Lainiotis filter implementation. via Chandrasekhar type algorithm


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1 Joural of Computatios & Modellig, vol.1, o.1, 2011, ISSN: prit, olie Iteratioal Scietific Press, 2011 Laiiotis filter implemetatio via Chadrasehar type algorithm Nicholas Assimais 1 ad Maria Adam 2 Abstract A implemetatio of the time ivariat Laiiotis filter usig a Chadrasehar type algorithm is preseted ad compared to the classical oe. he size of model determies which algorithm is faster; a method is proposed to apriori decide, which implemetatio is faster. I the ifiite measuremet oise case, the proposed method is always faster tha the classical oe. Mathematics Subject Classificatio : 93E11, 93C55, 68Q25 Keywords: Laiiotis filter, Chadrasehar algorithm 1 Itroductio Estimatio plays a importat role i may fields of sciece. he estimatio problem has bee solved by meas of a recursive algorithm based o Riccati 1 Departmet of Electroics, echological Educatioal Istitute of Lamia, Lamia 35100, Greece, 2 Departmet of Computer Sciece ad Biomedical Iformatics, Uiversity of Cetral Greece, Lamia 35100, Greece, Article Ifo: Revised : July 14, Published olie : August 31, 2011
2 116 Laiiotis filter implemetatio via Chadrasehar type algorithm type differece equatios. I the last decades, several authors have proposed faster algorithms to solve the estimatio problems by substitutig the Riccati equatios by a set of Chadrasehar type differece equatios [1, 5, 7, 8, 9]. he discrete time Laiiotis filter [6] is a well ow algorithm that solves the estimatio/filterig problem. I this paper, we propose a implemetatio of the time ivariat Laiiotis filter usig Chadrasehar type recursive algorithm to solve the estimatio/filterig problem. It is established that the classical ad the proposed implemetatios are equivalet with respect to their behavior. It is also developed a method to apriori before the Laiiotis filter s implemetatio decide which implemetatio is faster. his is very importat due to the fact that, i most realtime applicatios, it is essetial to obtai the estimate i the shortest possible time. he paper is orgaized as follows: I Sectio 2 the classical implemetatio of Laiiotis filter is preseted. I Sectio 3 the Chadrasehar type algorithm is preseted ad the proposed implemetatio of Laiiotis filter via Chadrasehar type algorithm is itroduced. I Sectio 4 the computatioal requiremets both implemetatios of Laiiotis filter are established ad comparisos are carried out. It is poited out that the proposed implemetatio may be faster tha the classical oe. I additio, a rule is established i order to decide which implemetatio is faster. 2 Classical Implemetatio of Laiiotis filter he estimatio problem arises i liear estimatio ad is associated with time ivariat systems described for 0 by the followig state space equatios: x +1 = F x + w z = Hx + v where x is the 1 state vector at time, z is the m 1 measuremet vector, F is the system trasitio matrix, H is the m output matrix, {w } ad {v } are idepedet Gaussia zeromea white ad ucorrelated radom processes, respectively, Q is the plat oise covariace matrix, R is the m m measuremet oise covariace matrix ad x 0 is a Gaussia
3 N. Assimais ad M. Adam 117 radom process with mea x 0 ad covariace P 0. I the sequel, cosider that Q, R are positive defiite matrices ad deote Q, R > O. he filterig problem is to produce a estimate at time L of the state vector usig measuremets till time L, i.e. the aim is to use the measuremets set {z 1, z 2,..., z L } i order to calculate a estimate value x L L of the state vector x L. he discrete time Laiiotis filter [6] is a well ow algorithm that solves the filterig problem. he estimatio x ad the correspodig estimatio error covariace matrix P at time are computed by the followig equatios, cosistig the Laiiotis filter, [ ] P = P + F I + P O P F 1 [ ] x = F I + P O x + [ ] + K + F I + P O P K m z +1 2 for 0, with iitial coditios P 0 0 = P 0 ad x 0 0 = x 0, where the followig costat matrices are calculated offlie: A = [ HQH + R ] 3 K = QH A 4 K m = F H A 5 P = Q K HQ = Q QH AHQ 6 F = F K HF = F QH AHF 7 O = K m HF = F H AHF 8 with F is a matrix, while K, K m are m matrices. he matrices P ad O are symmetric. Also, the m m matrix HQH + R is osigular, sice R > O, which meas that o measuremet is exact; this is reasoable i physical problems. Moreover, sice Q, R > O, A is a well defied m m symmetric ad positive defiite matrix as well as O is positive defiite. Furthermore, sice Q > O usig the matrix iversio Lemma 3 ad substitutig the matrix A by 3 i 6 we may rewrite P as P = Q QH [ HQH + R ] HQ = [ Q + H R H ] 9 3 Let A, C be osigular matrices, the holds: A + BCD = A A BC + DA B DA
4 118 Laiiotis filter implemetatio via Chadrasehar type algorithm from which it is clear that the symmetric P is a positive defiite matrix. Equatio 1 is the Riccati equatio emaatig from Laiiotis filter. I the case of ifiite measuremet oise R, we have A = O, K = O, K m = O, P = Q, F = F, O = O ad the Laiiotis filter becomes: P = P + F P F = Q + F P F 10 x = F x = F x 11 Equatio 10 is the Lyapuov equatio emaatig from Laiiotis filter. 3 Implemetatio of Laiiotis filter via Chadrasehar type algorithm For time ivariat systems, it is well ow [2] that if the sigal process model is asymptotically stable i.e. all eigevalues of F lie iside the uit circle, the there exists a steady state value P of the estimatio error covariace matrix. he steady state solutio P is calculated by recursively implemetig the Riccati equatio emaatig from Laiiotis filter 1 for = 0, 1,..., with iitial coditio P 0 0 = P 0. he steady state or limitig solutio of the Riccati equatio is idepedet of the iitial coditio [2]. he discrete time Riccati equatio emaatig from the Laiiotis filter equatios has attracted eormous attetio. I view of the importace of the Riccati equatio, there exists cosiderable literature o its recursive solutios [4, 7], cocerig per step or doublig algorithms. he Chadrasehar type algorithm has bee used [7, 8] to solve the Riccati equatio 1. he Chadrasehar type algorithm cosists of the recursio P = P + Y S Y usig recursios for the suitable quatities Y ad S. Hece, the algorithm is based o the idea of defiig the differece equatio ad its factorizatio δp = P P, 12 δp = Y S Y, 13
5 N. Assimais ad M. Adam 119 where Y is a r matrix ad S is a r r matrix, with For every = 0, 1,..., deotig 0 r = raδp 0. O = P + 14 we ote that O is a symmetric ad positive defiite matrix due to the presece of O, recallig that O i 8 is a positive defiite matrix ad P is a positive semidefiite as estimatio error covariace matrix. Also, sice O is a osigular matrix for every = 0, 1,..., the equatio 14 may be writte: P = I O 15 Usig the above otatios ad substitutig the equatios of the Laiiotis filter by a set of Chadrasehar type differece equatios, a recursive filterig algorithm is proposed, as established i the followig theorem, which presets computatioal advatage compared to the classical filterig algorithm, see 4 ad 5 statemets i the ext Sectio 4. heorem 3.1. Let the measuremet oise R be a positive defiite matrix, the plat oise Q be a positive defiite matrix ad P is a osigular matrix, for every = 0, 1, 2,... he set of the followig recursive equatios compose the ew algorithm for the solutio of the discrete time Laiiotis filter, O +1 = O + Y S Y 16 Y +1 = F O Y 17 S +1 = S S Y +1 Y S 18 P = P + Y S Y 19 x = F with iitial coditios: x + K + F P K m z+1, 20 P 0 0 = P 0 x 0 0 = x 0 O 0 = P Y 0 S 0 Y 0 = P + F [I + P 0 O ] P 0 F P 0 22 where F, O, K, K m, P are the matrices i 38.
6 120 Laiiotis filter implemetatio via Chadrasehar type algorithm Proof. Combiig 14 ad 12 we write O +1 = P = P O P = O + δp, i.e., for every = 0, 1, 2,..., holds δp = O +1 O, 23 i which substitutig δp by 13 the recursio equatio i 16 is obvious. Moreover, usig elemetary algebraic operatios ad properties we may write M + N = NM + N M, M N = N N MM, 24 whe M, N are osigular matrices, as well as [ I + P O ] P = [ ] P P + O P = [P + O ], 25 due to the osigularity of P for every = 0, 1,.... Combiig 12, 1, 25, the first equality i 24, 14 ad 15, we derive: δp +1 = P P = F [I + P+1 +1 O ] P [I + P O ] P F = F [P O ] [P + O ] F [ ] [ = F O P O P O P + O = F P+1 +1 [P O ] O P [P + = F P+1 +1 [P O ] P [P + O ] O = F P P O F = F O +1 O F P ] ] O F Usig the secod equality of 24 ad 23 the last equatio may be writte as: δp +1 = F = F = F = F +1 O +1 O O O O +1 δp O +1 δp δp O O F O F +1 δp F F O F F δp +1 δp F O F
7 N. Assimais ad M. Adam 121 From 13 the last equatio yields δp +1 = F Y S Y O F F i which settig the matrix Y +1 = F Y S Y +1 Y S Y Y by 17 immediately arises O F δp +1 = Y +1 S Y +1 Y +1 S Y +1 Y S Y By 13 we have δp +1 formulated as : = Y +1 S +1 Y+1, thus the equatio i 26 may be Y +1 S +1 Y +1 = Y +1 S S Y +1 Y S Y +1 Multiplyig with Y +1 o the left ad Y +1 o the right the last equality the recursio equatio i 18 is derived. Furthermore, rewritig x i 2 with differet way ad due to 14 we coclude [ ] x = F O I + P O x + [ ] + K + F O I + P O P K m z +1 [ ] = F O I + P O O x + [ ] + K + F O I + P O O P K m = F = F [ P + showig thus the equatio 20. ] x + K + F x + K + F P K m z+1 z +1 [ P + ] P K m z +1 Moreover, P 0 0 = P 0 ad x 0 0 = x 0 are give as the iitial coditios of the problem; by 14 O 0 is computed for = 0 ad the matrices Y 0, S 0 are computed by the factorizatio of the matrix P + F [I + P 0 O ] P 0 F P 0 i 22 i order to used as iitial coditios. Remar For the boudary values of r = raδp 0 we ote that: If r = 0, the, from 12 arises that the estimatio covariace matrix remais costat, i.e. P = P 0, ad equatio 23 yields O = O 0, for every = 0, 1, 2,... hus the algorithm of heorem 3.1 computes iteratively oly the estimatio x taig the form: x = F O O0 x + K + F O 0 P 0 K m z+1
8 122 Laiiotis filter implemetatio via Chadrasehar type algorithm If r = ad P 0 0 = P 0 = O, the we are able to use the iitial coditios Y 0 = I ad S 0 = P. 2. For the zero iitial coditio P 0 0 = P 0 = O, by 1 we derive P 1 1 = P ; recallig that by 9 holds P > O, it is evidet that for every = 1, 2,... arises P > O, that guaratees P be a osigular matrix. Hece heorem 3.1 is applicable for iitial coditio P 0 0 = P 0 = O; i this case by we are able to use the followig iitial coditios: O 0 = ad Y 0 S 0 Y 0 = P. 3.1 Ifiite measuremet oise R I the followig, the special case of ifiite measuremet oise is preseted. I this case P = Q, F = F ad O = O, the the Riccati equatio 1 becomes the Lyapuov equatio 10. Usig 12 ad combiig 10 with 13 we have δp +1 = P P = F P P F = F δp F = F Y S Y F, where settig Y +1 = F Y the above equality is formulated Y +1 S +1 Y δp +1 = Y +1 S Y+1 ad after some algebra arises : S +1 = S +1 = Sice the last equality of the matrices holds for every = 1, 2,..., without loss of geerality, we cosider a arbitrary r r symmetric matrix S = S, 27 with ras = r ad 0 < r. hus, usig 19, 11 ad 27 the followig filterig algorithm, which is based o the Chadrasehar type algorithm, is established. ad with iitial coditios: Y +1 = F Y P = P + Y SY x = F x, P 0 0 = P 0, x 0 0 = x 0, Y 0 SY 0 = Q + F P 0 F P 0
9 N. Assimais ad M. Adam 123 Sice i 27 the matrix S ca be arbitrarily chose, we propose as S the r r idetity matrix; thus we are able to establish the proposed algorithm, which is formulated i the ext theorem. heorem 3.2. Let R be the ifiite measuremet oise R, the plat oise Q be a positive defiite matrix ad F be a trasitio matrix. he set of the followig recursive equatios compose the algorithm for the solutio of the discrete time Laiiotis filter, for = 1, 2,..., with iitial coditios: Y +1 = F Y 28 P = P + Y Y 29 x = F x, 30 P 0 0 = P 0, x 0 0 = x 0, Y 0 Y 0 = Q + F P 0 F P 0 31 Remar 3.2. I the special case P 0 0 = P 0 = O, the equatio 31 becomes Y 0 Y 0 = Q. 4 Computatioal compariso of algorithms he two implemetatios of the Laiiotis filter preseted above are equivalet with respect to their behavior: they calculate theoretically the same estimates, due to the fact that equatios 12 are equivalet to equatios i heorem 3.1 i.e ad equatios are equivalet to equatios for the case of ifiite measuremet oise. he, it is reasoable to assume that both implemetatios of the Laiiotis filter compute the estimate value x L L of the state vector x L, executig the same umber of recursios. hus, i order to compare the algorithms, we have to compare their per recursio calculatio burde required for the olie calculatios; the calculatio burde of the offlie calculatios iitializatio process is ot tae ito accout. he computatioal aalysis is based o the aalysis i [3]: scalar operatios are ivolved i matrix maipulatio operatios, which are eeded for the implemetatio of the filterig algorithms. able 1 summarizes the calculatio burde of eeded matrix operatios.
10 124 Laiiotis filter implemetatio via Chadrasehar type algorithm able 1. Calculatio burde of matrix operatios Matrix Operatio Calculatio Burde A m + B m = C m m A + B = S S : symmetric I + A = B I : idetity A m Bm = C 2m A m Bm = S S : symmetric 2 m + m [A ] = B he per recursio calculatio burde of the Laiiotis filter implemetatios are summarized i able 2. he details are give i the Appedix. able 2. Per recursio calculatio burde of algorithms Implemetatio Noise Per recursio calculatio burde Classical R > O CB c,1 = m + 2m Classical R CB c,2 = Proposed R > O CB p,1 = r 2 Proposed R CB p,2 = 3 2 r From able 2, we derive the followig coclusios: 2r r m + 2m 1. he per recursio calculatio burde of the classical implemetatio depeds o the state vector dimesio. 2. he per recursio calculatio burde of the proposed implemetatio depeds o the state vector dimesio ad o r = raδp Cocerig the oifiite measuremet oise case R > O ad defiig q, r = CB c,1 CB p,1, from able 2 the respective calculatio burdes yield the relatio: q, r = r 2 + 2r 7 2 r 32 From Remar 3.1 the case r = 0 gives degeerated algorithm; thus cosider r 1 we ivestigate two cases : a r =, ad b r <. a 1 r =. I this case, it is obvious that q, = ad sice q, is a decreasig fuctio, we compute q, q1, 1 = 6 < 0. Hece, if r =, the the classical implemetatio is faster tha the proposed oe.
11 N. Assimais ad M. Adam 125 b 1 r <. I this case, we rewrite the equality i 32 as q, r = 6 8r r = fr,,33 6 with fr, = 8r r he discrimiat of fr, is ad its zeros are : = > 0, r 1 = , r 2 = Hece, the factorizatio of fr, is fr, = 8r r 1 r r 2, thus, the equality of q, r i 33 ca bee writte as q, r = 3r r 1 r r Also, it is easily proved that for = 1, 2,... holds > 42 12, from which immediately arises r 1 < 0 ad r 2 > 0; thus, due to the fact r 1, it is obvious r r 1 > 0. Cosequetly, i 35 the sig of q, r depeds o the sig of r r 2, with r 2 i 34, i.e., the choice of implemetatio of the suitable algorithm is related to the compariso of quatities r, r 2 ; if r > r 2 q, r < 0, thus the classical implemetatio is faster tha the proposed oe. if r < r 2 q, r > 0, thus the proposed implemetatio is faster tha the classical oe. 4. Figure 1 depicts the relatio betwee ad r that may hold i order to decide, which implemetatio is faster. I fact r is plotted as fuctio of usig r 2 i 34. he, we are able to establish the followig Rule of humb: the proposed Laiiotis filter implemetatio via Chadrasehar type algorithm is faster tha the classical implemetatio if the followig relatio holds: r <
12 126 Laiiotis filter implemetatio via Chadrasehar type algorithm Figure 1: Proposed algorithm may be faster tha the classical oe. hus, we are able to choose i advace the implemetatio of the faster algorithm comparig oly the quatities r ad by Cocerig the ifiite measuremet oise case R, the calculatio burde of the classical implemetatio is greater tha or equal to the calculatio burde of the proposed implemetatio; the equality holds for r =. hus, the proposed implemetatio is faster tha the classical oe. ACKNOWLEDGEMENS. he authors are deeply grateful to referees for suggestios that have cosiderably improved the quality of the paper. Refereces [1] Abdelhaim Aouche ad Fayçal Hamdi, Calculatig the autocovariaces ad the lielihood for periodic V ARMA models, Joural of Statistical Computatio ad Simulatio, 793, 2009, [2] B.D.O. Aderso, J.B. Moore, Optimal Filterig, Pretice Hall ic., [3] N. Assimais, M. Adam, Discrete time Kalma ad Laiiotis filters compariso, It. Joural of Mathematical Aalysis IJMA, , 2007, [4] N.D. Assimais, D.G. Laiiotis, S.K. Katsias, F.L. Saida, A survey of recursive algorithms for the solutio of the discrete time Riccati equatio, Noliear Aalysis, heory, Methods ad Applicatios, 30, 1997, [5] J.S. Baras ad D.G. Laiiotis, Chadrasehar algorithms for liear time varyig distributed systems, Iformatio Scieces, 172, 1979, [6] D.G. Laiiotis, Discrete Riccati Equatio Solutios: Partitioed Algorithms, IEEE rasactios o AC, AC20, 1975,
13 N. Assimais ad M. Adam 127 [7] D.G. Laiiotis, N.D. Assimais, S.K. Katsias, A ew computatioally effective algorithm for solvig the discrete Riccati equatio, Joural of Mathematical Aalysis ad Applicatios, 1863, 1994, [8] S. Naamori, A. HermosoCarazo, J. JiméezLópez ad J. LiaresPérez, Chadrasehartype filter for a widesese statioary sigal from ucertai observatios usig covariace iformatio, Applied Mathematics ad Computatio, 151, 2004, [9] S. Naamori, Chadrasehartype recursive Wieer estimatio techique i liear discretetime stochastic systems, Applied Mathematics ad Computatio, 1882, 2007,
14 128 Laiiotis filter implemetatio via Chadrasehar type algorithm Appedix Calculatio burdes of algorithms A Measuremet oise is a positive defiite matrix R > O A.1 Classical implemetatio of Laiiotis filter Matrix Operatio Matrix Dimesios Calculatio Burde P O I + P O + [I + P O ] [I + P O ] P F [I + P O ] P F [I + P O ] P F P = P + F [I + P O ] P F F [I + P O ] P K m m 2 2 m m K + F [I + P O ] P K m m + m m K + F [I + P O ] P K m z+1 m m 1 2m F [I + P O ] F [I + P O ] x x = F [I + P O ] x + + K + F [I + P O ] P K m z otal I idetity matrix symmetric matrix CB c,1 = m + 2m
15 N. Assimais ad M. Adam 129 A.2 Proposed implemetatio via Chadrasehar type algorithm Matrix Operatio Matrix Dimesios Calculatio Burde Y S r r r 2r 2 r Y S Y r r 2 r + r O +1 = O + Y S Y Y r 2 2 r r Y +1 = F O Y r 2 2 r r Y S r 2 2 r r S Y O +1 Y S r r r 2 + r 1 2 r2 + r S +1 = S S Y +1 Y S r r + r r 1 2 r2 + r P = P + Y S Y F O F O x F O P F O P K m m 2 2 m m K + F O P K m m + m m K + F O P K m z+1 m m 1 2m x = F x + + K + F O P K m z+1 otal CB p,1 = r 2 2r r m + 2m symmetric matrix
16 130 Laiiotis filter implemetatio via Chadrasehar type algorithm B Ifiite measuremet oise R B.1 Classical implemetatio of Laiiotis filter Matrix Operatio Matrix Dimesios Calculatio Burde F P F P F P = Q + F P F x = F x otal CB c,2 = symmetric matrix B.2 Proposed implemetatio via Chadrasehar type algorithm Matrix Operatio Matrix Dimesios Calculatio Burde Y +1 = F Y r 2 2 r r Y Y r r 2 r + r P = P + Y Y x = F x symmetric matrix otal CB p,2 = 3 2 r + 2 2
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