths appoach Standad statstcal valdaton tests ae applcable wthot extensve modfcatons Feenc doman appoach s also possble