A New Approach to Linear Filtering and Prediction Problems 1

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1 R. E. KALMAN Research Insue for Advanced Sudy, Balmore, Md. A New Approach o Lnear Flerng and Predcon Problems The classcal flerng and predcon problem s re-examned usng he Bode- Shannon represenaon of random processes and he sae ranson mehod of analyss of dynamc sysems. New resuls are: () The formulaon and mehods of soluon of he problem apply whou modfcaon o saonary and nonsaonary sascs and o growng-memory and nfnememory flers. () A nonlnear dfference (or dfferenal) equaon s derved for he covarance marx of he opmal esmaon error. From he soluon of hs equaon he coeffcens of he dfference (or dfferenal) equaon of he opmal lnear fler are obaned whou furher calculaons. (3) The flerng problem s shown o be he dual of he nose-free regulaor problem. The new mehod developed here s appled o wo well-known problems, confrmng and exendng earler resuls. The dscusson s largely self-conaned and proceeds from frs prncples; basc conceps of he heory of random processes are revewed n he Appendx. Inroducon AN IMPORTANT class of heorecal and praccal problems n communcaon and conrol s of a sascal naure. Such problems are: () Predcon of random sgnals; () separaon of random sgnals from random nose; () deecon of sgnals of known form (pulses, snusods) n he presence of random nose. In hs poneerng work, Wener [] 3 showed ha problems () and () lead o he so-called Wener-Hopf negral equaon; he also gave a mehod (specral facorzaon) for he soluon of hs negral equaon n he praccally mporan specal case of saonary sascs and raonal specra. Many exensons and generalzaons followed Wener s basc work. Zadeh and Ragazzn solved he fne-memory case []. Concurrenly and ndependenly of Bode and Shannon [3], hey also gave a smplfed mehod [] of soluon. Booon dscussed he nonsaonary Wener-Hopf equaon [4]. These resuls are now n sandard exs [5-6]. A somewha dfferen approach along hese man lnes has been gven recenly by Darlngon [7]. For exensons o sampled sgnals, see, e.g., Frankln [8], Lees [9]. Anoher approach based on he egenfuncons of he Wener- Hopf equaon (whch apples also o nonsaonary problems whereas he precedng mehods n general don ), has been poneered by Davs [] and appled by many ohers, e.g., Shnbro [], Blum [], Pugachev [3], Solodovnkov [4]. In all hese works, he objecve s o oban he specfcaon of a lnear dynamc sysem (Wener fler) whch accomplshes he predcon, separaon, or deecon of a random sgnal. 4 Ths research was suppored n par by he U. S. Ar Force Offce of Scenfc Research under Conrac AF 49 (638) Bellona Ave. 3 Numbers n brackes desgnae References a end of paper. 4 Of course, n general hese asks may be done beer by nonlnear flers. A presen, however, lle or nohng s known abou how o oban (boh heorecally and praccally) hese nonlnear flers. Conrbued by he Insrumens and Regulaors Dvson and presened a he Insrumens and Regulaors Conference, March 9 Apr, 959, of THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS. NOTE: Saemens and opnons advanced n papers are o be undersood as ndvdual expressons of her auhors and no hose of he Socey. Manuscrp receved a ASME Headquarers, February 4, 959. Paper No. 59 IRD-. Presen mehods for solvng he Wener problem are subjec o a number of lmaons whch serously cural her praccal usefulness: () The opmal fler s specfed by s mpulse response. I s no a smple ask o synhesze he fler from such daa. () Numercal deermnaon of he opmal mpulse response s ofen que nvolved and poorly sued o machne compuaon. The suaon ges rapdly worse wh ncreasng complexy of he problem. (3) Imporan generalzaons (e.g., growng-memory flers, nonsaonary predcon) requre new dervaons, frequenly of consderable dffculy o he nonspecals. (4) The mahemacs of he dervaons are no ransparen. Fundamenal assumpons and her consequences end o be obscured. Ths paper nroduces a new look a hs whole assemblage of problems, sdeseppng he dffcules jus menoned. The followng are he hghlghs of he paper: (5) Opmal Esmaes and Orhogonal Projecons. The Wener problem s approached from he pon of vew of condonal dsrbuons and expecaons. In hs way, basc facs of he Wener heory are quckly obaned; he scope of he resuls and he fundamenal assumpons appear clearly. I s seen ha all sascal calculaons and resuls are based on frs and second order averages; no oher sascal daa are needed. Thus dffculy (4) s elmnaed. Ths mehod s well known n probably heory (see pp and of Doob [5] and pp of Loève [6]) bu has no ye been used exensvely n engneerng. (6) Models for Random Processes. Followng, n parcular, Bode and Shannon [3], arbrary random sgnals are represened (up o second order average sascal properes) as he oupu of a lnear dynamc sysem exced by ndependen or uncorrelaed random sgnals ( whe nose ). Ths s a sandard rck n he engneerng applcaons of he Wener heory [ 7]. The approach aken here dffers from he convenonal one only n he way n whch lnear dynamc sysems are descrbed. We shall emphasze he conceps of sae and sae ranson; n oher words, lnear sysems wll be specfed by sysems of frs-order dfference (or dfferenal) equaons. Ths pon of vew s Transacons of he ASME Journal of Basc Engneerng, 8 (Seres D): Copyrgh 96 by ASME

2 naural and also necessary n order o ake advanage of he smplfcaons menoned under (5). (7) Soluon of he Wener Problem. Wh he sae-ranson mehod, a sngle dervaon covers a large varey of problems: growng and nfne memory flers, saonary and nonsaonary sascs, ec.; dffculy (3) dsappears. Havng guessed he sae of he esmaon (.e., flerng or predcon) problem correcly, one s led o a nonlnear dfference (or dfferenal) equaon for he covarance marx of he opmal esmaon error. Ths s vaguely analogous o he Wener-Hopf equaon. Soluon of he equaon for he covarance marx sars a he me when he frs observaon s aken; a each laer me he soluon of he equaon represens he covarance of he opmal predcon error gven observaons n he nerval (, ). From he covarance marx a me we oban a once, whou furher calculaons, he coeffcens (n general, me-varyng) characerzng he opmal lnear fler. (8) The Dual Problem. The new formulaon of he Wener problem brngs no conac wh he growng new heory of conrol sysems based on he sae pon of vew [7 4]. I urns ou, surprsngly, ha he Wener problem s he dual of he nose-free opmal regulaor problem, whch has been solved prevously by he auhor, usng he sae-ranson mehod o grea advanage [8, 3, 4]. The mahemacal background of he wo problems s dencal hs has been suspeced all along, bu unl now he analoges have never been made explc. (9) Applcaons. The power of he new mehod s mos apparen n heorecal nvesgaons and n numercal answers o complex praccal problems. In he laer case, s bes o resor o machne compuaon. Examples of hs ype wll be dscussed laer. To provde some feel for applcaons, wo sandard examples from nonsaonary predcon are ncluded; n hese cases he soluon of he nonlnear dfference equaon menoned under (7) above can be obaned even n closed form. For easy reference, he man resuls are dsplayed n he form of heorems. Only Theorems 3 and 4 are orgnal. The nex secon and he Appendx serve manly o revew well-known maeral n a form suable for he presen purposes. Noaon Convenons Throughou he paper, we shall deal manly wh dscree (or sampled) dynamc sysems; n oher words, sgnals wll be observed a equally spaced pons n me (samplng nsans). By suable choce of he me scale, he consan nervals beween successve samplng nsans (samplng perods) may be chosen as uny. Thus varables referrng o me, such as,, τ, T wll always be negers. The resrcon o dscree dynamc sysems s no a all essenal (a leas from he engneerng pon of vew); by usng he dscreeness, however, we can keep he mahemacs rgorous and ye elemenary. Vecors wll be denoed by small bold-face leers: a, b,..., u, x, y,... A vecor or more precsely an n-vecor s a se of n numbers x,... x n ; he x are he co-ordnaes or componens of he vecor x. Marces wll be denoed by capal bold-face leers: A, B, Q, Φ, Ψ, ; hey are m n arrays of elemens a j, b j, q j,... The ranspose (nerchangng rows and columns) of a marx wll be denoed by he prme. In manpulang formulas, wll be convenen o regard a vecor as a marx wh a sngle column. Usng he convenonal defnon of marx mulplcaon, we wre he scalar produc of wo n-vecors x, y as n x'y = x y = y'x = The scalar produc s clearly a scalar,.e., no a vecor, quany. Smlarly, he quadrac form assocaed wh he n n marx Q s, x'qx = xqjx n, j= We defne he expresson xy' where x' s an m-vecor and y s an n-vecor o be he m n marx wh elemens x y j. We wre E(x) = Ex for he expeced value of he random vecor x (see Appendx). I s usually convenen o om he brackes afer E. Ths does no resul n confuson n smple cases snce consans and he operaor E commue. Thus Exy' = marx wh elemens E(x y j ); ExEy' = marx wh elemens E(x )E(y j ). For ease of reference, a ls of he prncpal symbols used s gven below. Opmal Esmaes me n general, presen me. me a whch observaons sar. x (), x () basc random varables. y() observed random varable. x *( ) opmal esmae of x ( ) gven y( ),, y(). L loss funcon (non random funcon of s argumen). ε esmaon error (random varable). Orhogonal Projecons Y() lnear manfold generaed by he random varables y( ),, y(). x ( ) orhogonal projecon of x( ) on Y(). x ( ) componen of x( ) orhogonal o Y(). Models for Random Processes Φ( + ; ) ranson marx Q() covarance of random excaon Soluon of he Wener Problem x() basc random varable. y() observed random varable. Y() Z() lnear manfold generaed by y( ),, y(). lnear manfold generaed by y ( ). x*( ) x ( ) opmal esmae of x( ) gven Y(). error n opmal esmae of x( ) gven Y(). Opmal Esmaes To have a concree descrpon or he ype of problems o be suded, consder he followng suaon. We are gven sgnal x () and nose x (). Only he sum y() = x () + x () can be observed. Suppose we have observed and know exacly he values of y( ),..., y(). Wha can we nfer from hs knowledge n regard o he (unobservable) value of he sgnal a =, where may be less han, equal o, or greaer han? If <, hs s a daasmoohng (nerpolaon) problem. If =, hs s called flerng. If >, we have a predcon problem. Snce our reamen wll be general enough o nclude hese and smlar problems, we shall use hereafer he collecve erm esmaon. As was poned ou by Wener [], he naural seng of he esmaon problem belongs o he realm of probably heory and sascs. Thus sgnal, nose, and her sum wll be random varables, and consequenly hey may be regarded as random processes. From he probablsc descrpon of he random processes we can deermne he probably wh whch a parcular sample of he sgnal and nose wll occur. For any gven se of measured values η( ),..., η() of he random varable y() one can hen also deermne, n prncple, he probably of smulaneous occurrence of varous values ξ () of he random varable x ( ). Ths s he condonal probably dsrbuon funcon j Transacons of he ASME Journal of Basc Engneerng, 8 (Seres D): Copyrgh 96 by ASME

3 Pr[x ( ) ξ y( ) = η( ),, y() = η()] = F(ξ ) () Evdenly, F(ξ ) represens all he nformaon whch he measuremen of he random varables y( ),..., y() has conveyed abou he random varable x ( ). Any sascal esmae of he random varable x ( ) wll be some funcon of hs dsrbuon and herefore a (nonrandom) funcon of he random varables y( ),..., y(). Ths sascal esmae s denoed by X ( ), or by jus X ( ) or X when he se of observed random varables or he me a whch he esmae s requred are clear from conex. Suppose now ha X s gven as a fxed funcon of he random varables y( ),..., y(). Then X s self a random varable and s acual value s known whenever he acual values of y( ),..., y() are known. In general, he acual value of X ( ) wll be dfferen from he (unknown) acual value of x ( ). To arrve a a raonal way of deermnng X, s naural o assgn a penaly or loss for ncorrec esmaes. Clearly, he loss should be a () posve, () nondecreasng funcon of he esmaon error ε = x ( ) X ( ). Thus we defne a loss funcon by L() = L(ε ) L(ε ) when ε ε () L(ε) = L( ε) Some common examples of loss funcons are: L(ε) = aε, aε 4, a ε, a[ exp( ε )], ec., where a s a posve consan. One (bu by no means he only) naural way of choosng he random varable X s o requre ha hs choce should mnmze he average loss or rsk E{L[x ( ) X ( )]} = E[E{L[x( ) X ( )] y( ),, y()}] (3) Snce he frs expecaon on he rgh-hand sde of (3) does no depend on he choce of X bu only on y( ),..., y(), s clear ha mnmzng (3) s equvalen o mnmzng E{L[x ( ) X ( )] y( ),..., y()} (4) Under jus slgh addonal assumpons, opmal esmaes can be characerzed n a smple way. Theorem. Assume ha L s of ype () and ha he condonal dsrbuon funcon F(ξ) defned by () s: (A) symmerc abou he mean ξ : F(ξ ξ ) = F( ξ ξ) (B) convex for ξ ξ : F(λξ + ( λ)ξ ) λf(ξ ) + ( λ)f(ξ ) for all ξ, ξ ξ and λ Then he random varable x *( ) whch mnmzes he average loss (3) s he condonal expecaon x *( ) = E[x ( ) y( ),, y()] (5) Proof: As poned ou recenly by Sherman [5], hs heorem follows mmedaely from a well-known lemma n probably heory. Corollary. If he random processes {x ()}, {x ()}, and {y()} are gaussan, Theorem holds. Proof: By Theorem 5, (A) (see Appendx), condonal dsrbuons on a gaussan random process are gaussan. Hence he requremens of Theorem are always sasfed. In he conrol sysem leraure, hs heorem appears somemes n a form whch s more resrcve n one way and more general n anoher way: Theorem l-a. If L(ε) = ε, hen Theorem s rue whou assumpons (A) and (B). Proof: Expand he condonal expecaon (4): E[x ( ) y( ),, y()] X ( )E[x ( ) y( ),, y()] + X ( ) and dfferenae wh respec o X ( ). Ths s no a compleely rgorous argumen; for a smple rgorous proof see Doob [5], pp Remarks. (a) As far as he auhor s aware, s no known wha s he mos general class of random processes {x ()}, {x ()} for whch he condonal dsrbuon funcon sasfes he requremens of Theorem. (b) Asde from he noe of Sherman, Theorem apparenly has never been saed explcly n he conrol sysems leraure. In fac, one fnds many saemens o he effec ha loss funcons of he general ype () canno be convenenly handled mahemacally. (c) In he sequel, we shall be dealng manly wh vecorvalued random varables. In ha case, he esmaon problem s saed as: Gven a vecor-valued random process {x()} and observed random varables y( ),..., y(), where y() = Mx() (M beng a sngular marx; n oher words, no all co-ordnaes of x() can be observed), fnd an esmae X( ) whch mnmzes he expeced loss E[L( x( ) X( ) )], beng he norm of a vecor. Theorem remans rue n he vecor case also, provded we re- qure ha he condonal dsrbuon funcon of he n coord- naes of he vecor x( ), Pr[x ( ) ξ,, x n ( ) ξ n y( ),, y()] = F(ξ,,ξ n ) be symmerc wh respec o he n varables ξ ξ,, ξ n ξ n and convex n he regon where all of hese varables are negave. Orhogonal Projecons The explc calculaon of he opmal esmae as a funcon of he observed varables s, n general, mpossble. There s an mporan excepon: The processes {x ()}, {x ()} are gaussan. On he oher hand, f we aemp o ge an opmal esmae under he resrcon L(ε) = ε and he addonal requremen ha he esmae be a lnear funcon of he observed random varables, we ge an esmae whch s dencal wh he opmal esmae n he gaussan case, whou he assumpon of lneary or quadrac loss funcon. Ths shows ha resuls obanable by lnear esmaon can be beered by nonlnear esmaon only when () he random processes are nongaussan and even hen (n vew of Theorem 5, (C)) only () by consderng a leas hrdorder probably dsrbuon funcons. In he specal cases jus menoned, he explc soluon of he esmaon problem s mos easly undersood wh he help of a geomerc pcure. Ths s he subjec of he presen secon. Consder he (real-valued) random varables y( ),, y(). The se of all lnear combnaons of hese random varables wh real coeffcens = a y( ) (6) forms a vecor space (lnear manfold) whch we denoe by Y(). We regard, absracly, any expresson of he form (6) as pon or vecor n Y(); hs use of he word vecor should no be confused, of course, wh vecor-valued random varables, ec. Snce we do no wan o fx he value of (.e., he oal number of possble observaons), Y() should be regarded as a fnedmensonal subspace of he space of all possble observaons. Transacons of he ASME Journal of Basc Engneerng, 8 (Seres D): Copyrgh 96 by ASME

4 Gven any wo vecors u, v n Y() (.e., random varables expressble n he form (6)), we say ha u and v are orhogonal f Euv =. Usng he Schmd orhogonalzaon procedure, as descrbed for nsance by Doob [5], p. 5, or by Loève [6], p. 459, s easy o selec an orhonormal bass n Y(). By hs s mean a se of vecors e,, e n Y() such ha any vecor n Y() can be expressed as a unque lnear combnaon of e,, e and Ee e j = δ j = f = j (, j =,, ) (7) = f j Thus any vecor x n Y(). s gven by x = = and so he coeffcens a can be mmedaely deermned wh he ad of (7): Exe j = E ae e j = aeee j = aδj = a j (8) = = = I follows furher ha any random varable x (no necessarly n Y()) can be unquely decomposed no wo pars: a par x n Y() and a par x orhogonal o Y() (.e., orhogonal o every vecor n 'Y()). In fac, we can wre x = x + x = = a e ( Exe ) e + x Thus x s unquely deermned by equaon (9) and s obvously a vecor n Y(). Therefore x s also unquely deermned; remans o check ha s orhogonal o Y(): Exe = E( x x) e = Exe Exe Now he co-ordnaes of x wh respec o he bass e,, e, are gven eher n he form E xe (as n (8)) or n he form Exe (as n (9)). Snce he co-ordnaes are unque, Exe = E xe ( =,..., ); hence E xe = and x s orhogonal o every base vecor e ; and herefore o Y(). We call x he orhogonal projecon of x on Y(). There s anoher way n whch he orhogonal projecon can be characerzed: x s ha vecor n Y() (.e., ha lnear funcon of he random varables y( ),..., y()) whch mnmzes he quadrac loss funcon. In fac, f w s any oher vecor n Y(), we have E( x w) = E( x + x w) = E[( x x) + ( x w)] Snce x s orhogonal o every vecor n Y() and n parcular o x w we have (9) E( x w) = E( x x) + E( x w) E( x x) () Ths shows ha, f w also mnmzes he quadrac loss, we mus have E ( x w) = whch means ha he random varables x and w are equal (excep possbly for a se of evens whose probably s zero). These resuls may be summarzed as follows: Theorem. Le {x()}, {y()} random processes wh zero mean (.e., Ex() = Ey() = for all ). We observe y( ),, y(). If eher (A) he random processes {x()}, {y()} are gaussan; or (B) he opmal esmae s resrced o be a lnear funcon of he observed random varables and L(ε) = ε ; hen x*( ) = opmal esmae of x( ) gven y( ),, y() = orhogonal projecon x ( ) of x( ) on Y(). () These resuls are well-known hough no easly accessble n he conrol sysems leraure. See Doob [5], pp , or Pugachev [6]. I s somemes convenen o denoe he orhogonal projecon by x ) x *( ) = Eˆ[ x( ) Y()] ( The noaon Ê s movaed by par (b) of he heorem: If he sochasc processes n queson are gaussan, hen orhogonal projecon s acually dencal wh condonal expecaon. Proof. (A) Ths s a drec consequence of he remarks n connecon wh (). (B) Snce x(), y() are random varables wh zero mean, s clear from formula (9) ha he orhogonal par x ( ) of x( ) wh respec o he lnear manfold Y() s also a random varable wh zero mean. Orhogonal random varables wh zero mean are uncorrelaed; f hey are also gaussan hen (by Theorem 5 (B)) hey are ndependen. Thus = Ex ( ) = E[ x ( ) y( ),, y()] = E[x ( ) x ( ) y( ),, y()] = E[x ( ) y( ),, y()] x ( ) = Remarks. (d) A rgorous formulaon of he conens of hs secon as requres some elemenary noons from he heory of Hlber space. See Doob [5] and Loève [6 ]. (e) The physcal nerpreaon of Theorem s largely a maer of ase. If we are no worred abou he assumpon of gaussanness, par (A) shows ha he orhogonal projecon s he opmal esmae for all reasonable loss funcons. If we do worry abou gaussanness, even f we are resgned o consder only lnear esmaes, we know ha orhogonal projecons are no he opmal esmae for many reasonable loss funcons. Snce n pracce s dffcul o asceran o wha degree of approxmaon a random process of physcal orgn s gaussan, s hard o decde wheher Theorem has very broad or very lmed sgnfcance. (f) Theorem s mmedaely generalzed for he case of vecor-valued random varables. In fac, we defne he lnear manfold Y() generaed by y( ),..., y() o be he se of all lnear combnaons = m a y ( ) j j= of all m co-ordnaes of each of he random vecors y( ),, y(). The res of he sory proceeds as before. (g) Theorem saes n effec ha he opmal esmae under condons (A) or (B) s a lnear combnaon of all prevous observaons. In oher words, he opmal esmae can be regarded as he oupu of a lnear fler, wh he npu beng he acually occurrng values of he observable random varables; Theorem gves a way of compung he mpulse response of he opmal fler. As poned ou before, knowledge of hs mpulse response s no a complee soluon of he problem; for hs reason, no explc formulas for he calculaon of he mpulse response wll be gven. Models for Random Processes In dealng wh physcal phenomena, s no suffcen o gve an emprcal descrpon bu one mus have also some dea of he underlyng causes. Whou beng able o separae n some sense causes and effecs,.e., whou he assumpon of causaly, one can hardly hope for useful resuls. j Transacons of he ASME Journal of Basc Engneerng, 8 (Seres D): Copyrgh 96 by ASME

5 I s a farly generally acceped fac ha prmary macroscopc sources of random phenomena are ndependen gaussan processes. 5 A well-known example s he nose volage produced n a ressor due o hermal agaon. In mos cases, observed random phenomena are no descrbable by ndependen random varables. The sascal dependence (correlaon) beween random sgnals observed a dfferen mes s usually explaned by he presence of a dynamc sysem beween he prmary random source and he observer. Thus a random funcon of me may be hough of as he oupu of a dynamc sysem exced by an ndependen gaussan random process. An mporan propery of gaussan random sgnals s ha hey reman gaussan afer passng hrough a lnear sysem (Theorem 5 (A)). Assumng ndependen gaussan prmary random sources, f he observed random sgnal s also gaussan, we may assume ha he dynamc sysem beween he observer and he prmary source s lnear. Ths concluson may be forced on us also because of lack of dealed knowledge of he sascal properes of he observed random sgnal: Gven any random process wh known frs and second-order averages, we can fnd a gaussan random process wh he same properes (Theorem 5 (C)). Thus gaussan dsrbuons and lnear dynamcs are naural, muually plausble assumpons parcularly when he sascal daa are scan. How s a dynamc sysem (lnear or nonlnear) descrbed? The fundamenal concep s he noon of he sae. By hs s mean, nuvely, some quanave nformaon (a se of numbers, a funcon, ec.) whch s he leas amoun of daa one has o know abou he pas behavor of he sysem n order o predc s fuure behavor. The dynamcs s hen descrbed n erms of sae ransons,.e., one mus specfy how one sae s ransformed no anoher as me passes. A lnear dynamc sysem may be descrbed n general by he vecor dfferenal equaon and dx/d = F()x + D()u() y() = M()x() where x s an n-vecor, he sae of he sysem (he componens x of x are called sae varables); u() s an m-vecor (m n) represenng he npus o he sysem; F() and D() are n n, respecvely, n m marces. If all coeffcens of F(), D(), M() are consans, we say ha he dynamc sysem () s menvaran or saonary. Fnally, y() s a p-vecor denong he oupus of he sysem; M() s an n p marx; p n The physcal nerpreaon of () has been dscussed n deal elsewhere [8,, 3]. A look a he block dagram n Fg. may be helpful. Ths s no an ordnary bu a marx block dagram (as revealed by he fa lnes ndcang sgnal flow). The negraor n. Fg. Marx block dagram of he general lnear connuous-dynamc sysem 5 The probably dsrbuons wll be gaussan because macroscopc random effecs may be hough of as he superposon of very many mcroscopc random effecs; under very general condons, such aggregae effecs end o be gaussan, regardless of he sascal properes of he mcroscopc effecs. The assumpon of ndependence n hs conex s movaed by he fac ha mcroscopc phenomena end o ake place much more rapdly han macroscopc phenomena; hus prmary random sources would appear o be ndependen on a macroscopc me scale. u() x() x() D() F () M() y() () Fg. acually sands for n negraors such ha he oupu of each s a sae varable; F() ndcaes how he oupus of he negraors are fed back o he npus of he negraors. Thus f j () s he coeffcen wh whch he oupu of he jh negraor s fed back o he npu of he h negraor. I s no hard o relae hs formalsm o more convenonal mehods of lnear sysem analyss. If we assume ha he sysem () s saonary and ha u() s consan durng each samplng perod, ha s u( + τ) = u(); τ <, =,, (3) hen () can be readly ransformed no he more convenen dscree form. where [8, ] and x( + ) = Φ()x() + ()u(); =,, Φ() = exp F = = () = ( F /! (F = un marx) exp Fτ dτ ) D u() x( + ) x() un () delay Fg. Marx block dagram of he general lnear dscree-dynamc sysem See Fg.. One could also express exp Fτ n closed form usng Laplace ransform mehods [8,,, 4]. If u() sasfes (3) bu he sysem () s nonsaonary, we can wre analogously x( + ) = Φ( + ; ) + ()u() y() = M()x() bu of course now Φ( + ; ), () canno be expressed n general n closed form. Equaons of ype (4) are encounered frequenly also n he sudy of complcaed sampled-daa sysems []. See Fg. Φ( + ; ) s he ranson marx of he sysem () or (4). The noaon Φ( ; ) (, = negers) ndcaes ranson from me o me. Evdenly Φ(; ) = I = un marx. If he sysem () s saonary hen Φ( + ; ) = Φ( + ) = Φ() = cons. Noe also he produc rule: Φ(; s)φ(s; r) = Φ(; r) and he nverse rule Φ (; s) = Φ(s; ), where, s, r are negers. In a saonary sysem, Φ(; τ) = exp F( τ). As a resul of he precedng dscusson, we shall represen random phenomena by he model x( + ) = Φ( + ; )x() + u() (5) where {u()} s a vecor-valued, ndependen, gaussan random process, wh zero mean, whch s compleely descrbed by (n vew of Theorem 5 (C)) Eu() = for all ; Eu()u'(s) = f s Eu()u'() = G(). Φ ( + ; ) Of course (Theorem 5 (A)), x() s hen also a gaussan random process wh zero mean, bu s no longer ndependen. In fac, f we consder (5) n he seady sae (assumng s a sable sysem), n oher words, f we neglec he nal sae x( ), hen M() y() =,, (4) Transacons of he ASME Journal of Basc Engneerng, 8 (Seres D): Copyrgh 96 by ASME

6 x() = Φ(; r + )u(r). r= Therefore f s we have s Ex()x'(s) = Φ(; r + )Q(r) Φ'(s; r + ). r= Thus f we assume a lnear dynamc model and know he sascal properes of he gaussan random excaon, s easy o fnd he correspondng sascal properes of he gaussan random process {x()}. In real lfe, however, he suaon s usually reversed. One s gven he covarance marx Ex()x'(s) (or raher, one aemps o esmae he marx from lmed sascal daa) and he problem s o ge (5) and he sascal properes of u(). Ths s a suble and presenly largely unsolved problem n expermenaon and daa reducon. As n he vas majory of he engneerng leraure on he Wener problem, we shall fnd convenen o sar wh he model (5) and regard he problem of obanng he model self as a separae queson. To be sure, he wo problems should be opmzed jonly f possble; he auhor s no aware, however, of any sudy of he jon opmzaon problem. In summary, he followng assumpons are made abou random processes: Physcal random phenomena may be hough of as due o prmary random sources excng dynamc sysems. The prmary sources are assumed o be ndependen gaussan random processes wh zero mean; he dynamc sysems wll be lnear. The random processes are herefore descrbed by models such as (5). The queson of how he numbers specfyng he model are obaned from expermenal daa wll no be consdered. Soluon of he Wener problem Le us now defne he prncpal problem of he paper. Problem I. Consder he dynamc model x( + ) = Φ( + ; )x() + u() (6) y() = M()x() (7) where u() s an ndependen gaussan random process of n- vecors wh zero mean, x() s an n-vecor, y() s a p-vecor (p n), Φ( + ; ), M() are n n, resp. p n, marces whose elemens are nonrandom funcons of me. Gven he observed values of y( ),..., y() fnd an esmae x*( ) of x( ) whch mnmzes he expeced loss. (See Fg., where () = I.) Ths problem ncludes as a specal case he problems of flerng, predcon, and daa smoohng menoned earler. I ncludes also he problem of reconsrucng all he sae varables of a lnear dynamc sysem from nosy observaons of some of he sae varables (p < n!). From Theorem -a we know ha he soluon of Problem I s smply he orhogonal projecon of x( ) on he lnear manfold Y() generaed by he observed random varables. As remarked n he Inroducon, hs s o be accomplshed by means of a lnear (no necessarly saonary!) dynamc sysem of he general form (4). Wh hs n mnd, we proceed as follows. Assume ha y( ),..., y( ) have been measured,.e., ha Y( ) s known. Nex, a me, he random varable y() s measured. As before le y ( ) be he componen of y() orhogonal o Y( ). If y ( ), whch means ha he values of all componens of hs random vecor are zero for almos every possble even, hen Y() s obvously he same as Y( ) and herefore he measuremen of y() does no convey any addonal nformaon. Ths s no lkely o happen n a physcally meanngful suaon. In any case, y ( ) generaes a lnear manfold (possbly ) whch we denoe by Z(). By defnon, Y( ) and Z() aken ogeher are he same manfold as Y(), and every vecor n Z() s orhogonal o every vecor n Y( ). Assumng by nducon ha x*( ) s known, we can wre: x*( ) = Ê [x( ) Y()] = Ê [x( ) Y( )] + Ê [x( ) Z()] = Φ( + ; ) x*( ) + Ê [u( ) Y( )] + Ê [x( ) Z()] (8) where he las lne s obaned usng (6). Le = + s, where s s any neger. If s, hen u( l ) s ndependen of Y( ). Ths s because u( l ) = u( + s ) s hen ndependen of u( ), u( 3),... and herefore by (6 7), ndependen of y( ),..., y( ), hence ndependen of Y( ). Snce, for all, u( ) has zero mean by assumpon, follows ha u( l ) (s ) s orhogonal o Y( ). Thus f s, he second erm on he rgh-hand sde of (8) vanshes; f s <, consderable complcaons resul n evaluang hs erm. We shall consder only he case l. Furhermore, wll suffce o consder n deal only he case l = + snce he oher cases can be easly reduced o hs one. The las erm n (8) mus be a lnear operaon on he random varable y ( ): Ê [x( + ) Z()] = *() y ( ) (9) where *() s an n p marx, and he sar refers o opmal flerng. The componen of y() lyng n Y( ) s y ( ) = M()x*( ). Hence y ( ) = y() y ( ) = y() M()x*( ). () Combnng (8-) (see Fg. 3) we oban where x*( + ) = Φ*( + ; )x*( ) + *()y() () Φ*( + ; ) = Φ( + ; ) *()M() () Thus opmal esmaon s performed by a lnear dynamc sysem of he same form as (4). The sae of he esmaor s he prevous esmae, he npu s he las measured value of he observable random varable y(), he ranson marx s gven by (). Noce ha physcal realzaon of he opmal fler requres only () he model of he random process () he operaor *(). The esmaon error s also governed by a lnear dynamc sysem. In fac, y() x ( + ) = x( + ) x*( + ) = Φ( + ; )x() + u() Φ*( + ; )x*( ) *()M()x() Φ( + s; + ) y ( ) MODEL OF RANDOM PROCESS x*( ) x*( + ) un *() M() delay x*( + ) Φ ( + ; ) I Fg. 3 Marx block dagram of opmal fler x*( + s ) y ( ) Transacons of he ASME Journal of Basc Engneerng, 8 (Seres D): Copyrgh 96 by ASME

7 = Φ*( + ; ) x ( ) + u() (3) Thus Φ* s also he ranson marx of he lnear dynamc sysem governng he error. From (3) we oban a once a recurson relaon for he covarance marx P*() of he opmal error x ( ). Nong ha u() s ndependen of x() and herefore of x ( ) we ge P*( + ) = E x ( + ) x '( + ) = Φ*( + ; )E x ( ) x '( )Φ*' ( + ; ) + Q() = Φ*( + ; )E x ( ) x '( )Φ'( + ; ) + Q() = Φ*( + ; )P*()Φ'( + ; ) + Q() (4) where Q() = Eu()u'(). There remans he problem of obanng an explc formula for * (and hus also for Φ*). Snce, x ( + ) Z()) = x( + ) Ê [x( + ) Z()] s orhogonal o y ( ), follows ha by (9) ha = E[x( + ) *() y ( )] y '( ) = Ex( + ) y '( ) *()E y ( ) y '( ). Nong ha x ( + ) s orhogonal o Z(), he defnon of P() gven earler, and (7), follows furher = E x ( + ) y '( ) *()M()P*()M'() = E[Φ( + ; ) x ( ) + u( )] x '( )M'() *()M()P*()M'(). Fnally, snce u() s ndependen of x(), = Φ( + ; )P*()M'() *()M()P*()M'(). Now he marx M()P*()M'()wll be posve defne and hence nverble whenever P*() s posve defne, provded ha none of he rows of M() are lnearly dependen a any me, n oher words, ha none of he observed scalar random varables y l (),..., y m (), s a lnear combnaon of he ohers. Under hese crcumsances we ge fnally: *() = Φ( + ; )P*()M'()[M()P*()M'()] (5) Snce observaons sar a, x ( ) = x( ); o begn he erave evaluaon of P*() by means of equaon (4), we mus obvously specfy P*( ) = Ex( )x'( ). Assumng hs marx s posve defne, equaon (5) hen yelds *( ); equaon () Φ*( + ; ), and equaon (4) P*( + ), compleng he cycle. If now Q() s posve defne, hen all he P*() wll be posve defne and he requremens n dervng (5) wll be sasfed a each sep. Now we remove he resrcon ha = +. Snce u() s orhogonal o Y(), we have x*( + ) = Ê [Φ( + ; )x() + u() Y()] = Φ( + ; )x*( ) Hence f Φ( + ; ) has an nverse Φ(; + ) (whch s always he case when Φ s he ranson marx of a dynamc sysem descrbable by a dfferenal equaon) we have x*( ) = Φ(; + )x*( + ) If +, we frs observe by repeaed applcaon of (6) ha x( + s) = Φ( + s; + )x( + ) s + Φ( + s; + r)u( + r) (s ) r Snce u( + s ),, u( + ) are all orhogonal o Y(), x*( + s ) = Ê [x( + s) Y()] = Ê [Φ( + s; + )x( + ) Y()] = Φ( + s; + )x*( + ) (s ) If s <, he resuls are smlar, bu x*( s ) wll have ( s)(n p) co-ordnaes. The resuls of hs secon may be summarzed as follows: Theorem 3. (Soluon of he Wener Problem) Consder Problem I. The opmal esmae x*( + ) of x( + ) gven y( ),..., y() s generaed by he lnear dynamc sysem x*( + ) = Φ*( + ; )x*( ) + *()y() () The esmaon error s gven by x ( + ) = Φ*( + ; ) x ( ) + u() (3) The covarance marx of he esmaon error s cov x ( ) = E x ( ) x '( ) = P*() (6) The expeced quadrac loss s n = Ex ( ) = race P*() (7) The marces *(), Φ*( + ; ), P*() are generaed by he recurson relaons *() = Φ( + ; )P*()M'()[M()P*()M'()] Φ*( + ; ) = Φ( + ; ) *()M() P*( + ) = Φ*( + ; )P*()Φ'( + ; ) + Q() (8) (9) (3) In order o carry ou he eraons, one mus specfy he covarance P*( ) of x( ) and he covarance Q() of u(). Fnally, for any s, f Φ s nverble x*( + s ) = Φ( + s; + )x*( + ) ) = Φ( + s; + )Φ*( + ; )Φ(; + s ) x*( + s ) + Φ( + s; + ) *()y() (3) so ha he esmae x*( + s ) (s ) s also gven by a lnear dynamc sysem of he ype (). Remarks. (h) Elmnang * and Φ* from (8 3), a nonlnear dfference equaon s obaned for P*(): P*( + ) = Φ( + ; ){P*() P*()M'()[M()P*()M'()] P*()M()}Φ'( + ; ) + Q() (3) Ths equaon s lnear only f M() s nverble bu hen he problem s rval snce all componens of he random vecor x() are observable P*( + ) = Q(). Observe ha equaon (3) plays a role n he presen heory analogous o ha of he Wener-Hopf equaon n he convenonal heory. Once P*() has been compued va (3) sarng a =, he explc specfcaon of he opmal lnear fler s mmedaely avalable from formulas (9-3). Of course, he soluon of Equaon (3), or of s dfferenal-equaon equvalen, s a much smpler ask han soluon of he Wener-Hopf equaon. () The resuls saed n Theorem 3 do no resolve compleely Problem I. Lle has been sad, for nsance, abou he physcal sgnfcance of he assumpons needed o oban equaon (5), he convergence and sably of he nonlnear dfference equaon (3), he sably of he opmal fler (), ec. Ths can acually be done n a compleely sasfacory way, bu mus be lef o a fuure paper. In hs connecon, he prncpal gude and Transacons of he ASME Journal of Basc Engneerng, 8 (Seres D): Copyrgh 96 by ASME

8 ool urns ou o be he dualy heorem menoned brefly n he nex secon. See [9]. (j) By leng he samplng perod (equal o one so far) approach zero, he mehod can be used o oban he specfcaon of a dfferenal equaon for he opmal fler. To do hs,.e., o pass from equaon (4) o equaon (), requres compung he logarhm F* of he marx Φ*. Bu hs can be done only f Φ* s nonsngular whch s easly seen no o be he case. Ths s because s suffcen for he opmal fler o have n p sae varables, raher han n, as he formalsm of equaon () would seem o mply. By approprae modfcaons, herefore, equaon () can be reduced o an equvalen se of only n p equaons whose ranson marx s nonsngular. Deals of hs ype wll be covered n laer publcaons. (k) The dynamc sysem () s, n general, nonsaonary. Ths s due o wo hngs: () The me dependence of Φ( + ; ) and M(); () he fac ha he esmaon sars a = and mproves as more daa are accumulaed. If Φ, M are consans, can be shown ha () becomes a saonary dynamc sysem n he lm. Ths s he case reaed by he classcal Wener heory. (l) I s noeworhy ha he dervaons gven are no affeced by he nonsaonary of he model for x() or he fneness of avalable daa. In fac, as far as he auhor s aware, he only explc recurson relaons gven before for he growng-memory fler are due o Blum []. However, hs resuls are much more complcaed han ours. (m) By nspecon of Fg. 3 we see ha he opmal fler s a feedback sysem, and ha he sgnal afer he frs summer s whe nose snce y ( ) s obvously an orhogonal random process. Ths corresponds o some well-known resuls n Wener flerng, see, e.g., Smh [8], Chaper 6, Fg However, hs s apparenly he frs rgorous proof ha every Wener fler s realzable by means of a feedback sysem. Moreover, wll be shown n anoher paper ha such a fler s always sable, under very mld assumpons on he model (6 7). See [9]. The Dual Problem Le us now consder anoher problem whch s concepually very dfferen from opmal esmaon, namely, he nose-free regulaor problem. In he smples cases, hs s: Problem II. Consder he dynamc sysem x( + ) = Φˆ ( + ; )x() + Mˆ ()u() (33) where x() s an n-vecor, u() s an m-vecor (m n), Φˆ, Mˆ are n n resp. n m marces whose elemens are nonrandom funcons of me. Gven any sae x() a me, we are o fnd a sequence u(),..., u(t) of conrol vecors whch mnmzes he performance ndex T V[x()] = + x'(τ)q(τ)x(τ) τ= Where Qˆ () s a posve defne marx whose elemens are nonrandom funcons of me. See Fg., where = Mˆ and M = I. Probablsc consderaons play no par n Problem II; s mplcly assumed ha every sae varable can be measured exacly a each nsan, +,..., T. I s cusomary o call T he ermnal me ( may be nfny). The frs general soluon of he nose-free regulaor problem s due o he auhor [8]. The man resul s ha he opmal conrol vecors u*() are nonsaonary lnear funcons of x(). Afer a change n noaon, he formulas of he Appendx, Reference [8] (see also Reference [3]) are as follows: u*() = ˆ *()x() (34) Under opmal conrol as gven by (34), he closed-loop equaons for he sysem are (see Fg. 4) x( + ) = Φˆ *( + ; )x() and he mnmum performance ndex a me s gven by V*[x()] = x'()p*( )x() The marces ˆ *(), Φˆ *( + ; ), Pˆ *() are deermned by he recurson relaons: ˆ *() = [ Mˆ '() Pˆ *() Mˆ ()] Mˆ '() Pˆ *() Φˆ ( + ; ) (35) Φˆ *( + ; ) = Φˆ ( + ; ) Mˆ () ˆ *() Pˆ *( ) = Φˆ '( + ; ) Pˆ *() Φˆ *( + ; ) + Qˆ () Inally we mus se Pˆ *(T) = Qˆ (T + ). u*() PHYSICAL SYSTEM TO BE CONTROLLED Mˆ () Fg. 4 Marx block dagram of opmal conroller Comparng equaons (35 37) wh (8 3) and Fg. 3 wh Fg. 4 we noce some neresng hngs whch are expressed precsely by Theorem 4. (Dualy Theorem) Problem I and Problem II are duals of each oher n he followng sense: Le τ. Replace every marx X() = X( + τ) n (8 3) by Xˆ '() = Xˆ '(T τ). Then One has (35 37). Conversely, replace every marx Xˆ (T τ) n (35 37) by X'( + τ). Then one has (8 3). Proof. Carry ou he subsuons. For ease of reference, he duales beween he wo problems are gven n deal n Table. Problem I Table x() (unobservable) sae varables of random process. y() observed random varables. Problem II x() (observable) sae varables of plan o be regulaed. u() conrol varables 3 frs observaon. T las conrol acon. 4 Φ( + τ +; + τ) ranson Φˆ (T τ + ; T τ) ransmarx. on marx. 5 P*( + τ) covarance of opmzed esmaon error. 6 *( + τ) weghng of observaon for opmal esmaon. 7 Φ*( + τ + ; + τ) ranson marx for opmal esmaon error. 8 M( + τ) effec of sae on observaon. 9 Q( + τ) covarance of random excaon. x( + ) un delay Φˆ ( + ; ) I T (36) (37) Pˆ *(T τ) marx of quadrac form for performance ndex under opmal regulaon. ˆ *(T τ) weghng of sae for opmal conrol. Φˆ *(T τ + ; T τ) ranson marx under opmal regulaon. Mˆ (T τ) effec of conrol vecors on sae. Qˆ (T τ) marx of quadrac form defnng error creron. Remarks. (n) The mahemacal sgnfcance of he dualy beween Problem I and Problem II s ha boh problems reduce o he soluon of he Wener-Hopf-lke equaon (3). (o) The physcal sgnfcance of he dualy s nrgung. Why are observaons and conrol dual quanes? x() ˆ *() Transacons of he ASME Journal of Basc Engneerng, 8 (Seres D): Copyrgh 96 by ASME

9 Recen research [9] has shown ha he essence of he Dualy Theorem les n he dualy of consrans a he oupu (represened by he marx Mˆ () n Problem I) and consrans a he npu (represened by he marx Mˆ () n Problem II). (p) Applcaons of Wener's mehods o he soluon of nosefree regulaor problem have been known for a long me; see he recen exbook of Newon, Gould, and Kaser [7]. However, he connecons beween he wo problems, and n parcular he dualy, have apparenly never been saed precsely before. (q) The dualy heorem offers a powerful ool for developng more deeply he heory (as opposed o he compuaon) of Wener flers, as menoned n Remark (). Ths wll be publshed elsewhere [9]. Applcaons The power of he new approach o he Wener problem, as expressed by Theorem 3, s mos obvous when he daa of he problem are gven n numercal form. In ha case, one smply performs he numercal compuaons requred by (8 3). Resus of such calculaons, n some cases of praccal engneerng neres, wll be publshed elsewhere. When he answers are desred n closed analyc form, he eraons (8 3) may lead o very unweldy expressons. In a few cases, * and Φ* can be pu no closed form. Whou dscussng here how (f a all) such closed forms can be obaned, we now gve wo examples ndcave of he ype of resuls o be expeced. Example. Consder he problem menoned under Opmal Esmaes. Le x () be he sgnal and x () he nose. We assume he model: x ( + ) = φ ( + ; )x () + u () x ( + ) = u () y () = x () + x () The specfc daa for whch we desre a soluon of he esmaon problem are as follows: = + ; = Ex () =,.e., x () = 3 Eu () = a, Eu () = b, Eu () u () = (for all ) 4 φ ( + ; ) = φ = cons. A smple calculaon shows ha he followng marces sasfy he dfference equaons (8 3), for all : φc( ) *() = φ[ C( )] Φ*( + ; ) = a + φ b C( ) P*( + ) = b b where C( + ) = (38) a + b +φ b C( ) Snce was assumed ha x () =, neher x () nor x () can be predced from he measuremen of y (). Hence he measuremen a me = s useless, whch shows ha we should se C() =. Ths fac, wh he eraons (38), compleely deermnes he funcon C(). The nonlnear dfference equaon (38) plays he role of he Wener-Hopf equaon. If b /a <<, hen C() whch s essenally pure predcon. If b /a >>, hen C(), and we depend manly on x *( ) for he esmaon of x *( + ) and assgn only very small wegh o he measuremen y () ; hs s wha one would expec when he measured daa are very nosy. In any case, x *( ) = a all mes; one canno predc ndependen nose! Ths means ha φ * can be se equal o zero. The opmal predcor s a frs-order dynamc sysem. See Remark (j). To fnd he saonary Wener fler, le = on boh sdes of (38), solve he resulng quadrac equaon n C( ), ec. Example. A number or parcles leave he orgn a me = wh random veloces; afer =, each parcle moves wh a consan (unknown) velocy. Suppose ha he poson of one of hese parcles s measured, he daa beng conamnaed by saonary, addve, correlaed nose. Wha s he opmal esmae of he poson and velocy of he parcle a he me of he las measuremen? Le x () be he poson and x () he velocy of he parcle; x 3 () s he nose. The problem s hen represened by he model, x ( + ) = x () + x () x ( + ) = x () x 3 ( + ) = φ 33 ( + ; )x 3 () + u 3 () y () = x () + x 3 () and he addonal condons = ; = Ex () = Ex () =, Ex () = a > ; 3 Eu 3 () =, Eu 3 () = b. 4 φ 33 ( + ; ) = φ 33 = cons. Accordng o Theorem 3, x*( ) s calculaed usng he dynamc sysem (3). Frs we solve he problem of predcng he poson and velocy of he parcle one sep ahead. Smple consderaons show ha a a P*() = a a and *() = b I s hen easy o check by subsuon no equaons (8 3) ha P*() = b C( ) φ 33( ) φ33( ) φ ( ) ( ) ( ) + ( ) 33 φ33 φ33 C s he correc expresson for he covarance marx of he predcon error x ( ) for all, provded ha we defne C () = b /a C () = C ( ) + [ φ 33 ( )], I s neresng o noe ha he resuls jus obaned are vald also when φ 33 depends on. Ths s rue also n Example. In convenonal reamens of such problems here seems o be an essenal dfference beween he cases of saonary and nonsaonary nose. Ths msleadng mpresson creaed by he convenonal heory s due o he very specal mehods used n solvng he Wener-Hopf equaon. Inroducng he abbrevaon C () = C () = φ 33 ( ), and observng ha cov x ( + ) = P*( + ) = Φ( + ; )[cov x ( )]Φ'( + ; ) + Q() Transacons of he ASME Journal of Basc Engneerng, 8 (Seres D): Copyrgh 96 by ASME

10 he marces occurrng n equaon (3) and he covarance marx of x ( ) are found afer smple calculaons. We have, for all, C( ) Φ(; + ) *() = C( ) C ( ) C ( ) ( ) C Φ(; + )Φ*( + ; )Φ( + ; ) and C( ) C( ) = C( ) C( ) C( ) + C( ) C ( ) C ( ) C ( ) C ( ) C ( ) + C ( ) cov x ( ) = E x ( ) x b '( ) = C( ) 3 φ33c( ) φ33c ( ) + φ ( ) 33C To gan some nsgh no he behavor of hs sysem, le us examne he lmng case of a large number of observaons. Then C () obeys approxmaely he dfferenal equaon dc ()/d C () ( >> ) from whch we fnd C () ( φ 33 ) 3 /3 + φ 33 ( φ 33 ) + φ 33 + b /a Usng (39), we ge furher, Φ Φ*Φ and Φ * ( >> ) (39) ( >> ) Thus as he number of observaons becomes large, we depend almos exclusvely on x *( ) and x *( ) o esmae x *( + + ) and x *( + + ). Curren observaons are used almos exclusvely o esmae he nose x 3 *( ) y *() x *( ) ( >> ) One would of course expec somehng lke hs snce he problem s analogous o fng a sragh lne o an ncreasng number of pons. As a second check on he reasonableness of he resuls gven, observe ha he case >> s essenally he same as predcon based on connuous observaons. Seng φ 33 =, we have E x a b ( ) ( >> ; φ 3 33 = ) b + a / 3 whch s dencal wh he resul obaned by Shnbro [], Example, and Solodovnkov [4], Example, n her reamen of he Wener problem n he fne-lengh, connuous-daa case, usng an approach enrely dfferen from ours. Conclusons Ths paper formulaes and solves he Wener problem from he sae pon of vew. On he one hand, hs leads o a very general reamen ncludng cases whch cause dffcules when aacked by oher mehods. On he oher hand, he Wener problem s shown o be closely conneced wh oher problems n he heory of conrol. Much remans o be done o explo hese connecons. References N. Wener, The Exrapolaon, Inerpolaon and Smoohng of Saonary Tme Seres, John Wley & Sons, Inc., New York, N.Y., 949. L. A. Zadeh and J. R. Ragazzn, An Exenson of Wener's Theory of Predcon, Journal of Appled Physcs, vol., 95, pp H. W. Bode and C. E. Shannon, A Smplfed Dervaon of Lnear Leas-Squares Smoohng and Predcon Theory, Proceednqs IRE, vol. 38, 95, pp R. C. Booon, An Opmzaon Theory for Tme-Varyng Lnear Sysems Wh Nonsaonary Sascal Inpus, Proceedngs IRE, vol. 4, 95, pp J. H. Lanng and R. H. Ban, Random Processes n Auomac Conrol, McGraw-Hll Book Company, Inc., New York, N. Y., W. B. Davenpor, Jr., and W. L. Roo, An Inroducon o he Theory of Random Sgnals and Nose, McGraw-Hll Book Company, Inc., New York, N. Y., S. Darlngon, Lnear Leas-Squares Smoohng and Predcon, Wh Applcaons, Bell Sysem Tech. Journal, vol. 37, 958, pp G. Frankln, The Opmum Synhess of Sampled-Daa Sysems Docoral dsseraon, Dep. of Elec. Engr., Columba Unversy, A. B. Lees, Inerpolaon and Exrapolaon of Sampled Daa, Trans. IRE Prof. Grou.p on Informaon Theory, IT-, 956, pp R. C. Davs. On he Theory of Predcon of Nonsaonary Sochasc Processes, Journal of Appled Physcs, vol , pp M. Shnbro, Opmzaon of Trne-Varyng Lnear Sysems Wh Nonsaonary Inpus, TRANS. ASME, vol. 8, 958. pp M. Blum, Recurson Formulas for Growng Memory Dgal Flers, Trans. IRE Prof. Group on Informaon Theory. IT , pp V. S. Pugachev, The Use of Canoncal Expansons of Random Funcons n Deermnng an Opmum Lnear Sysem, Auomacs and Remoe Conrol (USSR), vol. 7, 956, pp ; ranslaon pp V. V. Solodovnkov and A. M. Bakov. On he Theory of Self- Opmzng Sysems (n German and Russan), Proc. Hedelberg Conference on Auomac Conrol, 956. pp J. L. Doob, Sochasc Processes, John Wley & Sons, Inc., New York, N. Y., M. Loève, Probably Theory, Van Nosrand Company, Inc., New York, N. Y., R. E. Bellman, I. Glcksberg, and. A. Gross, Some Aspecs of he Mahemacal Theory of Conrol Processes, RAND Repor R-33, 958, 44 pp. 8 R. E. Kalman and R. W. Koepcke, Opmal Synhess of Lnear Samplng Conrol Sysems Usng Generalzed Performance Indexes, TRANS. ASME, vol. 8, 958, pp J. E. Berram, Effec of Quanzaon n Sampled-Feedback Sysems, Trans. AlEE, vol. 77, II, 958, pp R. E. Kalman and J. E. Berram, General Synhess Procedure for Compuer Conrol of Sngle and Mul-Loop Lnear Sysems Trans. AlEE, vol. 77, II, 958, pp C. W. Merram, III, A Class of Opmum Conrol Sysems, Journal of he Frankln Insue, vol. 67, 959, pp R. E. Kalman and J. E. Berram, A Unfed Approach o he Theory of Samplng Sysems, Journal of he Frankln Insue, vol. 67, 959, pp R. E. Kalman and R. W. Koepcke, The Role of Dgal Compuers n he Dynamc Opmzaon of Chemcal Reacors, Proc. Wesern Jon Compuer Conference, 959, pp R. E. Kalman, Dynamc Opmzaon of Lnear Conrol Sysems, I. Theory, o appear. 5 S. Sherman, Non-Mean-Square Error Crera, Trans. IRE Prof. Group on Informaon Theory, IT-4, 958, pp V. S. Pugachev, On a Possble General Soluon of he Problem of Deermnng Opmum Dynamc Sysems, Auomacs and Remoe Conrol (USSR), vol. 7, 956, pp G. C. Newon, Jr., L. A. Gould, and J. F. Kaser, Analycal Desgn of Lnear Feedback Conrols, John Wley & Sons, Inc., New York, N. Y., J. M. Smh, Feedback Conrol Sysems, McGraw-Hll Book Company, Inc., New York, N. Y., R. E. Kalman, On he General Theory of Conrol Sysems, Proceedngs Frs Inernaonal Conference on Auomac Conrol, Moscow, USSR, 96. Transacons of he ASME Journal of Basc Engneerng, 8 (Seres D): Copyrgh 96 by ASME

11 A P P E N D I X RANDOM PROCESSES: BASIC CONCEPTS For convenence of he reader, we revew here some elemenary defnons and facs abou probably and random processes. Everyhng s presened wh he umos possble smplcy; for greaer deph and breadh, consul Lanng and Ban [5] or Doob [5]. A random varable s a funcon whose values depend on he oucome of a chance even. The values of a random varable may be any convenen mahemacal enes; real or complex numbers, vecors, ec. For smplcy, we shall consder here only real-valued random varables, bu hs s no real resrcon. Random varables wll be denoed by x, y,... and her values by ξ, η,. Sums, producs, and funcons of random varables are also random varables. A random varable x can be explcly defned by sang he probably ha x s less han or equal o some real consan ξ. Ths s expressed symbolcally by wrng Pr(x ξ) = F x (ξ); F x ( ) =, F x (+ ) = F x (ξ) s called he probably dsrbuon funcon of he random varable x. When F x (ξ) s dfferenable wh respec o ξ, hen f x (ξ) = df x (ξ)/dξ s called he probably densy funcon of x. The expeced value (mahemacal expecaon, sascal average, ensemble average, mean, ec., are commonly used synonyms) of any nonrandom funcon g(x) of a random varable x s defned by x ( ξ) = g( ξ) fx ( ξ) Eg( x) = E[ g( x)] = g( ξ) df dξ (4) As ndcaed, s ofen convenen o om he brackes afer he symbol E. A sequence of random varables (fne or nfne) {x()} =, x( ), x(), x(), (4) s called a dscree (or dscree-parameer) random (or sochasc) process. One parcular se of observed values of he random process (4), ξ( ), ξ(), ξ(), s called a realzaon (or a sample funcon) of he process. Inuvely, a random process s smply a se of random varables whch are ndexed n such a way as o brng he noon of me no he pcure. A random process s uncorrelaed f If, furhermore, Ex()x(s) = Ex()Ex(s) ( s) Ex()x(s) = ( s) hen he random process s orhogonal. Any uncorrelaed random process can be changed no orhogonal random process by replacng x() by x () = x() Ex() snce hen Ex ()x (s) = E[x() Ex()] [x(s) Ex(s)] = Ex()x(s) Ex()Ex(s) = I s useful o remember ha, f a random process s orhogonal, hen E[x( ) + x( ) + ] = Ex ( ) + Ex ( ) + (...) If x s a vecor-valued random varable wh componens x,, x n (whch are of course random varables), he marx [E(x Ex )(x j Ex j )] = E(x Ex)(x' Ex') = cov x (4) s called he covarance marx of x. A random process may be specfed explcly by sang he probably of smulaneous occurrence of any fne number of evens of he ype x( ) ξ,, x( n ) ξ n ; ( n ),.e., Pr[(x( ) ξ,, x( n ) ξ n )] = F x(),..., x( n)(ξ,, ξ n ) (43) where F x(),..., x( n) s called he jon probably dsrbuon funcon of he random varables x( ),, x( n ). The jon probably densy funcon s hen f x(),..., x( n)(ξ,, ξ n ) = n F n(),..., x(n)/ ξ,, ξ n provded he requred dervaves exs. The expeced value Eg[x( ),, x( n )] of any nonrandom funcon of n random varables s defned by an n-fold negral analogous o (4). A random process s ndependen f for any fne n, (43) s equal o he produc of he frs-order dsrbuons Pr[x( ) ξ ] Pr[x( n ) ξ n ] If a se of random varables s ndependen, hen hey are obvously also uncorrelaed. The converse s no rue n general. For a se of more han random varables o be ndependen, s no suffcen ha any par of random varables be ndependen. Frequenly s of neres o consder he probably dsrbuon of a random varable x( n + ) of a random process gven he acual values ξ( ),, ξ( n ) wh whch he random varables x( ),, x( n ) have occurred. Ths s denoed by Pr[x( n + ) ξ n + x( ) = ξ,, x( n ) = ξ n ] ξ n+ = f x( ),..., x( f ) x( ),..., x( ) n+ n ( ξ,..., ξ ( ξ,..., ξ n+ n ) dξ ) n+ (44) whch s called he condonal probably dsrbuon funcon of x( n + ) gven x( ),, x( n ). The condonal expecaon E{g[x( n + )] x( ),, x( n )} s defned analogously o (4). The condonal expecaon s a random varable; follows ha E[E{g[x( n + )] x( ),, x( n )}] = E{g[x( n + )]} In all cases of neres n hs paper, negrals of he ype (4) or (44) need never be evaluaed explcly, only he concep of he expeced value s needed. A random varable x s gaussan (or normally dsrbued) f ( ξ Ex ) f x (ξ) = exp [πe(x Ex) ] / E( x Ex) whch s he well-known bell-shaped curve. Smlarly, a random vecor x s gaussan f f x (ξ) = exp (ξ Ex) C (ξ Ex) (π) n/ (de C) / where C s he nverse of he covarance marx (4) of x. A gaussan random process s defned smlarly. The mporance of gaussan random varables and processes s largely due o he followng facs: Theorem 5. (A) Lnear funcons (and herefore condonal expecaons) on a gaussan random process are gaussan random varables. (B) Orhogonal gaussan random varables are ndependen. (C) Gven any random process wh means Ex() and covarances Ex()x(s), here exss a unque gaussan random process wh he same means and covarances. Transacons of he ASME Journal of Basc Engneerng, 8 (Seres D): Copyrgh 96 by ASME

12 Explanaon of hs ranscrpon, John Lukesh, January. Usng a phoo copy of R. E. Kalman s 96 paper from an orgnal of he ASME Journal of Basc Engneerng, March 96 ssue, I dd my bes o make an accurae verson of hs raher sgnfcan pece, n an up-o-dae compuer fle forma. For hs I was able o choose page formang and ype fon spacngs ha resuled n a documen ha s a close mach o he orgnal. (All pages sar and sop a abou he same pon, for example; even mos ndvdual lnes of ex do.) I used a recen verson of Word for Wndows and a recen Hewle Packard scanner wh OCR (opcal characer recognon) sofware. The OCR sofware s very good on plan ex, even dsngushng beween alc versus regular characers que relably, bu does no do well wh subscrps, superscrps, and specal fons, whch were que prevalen n he orgnal paper. And I found here was no pon n ryng o work from he OCR resuls for equaons. A lo of manual labor was nvolved. Snce I waned o make a fahful reproducon of he orgnal, I dd no make any changes o correc (wha I beleved were) msakes n. For example, equaon (3) has a P*()M() produc ha should be reversed, I hnk. I lef hs, and some oher hngs ha I hough were msakes n he orgnal, as s. (I ddn fnd very many oher problems wh he orgnal.) There may, of course, be problems wh my ranscrpon. The plan ex OCR resuls, whch ddn requre much edng, are prey accurae I hnk. Bu he subscrps ec and he equaons whch I coped essenally manually, are suspec. I ve revewed he resulng documen que carefully, several mes fndng msakes n wha I dd each me. The las me here were fve, four cosmec and one farly nconsequenal. There are probably more. I would be very pleased o know abou f any reader of hs fnds some of hem; jlukesh@delane.com.