Recursive linear estimation for discrete time systems in the presence of different multiplicative observation noises

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1 Recursve lear estmato for dscrete tme systems the resece of dfferet multlcatve observato oses C. Sáchez Gozález,*,.M. García Muñoz Deartameto de Métodos Cuattatvos ara la Ecoomía y la Emresa, Facultad de Cecas Ecoómcas y Emresarales, Uversdad de Graada, Camus Uverstaro de Cartuja, s/, 80, Graada, Sa Abstract hs aer descrbes a desg for a least mea square error estmator dscrete tme systems where the comoets of the state vector, measuremet equato, are corruted by dfferet multlcatve oses addto to observato ose. We show how kow results ca be cosdered a artcular case of the algorthm stated ths aer Keywords : State estmato, multlcatve ose, ucerta observatos * Corresodg author. el.: ; Fax: E-mals addresses: csachez@ugr.es (C. Sachez-Gozalez); tgarcam@ugr.es (.M. García-Muñoz).

2 . Itroducto It was back 960 whe R.E. Kalma [] troduces hs well kow flter. Assumg the dyamc system s descrbed through a state sace model, Kalma cosders the roblem of otmum lear recursve estmato. From ths evet much other research work was develoed cludg dfferet hyothess framework about system oses (Kalma [], Medtch [3], Jazwsk [4], Kowalsk ad Szyal [5]). I all studes above metoed the estmated sgal (state vector) measuremet equato s oly corruted by addtve ose. Rajasekara et al. [6] cosder the roblem of lear recursve estmato of stochastc sgals the resece of multlcatve ose addto to measuremet ose. Whe multlcatve ose s a Beroull radom varable, the system s called system wth ucerta observatos. he estmato roblem about these systems have bee extesvely treated (Nah [7], Hermoso ad Lares [8], Sachez ad García [9]). hs aer descrbes a desg for a least mea square error (LMSE) estmator dscrete tme systems where the comoets of the state vector, measuremet equato, are corruted by dfferet multlcatve oses addto to observato ose. he estmato roblems treated clude oe-stage redcto ad flterg. he reseted algorthm ca be cosdered as a geeral algorthm because, wth artcular secfcatos, ths algorthm degeerates kow results as Kalma [], Rajasekara ad Szyal [6], Nah [7],, Sachez ad García [0]. It ca also be fered that f multlcatve oses are Beroull radom varables, such stuato s ot, roerly seakg, a system wth ucerta observatos because the comoets of the state ca be reset the observato wth dfferet robabltes. herefore, the reseted algorthm solves the estmato roblems ths ew system secfcato wth comlete ucertaty about sgal.. Statemet ad Notato We ow troduce symbols ad deftos used across the aer. Let the followg lear dscrete-tme dyamc system wth elemets be the state vector x (

3 3 x( k + ) = Φ( k +, x( + Γ( k +, ω(, x(0) = x 0 ad m observato vector z ( be gve by k 0 (State Equato) z( = H ( ~ γ ( x( + v(, k 0 (Observato Equato) where Φ ( k +,, Γ ( k +, ad H ( are kow matrces wth arorate dmesos. Usual ad secfc hyothess regardg robablty behavor for radom varables are troduced to formalze the model as follows: (H.) x 0 s a cetered radom vector wth varace-covarace matrx P (0). (H.) { (, ω s cetered whte ose wth E[ ( ω ( ] = Q( γ (H.3) ~ ( γ ( k ) s a dagoal matrx γ ( ω. where { (, γ s a scalar whte sequece wth ozero mea m ( k ) ad varace (, =,...,. It s suosed that { (, γ j (, k 0 are γ ad { } correlated the same stat ad j ( Cov( γ (, γ j ( ) he ext matrx wll be used later o: (H.4) { (, m ( ) k M( = m( =,,j =,...,. v s a cetered whte ose sequece wth varace [ ( v ( ] R( E v =. (H.5) x, { ( k ), k 0 }, { v ( k ), k 0} 0 ω are mutually deedet. (H.6) he sequeces { (,, =,..., x, { ω (, ad{ v (,. 0 γ are deedet of tal state As we ca observe, the comoets of the state vector, the observato equato, are corruted by multlcatve ose addto to measuremet ose.

4 4 Let x ˆ( k / l) be the LMSE estmate of x ( gve observatos z ( 0),..., z( l). e( k / l) = x( xˆ( k / l) deote the estmato error, ad the corresodg covarace matrx s P( k / l) [ e( k / l) e ( k / l) ] =. he LMSE lear flter ad oe-ste ahead redctor of the state x ( are reseted the ext secto. 3. Predcto ad flter algorthm heorem. he oe-ste ahead redctor ad flter are gve by xk ˆ( + / =Φ ( k+, kxk ) ˆ( /, k 0 xˆ(0 / ) = 0 [ ] xk ˆ( / = xk ˆ( / k ) + Fk ( ) zk ( ) HkM ( ) ( kxk ) ˆ( / k ), k 0. he flter ga matrx verfes Fk Pk k MkH k k ( ) = ( / ) ( ) ( ) Π ( ) where ( ) ( ) ( ) Π k = H k S k H ( + H( M( P( k/ k ) M( H ( + R( wth Sk ( ) ( ks ) ( ( ks ) ( ( ks ) ( ( ks ) ( ( ks ) ( ( ks ) ( ( ks ) ( ( ks ) ( ( ks ) ( = S ( = I S( I where I = ( ) j j () Sk ( + ) =Φ ( k+, ksk ) ( ) Φ ( k+, +Γ ( k+, Q( Γ ( k+,, k 0 S(0) = P(0). he redcto ad flter error covarace matrces satsfy P( k + / = Φ( k +, P( k / Φ P(0 / ) = P(0) P( k / = P( k / k ) F( Π( F ( k +, + Γ( k +, Q( Γ (, k 0. ( k +,, k 0 Proof. By the state equato s easy to rove that the redctor Φ ( k +, xˆ( k / satsfes the Orthogoal Projecto Lemma (OPL) []. I the tal stat, the estmate of x (0) s ts mea, so that x ˆ (0 / ) = 0.

5 5 As a cosequece of the orthogoal rojecto theorem [], the state flter ca be wrtte as a fucto of the oe-ste ahead redctor as xˆ ( k / = xˆ( k / k ) + F( δ (, k 0 where δ ( = z( zˆ( k / k ) s the ovato rocess. Its exresso s obtaed below. Sce z ˆ( k / k ) s the orthogoal rojecto of z ( oto the subsace geerated by observatos{ z ( 0),..., z( k ) }, we kow that ths s the oly elemet that subsace verfyg E z( z ( α) = E z ˆ( k/ k ) z ( α), α = 0,..., k. he, by the observato equato ad the hyotheses (H.3)-(H.6), t ca be see that zk ˆ( / k ) = HkMkxk ( ) ( ) ˆ( / k ) ad the ovato rocess for the roblem we are solvg s gve by δ ( = z( H( M( xˆ ( k/ k ). o obta the ga matrx Fk ( ), we observe that, gve the OPL holds, [ ( k / z ( ] = 0 E e, ad we have E ek ( / k ) z ( = Fk ( ) Π( (3.) where Π ( are the covarace matrces of the ovato. From the observato equato ad the hyotheses (H.)-(H.6), t ca easly checked E ek ( / k ) z ( = Pk ( / k ) M( kh ) ( ad therefore Fk Pk k MkH k k ( ) = ( / ) ( ) ( ) Π ( ). o obta the covarace matrces of the ovato rocess, t ca be see that ( ) γ ( ) ˆ δ k = H( k x( + v( H( M( x( k/ k ) ad by addg ad subtractg HkMkxk ( ) ( ) ( ), he ( ) ( γ ( ) ) δ k = Hk ( ) k M( xk ( ) + vk ( ) + HkM ( ) ( kek ) ( / k ) Π ( = E δ ( z ( = HkE ( ) ( γ ( Mk ( )) xkz ( ) ( + Evkz ( ) ( + HkMkEek ( ) ( ) ( / k ) z(. Let us work out each of the terms revous exresso. By the observato equato we have that

6 6 ( ) ( ) ( γ ) H( E γ( M( x( z ( = H( E γ( M( x( x ( γ( H ( + HkE ( ) ( Mk ( ) xkv ( ) ( ad accordg to hyotheses (H.4)-(H.6) the secod term ca be cacelled. Addg ad subtractg ( γ ) ( γ ) H( E ( M( x( x ( M( H (, HkE ( ) ( Mk ( ) xkz ( ) ( = HkE ( ) ( γ( Mk ( )) xkx ( ) ( ( γ( Mk ( )) H ( + H( E ( γ ( M( ) x( x ( M( H ( where the secod term s zero by (H.3) ad (H.6). Accordg to (H.6), f we label S ( E[ x ( x ( ] herefore j = j for j,...,, =, we get ( γ ) ( γ ) Sk ( ) E ( Mk ( ) xkx ( ) ( ( Mk ( ) = ( γ ) ( γ ) ( γ ) ( γ ) E ( γ( m( ) x( ( γ( m( ) x( ( ks ) ( ( ks ) ( ( ks ) ( ( ks ) ( ( ks ) ( ( ks ) ( ( ks ) ( ( ks ) ( ( ks ) ( ( m( x( ( m( x( ( m( x( ( m( x( = ( γ ) HkE ( ) ( Mk ( ) xkz ( ) ( = HkSkH ( ) ( ) (. O the other had, by the observato equato ad (H.4)-(H.6) E v( z ( = E v( x ( γ ( H ( + E v( v ( = Rk ( ). By the same reasos H( M( E e( k/ k ) z ( = H( M( E e( k/ k ) x ( M( H ( = HkMkPk ( ) ( ) ( / k ) MkH ( ) (. I short, the covarace matrces of the ovatos rocess verfy Π ( = HkSkH ( ) ( ) ( + Rk ( ) + HkMkPk ( ) ( ) ( / k ) MkH ( ) (. o obta the comoets Sj ( k ) of the Sk ( ) S ( = I S( I j j, we oly eed to observe that

7 7 where Sk ( ) = E xkx ( ) ( ad I = ( ). he ext recursve exresso of ( ) whte ose sequece ad deedet of x (0) () Sk s mmedate gve that { (, ω s a S( k+ ) =Φ ( k+, S( Φ ( k+, +Γ ( k+, Q( Γ ( k+,, k 0 S(0) = P(0). he exresso of the redcto error covarace matrces P( k + / = Φ( k +, P( k / Φ ( k +, + Γ( k +, Q( Γ ( k +,. s mmedate sce ek ( + / =Φ ( k+, kek ) ( / +Γ ( k+, ω(. I the other had, gve that ek ( / = ek ( / k ) Fk ( ) δ ( the It ca be observed that P( k/ = P( k/ k ) E e( k/ k ) δ ( F ( FkE ( ) δ ( ke ) ( k/ k ) + Fk ( ) Π( kf ) (. E δ ( e ( k/ k ) = E z( e ( k/ k ) H( M( E x ˆ( k/ k ) e ( k/ k ) where the secod term cacels accordg to OPL ad by equato (3.) t s obtaed that E [ δ ( e ( k / k ) ] = Π( F ( ad the P( k / = P( k / k ) F( Π( F (. Next, we see how some kow results ca be cosdered as artcular secfcatos of the geeral model roosed ths aer: o If γ( = = γ ( = the state vector are ot corruted by a multlcatve ose, the γ ( = M( = I ( = 0, j j ad our algorthm degeerates Kalma algorthm []. where { ( ), 0} o If γ( = = γ ( = U( Uk k s a scalar whte sequece wth ozero mea mk ( ) ad varace k ( ), we ed u wth Rajasekara s [6] framework, γ ( = U( I, where the state vector (all comoets) s corruted by multlcatve ose. I ths case,

8 8 M( = m( I ( = (, j j ad the reseted algorthm collases Rajasekara s. o If γ( = = γ ( = γ( γ (, k 0 s a sequece of where { } Beroull deedet radom varable wth [ ] P γ ( = = (, the γ ( = γ ( I ad we ed u wth Nah s framework [7], where the state vector s reset the observato wth robablty (. I ths case, M( = ( I ( ) ( = ( (, j j ad the ew algorthm collases Nah s. o If γ( = = γ ( = ad γ + ( = = γ ( = γ( where { (, γ s a sequece of Beroull deedet radom varable wth [ ] P γ ( = = (, the observatos ca clude some elemets of the state vector ot beg esure the resece of the restg others (Sachez ad García s framework [0]). I ths case I 0 ( ) M( = 0( ) I( ) ( ) 0,, j 0,, j > j ( = 0, >, j ( ( ( ),, j > ad the ew algorthm degeerates Sachez ad García s. Aother terestg stuato aears whe some of the comoets the state vector are reset the observato but aear wth dfferet robabltes. Such a stuato s ot a system wth ucerta observatos. he reset algorthm solves estmato roblems ths tye of system, t s oly ecessary to suose that the multlcatve oses are dfferet Beroull radom varables.

9 9 4. Some umercal smulato examles We show ow some umercal examles to llustrate the flterg ad redcto algorthm reseted heorem. Examle We cosder the followg lear system descrbed by the dyamc equato: x ( k+ ) x ( 0.0 = + ω(, k 0 (4.) x( k ) x( x (0) x0 = x(0) x0 γ ( 0 x ( zk ( ) vk k where { (, (4.) ( ) = ( ), 0 (4.3) 0 γ ( x( ω s cetered Gaussa whte ose wth Qk ( ) =.89 ; x 0 ad x 0 are cetered Gaussa radom varables wth varaces equal to 0.5; { γ (, k 0} ad { (, k 0} γ are Gaussa whte ose wth meas ad 3 ad varaces ad { γ (, k 0} are deedet; { (, ose wth varace R = 0.., resectvely; { (, k 0} γ ad v s cetered Gaussa whte Usg the estmato algorthm of heorem, we ca calculate the flterg estmate xˆ( k/ k ) of the state recursvely. Fg. ad Fg. llustrate the state x ( k ) ad the flter xˆ ( k/ k ), for =,, vs. k for the multlcatve Gaussa observato oses γ N (, 0.5) ad γ N ( 3, 0.) rereseted wth black ad the flter wth red color.. he state s

10 0 Fg.. x ( k ) ad xˆ ( k/ k ) vs. k Fg.. x ( k ) ad xˆ ( k/ k ) vs. k ables ad shows the mea-square values (MSVs) of the flterg errors x ( xˆ ( k/ for =, ad k =,,..., 00 corresodg to multlcatve whte observato oses: γ : N(, 0.), N(, 0.5), N(, ) γ : N(3, 0.), N(3, 0.5), N(3, ).

11 able. MSV of flterg errors x ˆ ( k ) x ( k / k ), k =,,..., 00 = 0. = 0.5 = = = = able. MSV of flterg errors x ˆ ( x( k/, k =,,..., 00 = 0. = 0.5 = = = = Examle We cosder a lear system descrbed by equatos (4.)-(4.3) where { γ (, k 0} ad { (, k 0} γ are sequeces of deedet Beroull radom varables beg wth robabltes ad, resectvely. Fg. 3 ad Fg. 4 llustrate the state x ( k ) ad the flter xˆ ( k/ k ), for =,, vs. k for the multlcatve observato oses Beroull ( ) γ ( ) Beroull color. γ 0.5 ad. he state s rereseted wth blue ad the flter wth gree

12 Fg. 3. x ( k ) ad xˆ ( k/ k ) vs. k Fg. 4. x ( k ) ad xˆ ( k/ k ) vs. k ables 3 ad 4 shows the mea-square values (MSVs) of the flterg errors x ( xˆ ( k/ for =, ad k =,,...,00 corresodg to multlcatve whte observato oses: γ : Beroull(0.), Beroull(0.5), Beroull() γ : Beroull(0.), Beroull(0.5), Beroull().

13 3 able 3. MSV of flterg errors x ˆ ( k ) x ( k / k ), k =,,..., 00 = 0. = 0.5 = = = = able 4. MSV of flterg errors x ˆ ( x( k/, k =,,..., 00 = 0. = 0.5 = = = = As we ca observe, the smulato grahs ad the MSV of the flterg both examles show the effectveess of the ew algorthm. Refereces [] R.E. Kalma, A ew aroach to lear flterg ad redcto roblems, ras. ASME Ser. D: J. Basc Eg., Vol. 8, 960, [] R.E. Kalma, New methods Weer flterg theory, Proc. Sym. Eg. Al. radom fucto theory ad robablty (Wley, New Yor, 963, [3] J.S. Medtch, Stochastc otmal lear estmato ad cotrol, McGraw- Hll, New York [4] A.H. Jazwsk, Stochastc rocesses ad flterg theory, Academc Press, New York, 970. [5] A. Kowalsk ad D. Szyal, Flterg dscrete-tme systems wth state ad observato oses correlated o a fte tme terval, IEEE ras. Automatc Cotrol Vol. AC-3 (4), 986, [6] P.K. Rajasekara, N. Satyaarayaa ad M.D. Srath, Otmum lear estmato of stochastc sgals the resece of multlcatve ose,

14 4 IEEE ras. o aerosace ad electroc systems, Vol. AES-7, No. 3, 97, [7] N. Nah, Otmal recursve estmato wth ucerta observatos, IEEE rasacto Iformato heory, Vol. I-5, 969, [8] A. Hermoso ad J. Lares, Lear estmato for dscrete-tme systems the resece of tme-correlated dsturbaces ad ucerta observatos, IEEE ras. Automatc Cotrol, Vol. 39 (8), 994, [9] C. Sachez-Gozalez ad.m. García-Muñoz, Lear estmato for dscrete systems wth ucerta observatos: a alcato to the correcto of declared comes qury, Aled Mathematcs ad Comutato, Vol. 56, 004, [0] C. Sachez-Gozalez ad.m. García-Muñoz, Lear estmato of dscrete dyamc systems wth comlete sgals, Iteratoal Mathematcal Joural, I ress. [] A.P. Sage, ad J.L. Melsa, Estmato theory wth alcatos to commucatos ad cotrol, McGraw-Hll, New York, 97.