Stability of the SDDRE based Estimator for Stochastic Nonlinear System

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1 26 ISCEE Inernainal Cnference n he Science f Elecrical Engineering Sabiliy f he SDDRE based Esimar fr Schasic Nnlinear Sysem Ilan Rusnak Senir Research Fellw, RAFAEL (63, P.O.Bx 225, 322, Haifa, Israel.; als Adjunc Senir Lecurer, Faculy f Elecrical Engineering, echnin, Haifa 32, Israel. ( ilanru@rafael.c.il Absrac Bunded-Inpu-Bunded-Sae and Bunded- Inpu-Bunded-Oupu sabiliy f he Sae Dependen Differenial Riccai Equain (SDDRE based esimar f nnlinear schasic ime-varying sysems is prved by hree appraches. Each prf highlighs differen aspec f he SDDRE based esimar sabiliy. he Sae Dependen Cefficien (SDC frm represenain f nnlinear schasic sysem is used. In he firs apprach hery f linear ime-varying schasic sysems is used. In secnd apprach i is shwn ha he sae f he SDDRE based esimar is always bunded by defining a Lyapunv funcin and usage he Barkana's invariance principle. he expnenial sabiliy f he sae ransiin marix f he esimar and cmpuain f bunds n he sae, under specific cndiins, is prved by he hird apprach. I is shwn ha unifrm cmplee bservabiliy and cnrllabiliy, and respecive bundedness cndiins alng he esimar's rajecries is sufficien fr he resuls abve. Keywrds SDDRE, SDC, Esimars, Observers, Sabiliy, nn-linear sysems, Schasic sysems I. INRODUCION he research f esimars f nnlinear dynamic sysems is a vibran area f research. In curse f his wrk he auhr gahered mre han niney, and cuning, differen appraches esimain and smhing f nnlinear sysems. he issues deal wih in hese researches and are impran in implemenain are: perfrmance, sabiliy, rbusness, cmplexiy, numerical sabiliy, implemenain issues and mre. he Exended Kalman Filer (EKF [] is ne f he ms cmmn esimar f nnlinear sysem ha is widely used is. Hwever he sabiliy and unbiasness f he EKF is n guaraneed. he available herems f sabiliy can guaranee as fllws (lsely: If he iniial cndiins f he esimarbserver are sufficienly clse he iniial cndiins f he esimaed sysem and he measuremen and sysem driving nises (he mdelling errrs f he bserver are sufficienly small hen he EKF esimar-bserver will n diverge. ha is, neiher unifrm glbal sabiliy nr unbiasness can be guaraneed by he exising hery. In spie f is deficiency, he EKF is widely implemened. Recenly, he Sae Dependen Differenial Riccai Equain (SDDRE based apprach gained is place in research and applicain f esimars-bservers f nnlinear sysems [2, 3,4, 5,6]. he SDDRE apprach is based n exended linearizain [7] f he prcess dynamics and bservains. his mehd uses he marices A( and creaed when he dynamics f he nnlinear ime varying sysem are represened in he Sae Dependen Cefficien (SDC frm x. y = x Such represenain always exiss, albei i is n unique. Selecin f hese marices is knwn as he parameerizain prblem [8] and may affec he perfrmance f he esimar. An apprach fr slving he parameerizain prblem by crdinae basis selecin via sabiliy measure can be fund in [9]. he EKF and SDDRE filers are subpimal. he sabiliy f he SDDRE based esimar f nnlinear discree schasic sysems is cnsidered in [5] and f cninuus deerminisic sysems in [6] under he assumpin f he Lipschizian cndiins f he sysem's nnlinear funcins. he biasness is n reaed in hese references. hese resuls are based n []. he main issue deal wih in his paper is he design and implemenain f esimars f schasic nnlinear sysems, and heir perfrmance in erms f sabiliy and biasness wihu he requiremen f he Lipschizian cndiins f he sysem's nnlinear funcins. he Lipschizian cndiins are replaced by much milder cndiin f bundedness f he marices f he SDC frm represenain. In [4] he issue f implemenain f esimars f schasic nnlinear sysems is discussed wih respec he Sae Dependen Algebraic Riccai Equain (SDARE based bserver f nnlinear sysem. hey sae ha a sufficien cndiin fr sluin f he Riccai equain (and hus he sabiliy f he bserver-esimar, is ha he pair [A(ξ(,,ξ(,] mus be cninuusly bservable lcally fr all ime, i.e., he rank f he bservabiliy es marix O = [,..., (A( n ] mus be f full rank alng every bserver's rajecry ξ( ha he bserver aains. hey als cmmen ha: (i in sme cases, his sufficien cndiin can impse an "verly resricive requiremen n he bservabiliy f he nnlinear sysem in quesin"; and (ii his rank cndiin "may n hld in sme sysems ha are neverheless bservable in he mre general sense". he sabiliy f he SDDRE based bserver fr Deerminisic Nnlinear Sysems wihu he Lipschizian cndiins is deal wih []. his paper presens prfs f sabiliy f he SDDRE based esimar-bservers wihu he requiremen f he /6/$3. 26 IEEE

2 26 ISCEE Inernainal Cnference n he Science f Elecrical Engineering Lipschizian cndiins f he sysems nnlinear funcins by hree appraches. hus differen aspecs he sabiliy prpery are highlighed. hese are: Sabiliy f linear ime-varying sysems based prf; 2 Purely Lyapunv based analysis based prf by he use f Barkana's Invariance Principle [2]; 3 Parial Lyapunv based analysis and bunds cmpuain. he main nvely and imprance f he paper is presenain f cndiins fr unifrm glbal Bunded-Inpu- Bunded-Oupu (BIBO and Bunded-Inpu-Bunded-Sae (BIBS sabiliy (sufficien and necessary. I is shwn in he paper ha subjec unifrm glbal bservabiliy and cnrllabiliy, and cerain bundedness cndiins alng he esimar's rajecries he SDDRE based esimar-bserver is BIBS and BIBO sable and he iniial cndiins decay expnenially. Simulain f SDDRE based esimar f he Van der Pl equain is presened. II. HE SAE DEPENDEN COEFFICIEN FORM he fllwing nnlinear schasic sysem is cnsidered = f ( + B( u + G( w( y = g( + v( where, is he sae, u( is he inpu (measured, is he upu (measured, v( is he measuremen nise and w( represens he sysem driving nise and imbeds he difference beween he realiy and he mdel (. I is assumed ha w( and v( are bunded. Assuming f (, =, g(, = (2 he Sae Dependen Cefficien (SDC frm represenain f he nnlinear sysem is x + B( u + G( w( y = x + v( I is assumed here ha all marices A(,, B(, G( are knwn piecewise cninuus and unifrmly bunded wih respec all variables. Such represenain always exis albei is n unique (N all nnlinear sysem can be represened in he SDC frm wih unifrmly bunded marices. An impran prpery f he SDC represenain ha will be assumed in paper is i's he unifrm cmplee bservabiliy and cnrllabiliy, as a ime varying sysem (5, alng he rajecries f he SDDRE based esimar-bserver. All marices and vecrs are f he respecive dimensin. III. PROBLEM SAEMEN A nnlinear schasic sysem is cnsidered. In his case here are he sysem driving nise and he measuremen nise. he SDC represenain f he schasic nnlinear sysem is (3. In his case he sae,, f he sysem's mdel is unknwn. he prblem cnsidered here is he "esimain" f he sysem's sae by he SDDRE based esimar. ( (3 IV. HE SDDRE BASED ESIMAOR In his paper he SDDRE based [2,4,3], esimar f (,2 is ξ ( y ξ = C ( N ( Q ( + A ( + W ( C ( N (, = Q where: W is he specral densiy f he sysem driving nise, N> is he specral densiy f he measuremen nise; Nice ha wih sligh abuse f nain alng he aainable bserver's-esimar's rajecries we use he nain A ( ξ,, =, =, =. (5 his nain is used inerchangeably hrughu he paper. he esimar equains (4 can be slved as up he curren ime, ξ(, he esimaed sae, is perfecly knwn ime funcin, and he inpu and upu f he real sysem are measured. V. SABILIY PROOFS OVERVIEW hree prfs f BIBO and BIBS sabiliy f he SDDRE based esimar will be presened. Each f hem highlighs differen aspec he sabiliy prpery. hese are: Prf based n he sabiliy f linear ime-varying sysems; 2 Purely Lyapunv based analysis based prf by he use he Barkana's Invariance Principle [2]; 3 Lyapunv based analysis and bunds cmpuain. he prfs give cndiin n he BIBO and BIBS sabiliy f he SDDRE based esimar, hwever unbiasness, i.e. zer mean esimain errr, fr he schasic esimar, cann be guaraneed a his sage f he research, and is an issue f nging invesigain. VI. SABILIY PROOF BASED ON HE SABILIY OF LINEAR IME-VARYING SYSEMS he sabiliy herems fr linear ime-varying are based n [4]. he secin uses bservabiliy and cnrllabiliy as deal wih in [4,5]. he fac ha he esimar's rajecries, ξ(, are perfecly knwn up he curren ime, i.e. he esimar's rajecry is perfecly knwn funcin f ime is used. A. Sabiliy f he Esimar Alng he aainable rajecries f he esimar (4 he esimar can be wrien as (wihu he exernal inpu u(, ha is immaerial he sabiliy issue (4

3 26 ISCEE Inernainal Cnference n he Science f Elecrical Engineering ξ ξ + ( y ξ = C ( N ( Q ( + A ( + W ( C ( N (, = Q he fllwing herems f sabiliy frmalize he resuls. herem VI.: Cnsider he sysem in (6. Suppse ha A( is cninuus and bunded, ha, G(, W( and N( are piecewise cninuus and bunded, and furhermre ha W( αi, N(>βI, fr all and α,β> hen if he sysem in (6 is bh unifrmly cmpleely recnsrucible and unifrmly cmpleely cnrllable hen he sluin f he Riccai equain in (6 wih he iniial cndiin =Q cnverges Q ( as fr any Q. he prf is direc ucme f [4, herem 4.9]. herem VI.2: Cnsider he sysem (6. Suppse ha he sae exciain, w(, and bservain nises, v(, are uncrrelaed. Suppse ha he cninuiy, bundedness and psiive-definieness cndiins f herem VI. cncerning A(,, G(, W( and V( are saisfied. hen he seady-sae esimar ˆ( ξ ˆ( ξ + K ( [ ˆ( ξ ], (8 where K ( = Q ( C ( N ( is expnenially sable. Here Q ( is as defined in herem VI.. he prf is direc ucme f [4, herem 4.]. herems VI. and VI.2 give he cndiins ha he SDDRE esimar des n diverge (sable. Unfrunaely he biasness f he esimain errr cann be guaraneed. VII. SABILIY PROOF BASED ON BARKANA'S INVARIANCE PRINCIPLE his apprach defines Lyapunv funcin and uses he Barkana's Invariance Principle [2] shw ha he sae f he SDDRE based esimar is always bunded. Up dae ms prfs f sabiliy f nnlinear nnaunmus sysems f he frm ( (i.e. sysems where he ime variable explicily appears are based n s-called Barbala s Lemma [6, pp. 22]. his lemma seems require unifrm cninuiy f he Lyapunv derivaive and hus, f he sysem velciy. LaSalle s Invariance Principle [7] managed miigae his preliminary cndiin ha was apparenly needed fr he prf f sabiliy. Neverheless, because is frmulain uses he 'mdulus f cninuiy' μ(β α, i sill apparenly requires cninuiy f he rajecries. Hwever, because even his remaining cninuiy cndiin is n needed fr he prf f sabiliy [2], i is sufficien check saisfacin f ne f he fllwing w milder assumpins alng he rajecries f he sysem under invesigain: i. f (, is bunded fr any bunded x; r (2 (6 ii. β f τ, τ dτ α ( is bunded fr any finie ime inerval γ=β α. (3 his cndiin allws he presence f discninuiies and even impulse funcins and nly implies ha he rajecry cann pass an infinie disance in finie ime. herem VII.2 (he Barkana's Invariance Principle: Cnsider he nnlinear nnaunmus sysem (. Assume ha here exiss a psiive definie Lyapunv funcin V(x and ha is derivaive V (, alng he rajecries f Equain ( is negaive semi-definie, i.e., saisfies V (,. Fr any iniial cndiin x ( = = x define he dmain Ω = { x V ( x V ( x }. Because he Lyapunv derivaive is negaive semi-definie, i is clear ha all sysem rajecries are bunded and cnained wihin he dmain Ω. Nw, define he dmains Ω = { x e ( = } and Ω = { x i ( } (i.e, in Ω he Lyapunv derivaive is i idenically zer, namely, n jus equal zer. hen, if ne f he assumpins (24 r (25 hlds, all limi pins f he bunded rajecry belng he dmain Ω f = Ω Ω. e In paricular, limi pins ha are als equilibrium pins f belng Ω f = Ω Ω [2]. i he fllwing herem deals wih he BIBO sabiliy f he SDDRE based esimars. herem VII.3: Subjec he bundedness assumpins in secin II and herem VI. he SDDRE based esimar (4 is BIBO sable. Prf: he sabiliy f he aunmus par f he esimar (4, ξ = ( A( ξ. (4 is cnsidered alng he esimar's rajecries. A. he esimar's gain is bunded due he unifrm cmplee bservabiliy and cnrllabiliy assumpin f he sysem (2 alng he esimars rajecries. hen he requiremens f Barkana's invariance principle are saisfied. Furher, shw ha his unfrced sysem is asympically sable, we chse he Lyapunv funcin V = ξ Q ( ξ (5 hen i can be shwn ha ( C ( N ( + W( Q ( ξ V = ξ Q ( (6 and i is negaive definie, which implies ha he unfrced sysem is BIBS asympically sable. As A(, are bunded and N( βi, W( αi, α,β> are bunded and psiive definie he sysem is BIBO sable. B. he bundedness f he sysem sae wih he exernal signal cmmands alng he sysem's rajecries is cnsidered, i.e. he sabiliy f ξ ( ξ (7 ξ = ( A( ξ (8

4 26 ISCEE Inernainal Cnference n he Science f Elecrical Engineering hen fr he Lyapunv funcin (5 ( ξ, C ( ξ, N ( ξ, ξ, ξ, + W( = ξ Q ( ξ, + ξ Q ( ξ, B( u + u B( Q ( ξ, ξ + ξ Q ( ξ, ξ, y + y K ( ξ, Q ( ξ, ξ Q ( ξ, ξ (9 Assuming ha u( and are bunded and as all marices are bunded fr bunded han if ξ ends increase, he firs negaive erm ha is quadraic in ξ becmes dminan and s he Lyapunv derivaive becmes negaive and i guaranees bundedness f ξ. VIII. SABILIY BASED ON HE EXPONENIAL SABILIY OF HE SAE RANSIION MARIX he expnenial sabiliy f he sae ransiin marix f he esimar is prved and bunds n he sae are cmpued. herem VIII.: he sae ransiin marix f he SDDRE based bserver-esimar (6, subjec assumpin f herem VI., is asympically/expnenially sable and BIBO sable. Prf: A. he sabiliy f he aunmus sysem alng he sysem's rajecries he unfrced sysem (26 alng he sysem's rajecries wih he nain (7 is ξ = [ A( K ( ]ξ. (2 shw ha he aunmus sysem is asympically sable, we chse he Lyapunv funcin V = ξ Q ( ξ = ξ Q ( ξ (2 hen = e Q [ W ( + QC ( N ( Q] Q ( e <, fr all e (22 his immediaely guaranees asympic sabiliy. If he respecive cndiins are guaraneed his give expnenial sabiliy. his is furher elabraed. Fr he ime varying sysem his means ha he sae ransiin marix [ A( K ( ],, I Φ ( = (23 = is asympically/expnenially sable, ha is (24 shw expnenial sabiliy he cndiins f herem VI. are required. Dene, fllwing [8, herem ], = ξ Mξ Μ = Q [ W ( + QC ( N ( Q] Q ( > (25 and we require ha fr all < here exis α and β such ha α I Μ β I <, < α < β < (26 < i.e. M is unifrmly psiive definie and unifrmly bunded. hen β x M = α (27 ( Furher we assume ha here exis sme < a < b such ha i.e. V is bunded (guaraneed frm herem VI. a V b, < a < b, (28 hen ( a V ( V ( < α = α (29 Frm Grnwall-Bellman Lemma ( a V ( V (exp α (3 ha is, expnenial sabiliy alng he rajecries ξ(. he + expnenial sabiliy means ha here exis c, c2 R such ha Φ (3 c2 ( ( < c e B. Quaniaive esimae-bund n he BIBO sabiliy - Cmpuain f he H 2 gains he nn-aunmus esimar is [ y ξ ] ξ (32 and alng he aainable rajecries i may be wrien as [ y ξ ] ξ u + K ( hen we have τ τ ] ξ ( = ξ + τ + dτ (49 where Φ (, = I hen ξ ( = ξ + ξ + τ τ τ + dτ τ + dτ (33 (34 (35 Given ha he cndiins fr expnenial sabiliy f he sae ransiin marix presened in he previus secin are saisfied hen ξ ( c τ c2 ( c2 ( τ e ξ + c e + dτ (36 If [ ( τ B + is unifrmly glbally bunded, hen if here exiss finie M R, M < such ha [ B ( u( τ + < M τ. (37

5 26 ISCEE Inernainal Cnference n he Science f Elecrical Engineering Ne: u( and are he measured inpu and upu f he real plan and hus inherenly bunded. hus he requiremen is ha B( is unifrmly bunded by design and = is unifrmly bunded as i is he sluin f he ime-varying Riccai equain (4 under he respecive unifrm cmplee bservabiliy and cnrllabiliy cndiins. IX. ESIMAOR OF HE VAN DER POL EQUAION - SIMULAION he Van der Pl equain [6, pp. 8] is simulaed. he deails will be presened in he full paper. Figures presen he phase plane pl f he Van der Pl scillar sae, he esimaed sae and is esimaed errrs, respecively. esimaed velciy esimaed velciy errrs esimaed psiin real sae esimaed sae real and esimaed sae esimain errrs esimaed psiin errrs Figure : Phase plane pls f he Van der Pl scillar rajecries and is sae esimain errr. I. Biasness f he esimaed sae We shwed ha he esimaed sae, ξ(, is bunded (nn diverging. Hwever he mre ineresing issue if his esimae is unbiased. ha is if E[ ξ ( ] = E[ e( ] =. (37 where e( is he esimain errr. I is difficul analyze his issue fr nnlinear sysems. I is hined in [5,6] ha addiinal requiremens n he nnlineariies like Lipschizian cndiins (and may be sme addiinal cndiins can lead unbiased esimar. his is an addiin he requiremen ha all marices A(,, B(, G( are knwn piecewise cninuus and unifrmly bunded wih respec all variables. REFERENCES [] Gelb, A. Ed.: Applied Opimal Esimain, he MI press, 974. [2] Mracek,C.P., Cluier, J.R., and D'Suza, C.A.: A New echnique fr Nnlinear Esimain, Prceedings f he 996 IEEE Inernainal Cnference n Cnrl Applicains, Dearbrn, MI, Sepember 5-8, 996. [3] Xin, M. and Balakrishnan, S.N.: A New Filering echnique fr a Class f Nnlinear Sysems, Prceedings f he 4s IEEE Cnference n Decisin and Cnrl, Las- Vegas, Nevada USA, December 22. [4] Haessig, D.A. and Friedland, B.: Sae Dependen Differenial Riccai Equain fr Nnlinear Esimain and Cnrl, 22 IFAC, 5h riennial Wrld Cngress, Barcelna, Spain. [5] Beikzadeh, H. and aghirad, H.D.(29: Sabiliy Analysis f he Discree-ime Difference SDRE Sae Esimar in a Nisy Envirnmen 29 IEEE Inernainal Cnference n Cnrl and Aumain Chrischurch, New Zealand, December 9-, 29. [6] Beikzadeh, H. and aghirad, H.D.(22: Expnenial Nnlinear Observer Based n he Differenial Sae-Dependen Riccai Equain, Inernainal Jurnal f Aumain and Cmpuing, 9(4, Augus 22, DOI:.7/s y [7] Williams, D., B. Friedland and A. N. Madiwale (987: Mdern Cnrl hery fr Design f Aupils fr Bank-- urn Missiles, AIAA Jurnal f Guidance, Cnrl, and Dynamics, Vl., N. 4, pp [8] Cluier, J. R., C. N. D Suza and C. P. Mracek, (996 Nnlinear Regulain and Nnlinear H Cnrl via he Sae- Dependen Riccai Equain echnique: Par 2, Examples, Firs In. Cnf n Nnlinear Prblems in Aviain and Aerspace, Dayna,Beach, FL. [9] Weiss, H. and Mre, J.B.: Imprved Exended Kalman Filer Design fr Passive racking, IEEE ransacins n Aumaic Cnrl, Vl. AC-25, N. 4, Augus 98, pp [] arn,. J. and Rasis, Y. Observers fr Nnlinear Schasic Sysems, IEEE rans. n Aumaic Cnrl, Vl. 2, N. 4, pp , 976. [] Rusnak, I. and Barkana, I.: Sabiliy f he SDDRE based Observer fr Deerminisic Nnlinear Sysems, ECC 26, Eurpean Cnrl Cnference, June 29 July, 26, Aalbrg, Denmark. [2] Barkana, I.: Defending he beauy f he Invariance Principle, Inernainal Jurnal f Cnrl, 24, Vl. 87, N., 86-26, hp://dx. DOI:.8/ [3] Çimen,. and Merpçu lu, A. O.: Asympically Opimal Nnlinear Filering: hery and Examples wih Applicain arge Sae Esimain, Prceedings f he 7h Wrld Cngress, he Inernainal Federain f Aumaic Cnrl, Seul, Krea, July 6-, 28. [4] Kwakernaak, H. and Sivan, R.: Linear Opimal Cnrl Sysems, Wiley-Inerscience, 972. [5] Chen, C..: Linear Sysem hery and Design, Hl Rinehar and Winsn, Inc, 984. [6] Sline, J.J., & Li, M.: Applied nnlinear cnrl. Englewd Cliffs, NJ: Prenice Hall. 99. [7] LaSalle, J.P. he sabiliy f dynamical sysems. New Yrk: SIAM [8] Vidyasagar, M.: Nnlinear Sysems Analysis, Prenice Hall,2nd Ediin, 993.