Journal of Electrcal Engneerng NONLINEAR SAE ESIMAION OF VAN DER POL OSCILLAOR USING PARICLE FILER WIH UNSCENED KALMAN FILER AS PROPOSAL D. JAYAPRASANH, S. KANHALAKSHMI Department of Instrumentaton and Control systems Engneerng, PSG College of echnology aml Nadu, Inda Emal: djp@ce.psgtech.ac.n, sl@ce.psgtech.ac.n Abstract: State estmaton s a major problem n ndustral systems. he accurate estmaton of states leads to effectve montorng of system, fault dagnoss and good control performance. he partcle flter s potentally suted for better estmaton of hghly nonlnear and non-gaussan system. he selecton of a sutable mportance proposal densty s a crucal step n the desgn of partcle flter. he unscented Kalman flter (UKF) provdes better state estmates for a nonlnear system than the well nown etended Kalman flter (EKF). he partcle flter usng an UKF to generate proposal densty s referred as unscented partcle flter (). he potental advantage of s that the UKF allows the partcle flter to ncorporate the latest measurements n to a pror updatng routne. hs paper proposes an applcaton of n the feld of electrcal engneerng, wth specal emphass on hghly nonlnear Van der Pol oscllator (VPO). Smulaton tests were carred out on VPO system to assess the state estmaton performance of the samplng mportance resamplng partcle flter () and under varous condtons such as ntal state estmate msmatch, large measurement nose and model error. he results ndcate that the s hghly robust and provdes accurate estmaton of states than the. Key words: Van der Pol oscllator, state estmaton, partcle flter, mportance proposal densty, unscented partcle flter.. Introducton It s well nown that the oscllator plays a vtal role n the development of ndustral electroncs. An oscllator consdered n ths wor s a nonlnear Van der Pol oscllator (VPO). he VPO s the eystone for studyng systems wth lmt cycle oscllatons due to ts unque nature. It s also a wdely used eample n the lterature because of ts nterestng behavor [-]. It can ehbt both stable lmt cycle and unstable lmt cycle dependng on the drecton of tme [4]. In the recent years, many research wors have been carred out related to the control of chaotc systems [5-8]. As the reported wor on the VPO s etensve n the lterature, only a very small part of t deals wth the probablstc aspects of ths oscllator,.e., the state estmaton performance when the VPO s subjected to stochastc noses. Over the last few decades, state estmaton has been largely appled to many engneerng problems for estmatng the states of the dynamcal system usng a sequence of nosy measurements. Currently, the state estmaton s becomng an mportant aspect n the feld of electrcal engneerng also for fault dagnoss, control and performance montorng applcatons [9- ]. he Kalman flter (KF) s an optmal state estmator for lnear dynamcal systems subject to Gaussan nose [,]. Most of the practcal systems by nature ehbt some degree of nonlnearty. One of the generalzatons of the KF s the etended Kalman flter (EKF) whch uses nonlnear models drectly n order to estmate the states of a nonlnear system [4]. It adopts frst order aylor seres epanson to provde a local lnearzaton of the system around the operatng pont at each tme nstant [9]. hus the EKF nether has a proof of ts convergence nor a proof that the resultng estmaton satsfes optmalty crtera []. he unscented Kalman flter (UKF) addresses the appromaton ssues of the EKF by determnstcally choosng a mnmal set of sample ponts and propagatng t through the true nonlnear system n a smple and most effectve way wthout mang any lnear appromatons. Hence, the UKF can provde better state estmates than the EKF [5,6]. Le EKF, the UKF also assumes Gaussan posteror densty but both these flters do not address non-gaussan dstrbutons. In order to overcome the lmtatons of the Kalman flter- based estmators, the partcle flter whch can perform equally well for Gaussan and non-gaussan dstrbutons has been proposed for state estmaton of a nonlnear system [7]. he mportance proposal densty s a vtal desgn choce of the partcle flter that wll sgnfcantly affect ts performance. In the sequental mportance resamplng partcle flter (SIR- PF), the transton pror s chosen as proposal densty whch does not mae use of current measurement to propose new partcles. So due to such weaer assumptons, the may become neffcent [8]. he UKF algorthm whch does not nvolve analytcal lnearzaton and computaton of Jacobans s used as proposal for the partcle flter n ths wor to develop an effcent partcle flterng algorthm nown as unscented partcle flter () [9]. Unle, the maes use of the measurement model and the nformaton present n the current measurement to propose new partcles [-]. Many varants of partcle flters have been
Journal of Electrcal Engneerng developed n the recent years and appled to varous felds of engneerng [-5]. here has, however, been a lmted applcaton of partcle flters n the feld of electrcal engneerng [4]. Kalman flter based estmators such as EKF and UKF for estmatng the states of hghly nonlnear Van der Pol oscllator (VPO) was reported n [,6]. Sajeeb et al. [] provdes mprovement over earler wors by usng the partcle flter for VPO. But n [], a sem-analytcal partcle flter used for the state estmaton problem of nonlnear VPO nvolves the comple analytcal appromaton approach n the partcle flterng algorthm whch s cumbersome. he wor carred out n ths paper addresses the above ssue for the nonlnear oscllator by usng the concept of (determnstc) sample statstcs n the partcle flter approach whch s free from analytcal calculatons. As the determnstc samplng technque s used n the partcle flter framewor, the chances of error beng ntroduced n the desgn of such flter for VPO can be very mnmum. herefore, ths paper focuses on usng the for nonlnear state estmaton of VPO and ts estmaton performance s evaluated n comparson wth the samplng mportance resamplng partcle flter (). he robustness of the s also tested under dfferent condtons such as ntal state msmatch, large measurement nose and model error.. Van der Pol Oscllator he study of nonlnear oscllators has been mportant n the development of the theory of nonlnear dynamcal systems []. Unle lnear oscllator, the nonlnear oscllator s structurally stable and the ampltude of oscllaton s ndependent of ntal condtons. Earler, Van der Pol nvestgated electrcal crcuts employng vacuum tubes and found that they have stable oscllatons and also constructed a crcut model of the human heart to study the range of stablty of heart dynamcs, whch later came to be nown as Van der Pol Oscllator [4]. he Van der Pol oscllator (VPO), an oscllator wth nonlnear dampng s a hghly nonlnear system [7]. In the harmonc oscllator, there s a contnuum of perodc orbts but n the case of VPO, there s only one solated perodc orbt whch s called as lmt cycle. A lmt cycle s a closed trajectory n phase plane havng the property that at least one other trajectory sprals n to t ether as tme approaches nfnty or as tme approaches negatve nfnty. he lmt cycle s used to descrbe the perfect behavor of the VPO. If all the neghborng trajectores approach the lmt cycle as tme approaches nfnty (forward tme), t s called as stable lmt cycle. Instead, f all the neghborng trajectores approach the lmt cycle as tme tends to negatve nfnty (reverse tme) then t s called as unstable lmt cycle [].. Mathematcal model of Van der Pol oscllator he VPO for analyzng the nonlnear oscllatons can be regarded as a RLC crcut wth a negatvenonlnear resstor whch has the ablty to pump energy n to the system or a parallel RLC crcut lned to a trode valve as the amplfer, wth the anode current n the trode beng a nonlnear functon of the lumped voltage [,7]. he dynamcs of VPO n electrcal crcuts s governed by a second order nonlnear dfferental equaton as ( ) () where s the poston coordnate whch s a functon of tme and s a control parameter that reflects the degree of nonlnearty of the VPO system. For eample, the system nonlnearty ncreases for ncrease n the value of. he Van der Pol equaton n () has become a staple model for most of the oscllatory processes n ndustres [6]. It can descrbe selfsustaned oscllatons n the form of lmt cycles. he system wll enter n to a stable lmt cycle for and an unstable lmt cycle for. hs dfferental equaton possesses a perodc soluton that attracts other soluton ecept the trval one at the unque equlbrum pont. he state equatons of the VPO, wth n () are as follows:, ( ), () he measurement equaton s gven as y [ ] () As the Van der Pol equaton s commonly used to model the processes nvolvng nonlnear oscllatons, the studes on state estmaton of VPO s generally carred out n the lterature by consderng ts states as, and the same s followed n ths wor [,,6]. A phase portrat s usually constructed to study the nature of the VPO system. hs phase portrat depends on the selecton of the value of. he mportance of the control parameter n the VPO s further provded by consderng the three cases as follows: When, there s no dampng functon and the system functons as the smple harmonc oscllator. When, the system wll enter a stable lmt cycle where energy contnues to be conserved. When, the system wll be damped and ehbt an unstable lmt cycle.
Journal of Electrcal Engneerng. Partcle Flter he partcle flter based on sequental Monte Carlo (SMC) method generates a large number of samples (partcles) to appromate the posteror probablty of the states [7]. If the state and measurement functons are nonlnear, and the process and measurement noses are non-gaussan then the partcle flter has the ablty to gve superor performance than the EKF and UKF [4]. For nonlnear systems, even when the process and measurement nose are ntally assumed to be Gaussan, the dstrbuton becomes non-gaussan after they pass through the nonlnear dynamc system []. he basc framewor for the partcle flter nvolves the state estmaton of a stochastc nonlnear dynamc system gven by (4) and (5), f (, u, v ) (4) y g(, n ) (5) where s the tme nde, represents state of the system, u sgnfes system nput, y represents nosy measurement of the system, v and n are consdered as process nose wth covarance Q and measurement nose wth covarance R respectvely. It s assumed that the stochastc noses v and n are uncorrelated. he functon f (.) and g(.) represents the nonlnear state and measurement functons respectvely. he man dea of partcle flter s to appromate the requred posteror probablty densty p( y: ) of the state by a large number of random partcles {,,..., N } wth assocated weghts p { w,,..., N } and to compute the state estmates p based on these partcles and weghts [8]. N p refers to the number of partcles. he posteror probablty densty can then be appromated by the followng emprcal densty functon as N p : p( y ) w ( ) (6) where ( ) s the Drac delta functon whch s equal to unty f ; otherwse t s equal to zero. Hence, the vtal step s to draw random partcles from the posteror densty p( y: ) but snce p( y: ) s not of the conventonal form such as Gaussan pdf, t becomes mpossble to draw partcles. herefore, the partcle flter reles on mportance samplng method and uses mportance proposal densty ( y: ) and drawng partcles from the proposal densty would be equvalent to drawng partcles from the posteror densty [8]. So the selecton of the proposal densty ( y: ) s one of the most crtcal desgn ssue n the partcle flter algorthm because the ncorrectly chosen proposal densty may result n poor flter estmaton response [9]. he uses the pror densty (transton pror) p( ) as proposal densty and employs the resamplng technque for elmnatng the partcles wth smaller weght and creatng copes of partcles wth hgher weght, thereby, avodng the degeneracy phenomenon [8]. he proposal densty for such varant of partcle flter s defned as (, y ) p( ) (7) he mportance weghts w for ths choce of proposal s obtaned as w p( y ), (8) and then the weghts obtaned from (8) are normalzed to w before the resamplng stage. hus n the, a new partcle set s regenerated by samplng wth replacement from the orgnal set {,,..., N } wth j probablty p( ) w. he nde j ndcates the partcle nde after resamplng [5]. herefore, the resultng partcles are consdered as ndependent and dentcally dstrbuted (..d.) partcles from an appromated dscrete densty functon gven n (6) and ts correspondng normalzed mportance weghts are assumed to be unform whch can be epressed as w (9) N p herefore, the estmated state ˆ usng the s calculated as ˆ N p j () N p j As the proposal densty for the s ndependent of the current measurement y, the states are estmated wthout any nowledge of the measurements. Hence, ths flter becomes senstve to outlers and can be neffcent because the assumptons made n ths flter are very wea. Also the resamplng step appled recursvely at every teraton n the SIR flter can result n rapd loss of dversty n partcles. An alternatve approach s to use a specal varant of the partcle flter whch nvolves the proposal densty dependant on the most recent measurements [8]. Flters wth such an mportance densty are generally p
Journal of Electrcal Engneerng nown as local lnearzaton partcle flters. It nvolves the Kalman flter based estmators such as EKF or UKF as proposal [9]. he partcle flter usng EKF as proposal s referred as etended Kalman partcle flter (EKPF). But the EKF may perform poorly for hghly nonlnear systems [,]. It can also ntroduce an nstablty problem due to analytc local lnearzaton approach whch nvolves the computaton of Jacobans. herefore, the partcle flter wth proposal densty generated by the EKF s not always relable [9]. 4. Unscented Partcle Flter A more relable proposal densty for the partcle flter was proposed n [9]. he UKF used as proposal wth n a partcle flter framewor s called the unscented partcle flter (). he has the ablty to solve the state estmaton problem n a stochastc hghly nonlnear system. he proves to be more robust n estmatng the states of a system under hgh plant-model msmatch []. he UKF uses the unscented transformaton (U) method to pc a mnmal set of sample ponts called as sgma ponts around the mean. he U forms the algorthmc core of the UKF and t s based on the prncple that t s easer to appromate a Gaussan dstrbuton than an arbtrary nonlnear functon [5]. he UKF also referred as the so-called dervatve free Kalman flter s accurate up to second order for any nonlnearty n estmatng the mean and covarance of the states [6]. herefore, the UKF has the potental of generatng proposal densty for the partcle flter that matches the true posteror densty more closely and also has the capablty to control the appromaton errors n the hgher order moments of the proposal densty, allowng for heaver taled dstrbutons than the EKF [9,]. he dea of s to use a separate UKF to generate a Gaussan proposal densty and allowng each partcle to propagate through t,.e., (, y ) N ( ; ˆ, P ) () where ˆ and P are the estmate of the mean and covarance of a partcle respectvely. he symbol N represents that the UKF assumes Gaussan dstrbuton. In summary, the algorthm for the tme nde s as follows [9]: a) For : Np Run UKF Algorthm (for each partcle ) [ ˆ, P ] UKF[, P, y ] Draw a sample from the proposal densty,.e. N ( ; ˆ, P ) Calculate mportance weght, p( y ) p( ) w (, y ) End b) Normalze the mportance weghts c) Resample to get an updated partcle j j N set{, } p j, where j refers to the nde of the partcle after resamplng. d) For : Np End Assgn Covarance: P j j P he output of the algorthm s the mean ˆ of the updated partcle set whch s computed as n (). 5. Smulaton Results and Analyss Smulatons of VPO system and ts state estmaton wth and have been carred out usng MALAB program n an open loop condton. It should also be noted that n order to now effectvely the estmaton performance of the flters, performance comparson of the and s realzed n a smulaton envronment by tang nto consderaton ther estmaton errors under the same test condtons. For eample, the ntal state estmate, ntal state covarance, and process and measurement nose covarance are chosen to be the same for both the partcle flters consdered for estmaton n ths wor. Unle SIR flter, the propagates the partcles towards the lelhood functon as a result of whch very mnmum number of partcles can be consdered n order to acheve better estmaton performance [9]. Hence n ths smulaton study, the partcle count N p s chosen as 5 for and only 5 for whch s about ten tmes lesser than the other. Smulatons have been carred out for 5 samplng nstances wth a samplng nterval of. sec. he root mean square error (RMSE) gves the measure of the estmaton performance of the flters because t facltates quanttatve comparson. he RMSE values for a Monte Carlo run s defned as t / t () RMSE= ˆ where and ˆ are true and estmated state at the tme step respectvely and t ndcates the total number of tme steps. o further nvestgate the robustness of the over to the random perturbatons, Monte Carlo smulatons are performed wth dfferent process and measurement nose realzatons of same varance. hus an average value of the calculated 4
Journal of Electrcal Engneerng RMSE for each nose realzaton s taen as the performance nde for flters. 5. Desgn of partcle flters for VPO system hs secton focuses on the equatons and the parameters consdered for the process and flters used n ths wor. he true states of the VPO system s computed at each samplng nstant by solvng the state equaton gven n () usng the ode solver functon n MALAB. ( ) () As the VPO n ths wor s consdered to be a stochastc nonlnear dynamc system, the random noses are assumed to be present n both the state and measurement equatons of the true (actual) system. So the true state and the actual measurement y s represented as v y n (4) where the states and n (4) ndcates the states obtaned at the nstant. v s the zero mean whte process nose wth covarance Q and n s the zero mean whte measurement nose wth covarance R. Both these noses are assumed to be ndependent of past and current state. he randn functon n MALAB s used to generate random values of order for the process nose and measurement nose. he values of the ntal state and control parameter consdered for each case of VPO s hghlghted n the secton 5. and secton 5.. he desgn of UKF forms the core n the development of the for VPO system whch s dscussed here. Intally, random number of partcles {,,..., Np} are generated and then each partcle s made to propagate through the UKF algorthm. As a result, the mean ˆ and covarance P of the partcle s obtaned from the UKF whch s the proposal densty n ths partcle flterng desgn [9]. he UKF algorthm for the tme nde and ts desgn for VPO are as follows: Calculaton of sgma ponts: he estmated state ˆ at the prevous tme nde s augmented wth the mean of the process nose v and measurement nose n as ˆ [ ] [ ] (5) a E v E n In the unscented partcle flterng framewor, the mean th of the partcle at the prevous nstant, ˆ s augmented wth the mean of the noses and so the above equaton s modfed as a ˆ (6) a where the augmented term s of the order 6 n ths wor. he ntal state estmate ˆ chosen for each case of VPO s shown n the sectons below. Smlarly, the covarance matr P of the th partcle s also augmented wth the process nose covarance Q and measurement nose covarance R as P a P Q R (7) he values of covarance matrces P, Q and R of the order are shown n the followng sectons from whch the augmented covarance matr P a of the order 66 s calculated. Calculate Lsgma ponts from the augmented state and covarance to form the sgma pont matr as a a a ( ) a L P (8) where L L L L s the augmented state v n dmenson and ( L) L s a scalng parameter. and are tunng parameters. Choose to guarantee postve sem-defnteness of the covarance matr. s a factor determnng the spread of sgma ponts around the mean. Choose and t should deally be a small number to mnmze hgher order effects when the nonlneartes are strong. hs flter desgn for the -dmensonal VPO system consders the augmented state dmenson L 6 from whch sgma ponts are calculated to form the sgma pont matr as n (8). he flter tunng parameters are chosen as. and from whch the scalng parameter s computed. he sgma pont matr calculated n (8) can also be represented as 5
Journal of Electrcal Engneerng a v n (9) where the superscrpts, v and n refer to a partton conformal to the dmenson of the state, process nose and measurement nose respectvely. me update equatons: ransform the sgma ponts through the nonlnear system functon, f (, ) () v where the above nonlnear functon f s the state functon of nonlnear VPO system gven n (). Calculate the pror estmate of the state and covarance as L ( ), ˆ W m () L ˆ ˆ ( c) [, ][, ] P W he weghts W ( m) and W( c) are defned as W( m), L W L ( c) ( ), W( m) W( c),,..,l ( L ) () () Choose the parameter whch s a nonnegatve weghtng term. he parameter can be used to control the error n the urtoss (hgher order moments) whch affects the heavness of the tals of the posteror dstrbuton. he value of s chosen n ths desgn. hen the selected flter tunng parameters are substtuted n () to evaluate the weghts W ( m) and W( c) for the mean and covarance respectvely. Measurement update equatons: ransform the sgma ponts through the measurement functon, (, n g ) (4) where the above functon g s the measurement functon of VPO system whch s consdered to be lnear n ths wor. Calculate the predcted measurement and ts covarance as yˆ L W( m), (5) L ˆ ˆ y ( ) [, ][, ] y c (6) P W y y he state-measurement cross covarance matr s calculated as L ˆ ˆ ( ) [, ][, ] y c (7) P W y he Kalman gan s gven as K P P (8) y y y herefore, the posteror estmates of the state and covarance s computed as ˆ ˆ K ( y yˆ ) (9) P P K P K () yy Hence, the mean ˆ and covarance P of the partcle s calculated usng UKF algorthm from (9) and () respectvely. In the same manner, ˆ and P are obtaned for all the partcles. hen the mportance weght w whch depends on the lelhood functon p( y ) s calculated for each partcle. he lelhood functon consdered n ths desgn s as follows: ( ) ( ) p( y ) ep r R r ( ) det( R) where () r () s the predcton error based on the th partcle whch corresponds to the dfference between the actual measurement y and predcted measurement yˆ of the VPO system wth -dmensonal state vector and R s the measurement nose covarance. he mportance weght w s then normalzed as 6
Journal of Electrcal Engneerng w w N p w () he resamplng technque s net carred out to select only the partcles and ts assocated covarance wth hgher mportance weght. Fnally, average of these updated partcles from ths algorthm s taen to obtan the state estmate of the VPO system at the nstant. he updated partcles and ts covarance matrces are used for the net samplng nstant and the same procedure s recursvely carred out for all the samplng nstants. 5. State estmaton of VPO under stable lmt cycle usng partcle flters In ths secton, the state estmates of a VPO system ehbtng stable lmt cycle usng the and are compared and the flter responses are analysed under dfferent condtons. he stable lmt cycle of the VPO system has the property that all trajectores n the vcnty of the lmt cycle ultmately tend towards t as t. In the case of stable lmt cycle, any non-zero ntal state converges to a stable lmt cycle. he estmaton performance of the and are analysed below for dfferent cases. able State estmaton performance of flter under dfferent condtons for VPO system ehbtng stable lmt cycle Average RMSE Condtons Estmates Estmates ˆ ˆ ˆ ˆ Normal system operatng condtons.47.95.45.4 Intal state msmatch and model error Intal state msmatch and large measurement nose.5646.596.499.489.49.977.994.67 state ( ) and the ntal state estmate ( ˆ ) are chosen as.. he ntal state covarance( P ), process nose covarance ( Q ) and measurement nose covarance ( R ) are chosen as P, R (.5) State State - - Q (.5) and - 5 5 5 Samplng Instants - - - 5 5 5 Samplng Instants () Fg.. Evoluton of true and estmated states of VPO n forward tme usng and under normal operatng condtons. - Case : Normal system operatng condtons In the frst case, t s assumed that the states of the VPO system wth stable lmt cycle are estmated under the normal operatng condtons n the absence of ntal state and model parameter msmatch. he value of the control parameter s chosen as.4. he ntal - - -.5 - -.5 - -.5.5.5 Fg.. Phase portrat of estmates under normal operatng condtons for VPO ehbtng stable lmt cycle. 7
Journal of Electrcal Engneerng - - - - - - Fg.. Phase portrat of estmates under normal operatng condtons for VPO ehbtng stable lmt cycle. he true and estmated states usng the and are shown n Fg.. he estmates tracs closely the true state trajectory as compared to that of. he reason s that the pror densty consdered as proposal densty n the does not allow the partcles n the pror to move to the regons of hgh lelhood.he phase portrat of and estmates representng the closed trajectory of the VPO system n the phase plane are shown n Fgs. and respectvely. he phase portrats of the depcts that ths flter response very closely follows the trajectory of the true system subjected to nose. he average RMSE values of both the flter estmates under normal operatng condtons are lsted n the frst row of able. Case : Intal state and model parameter msmatch he second case s consdered to evaluate the robustness of the over under the presence of model error and ntal state estmate msmatch. In the true system and the state estmator model, the value of s chosen as.4 and.6 respectvely to ntroduce model error. he ntal state s chosen as.. he ntal state estmate ( ˆ ), ntal state covarance ( P ), process nose covarance( Q ) and measurement nose covarance ( R ) are chosen as ˆ, State State - - - 5 5 5 Samplng Instants - - - 5 5 5 Samplng Instants Fg. 4. Evoluton of true and estmated states of VPO n forward tme usng and wth large ntal state msmatch and model error. It s observed from Fg. 4 that when there s a sgnfcant model error and large ntal state msmatch, the s more robust and thereby, gvng sgnfcantly better estmaton results but the estmates are not able to converge to the true states. he phase portrat of and estmates n the presence of ntal state msmatch and model error are shown n Fgs. 5 and 6 respectvely. he average RMSE values calculated for ths case are lsted n the second row of able. P, Q (.5) and - R (.5) (4) he choce of P s reasonable here as the ntal state estmate s far from the true ntal state. - - - - - Fg. 5. Phase portrat of estmates wth large ntal state msmatch and model error for VPO ehbtng stable lmt cycle. 8
Journal of Electrcal Engneerng - - State - - - - - - Fg. 6. Phase portrat of estmates wth large ntal state msmatch and model error for VPO ehbtng stable lmt cycle. Case : Intal state msmatch and large measurement nose he flter responses under the assumpton of ntal state msmatch and large measurement nose are dscussed here. he parameter for both the system and the state estmator model s chosen as.4. he ntal state s chosen as.. he ntal state estmate ( ˆ ), ntal state covarance( P ), process nose covarance ( Q ) and measurement nose covarance ( R ) are chosen as ˆ, P, R (.) Q (.5) and (5) It s reasonable to assume hgher value of measurement nose covarance for both the flters as ths case deals wth large measurement nose. he nterestng fndng from Fg. 7 s that n spte of consderng large measurement nose and ntal state vector msmatch, the estmates converge to the true states and follow ts trajectory, whereas the estmates could not converge to the true states as the estmates are obtaned at each tme nstant wthout any nowledge of the measurements.he phase portrat of and estmates subjected to large measurement nose covarance are shown n Fgs. 8 and 9 respectvely. he average RMSE values obtaned for ths condton are lsted n thrd row of able. State - 5 5 5 Samplng Instants - - - 5 5 5 Samplng Instants Fg. 7. Evoluton of true and estmated states of VPO n forward tme usng and wth large ntal state msmatch and measurement nose. - - - - - - Fg. 8. Phase portrat of estmates wth large ntal state msmatch and measurement nose for VPO ehbtng stable lmt cycle. - - - - - - Fg. 9. Phase portrat of estmates wth large ntal state msmatch and measurement nose for VPO ehbtng stable lmt cycle. 9
Journal of Electrcal Engneerng 5. State estmaton of VPO under unstable lmt cycle usng partcle flters he unstable lmt cycle of the VPO system has the property that all trajectores startng from ponts arbtrarly close to the lmt cycle tend away from t as t. In the case of an unstable lmt cycle, f the ntal state s nsde the lmt cycle, t converges to zero as tme progresses. But nstead, f the ntal state s outsde the lmt cycle, t dverges. Intally, the smulaton study has been carred out to understand the behavour of VPO system ehbtng unstable lmt cycle n the absence of process and measurement nose. Frst, let us consder the ntal state as.7 whch s well nsde the lmt cycle and.. Fg. shows that both the states of the system converge to zero as tme progresses for the above assumed ntal state. he phase portrat of the system under states convergng condton s shown n Fg...4. -. -.4 -.6 -.6 -.4 -...4.6.8 Fg.. Phase portrat of VPO ehbtng unstable lmt cycle wth an ntal state nsde the lmt cycle..5 7 State.8.6.4. -. -.4 -.6 5 5 5 Samplng Instants State.5.5 5 5 5 Samplng Instants.5 State.4. -. -.4 -.6 5 5 5 Samplng Instants Fg.. Response of VPO n reverse tme wth an ntal state nsde the lmt cycle. State.5.5 5 5 5 Samplng Instants Fg.. Response of VPO n reverse tme wth an ntal state outsde the lmt cycle. On the other hand, consder the ntal state as.5 whch s outsde the lmt cycle and the parameter.. It can be observed n Fg. that the states of the system dverge for the above assumed ntal state and the phase portrat of the system under dvergng condtons s shown n Fg..
Journal of Electrcal Engneerng.5.5.5 State -.5.5 -.5.5.5 7 Fg.. Phase portrat of VPO ehbtng unstable lmt cycle wth an ntal state outsde the lmt cycle. he state estmaton performance of and SIR- PF for a VPO system wth unstable lmt cycle n the presence of stochastc process and measurement nose are analysed below for dfferent condtons. Case : Normal system operatng condtons he states of the system are estmated n ths case wthout the ntal state vector msmatch and model error. he value of the parameter s chosen as.. he ntal state ( ) and the ntal state estmate ( ˆ ) are chosen as.7 whch s nsde the lmt cycle. he ntal state covarance ( P ), process nose covarance ( Q ) and measurement nose covarance ( R ) are chosen as P, Q (.5) and R (.5) (6) It s nferred from Fg. 4 that the estmates captures the trajectory of the true states more closely than the estmates whch does not converge to the true state as tme progresses. State.5.5 -.5-5 5 5 Samplng Instants -.5 5 5 5 Samplng Instants Fg. 4. Estmaton performance of and wth an ntal state estmate nsde the lmt cycle for VPO n reverse tme. Case : Intal state and model parameter msmatch In the second case, the parameter for the system and the state estmator model s chosen as. and.5 respectvely and thereby, ntroducng model parameter msmatch. he ntal state s chosen as.7 whch s nsde the lmt cycle. he ˆ.5 whch ntal state estmate s chosen as s outsde the lmt cycle. State State.5.5 -.5 - -.5-5 5 5 Samplng Instants - - - 5 5 5 Samplng Instants Fg. 5. Estmaton performance of and wth an ntal state estmate far from the lmt cycle and wth model error for VPO n reverse tme. he ntal state covarance ( P ), process nose covarance( Q ) and measurement nose covarance( R ) are chosen as
Journal of Electrcal Engneerng P, Q (.5) and he ntal state covarance ( P ), process nose covarance ( Q ) and measurement nose covarance ( R ) are chosen as R (.5) (7) Fg. 5 llustrates that the attans the superor estmaton results than the whch results n large estmaton error. It s also clear from Fg. 5 that the SIR flter estmates are more senstve to the model error but the provdes hgher degree of robustness to the model error. It s also noted that under hgh ntal state msmatch, the estmates converge at a faster rate as ts state covarance decreases faster. Case : Intal state msmatch and large measurement nose It s assumed under ths condton that for the system and estmator model, the value of s chosen. he ntal state s chosen as as..7 whch s nsde the lmt cycle. he ntal state estmate s chosen as ˆ.5 whch s outsde the lmt cycle. State State.5.5 -.5 - -.5 5 5 5 Samplng Instants.5.5 -.5 - -.5 5 5 5 Samplng Instants Fg. 6. Estmaton performance of and wth an ntal state estmate far from the lmt cycle and wth large measurement nose for VPO n reverse tme. P, R (.) Q (.5) and (8) It s observed from Fg. 6 that the s much better n handlng both ntal state msmatch and large measurement nose and hence, acheves better state estmates than the. It also shows that the SIR flter s unable to trac the true state as ts partcles are not able to le n the regon of the true state. able State estmaton performance of flter under dfferent condtons for VPO ehbtng unstable lmt cycle Average RMSE Condtons Estmates Estmates ˆ ˆ ˆ ˆ Normal system operatng condtons.7.594.47.475 Intal state msmatch and model error Intal state msmatch and large measurement nose.7776.86.48.5.569.5854.8.96 he average RMSE values of the and estmates under the above three condtons for VPO system ehbtng unstable lmt cycle are lsted n able to assess the estmaton performance. 6. Concluson hs paper has demonstrated the effectveness of the for estmatng the states of a hghly nonlnear Van der Pol oscllator (VPO) and the results are compared wth the. It s found through smulaton studes that the UKF s better suted as proposal densty n the partcle flter because t ncorporates the latest measurements before the evaluaton of mportance
Journal of Electrcal Engneerng weghts. Hence, ths UKF proposal allows the partcles to move towards the hgh lelhood regon. Even though the UKF whch assumes Gaussan dstrbuton s used for generatng the proposal, the serves to brng the partcles closer to the true state and retans ts ablty to estmate non-gaussan state dstrbutons. From the results, t can be nferred that the outperforms the under normal system operatng condtons, hgh ntal state msmatch and large measurement nose. he proves to be more robust to the error nduced n the state estmator model of VPO. he flter performances are also evaluated for ths nonlnear oscllator by calculatng the RMSE values under dfferent operatng condtons. Hence, t s found that can be the good choce for VPO because of ts ablty to provde more accurate estmaton of states. References. R. S. Barbosa, J. A.. Machado, B. M. Vnagre, and A. J. Calderon, Analyss of the Van der Pol oscllator contanng dervatves of fractonal order, J. Vb. Control, Vol., No. 9-, pp. 9-, Sep. 7.. R. Kandepu, B. Foss, and L. Imsland, Applyng the unscented Kalman flter for nonlnear state estmaton, J. Process Contr., Vol. 8, No. 7-8, pp. 75-768, Aug. 8.. R. Sajeeb, C. S. Manohar, and D. Roy, A semanalytcal partcle flter for dentfcaton of nonlnear oscllators, Probablst. Eng. Mech., Vol. 5, No., pp. 5-48, Jan.. 4. H. K. Khall, Nonlnear Systems, Prentce Hall, pp. 4-47,. 5. I. M. Gnarsa, A. Soeprjanto, and M. H. Purnomo, Controllng chaos and voltage collapse usng an ANFIS-based composte controller-statc var compensator n power systems, Int. J. Elec. Power, Vol. 46, pp. 79-88, March. 6. R. E. Precup, M. L. omescu, and C. A. Dragos, Stablzaton of Rossler chaotc dynamcal system usng fuzzy logc control algorthm, Int. J. Gen. Syst., Vol. 4, No. 5, pp. 4-4, Feb. 4. 7. S. K. Choudhary, LQR based optmal control of chaotc dynamcal systems, Int. J. Model. Smul., Vol. 5, No. -4, pp. 4-, Dec. 5. 8. A. Bouzerba, A. Boulroune, and. Bouden, Projectve synchronzaton of two dfferent fractonalorder chaotc systems va adaptve fuzzy control, Neural Comput. Appl., Vol. 7, No. 5, pp. 49-6, July 6. 9. M. Barurt, S. Bogosyan, and M. Gasan, Speedsensorless estmaton for nducton motors usng etended Kalman flter, IEEE rans. Ind. Electron., Vol. 54, No., pp. 7-8, Feb. 7.. S. Kumar, J. Praash, and P. Kanagasabapathy, A crtcal evaluaton and epermental verfcaton of etended Kalman flter, unscented Kalman flter and neural state flter for state estmaton of three phase nducton motor, Appl. Soft Comput., Vol., No., pp. 99-8, Aprl.. P. erwesch and M. Agarwal, A dscretzed nonlnear state estmator for batch processes, Comput. Chem. Eng., Vol. 9, No., pp. 55-69, Feb. 995.. R. E. Kalman, A new approach to lnear flterng and predcton problems, J. Basc Eng.- ASME, Vol. 8, No., pp. 5-45, 96.. A. H. Jazwns, Stochastc Process and Flterng heory, Academc Press, pp. 4-58, 97. 4. A. Gelb, Appled Optmal Estmaton, M.I.. Press, pp. -7, 974. 5. E. A. Wan and R. Van der Merwe, he unscented Kalman flter for nonlnear estmaton, n Proceedngs of IEEE Symposum (AS-SPCC), Lae Louse, Alberta, Canada, pp. 5-58,. 6. S. J. Juler and J. K. Uhlmann, Unscented flterng and nonlnear estmaton, n Proceedngs of IEEE, Vol. 9, No., pp. 4-4, March 4. 7. N. Gordon, D. Salmond, and A. F. M. Smth, Novel approach to nonlnear and non-gaussan Bayesan state estmaton, IEE Proc.-F, Vol. 4, No., pp. 7-, 99. 8.. Chen, J. Morrs, and E. Martn, Partcle flterng for state and parameter estmaton n batch processes, J. Process Contr., Vol. 5, No. 6, pp. 665-67, Sep. 5. 9. R. Van der Merwe, A. Doucet, N. de Fretas and E. Wan, he Unscented Partcle Flter, ech. Rep. CUED/F- INFENG/R 8, Department of Engneerng, Unversty of Cambrdge, UK,.. X. Nng and J. Fang, Spacecraft autonomous navgaton usng unscented partcle flter based celestal/doppler nformaton fuson, Meas. Sc. echnol., Vol. 9, No. 9, pp. -8, 8.. A. V. Shenoy, J. Praash, K. B. McAuley, V. Prasad V, and S. L. Shah, Practcal ssues n the applcaton of the partcle flter for estmaton of chemcal processes, n Proceedngs of 8 th IFAC World Congress, Mlano, Italy, pp. 77-778,.. Y. Shen, Hybrd unscented partcle flter based stateof-charge determnaton for lead-acd batteres, Energy, Vol. 74, No., pp. 795-8, 4.. G. G. Rgatos, Partcle flterng for state estmaton n nonlnear ndustral systems, IEEE rans. Instrum. Meas., Vol. 58, No., pp. 885-9, Nov. 9. 4. O. Aydogmus and M. F. alu, Comparson of Etended-Kalman-and partcle-flter-based sensorless speed control, IEEE rans. Instrum. Meas., Vol. 6, No., pp. 4-4, Feb.. 5. J. W. Lee, Y. S.. Hong, C. Suh, and H. S. Shn, Onlne nonlnear sequental Bayesan estmaton of a bologcal wastewater treatment process, Boproc. Bosyst. Eng., Vol. 5, No., pp. 59-69, July. 6. F. Kwasno, Estmaton of nose parameters n dynamcal system dentfcaton wth Kalman flters,
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