A Note on Sampling and Parameter Estimation in Linear Stochastic Systems

Transcription

1 22 IEEE RASACIOS O AUOMAIC COROL, VOL. 44, O., OVEMBER 999 REFERECES [] M. O. Ahma an J. D. Wang, An analytcal least square soluton to the esgn roblem of two-mensonal FIR flters wth quarantally symmetrc or antsymmetrc frequency resonse, IEEE rans. Crcuts Syst., vol. 36, , 989. [2] S. C. Pe an J. J. Shyu, Fast esgn of 2-D lnear-hase comlex FIR gtal flters by analytcal least squares methos, IEEE rans. Sgnal Processng, vol. 44, , 996. [3] W. H. Kwon an O. K. Kwon, FIR flters an recursve forms for contnuous tme-nvarant state-sace moels, IEEE rans. Automat. Contr., vol. 32, , 987. [4] W. H. Kwon, K. S. Lee, an O. K. Lee, Otmal FIR flters for tmevaryng state-sace moels, IEEE rans. Aeros. Electron. Syst., vol. 26,. 2, 99. [5] R. E. Kalman an R. S. Bucy, ew results n lnear flterng an recton theory, rans. ASME J. Basc Eng., vol. 83,. 95 8, 96. [6] A. H. Jazwnsk, Lmte memory otmal flterng, IEEE rans. Automat. Contr., vol. 3, , 968. [7] R. J. Ftzgeral, Dvergence of the Kalman flter, IEEE rans. Automat. Contr., vol. 6, , 97. [8] A. M. Brucksten an. Kalath, Recursve te memory flterng an scatterng theory, IEEE rans. Inform. heory, vol. 3, , 985. [9] S. H. Park, W. H. Kwon, O. K. Kwon, an M. J. Km, Short-tme Fourer analyss usng otmal harmonc FIR flters, IEEE rans. Sgnal Processng, vol. 45, , 997. [] W. H. Kwon, P. S. Km, an P. Park, A receng horzon Kalman FIR flter for screte tme-nvarant systems, IEEE rans. Automat. Contr., vol. 44, , Set [] W. H. Kwon an A. E. Pearson, A mofe quaratc cost roblem an feeback stablzaton of a lnear system, IEEE rans. Automat. Contr., vol. 22, , 977. [2] W. H. Kwon, Avances of rectve controls: theory an alcatons, n Proc. st Asan Contr. Conf., okyo, Jaan, Oct , 994. [3] P. G. Kamnsk, A. E. Bryson, an S. F. Schmt, Dscrete square root flterng: A survey of current technques, IEEE rans. Automat. Contr., vol. 6, , 97. [4] B. D. O. Anerson an J. B. Moore, ew results n lnear system stablty, Sam J. Contr., vol. 7, 969. [5]. Mname, M. Ohno, an H. Imanaka, Recursve solutons to eabeat observers for tme-varyng multvarable systems, Int. J. Contr., vol. 45, 987. [6] K. J. Åström an B. Wttenmark, Comuter Controlle Systems. Englewoo Clffs, J: Prentce-Hall, 984. A ote on Samlng an Parameter Estmaton n Lnear Stochastc Systems. E. Duncan, P. Manl, an B. Pask-Duncan Abstract umercal fferentaton formulas that yel consstent least squares arameter estmates from samle observatons of lnear, tme nvarant hgher orer systems have been ntrouce revously by Duncan et al. he formulas gven by Duncan et al. have the same tng system of equatons as n the contnuous tme case. he formula resente n ths note can be characterze as reservng asymtotcally a artal ntegraton rule. It leas to tng equatons for the arameter estmates that are fferent from the contnuous case, but they agan mly consstency. he numercal fferentaton formulas gven here can be use for an arbtrary lnear system, whch s not the case n the revous aer by Duncan et al. Inex erms Estmaton, lnear stochastc systems, numercal fferentaton for stochastc systems, samlng. I. IRODUCIO In a revous aer by the authors [2], the followng arameter estmaton roblem n lnear stochastc fferental equatons s consere. Let (X(t); t ) be an n-mensonal rocess satsfyng X () (t) = where f ()X () (t) +g()u (t) = X () (t) = t X() (t); =; 2; ; t + W (t) X () (t) =X(t) (2) where (U (t); t ) s a nonantcatve nut rocess, (W (t); t ) s an n-mensonal Wener rocess wth the local varance matrx h; =( ; ; ) s a -mensonal unknown arameter f () =f + j f j ; g() =g + j g j : j= j= In these escrtons, f j; g j for ; j = ; ; are known matrces. he true value of the unknown arameter s enote by = ( ; ; ). he least squares estmaton of from the observaton of () (X t ;U t ; t 2 [; ]) s etermne by mnmzng the formal quaratc functonal X () = f ()X () g()u 2 L X () f ()X () g()u X () LX () t = (3) Manuscrt receve January 26, 998. Recommene by Assocate Etor, J. C. Sall. hs work was suorte n art by the atonal Scence Founaton uner Grant DMS E. Duncan an B. Pask-Duncan are wth the Deartment of Mathematcs, Unversty of Kansas, Lawrence, KS 6645 USA (e-mal: uncan@math.ukans.eu). P. Manl s wth the Deartment of Probablty an Mathematcal Statstcs, Charles Unversty, Prague, Czech Reublc. Publsher Item Ientfer S (99) /99$. 999 IEEE

2 IEEE RASACIOS O AUOMAIC COROL, VOL. 44, O., OVEMBER where L s a ostve semefnte matrx. Prme enotes the transoston of vectors an matrces. he unefne term X () LX () s cancelle, an X () t = X () n (3). By mnmzng (3), the followng famly of equatons for the least squares estmate 3 ( )=( 3 ( ); ; 3 ( )) of s obtane: k= 2 L = h= = f jx () + g ju f hk X (h) + g k U t 3k ( ) = 2 L X () f jx () + g ju h= f h X (h) + g U j =; ;: (4) It s assume n [2] that screte observatons of (X(t);t 2 [;]) an (U (t); t2 [;]) wth the unform samlng nterval > are only avalable yelng the observe ranom varables X m; = X(m); U m; = U (m); m =; ;+ n: (5) he notaton U s use to nclue the case when the rouct g()u (t) eens only on some coornates of U. It s assume that all of these coornates are observe. o aroxmate (4) usng only ranom varables n (5), a substtuton for the ervatves X () (m) by the forwar fferences D X m; = D X m+; D X m; ; =; 2; ; (6) s erforme an some numercal ntegraton formulas are use to evaluate the ntegrals. Denotng by ^ = ^ ; ; ^ the estmate so obtane by these substtutons t s esrable that the consstency roerty! t! ^ = (7) s satsfe. In ths exresson! ^ enotes the t n robablty, whch uner arorate hyotheses s a nonranom quantty. In [4] an [6] t s note that the forwar an backwar Euler aroxmatons cannot be use, but t s shown how to mofy the aroxmaton for the hghest ervatve to satsfy (7). In [6] a secfc aroxmaton of the elta oerator s gven for the samle ata. It s shown n [] for n =2an n [2] for n 2 that (7) oes not hol unless a correcton term s ntrouce nto the equatons for ^ or unless (6) s mofe. A numercal fferentaton formula that etermnes estmates satsfyng (7) s gven. In ths note, a fferent metho s gven for the numercal evaluaton of (4) that satsfes (7). he metho s base only on the ranom varables (5) an t emloys together wth (6) the backwar fferences BX m; =(X m; X m; )= B X m; = B X m; B X m; : (8) Whle the metho n [2] estmate the fference from the case of contnuous observatons, the metho resente here exlots the nfntesmal roertes of the covarance functon of the rocess. he metho s more general n the sense that t ales to the case when the fferentaton s subjecte to whte nose. hs generalzaton has the followng motvaton. It s known (see, e.g., [3]) that a statonary Gaussan rocess (X(t);t 2 ) wth the sectral ensty h f () = j() () j 2 can be reresente as a soluton of the stochastc fferental equaton X () = X () + 2 X () + + X () t + hw where (W (t); t ) s a stanar Wener rocess. Equaton (9) s a artcular case of (). More generally, a Gaussan rocess wth a ratonal sectral ensty f () = jb () + + b 2 + b j 2 j() () j 2 satsfes (9) where X () (t); = ; ;, enote the ranom rocesses satsfyng wth X () = X () t + W; =; ; = b ; = b + + j= j +j+ : hus, t s mortant to conser the followng generalzaton of (), (2) X () (t) =X () (t) t + W () (t); =; ; () where ((W () (t); ;W () (t);w(t)) ;t ) s a n- mensonal Wener rocess wth local varance matrx H. II. PARAMEER ESIMAIO Assume that () an () are satsfe wth =, an let the q-mensonal rocess (U (t); t ) be the soluton of the lnear stochastc fferental equaton U (t) =cu (t) t + W (t); U() = U where c s a constant matrx, an (W (t); t ) s a q-mensonal Wener rocess wth local varance matrx h that s neenent of (W (t); W () (t); ;W () (t); t ). o escrbe the evoluton of the entre moel, ntrouce the state vector (t) 2 n+q an the matrces F; H that are escrbe n block form accorng to the artton of (t) = F = X () (t). X () (t) U (t) I I I f f 2 f 3 f g c H = H () h where I enotes the entty matrx n n, an f = f ( ); =; ;; g= g( ): (9)

3 222 IEEE RASACIOS O AUOMAIC COROL, VOL. 44, O., OVEMBER 999 It easly follows that (t) =F (t) t + (t); X() = X where ( (t);t ) s a n + q-mensonal Wener rocess wth local varance matrx H. he followng assumton s mae. Assumton : F s a stable matrx. hs assumton mles that (t) has a tng Gaussan strbuton as t!wth zero mean an varance matrx, whch s the soluton of the Lyaunov equaton Proof: From the theory of lnear stochastc systems, t follows that for > R() = k= k k! F k (9) where (A) j enotes the block a j of the matrx A usng the arttonng n (). Consequently, R k =(F k ) ; k =; ; : (2) he artton of F + F + H =: (2) nto the blocks r j as =(r j ) (3) Obvously, R = r, an erformng successve multlcatons on by F, t follows that R = r 2 ; ;R = r : as ntrouce n () s use. he matrx on the left-han se of the system of equatons (4) that acts on 3 ( ) can be wrtten as for j; k = ; ;where F j =(f j ; ;f j ;g j ); F j LF k t j =; ;: By Assumton, ths famly of matrces nexe by > converges (n quaratc mean) as!to the matrx Q =(tr(f j LF k )) (4) Furthermore, usng (8), t follows that Hence, F j+ F = = F F j+ = R +j : (2) R. R cr +; cr +;+ hs equalty yels (8) for k =. By the Lyaunov equaton (2) : for j; k =; ;where tr( ) enotes the trace oerator. Assumton an the assumton of nonsngularty of Q guarantee the consstency of the famly ( 3 ( ); >) as!, that s, cr +;j = r j+;+ = r +;j+ ; Consequently, j =; ;+:! 3 ( )= a.s. c k r +; =() k r +;k+ ; k =; ; : (22) he screte aroxmaton of (4) resente here leas n the t to a fferent system of equatons. he assocate matrx has a form smlar to (4) wth the matrx n (2) relace by a matrx S whch s efne subsequently. Whle the methos ntrouce n [2] estmate the fference from the case of the contnuous observatons, the metho resente here exlots the nfntesmal roertes of the covarance functon. For > efne an thus R() = t! EX(t + )X(t) (5) R() =R() = t! EX(t )X(t) : Lemma : Let Assumton be satsfe. For > the followng equalty s satsfe: where R() = k= k k! R k (6) R k = r k+; ; k =; ; ; (7) R k = f R k + f 2R k+ + + f R +k +() k gr +;k+ ; k = ; +; ; 2 : (8) Usng (2) an (22), t follows that R +k =(F (F k = F )) R k. c k r +; = f R k + + f R +k + gc k r +; : hs equalty an (22) yel (8) for k =; ;. Conser next the observe ranom varables (5), an note that t follows from (6) an (8), resectvely, that D r X m; = r B q X m; = q r k= q k= () k r k X m+rk; (23) () k q k X mq+k; : (24) Lemma 2: Let Assumton be satsfe. For r; s = ; ; ; the followng equalty s satsfe:!! r m=s D r X m; B s X m; = R r+s : (25)

4 IEEE RASACIOS O AUOMAIC COROL, VOL. 44, O., OVEMBER Proof: Usng the Law of Large umbers an (5), (9) t follows that, for j; k =; ; ;! j m=k X m+j; X mk; = e (j+k)f : From (2), (23), an (24), t follows that! = r+s r D r X m; B s X m; m=s e F r I e F s I =(F r+s ) + O() = R r+s + O(): From ths equalty, (22) follows. Recall that U (t) s forme by those coornates of U (t) for whch g()u (t) eens. Further enote r u = EX () (t)u (t) ; =; ; t! r uu = EU(t)U (t) ; t! r uu = EU (t)u (t) : t! ote that r u = r u ; r uu = r uu. Lemma 3: Let Assumton be satsfe. Let X(t) be r-tmes fferentable an U (t) be s-tmes fferentable (n quaratc mean), an let r + s. hen!!!! m= m= D X m; U m; = r +; u ; =; ; (26) D X m; U m; = f r u + + f r u + gr uu : (27) Proof: Wthout loss of generalty, t can be assume that the coornates of U (t) are escrbe n the followng way: U (t) = U (t); U () (t); ; U (s) (t); U 3 (t) (28) where U () (t) enotes the ervatve of orer of U (t), an U 3 (t) contans the rest of the coornates. hen the matrces c; h have the form c = I I c 3 h = h 3 : (29) From the artton of the matrces ;F;etc., nto block matrces a fner artton s also emloye, wth nces ; j = +; ;+ s +2 referrng to the comonents of (28). In artcular, r u = r + ; = ; ;. Conser (26). Let v s be such that v r. Usng artal summaton f v > t follows that!! =!! m= D X m; U m; m= D v X m; B v U m; =() v (F v ) ;++v =() v r v+;++v : (3) From (29) an (2) t follows that r j+; =(F so that, ) j = r j; + ; =; ; ; j = +; ;+ q () v r v+;++v = r +;+ = r +; u whch yels (26). Smlarly, to verfy (27), t follows as n (3) that!! m= D X m; U m; =() v (F v ) ;++v =() v (F ) v; ++v : For v =, the last term n the equalty conces wth the rght-han se of (26). For v>, t follows from (2) that (F ) v+;+v = (F ) +v; v+ Reeatng ths argument, t follows that = r +v+;v+ = r v+;+v+ = (F ) v; +v+ : () v (F ) v; ++v =(F ) ; + whch establshes (27). ow the screte observaton verson of (4) s ntrouce by lettng the estmate ^ of be the soluton of k= 2 L = m= h= m= = f jb X m; + g j U m; f hk D h X m; + g k U m; = 2 L D X m; ^ k f j B X m; + g j U m; h= f h D h X m; g U m; (3) for j =; ;. o guarantee the consstency of ^ as! an!, a matrx ^Q that s analogous to Q n (4) s ntrouce by relacng by a matrx S. Let R ; ;R 2 ;r +; ; ;r +; be gven by (3), (7), an (8). Defne S := R R R r ;+ R R 2 R r 2;+ R R R 2 r ; + r +; r +; 2 r +; r +;+ he matrx ^Q s gven n the followng assumton. Assumton 2: he matrx : (32) ^Q := (tr(f j LF k S)); j; k =; ; (33) s nonsngular. heorem : Let Assumtons an 2 be satsfe, an let ^ be the soluton of (3). Let (X(t); t ) be r-tmes fferentable an (U (t); t ) be s-tmes fferentable (n quaratc mean), an let r + s. hen!! ^ = : (34)

5 224 IEEE RASACIOS O AUOMAIC COROL, VOL. 44, O., OVEMBER 999 Fg. Convergence of the estmator. Proof: It s suffcent to rove that the system of equatons, whch s obtane by erformng the two assages to the t, that s,!an! n (3), has the unque soluton. If!an! n (3), then the followng famly of lnear equatons for the estmate ^; s obtane by Lemmas 2 an 3. k= tr =tr L L h; + f hk R h+2f j + h f hk r hu g j g k r u f j + g k r uu g j ^ k R +f j +(f r u + + f r u + g k r uu )g j f h R h+2f j g k r u f j g r u f j ; j =; ;: (35) It s seen that ^Q s the matrx of the left han se for the system (35) so ths system of equatons has a unque soluton by Assumton 2. It only remans to verfy that (35) s satsfe wth ^ = ; ; ^ = : (36) Insertng the values n (36) nto the left-han se of (35) an recallng that f h = f h ( );g= g( ), the left-han se of (35) s tr L h; + h f h R h+f j + gr u f j f h r hu g j + gr uu g j ; j =; ;: (37) By Lemma, the frst two sums n the square bracket equals R +f j: Consequently, (37) s equal to the rght-han se of (35). he quaratc mean fferentablty of a rocess s etermne by the fferentablty of ts covarance functon Examle: Let (X(t); t 2 ) be a statonary Gaussan rocess wth the covarance functon R() =R e ajj cos b (38) where a; b are ostve, unknown constants. he sectral ensty corresonng to (38) s j + a 2 + b 2 j 2 f () =2R a j() 2 +2a + a 2 + b 2 j : (39) 2 Furthermore, t follows from (38) or (39) that X = X () t + W X () = Xt+ 2 X () t + 2 W where = (a 2 + b 2 ); = 2R a; 2 = 2a 2 = 2R a( a 2 + b 2 2a): (4) (4) he arameter =( ; 2 ) s estmate by (4) an (35) an 2 s estmate by the quaratc varaton of (X(t); t ). Estmates of a; b an R are obtane from (4) an (4). Exanng R() n terms of t follows that R() =R ajj + 2 (a2 b 2 ) (3ab2 a 3 )jj 3 + : Lettng L = I n (3), t s exresse as X (2) X 2 X () 2 2 (2) X t

6 IEEE RASACIOS O AUOMAIC COROL, VOL. 44, O., OVEMBER so (4) s (X) 2 t 3 ( )+ XX () t 32 ( ) = XX () (42) XX () t 3 ( )+ X () 2 t 32 ( ) = X () X () : (43) In the t as!, there are the two equatons R 3 () ar 32 () =R ar 2 (44) ar 3 () +r () = ar + 2 r 22: (45) For the scretze verson (3) of (42), (43), the t as! an! s fferent from (44), (45), secfcally R ^ () ar ^ 2 () =R (a 2 b 2 ) (46) ar ^ () +R (a 2 b 2 )^ 2 () =R (3ab 2 a 3 ): (47) he soluton s ^ =. he quaratc varaton can be use to estmate 2 by ^ 2 = m (X m+; X m; ) 2 : A numercal examle s escrbe grahcally (see Fg. ) where a =:5; b=2:, an R =so that = 4:25 an 2 = :. he convergence of famly of estmates s relatvely fast. If a longer tme nterval s use, then the famly of estmates for are closer to the true value. ACKOWLEDGME he authors thank Y. Yan for erformng the comutatons for the numercal examle. he authors also thank the referees for useful comments that mrove the note. Stochastc Control of Dscrete Systems: A Searaton Prncle for Wener an Polynomal Systems M. J. Grmble Abstract A new searaton rncle s establshe for systems reresente n screte frequency-oman Wener or olynomal forms. he LQG or H 2 otmal controller can be realze usng an observer base structure estmatng nose free outut varables that are fe back through a ynamc gan control block. Surrsngly, there are also two searaton rncle theorems, eenng uon the orer n whch the eal outut otmal control an the otmal observer roblems are solve. Inex erms Otmal control, olynomal systems, Wener theory. I. IRODUCIO he searaton rncle of stochastc otmal control theory has often been utlze for systems reresente n state equaton form. However, no such results have been establshe for systems reresente n transfer-functon or olynomal matrx form. he frequency oman aroach to otmal control an estmaton was ntate by Wener [], but two semnal contrbutons later establshe the man tools for synthess. hese contrbutons were unertaken n the same ero by Youla et al. [2] an by Kucera [3]. he searaton rncle that s well known n state-sace LQG synthess was not use n the frequency-oman solutons, although Kucera [4] rove neenent solutons of the LQ state feeback control an the Kalman flterng roblems. hus, n ths case, f the olynomal moels are relate back to a system escrbe n state equaton form, t s ossble to use the olynomal solutons to calculate the constant control an flter gans. he state-sace searaton rncle results can then be nvoke to obtan the LQG outut feeback controller. he searaton rncle was not, however, establshe n the olynomal settng. Moreover, there was no attemt to generalze the results to the case where the control law feeback nclue a reuce set of varables, such as lant outut estmates. he objectve of the analyss that follows s to use frequency oman moels an analyss, to establsh a new searaton rncle result for systems reresente n frequency oman matrx fracton form. REFERECES []. E. Duncan, P. Manl, an B. Pask-Duncan, On statstcal samlng for system testng, IEEE rans. Automat. Contr., vol. 39,. 8 22, 994. [2], umercal fferentaton an arameter estmaton n hgher orer stochastc systems, IEEE rans. Automat. Contr., vol. 4, , 996. [3] R. S. Ltser an A.. Shryaev, Statstcs of Ranom Processes, Vol. 2: Alcatons. ew York: Srnger-Verlag, 978. [4]. Söerström, H. Fan, S. Beg, an B. Carlsson, Can a least-squares ft be feasble for moelng contnuous-tme autoregressve rocesses from screte-tme ata?, n Proc. 34th IEEE Conf. Decson Contr., ew Orleans, 995, [5]. Söerström, H. Fan, B. Carlsson, an S. Beg, Least squares arameter estmaton of contnuous-tme ARX moels from scretetme ata, IEEE rans. Automat. Contr., vol. 42, , 997. [6]. Söerström, H. Fan, B. Carlsson, an M. Mossberg, Some aroaches on how to use the elta oerator when entfyng contnuoustme rocesses, n Proc. 36th IEEE Conf. Decson Contr., San Dego, CA, 997, II. POLYOMIAL SYSEM DESCRIPIO he lnear tme-nvarant screte-tme multvarable, fntemensonal system of nterest s llustrate n Fg.. he nose free system outut sequence s enote by fy(t)g, where y(t) 2 R r, an the observatons sgnal s enote by fz(t)g. he whte rvng nose sgnals f(t)g an fv(t)g reresent the sturbance, an measurement nose sgnals, resectvely. hese sgnals are statstcally neenent an the covarance matrces (R f > ): cov[(t); ( )] = I q t an cov[v(t); v( )] = R f t : () Manuscrt receve January 22, 999. Recommene by Assocate Etor, G. Gu. hs work was suorte by the Engneerng an Physcal Scences Research. he author s wth the Inustral Control Centre, Unversty of Strathclye, Glasgow G QE, U.K. (e-mal: m.grmble@eee.strath.ac.uk). Publsher Item Ientfer S (99) /99$. 999 IEEE