Proceedngs of he 7h World Congress The Inernaonal Federaon of Auomac Conrol Seoul, Korea, July 6-, 8 Recursve parameer esmaon by means of he SG-algorhm Magnus Evesed Alexander Medvedev Deparmen of Informaon Technology, Uppsala Unversy, P O Box 337, SE-75 5, SWEDEN magnusevesed@uuse; alexandermedvedev@uuse Absrac: Recursve parameer esmaon n lnear regresson models by means of he Senlund-Gusafsson algorhm s consdered The manfold of saonary soluons o he parameer updae equaon s parameerzed n erms of excaon properes I s shown ha he parameer esmaon error vecor does no dverge under lack of excaon, herefore achevng he purpose of an-wndup Furhermore, an elemenwse form of he parameer vecor esmae s suggesed revealng he effec of ndvdual marx enres n he Rcca equaon on he parameer esmaon updaes Smulaons are performed o llusrae he loss of convergence rae n he esmaes versus he decrease of compuaonal power needed for wo specfc approxmaons of he Rcca equaon n he elemenwse form Keywords: Recursve denfcaon, Parameeresmaon, Kalman fler INTRODUCTION The ask of a recursve parameer esmaon algorhm s o rack dynamcal properes of sgnals and sysems A number of mehods have been suggesed n he pas and he mos common ones are he recursve leas squares wh forgeng facor RLS and normalzed leas mean squares N-LMS These are shown n Ljung and Gunnarsson [99] o be ad hoc varans of he opmal Kalman fler algorhm A sgnfcan complcaon n he praccal use of he Kalman fler n parameer esmaon s s sensvy o he sysem npu excaon If he npu does no provde suffcen nformaonfor parameer denfcaon, a phenomenon called covarance wndup occurs Ad hoc soluons o hs ssue have been proposed eg n Ban e al [99], Hägglund [983] An neresng specalzaon of he Rcca equaon wh an-wndup properes, n he sequel referred o as he Senlund-Gusafsson SG algorhm, had been suggesed n Senlund and Gusafsson [] Laer, a varaon of hs dea was presened n Cao and Schwarz [4] The specal form of he Rcca equaon n he SGalgorhm has made possble o show s non-dvergence under lack of excaon n Medvedev [4], compleely parameerze s saonary behavour n Evesed and Medvedev [6a] and also enabled elemenwse decouplng and convergence analyss n Medvedev and Evesed [8] However, he properes of he parameer esmaon equaon of he SG-algorhm have no been addressed so far Now, he fac ha boh he Rcca equaon and he parameer esmae of he SG algorhm are governed by a ceran marx operaor elemenary marx ransformaon, faclaes he analyss snce some of he resuls from he Rcca equaon analyss carry over o he parameer esmaon equaon The advanageous an-wndup properes of he SGalgorhm make a good canddae for engneerng ap- Ths work has been carred ou wh fnancal suppor by Swedsh Research Councl plcaons such as acve vbraon conrol Olsson [5], acousc echo cancellaon Evesed e al [5] and change deecon Evesed and Medvedev [6b] In hs paper, he close relaonshp beween he recursve parameer esmaon and he Rcca equaon of he SG-algorhm s exploed o formulae new resuls concernng saonary soluons, convergence properes and elemenwse decouplng of he parameer esmae The elemenwse decouplng propery descrbed n Medvedev andevesed [8] s ulzedo lowerhe compuaonal power demands of he SG-algorhm for he case of perodc regressor and s llusraed by smulaon PRELIMINARIES The focus of hs paper s on lnear regresson models of he followng ype y = ϕ T θ + e where y s he scalar oupu measured a dscree me nsances = [,, ϕ R n s he regressor vecor, θ R n s he parameer vecor o be esmaed and he scalar e s he dsurbance If e s whe and he parameer vecor s subjec o he random walk model drven by a zero-mean whe sequence w θ = θ + w hen he opmal, n he sense of mnmum of he a poseror parameer error covarance marx, esmae s yelded by ˆθ = ˆθ + K y ϕ T ˆθ = I Kϕ T ˆθ + Ky wh he Kalman gan P ϕ K = r + ϕ T P ϕ and P, = [,, he soluon o he Rcca equaon 3 978--34-789-/8/$ 8 IFAC 8 38/876-5-KR-487
7h IFAC World Congress IFAC'8 Seoul, Korea, July 6-, 8 P=P P ϕϕt P r + ϕ T +Q 4 P ϕ for some P = P T, P descrbng he covarance of he nal guess of ˆθ, = Opmaly of he esmae s guaraneed only when Q s he covarance marx of w and r = var e, Ljung and Gunnarsson [99] Inroducng he marx-valued marx funcon A X = I + r Xϕϕ T can be rewren as ˆθ = A P ˆθ + Ky 5 Apparenly, he geomercal properes of he funcon A are of fundamenal mporance for he dynamcs of he recursve denfcaon algorhm The regressor vecor sequence s called perssenly excng Södersröm and Soca [989], f here exss a c R + and neger m > such ha for all +m ci ϕkϕ T k 6 k= Ths condon s mporan for he dynamc behavor of 4 as when s no sasfed, some egenvalues of P ncrease lnearly Ths s usually referred o as covarance wndup The SG-algorhm In he approach aken n Senlund and Gusafsson [], a specal choce of Q s suggesed o deal wh he wndup problem and o conrol he convergence pon of he soluon o 4 Q = P dϕϕ T P d r + ϕ T P d ϕ where P d R n n, P d > The dfference E = P P d s shown, n Senlund and Gusafsson [] for non-sngular P and n Evesed and Medvedev [6a] for a general case, o obey he recurson E + = A PEA P d 7 or, n a vecorzed form vec E + = MP d, Pvec E 8 MP d, P = A P A P d 9 where denoes Kronecker produc Thus he marx equaon n he SG-algorhm can be rewren as a lnear dscree me-varyng sysem whch form faclaes s analyss In Medvedev [4], hs srucure s ulzed o show non-dvergence of he algorhm and n Evesed and Medvedev [6a] o examne s saonary properes under lack of excaon The fac ha he marx funcon A X appears boh n he Rcca equaon of he SG-algorhm and n he parameer esmae s srkng I was a key o he analyss of he Rcca equaon and wll n he sequel also be proved o faclae he analyss of he parameer updae equaon 3 STATIONARY POINTS Usng regresson model, esmaor can be wren asˆθ = ˆθ + Ke + ϕ T θ ˆθ Subracng θ from boh sdes of he equaon and defnng θ ˆθ θ yelds θ = θ + Ke ϕ T θ = A P θ + Ke In order o separae he drecon of excaon a each parcular me nsan from he excaon nensy, nroduce a re-paramerzaon of he marx funcon A X A X = I + ρxu where ρ = r ϕ T ϕ and U = ϕϕt ϕ T ϕ The marx U s a Hermanprojeconwh ranku = Defne he normalzed egenvecors of U as ξ, =,,n, where ξ corresponds o he un egenvalue of U and ξ,, ξ n correspond o he zero egenvalues of U Then ρ descrbes he energy n he regressor vecor a me and ξ characerzes he drecon Excaon s called suffcen a me when he followng rank condon s sasfed rank [ξ + n ξ ] = n whch s a srcer condon han perssen excaon snce demands ha each sequence of n consequen regressor vecors s lnearly ndependen Now, assumng e =, consder a saonary pon of e ˆθ = ˆθ = ˆθ = cons The proposon below characerzes he space of all possble saonary soluons Proposon For e =, any saonary soluon of can be decomposed as n ˆθ = θ + m k ξ k 3 for some scalars m,, m n Proof: Omed k= Noce ha he resul above s vald even when condon 6 for perssen excaon s no sasfed The acual saonary soluon s dependen on he curren excaon properes of he regressor vecor For he case of perssen excaon follows ha ˆθ = θ 4 DYNAMICS OF THE PARAMETER ESTIMATE The smlares beween 5 and 7 sugges he use of he Lyapunovransformaon, earlerulzednmedvedev [4] for analyss of he Rcca marx equaon, n order o brng he parameer esmaon equaon o a srucure revealng form 4 Lyapunov Transformaon Suppose ha a each τ =, n+, n+,, he sequence {U} s known n seps n advance If he sequence s suffcenly excng on eachnerval of n consecuve seps, a marx Tτ can be defned as follows detτ Tτ = [ξ τ ξ τ + n ] If, however, he sequence s no suffcenly excng such ha he se {ξ, = τ,τ + n } ncludes only k < n lnearly ndependen vecors, he marx T mus be 8
7h IFAC World Congress IFAC'8 Seoul, Korea, July 6-, 8 consruced dfferenly Le he marx T k τ conss of he k lnearly ndependen ξ on [τ,, τ + n ] and µ, = k +,,n form an orhonormal bass of he lef nullspace of T k The ransformaon hen has he followng form Tτ = [T k τ µ k+ µ n ] The marx Tτ s a Lyapunov ransformaon and preserves he sably properes of he orgnal dynamc sysem The columns of he ransformaon marx are denoed as T = [ξ ξn ] wh me varable dropped when approprae o save space Inroduce he sae marx Z = T T ET 4 and denoe he elemens of Z by Z = {z kl, k =,,n; l =,, n} Snce E s a symmerc marx, he ransformedmarx Z s also symmerc Le he column poson n T of he curren drecon of excaon, e ξ be denoed by Then, for some X R n n, consder X he vecors ] d T [ξ X = T Xξ ξ nt Xξ D T X = [D X D n X] whch are relaed o each oher as ρ d X D X = + ρ ξt Xξ I s shown n Medvedev and Evesed [8] ha f P s a soluon o 7, hen ρ ξ k T D k P = Pd ξ + z k 5 + ρ ξ k T Pd ξ + z Afer he ransformaon θ = T T ˆθ of lnear mevaryng dscree sysem, he ransformed sysem marx becomes Ā = T T A T T The sysem marx s calculaed n Medvedev [4] o be Ā X = I [ n D X n n ] 6 Thus, a each sep =,,n, Ā X s he sum of a un marx and a marx whose nonzero elemens are all n one column The poson of he column vecor D X n 6 s defned by he curren excaon drecon ξ In he sequel, he noon of vecor elemen n he drecon of excaon comes n handy Consder he vecor x = [x x x n ] T, recursvely updaed accordng o x = Ā x = x x D 7 Ths means ha a each sep, he vecor x s updaed by he vecor D weghed by he vecor elemen x The parcular elemen s defned by he curren drecon of excaon and hus, he scalar x s defned as he vecor elemen n he drecon of excaon 4 Non-dvergence of Parameer Esmae For a parameer esmaon algorhm o perform well when excaon lacks n some drecons, s desrable ha he esmaon error does no dverge The followng proposons show, ulzng he ransformaon marx T, ha hs s he case for he SG-algorhm as well as for any algorhm of he form, 3 Proposon In he deal case of perfec measuremens, e e =, he parameer esmaon error, θ s nondvergng Proof: Omed The nex proposon shows ha he ncrease n he parameer error vecor elemens ousde he curren excaon drecon s bounded a each sep and he value of he bound s defnedbyhe elemenofhe parameeresmaonerror n he curren drecon of excaon Proposon 3 Defne he ransformed parameer error vecor v = T T θ If e =, for each elemen v k of v, k, he followng nequaly apples v k + v k v 8 Proof: Omed 5 ELEMENTWISE FORM In, s no clear how he ndvdual elemens of he marx P n he Rcca equaon of he SG-algorhm affec he parameer updaes Ths s clarfed by he followng proposon Inroduce a new parameer sae vecor as θ = T T ˆθ Proposon 4 The elemenwse parameer updae equaon can be wren as θ k = θ ρ ξ k T Pd ξ + z k k 9 + ρ ξ k T Pd ξ + z θ + y ϕ or, for he specal case of k = θ θ = + ρ ξ T Pd ξ + z + ρ ξt Pd ξ + z y ϕ + ρ ξ T Pd ξ + z Proof: See Appendx A As can be seen above, for each me nsan, he dynamcs of each ndvdual parameer updae s only dependen upon wo elemens of he ransformed Rcca equaon, z and z k, n he row correspondng o he curren drecon of excaon 5 Comparson o he Normalzed Leas Mean Squares N-LMS Consder he case when P = P d n he SG-algorhm Then he gan marx n becomes P d ϕ K = r + ϕ T P d ϕ In Ljung and Gunnarsson [99], s shown ha he N-LMS s a specal case of he Kalman fler wh he choces P d = µi, µ R + and r = A general choce of P d yelds a marx sep-sze N-LMS algorhm 83
7h IFAC World Congress IFAC'8 Seoul, Korea, July 6-, 8 3 5 5 5 3 35 4 45 5 3 5 5 5 3 35 4 45 5 3 v k v k v 5 5 5 3 35 4 45 5 Fg Bound 8 on he change of he esmaed parameer vecor elemens Robusness condons of such an algorhm are gven n Rupp and Cezanne [] Ways of choosng he gan marx are suggesed n Mokno [993], Gay [998] In 9, s clear ha P d works as a weghng marx of he regressor drecons affecng he updaes of he ndvdual parameer esmae vecor elemens 6 SIMULATION In hs secon, smulaons are performed o llusrae he possbles ofdecreasnghe compuaonalcomplexyof recursve parameer esmaon by employng he elemenwse calculaons Two ypes of approxmaons are made and analyzed n erms of performance loss compared o he full SG-algorhm 6 Perodc Excaon In order o faclae he mplemenaon of he elemenwse SG-algorhm, he specal case of perodc regressor vecors s consdered Ths assumpon s ofen made n he leraure, Ramos e al [7], Akçay and A [6] and s reasonable n some engneerng applcaons Here he number of esmaed parameers equals he npu perod as n Akçay and A [6] For he SG-algorhm, means ha he ransformaon T s a consan marx 6 Bound on Parameer Error Increase The bound saed n Proposon 3 s llusraed n Fg for a smulaon of a sysem wh n = 3 The subplos correspond o dfferen elemens n he parameer error vecor As can be seen, boh he bound and he esmaes converge a wha seems o be exponenal rae 63 Band Marx Approxmaon Accordng o Medvedev and Evesed [8], ransformed Rcca equaon 4 can be updaed elemenwse as ξ T l Pd ξ z kl + = z kl + ρ ξk T Pd ξ + ρ ξt Pd ξ ξ T l Pd ξ + z lξ k T Pd ξ + z k + ρ ξt Pd ξ + z k= k= k=3 or, for he specal case of l = z k + = + ρ ξ T Pd ξ z k ρ ξ k T Pd ξ z + ρ ξt Pd ξ + ρ ξt Pd ξ + z whch reads for k = l = as z + = z + ρ ξt Pd ξ + ρ ξt Pd ξ + z The equaons above allow o choose whch elemens of he Rcca equaon o employ n he parameer esmaon algorhm and s empng o explo hs possbly n order o decrease s compuaonal complexy One obvous approach s o only use a lmed number of super- and subdagonals n he marx Z for updang he parameer esmaes These elemens are guaraneed o be non-dvergng snce hey obey he same equaons as he full SG-algorhm All oher elemens n Z are consdered o be zero, e he correspondng elemens of P are assumed o already have converged o he elemens of P d a hese posons, makng Z a band marx z z z q z z z 3 z q+ z 3 z 33 Z q = z q zn q+n z q+ z n q+n z nn q+ z nn q+ z nn 3 The seleced z kl can eher be compued onlne or, f he regressor sequence s known, n advance The procedure for updang he parameer vecor beween = τ + τ + n s he followng Algorhm Selec q as he number of requred dagonals Le = τ + 3 Transform he parameer vecor by θ = T T ˆθ 4 Updae he parameer esmaes accordng o 9 5 Updae he Rcca equaon elemens accordng o and 3 6 Increase = + 7 f τ + n goo 3 8 Transform he parameer vecor by ˆθτ + n = T T θτ + n Ths procedure s smlar o ha of AKFA Average Kalman Fler Algorhm n Wgren [998], where he adapaon gans n he Kalman fler are he soluon of anaverageddagonalrccaequaon, breaknghe laer down o a small number of scalar equaons Here s however possble o add super- and subdagonals o mprove he esmaon performance An exensve comparson of he SG-algorhm o oher parameer esmaon mehods s gven n Evesed e al [5] A 3-dmensonal sysem was smulaed wh 3-perodc npu sgnal u The oupu sgnal was consruced 84
7h IFAC World Congress IFAC'8 Seoul, Korea, July 6-, 8 5 5 SG full 5 5 SG dagonal AKFA 5 SG dagonal 5 5 5 5 5 5 3 Fg Comparson beween AKFA and he dagonal SGalgorhm wh q = wh, ϕ = [u u u 3] and whe measuremen nose SNR= 5dB The marces P d and P n he SG-algorhm were seleced randomly and no unng oher han ha descrbed n Wgren [998] was performed for he AKFA In Fg, can be seen ha boh esmaon algorhms reach he rue parameer vecor values bu he dagonal elemen q = SG-algorhm converges faser a he cos of larger esmae varance In general, AKFA seems o be more sensve o he acual excaon drecons used n he consrucon of he regressor vecors The performance of he SG-algorhm s on he oher hand hghly dependen on he choce of P d n he same way as he Kalman fler algorhm s dependen on r and Q The man dfference beween employng he full SGalgorhm Rcca equaon and he super- subdagonal ones les n he ransen perod When P has converged o P d, he ransformed marx Z becomes zero whch gves he same parameer esmaon updaes n boh cases In Fg 3 llusraes hs for he man dagonal SG-algorhm By ncreasng he number of dagonals used, he devaon from he full SG-algorhm s decreased For a hgher dmensonalsysem, a smalldfference persss over a longer perod of me 64 Small Marx Elemens Approxmaon Ye anoher approxmaon of he SG-algorhm s provdedn MedvedevandEvesed [8] There s argued ha mples fas convergence of he dagonal elemens of Z n he drecon of excaon Assumng small z s hus jusfed and an approxmaon of s gven as z z + = + ρ ξt Pd ξ and for he off-dagonal elemens of Z n he drecon of excaon z k z k + = + ρ ξt Pd ξ Ths means ha also z k, k =,, n, k are small Then becomes z kl + = z kl, k, l The convergence of Z o zero s guaraneed by he naure of he algorhm 5 5 5 3 35 Fg 3 Comparsonbeweenhe parameeresmaes usng he full SG-algorhm and only he man dagonal, q = Asersks gve he rue values of he parameers Combnng he above approxmaons no marx form yelds Z + = 4 z z z z n + ρ ξt Pd ξ z z z 3 z 3 z 33 z n z + ρ ξt Pd ξ z n + ρ ξ T Pd ξ z nn The adapaon gans can be calculaed beforehand f ρ and P d are known The algorhm becomes Algorhm Le = τ + Transform he parameer vecor by θ = T T ˆθ 3 Updae he parameer esmaes accordng o 9 4 Updae he Rcca equaon elemens accordng o and 4 5 Increase = + 6 f τ + n goo 7 Transform he parameer vecor by ˆθτ + n = T T θτ + n The ransen employng hs procedure s shown n Fg 4 for he 3-dmensonal sysem descrbed above 7 CONCLUSIONS The parameer esmaon by means of he SG-algorhm s suded for suffcenly and nsuffcenly excng regressor vecorsequences In absence of measuremendsurbance, he saonary soluon o he parameer updae equaon s shown o belong o a manfold defned by he properes ofhe regressorvecorsequence The parameererrorvecor s proved o be non-dvergen under lack of excaon 85
7h IFAC World Congress IFAC'8 Seoul, Korea, July 6-, 8 5 5 5 5 SG full Small Z elemen approxmaon SG 5 5 5 3 35 Fg 4 Comparsonbeween he parameeresmaes usng he full SG-algorhm and he small marx elemens approxmaon Asersks gve he rue values of he parameers An elemenwse represenaon of he parameer updaes and he relaed Rcca equaon s ulzed o decrease he compuaonal load for he case of perodc regressor a he prce of a small decrease n he denfcaon algorhm performance REFERENCES H Akçay and N A Membershp se denfcaon wh perodc npus and orhonormal regressors Sgnal Processng, 86:3778 3786, 6 W J Beyn and L Elsner Infne producs and paraconracng marces The Elecronc Journal of Lnear Algebra, : 8, 997 S Ban, P Bolzern, and M Camp Convergence and exponenal convergence of denfcaon algorhms wh dreconal forgeng facor Auomaca, 65: 99 93, 99 L Cao and H M Schwarz Analyss of he Kalman fler based esmaon algorhm: An orhogonal decomposon approach Auomaca, 4:5 9, 4 I Daubeches and J C Langaras Ses of marces all fne producs of whch converge Lnear Algebra and Is Applcaons, 6:7 63, 99 I Daubeches and J C Langaras Corrgendum/addendum o: Ses of marces all fne producs of whch converge Lnear Algebra and Is Applcaons, 37:69 83, M Evesed and A Medvedev Saonary behavor of an an-wndup scheme for recursve parameer esmaon under lack of excaon Auomaca, 4:5 57, January 6a M Evesed and A Medvedev Model-based sloppng monorng by change deecon In Proceedngs of IEEE Conference on Conrol Applcaons, Munch, Germany, 4-6 Ocober 6b M Evesed, A Medvedev, and T Wgren Wndup properes of recursve parameer esmaon algorhms nacousc echocancellaon InProceedngs of he 6h IFAC World Congress, Prague, Czech Republc, July 5 S L Gay An effcen, fas convergng adapve fler for nework echo cancellaon In Proceedngs of Aslomar Conference, Monerey, USA, November 998 T Hägglund New Esmaon Technques for Adapve Conrol Lund Insue of Technology, Sweden, 983 D J Harfel Nonhomogeneous Marx Producs World Scenfc Publshng, L Ljung and S Gunnarsson Adapaon and rackng n sysem denfcaon - A survey Auomaca, 6: 7, 99 A Medvedev Sably of a Rcca equaon arsng n recursve parameer esmaon under lack of excaon IEEE Transacons on AuomacConrol, 49:75 8, December 4 A Medvedev and M Evesed Elemenwse decouplng of he rcca equaon n he SG-algorhm wh convergence esmaes In Submed o he 7h IFAC World Congress, Seoul, Korea, July 8 S Mokno Exponenally weghed sepsze NLMS adapve fler based on he sascs of a room mpulse response IEEE Transacons on Speech and Audo Processng, : 8, January 993 C Olsson Acve vbraonconrolofmulbody sysems Ph D Thess Dgal Comprehensve Summares of Uppsala Dsseraons from he Faculy of Scence and Technology: 59, Uppsala Unversy, 5 P Ramos, R Torruba, A López, A Salnas, and E Masgrau Sep sze bound of he sequenal paral updae LMS algorhm wh perodc npu sgnals EURASIP Journal on Audo, Speech, and Musc Processng, 7, 7 M Rupp and J Cezanne Robusness condons of he LMS algorhm wh me-varan marx sep-sze Sgnal Processng, 8:787 794, T Södersröm and P Soca Sysem Idenfcaon Prence Hall, 989 B Senlund and F Gusafsson Avodng wndup n recursve parameer esmaon In Preprns of Reglermöe, pages 48 53, Lnköpng, Sweden, May T Wgren Fas convergng and low complexy adapve flerng usng an averaged Kalman fler IEEE Transacons on Sgnal Processng, 46:55 58, February 998 Appendx A PROOF OF PROPOSITION 4 Afer he ransformaon of 5 by θ = T T ˆθ he sysem marx becomes Ā Furhermore, he npu marx s ransformed as T T K = = ξ T ξ n T P ϕ r + ρ ξ T P ξ ϕ D P Thus he ransformed sysem equaons can be wren n erms of ndvdual elemens as θ k = θ k D k P θ + y ϕ or, for he specal case of k = θ = DP θ + DP y ϕ The above equaon ogeher wh 5 yelds he desred resul 86