State and Parameter Estimation of The Lorenz System In Existence of Colored Noise

Transcription

1 Sae ad Parameer Esimaio of he Lorez Sysem I Eisece of Colored Noise Mozhga Mombeii a Hamid Khaloozadeh b a Elecrical Corol ad Sysem Egieerig Researcher of Isiue for Research i Fudameal Scieces (IPM ehra Ira b Faculy of Elecrical & Compuer Eg. K.N. oosi Uiv. of echol. ehra Ira mzh.mombeii@ipm.ir a H_Khaloozadeh@u.ac.ir b Absrac: May researchers are ieresed o use Eeded Kalma Filer (EKF for sae esimaio of comple oliear dyamics ih uceraiies hich modeled ih hie oises. O he oher had behavior of he chaoic sysems i ime domai iself is similar o oise oo. I his paper saes of he chaoic Lorez sysem ha is uceraiies modeled ih colored oise o saes ad also o oupu are cosidered. For boh cases he case ha parameers of Lorez sysem are o ad he case ha parameers of he Lorez sysem are uo EKF is used o esimae he saes. I he case ha parameers are uo usig a sochasic viepoi parameers of he sysem ad parameers of he firs order filer of he colored oise are esimaed. Efficiecy of he mehod is sho ih simulaio. Keyords: Colored oise EKF Esimaio Lorez Uceraiy.. Iroducio Eeded Kalma Filer (EKF is a poerful ool for esimaio of oliear sysems saes i presece of oise ad used i may oliear cases. I some oliear sysems by chagig parameers of he sysems hey ge o period doublig ad coiuig he parameer chagig he sysem becomes chaoic. his procedure deermies he domai i hich sysem is chaoic here behavior of he sysem i ime domai is similar o oise ad also sysem is sesiive o iiial codiio. he accordig he oise lie behavior of he chaoic sysems efficiecy of he EKF for sae ad parameer esimaio of hese sysems is ivesigaed i some ors [-]. I his paper efficiecy of he EKF for separaio of chaoic saes from oise is ivesigaed for Lorez sysem ih o parameers ad for Lorez sysem ih uo parameer. Where i boh cases he eise oise is a colored oise ad EKF is used o esimae he filer parameers oo.

2 . Discree modelig ad differe uceraiies For coiuous model of sysem i sae space ih & f ( θ z H( m Where ε R deoed o sae variables θ deoe o sysem parameers zε R is oupu ad H measureme mari is a m mari a forard differece ( θ ( θ approimaio for derivaios of he saes is &. Choosig small eough as sample ime ( θ ad ( θ resul i f ( θ or equivalely a discree model as F( θ hich ca be simulaed ih compuer codes. Noe ha for ε he forard approimaio of derivaives of each sae should be calculaed separaely. he m realizaio of discree model ih measured oupu zε R i sae space is F( θ ( z H. Whie oise ype uceraiy Whie oises are some uceraiies ha are idepede ime series ad disribued ideically hich meas o auo correlaio beee hem. I paricular case he "Gaussia hie oise" has ormal disribuio ih zero mea ad sadard variaioσ. For sysem( ih Gaussia hie oise { } o saes ad Gaussia hie oise { v } o oupu i chage as folloig F( G θ z H v. ( Where mea vecor ad auocorrelaio mari of ad v are µ E{ } R E{ } σ I E{ v} R vv { vv } σ I E E is epeced value operaor I is ideiy mari ad v G is a mari or vecor ha is dimesio maches o size of oher elemes of he equaio. µ v R

3 . Colored oise ype uceraiy Colored oises are some uceraiies ha are depede o heir pas saes he have auo correlaio. I paricular case passig a Gaussia hie oise from a firs order filer resuls i a colored oise. Model of a firs order filer hich resul i { } colored oise o saes is c e ( Where { e } is Gaussia hie oise ad c is a real cosa. Similarly by defiig dv e v (4 Where { e } is Gaussia hie oise ad d is a real cosa{ v } is realizaio for colored oise o measured oupu.. Ucerai parameers For mos mahemaical modelig of sysems here are uceraiies i parameers of he model. Some are ihere resul of he accuracy i modelig some are because of variaio i realisic codiio of he sysem amog he ime ad ec. he alhough heθ parameers of he sysem ( are cosidered cosa i observaio scale ime bu hey may have chages amog he ime. I oher ord i bigger scale imes hese ucerai parameers are variable. Here usig sochasic perspecive for a uo parameer θ by variaio i he rage [ θ θ θ θ ] he parameer is cosider he as a more geeral θ variace value or equivalely case a hie oise ih θ mea value ad θ θ ϑ ϑ ~ N( θ. Eeded Kalma Filer (EKF Eeded Kalma Filer is a mahemaical ool for sae esimaio of oliear sysems ad he mehod is used i may cases [][]. I called Firs- Order Filer oo because i uses a approimaio of he sysem ih epadig i i aylor series geig he firs order erms ad higher order erms (H.O.. are cosidered egligible. I shor descripio for he oliear model ih Gaussia oise F( G( θ y H ( v. (6 ~ N( Q v ~ N( R N( ~ Q (

4 Where ε R deoed o sae variables m zε R is oupu ad v are hie oises ih zero mea ad sadard deviaio Q R liearizaio of (6 a every sep resuls i A F( θ C H ( G G Eeded alma filer recursively esimaes he saes i o phases forecas ad correcio.. EKF for sae esimaio For sysem (6 ih saes ad θ o cosa parameers forecas phase is ( ˆ P ˆ F( ˆ θ A P A G QG P Q (7 Ad correcio phase is ˆ L ˆ L ( y L C P H ( ˆ P C ( C P C R P P (8 Where ˆ is he esimaed value for he saes.. EKF for sae ad parameers esimaio A mehod is iroduced i lieraure o eed applicaio of he EKF mehod for esimaio of ucerai parameers oo. Essece of he mehod is o cosider he uo parameer θ as a addiioal sae [-]. I his paper he mehod is used for parameer esimaio of he discree model. Cosider he discree model ( ihθ ucerai parameers. By cosiderig his uceraiy o parameers as a oise model ( ad defiig augmeed vecor for e saes as ζ [ θ ] sae space model ( chages o folloig equaio

5 v H y I G F G W F ϑ θ θ ξ θ ξ ( ( (9 ( ~ ( ~ ( ~ θ θ ϑ N R N v Q N his e descripio of he sysem ih uo parameers is i he class of sysem (6 ad EKF mehod ca be used for e saes [ ] ζ θ ih liearizaio of (9 a every samplig ime isa. 4. Effec of uceraiy o Lorez sysem he Lorez sysem is a chaoic sysem ha iroduced i 96 ih equaios ( b r & & & σ ( he sysem is chaoic i rage of parameers here cosidered ih 8. r b σ. Discree model ( for sysem ( is ( ( ( b r σ ( I his par usig simulaio effec of he differe uceraiies o Lorez sysem rajecory is sho. o quaify differece beee saes ad esimaio or perurbed saes error fucio hich is used here is N y N y error ( ( Where ad y are compared saes N is oal umber of samples of he sysem evoluio ad. is orm-o operaor. o model effec of oise o saes ad parameers he values of G ad H mari of he model ( are chose as [ ] G [ ] H ad.

6 error fucio is calculaed ih 7 d. or equivalely N 7. All models are ru ih same iiial codiio ( for saes. 4. Lorez sysem ih hie oise uceraiy Lorez sysem ( ih hie oise uceraiy chages o σ ( ( ( r b z v ~ N( Q v ~ N( R ( Figures ( ( sho rajecories of he ideal model ( ad oisy model ( ihq. ad R. i ime domai ad phase domai respecively. Deviaio i sysem rajecory ih addig he hie oise o saes is obvious i hese figures. Accordig o figures addig hie oise o he saes sysem s behavior chages from he ideal model hich is ihou ay oise. Value of he error ( is error ( 6. 4 i simulaio. 4- Lorez sysem ih colored oise Accordig o ( (4 Lorez sysem ( ih colored oise uceraiy chages o σ ( v z c e dv e v ( ( r b e ~ N( Q e ~ N( Q Where he parameers of firs order filer are chose as c. d. Q. adq.. Figures ( (4 sho rajecories of he real model ( ad oisy model (4 i ime domai ad phase domai respecively. Accordig o figures addig colored oise o he saes sysem s behavior chages from he ideal model ( hich is ihou ay oise. Value of he error ( is error i simulaio. ( 7.98 (4

7 4 6 7 Effec of Whie Noise o Saes Noisy ihou Noise - - Fig. for ideal sysem ( ad for oisy sysem ( Effec of Whie Noise i Phase Space Noisy - - ihou Noise for sysem ( ad Fig. for oisy sysem (

8 4 6 7 Effec of Colored Noise o Saes Noisy ihou Noise - - Fig. for ideal sysem ( ad for oisy sysem (4 Effec of Colored Noise i Phase Space Noisy - - ihou Noise for sysem ( ad Fig.4 for oisy sysem (4

9 4. Lorez sysem ih ucerai parameers Alhough he chaoic sysems are o ih propery of sesiiviy o iiial codiio his class of oliear sysems is sesiive o parameers chages oo. I his par his propery is ivesigaed o folloig ucerai Lorez sysem σ z ( ( r ( ( b σ σ ± σ b b ± b r r ± r c c ± c d d ± d Where σ b r σ b r are omial or mea value of he uo parameers ad are rage of he parameers variaios. Figures ( (6 sho rajecories of he model ( ih omial parameers ad ih small variaio of σ parameer from o. i ime domai ad phase domai respecively. Accordig o figures ih a small perurbaio σ. sysem s behavior chages from he omial model. Value of he error ( is error ( i simulaio.. EKF esimaio of Lorez sysem I his par problem of esimaio of differe ucerai models are solved by chagig hem o model (6 hich EKF is applicable for esimaio of is saes ad parameers. For realizaio [.] θ [.] Q []. vecors are defied i his par. Where is he sae vecor ha EKF esimaes θ is vecor of o parameers Q is a auiliary vecor hich is arrays are variace of he uceraiies ha are cosidered as hie oises i he sysem.. EKF for sae esimaio of Lorez sysem ih hie oise o saes Realizaio of he Lorez sysem ( ih hie oise i he form (6 resuls i he folloig parameers ] θ [ σ b r] Q Q R [ [ ] Figures (7 (8 sho rajecories of he oisy model ( ad esimaed value [ ˆ ˆ ˆ ] ih EKF i ime domai ad phase domai respecively. ˆ Accordig o figures EKF esimaed he real value of correcly. Value of he error is error( ˆ. i simulaio.

10 4 6 7 Effec of Sigma Parameer Perurbaio o Saes ih Sigma Perurbaio ihou Perurbaio - Fig. for ideal sysem ( ad perurbed sysem ( ih σ. Effec of Sigma Parameer Perurbaio i Phase Space Sigma Perurbed - - ihou Perurbaio for sysem ( ad Fig.6 σ. for perurbed sysem ( ih

11 . EKF for sae esimaio of sysem ih colored oise Realizaio of he Lorez sysem (4 ih hie oise i he form (6 resuls i he folloig parameers v ] θ [ σ b r c d] Q [ Q ] [ R Figures (( sho rajecories of he oisy model(4 ad esimaed value ˆ [ ˆ ˆ ˆ ] ih EKF i ime domai ad phase domai respecively. Value of he error is error( ˆ. 7 i simulaio. EKF esimaio for ih hie Noise Real ih Whie Noise Esimaed Fig.7 for sysem ( ad ˆ esimaed value ih EKF EKF Esimaio of - ih hie Noise - iou Noise - ih Colored Noise Fig.8 for sysem ( ad ˆ ˆ esimaed values ih EKF

12 4 6 7 EKF esimaio for ih Colored Noise Real ih Colored Noise Esimaed - - Fig.9 for sysem (4 ad ˆ esimaed value ih EKF EKF Esimaio of - ih Colored Noise - iou Noise - ih Colored Noise Fig. for sysem (4 ad ˆ ˆ esimaed values ih EKF

13 . EKF for sae esimaio of sysem ih colored oise ad uo parameers Realizaio of he Lorez sysem ih hie oise ad ucerai parameers resuls i σ ( v c e z v dv e ( r ( b e ~ N( Q ~ N( Q e σ σ ± σ b b ± b r r ± r c c ± c d d ± d For realizaio i he form (7 parameers are v c d σ b r] θ [] [ [ Q Q c d σ b r] Q (6 Figures ( ( sho rajecories of he oisy model(6 ad esimaed value ih EKF i ime domai ad phase domai respecively. Accordig o figures EKF esimaed he real value of correcly. Value of he error is error( ˆ [ ˆ σ bˆ rˆ].94 i simulaio. Figure ( shos esimaed value ih EKF. Accordig o figures EKF esimaed he real value of he θ ˆ θ sysem parameers correcly. Value of he error is error(. 9 i simulaio.

14 4 6 7 EKF esimaio of ih Colored Noise ad Parameer Perurbaio Real ih Noise ad Parameer Perurbaio Esimaed - Fig. for sysem (6 ad ˆ esimaed value ih EKF EKF Esimaio of - ih Colored Noise ad Parameer Perurbaio Real - ih Noise ad Parameer Perurbaio Esimaed Fig. for sysem (6 ad ˆ ˆ esimaed values ih EKF

15 4 6 7 Sigma Esimaio Sigmaha b Esimaio.4 bha. rha r Esimaio ˆ bˆ rˆ Fig. σ esimaio for sysem (6 ih EKF 6. Coclusio he simulaio resuls saisfyig he efficiecy of he EKF a he firs order filer hich is resul of aylor epasio ad eglecig higher order erms of he derivaives of he sysem equaios for esimaio of he saes ad he parameers of he Lorez sysem hich is highly oliear chaoic sysem ih oise lie behavior. Refreces. J.. Ambada Y. ag. Sigma-Poi Kalma Filer daa assimilaio mehods for srogly oliear sysems Joural of he Amospheric Scieces vol P. Li R. GoodallV. Kadiramaaha Parameer Esimaio of railay Vehicle Dyamic Model Usig Rao-Blacellised Paricle Filer I (CDROM Proceedigs of he UKACC ieraioal coferece o corol 4 Uiversiy of Bah UK 4.. X. Su L. Ji M. Xiog Eeded Kalma filer for esimaio of parameers i oliear sae-space models of biochemical eors Plosoe vol. Is. e788.