UNCLASSIFIED Nationaal Lucht--- en Ruimtevaartlaboratorium National Aerospace Laboratory NLR Executive summary Exact Bayesian and particle filtering of stochastic hybrid systemssystemssystems Problem area Description of work Report no. In literature on Bayesian filtering of This report considers filtering of NLR-TP-2006-691 stochastic hybrid systems most stochastic hybrid systems that go studies are limited to Markov jump beyond the well known Markov Author(s) systems. The main exceptions are jump system. First the non- H.A.P. Blom, E.A. Bloem approximate Bayesian filters for Markovian jump system studied in Report classification semi-Markov jump linear systems. this report is formally defined. Next Unclassified These studies showed that nonlinear the exact Bayesian filter recursion filtering becomes much more for this system is developed and the Date challenging under non-Markov implication of going beyond September 2006 jumps. This challenge however does Markov jumps is shown. not apply to particle filtering of Subsequently a novel particle filter Knowledge area(s) stochastic hybrid systems. In for jump nonlinear systems is Advanced (sensor-) information practice, non-Markov jumps rather developed which is referred to as processing are the rule, not the exception. For the Interacting Multiple Model example, on an airport, the particle filter (IMMPF). For probability at which a taxiing comparison Monte Carlo aircraft makes a maneuver depends simulations are performed. heavily on its position; e.g. when Descriptor(s) taxiing near a crossing on the Results and conclusions Stochastic hybrid systems airport, the probability of starting a Through Monte Carlo simulations Bayesian filtering turn is relatively high, whereas IMMPF has been tested and Particle filtering outside these areas this probability compared with standard Particle State dependent switching may be very small. A similar Filter (PF) and IMM. The results Jump-nonlinear systems difference applies to the probability show that the IMMPF performs Non-Markov jumps Maneuvering target tracking very well, even in cases where the of an aircraft making a turn when it is flying near a waypoint versus standard PF or IMM have problems. flying halfway two waypoints. Similar non-Markov jump behavior Applicability also applies to other traffic The applicability of the work modalities, and to any other comprises the implementation of intelligently controlled system. the resulting filtering algorithms in Nevertheless in target tracking, a multitarget tracker, in particular particle filtering studies have the Advanced suRveillance Tracker continued to focus on Markov jump And Server ARTAS. systems Nationaal Lucht- en Ruimtevaartlaboratorium, National Aerospace Laboratory NLR Anthony Fokkerweg 2, 1059 CM Amsterdam, P.O. Box 90502, 1006 BM Amsterdam, The Netherlands UNCLASSIFIED Telephone +31 20 511 31 13, Fax +31 20 511 32 10, Web site: www.nlr.nl Nationaal Lucht- en Ruimtevaartlaboratorium National Aerospace Laboratory NLR NLR-TP-2006-691 Exact Bayesian and particle filtering of stochastic hybrid systems H.A.P. Blom and E.A. Bloem This report contains an article to appear in IEEE Transaction on Aerospace and Electronic Systems. This report may be cited on condition that full credit is given to NLR and the authors. Customer: National Aerospace Laboratory NLR Working Plan number: 2005 AT.1.C Owner: National Aerospace Laboratory NLR Division: Air Transport Distribution: Unlimited Classification title: Unclassified September 2006 Approved by: Author Reviewer Managing department Anonymous peer reviewers Exact Bayesian and Particle Filtering of Stochastic Hybrid Systems Henk A.P. Blom Senior Member IEEE & Edwin A. Bloem challenging under non-Markov jumps. This challenge Abstract —The standard way of applying particle filtering to however does not apply to particle filtering of stochastic stochastic hybrid systems is to make use of hybrid particles, hybrid systems [5]. Hence there is no reason to continue where each particle consists of two components, one assuming focusing on Markov jump systems. In practice, non-Markov Euclidean values, and the other assuming discrete mode values. This paper develops a novel particle filter for a jumps rather are the rule, not the exception. For example, discrete-time stochastic hybrid system. The novelty lies in the on an airport, the probability at which a taxiing aircraft use of the exact Bayesian equations for the conditional mode makes a maneuver depends heavily on its position; e.g. probabilities given the observations. Therefore particles are when taxiing near a crossing on the airport, the probability needed for the Euclidean valued state component only. The of starting a turn is relatively high, whereas outside these novel particle filter is referred to as the Interacting Multiple areas this probability may be very small. A similar Model (IMM) particle filter because it incorporates a filter step which is of the same form as the interaction step of the difference applies to the probability of an aircraft making a IMM algorithm. Through Monte Carlo simulations, it is turn when it is flying near a way-point versus flying halfway shown that the IMM particle filter has significant advantage two way-points. Similar non-Markov jump behavior also over the standard particle filter, in particular for situations applies to other traffic modalities, and to any other where conditional switching rate or conditional mode intelligently controlled system. Nevertheless in target probabilities have small values. tracking, particle filtering studies have continued to focus on Markov jump systems [6] – [19]. Keywords: Stochastic hybrid systems, Bayesian filtering, Particle filtering, state dependent switching, jump-nonlinear systems, non-Markov jumps, maneuvering target tracking. In nonlinear filtering studies, the Sampling Importance Resampling (SIR) based approach [20] has surfaced as the I. INTRODUCTION baseline particle filter. It has shown to form an elegant and The paradigm of particle filtering has stimulated a general approach towards the numerical evaluation of the renewed interest in exact Bayesian filtering of non-linear conditional density solution of the Chapman-Kolmogorov- stochastic systems. In line with this, [1] recently developed Bayes (CKB) filter recursion [21] – [24]. For a large class the exact Bayesian filter for Markov jump non-linear of problems it has been shown that increasing the number of systems. The aim of the current paper is to extend the exact particles ensures weak sense convergence of the Bayesian filter characterization to the much larger class of approximation density to the exact conditional density at a non-Markov jump non-linear systems, and subsequently to rate that is independent of or increases linearly with the exploit this exact Bayesian characterization for the state dimension of the process to be estimated [25] – [27]. development of a novel particle filter for stochastic hybrid This means that the SIR particle filter significantly relaxes systems. the curse of dimensionality of CKB filtering. Practically, In literature on Bayesian filtering of stochastic hybrid the SIR particle filter has found a large variety of useful systems most studies are limited to Markov jump systems. applications, including some where established non-linear The main exceptions are approximate Bayesian filters for filtering approaches do not work at all, such as the visual semi-Markov jump linear systems [2] – [4]. These studies tracking example of [28] and the track-before-detect showed that nonlinear filtering becomes much more example of [29]. Because of its generality and theoretical validity, the SIR particle filter has become a useful reference in numerically approximating CKB filter Manuscript received October 25, 2004; revised October 7, 2005; performance [18]. released for publication March, 2006 This work was supported in part by the European Commission project HYBRIDGE, IST-2001-32460. The SIR particle filter is also capable in approximating Both authors are with the National Aerospace Laboratory NLR, P.O. Box the CKB equations of a stochastic hybrid Markov process 90502, 1006BM Amsterdam, The Netherlands (phone: +31 20 5113544; θ n θ fax: +31 20 5113210; e-mail: [email protected], [email protected]). {xt , t } , with xt assuming values in R , and t assuming 1 θ values in a finite set M of possible modes [5]. Let {xt , t } work well without the need of ordering the samples be the hidden state process to be estimated from noisy according to their weights. • Boosting diversity by regularization, i.e. prior to observations, then the SIR particle filter use NP particles, resampling, replace each Dirac measure in the jθ j particle j of which has two components (xt , t ) at moment empirical distribution by an absolutely continuous j θ j t, with xt assuming an Euclidean value and t assuming a distribution. This approach has largely been mode value. Each SIR particle filter cycle from t-1 to t developed for particle filtering of stochastic consists of three steps: processes that, without regularization, do not satisfy the regularity conditions under which the • Evolution: For each of the N particles at moment t-1, p weak convergence to the exact conditional density jθ j is satisfied [12]. A valuable alternative is to insert draw a new hybrid particle (xt , t ) according to the Chapman-Kolmogorov transition kernel; a Monte Carlo Markov Chain (MCMC) move in each SIR cycle [31], [11]. • Correction: For each of the N particles evaluate µ j p t • Importance density based resampling. This means as the likelihood of the measurement at moment t; that particle resampling anticipates how the jθ j µ j given (xt , t ) and normalize the resulting t ’s. weights per particle will evolve over the next prediction and correction steps. Each new particle • Resampling: Draw N independently identically p µ j sample receives a weight t that compensates for jθ j distributed (i.i.d.) hybrid particle values (xt , t ) , with this importance resampling. The optimal µ j importance density has been characterized by from the sum of t weighted Dirac measures at Doucet et al. [32]. Because exact implementation (x j ,θ j ) . t t is hard, approximations are needed.