A Nonlinear Filter based on Fokker Planck Equation and MCMC Measurement Updates

Transcription

1 A Nonlinear Filter based on Fokker Planck Equation and MCMC Measurement Updates Mrinal Kumar* and Suman Chakravorty** Abstract This paper presents a nonlinear filter based on the Fokker-Planck equation (FPE) for uncertainty propagation, coupled with a fast measurement update step. The measurement update is implemented as a function approximation performed over a Markov chain Monte Carlo (MCMC) sample of the unnormalized posterior obtained from the Bayes rule. MCMC sampling also results in fast computation of the normalization factor of the posterior, which is typically a computationally heavy step. A previously developed semianalytical meshless tool is employed to solve FPE for high dimensional systems in real time. Performance of the filter is studied for dynamical systems with 2 and 4 dimensional state spaces. I. INTRODUCTION Nonlinear filtering has been a widely researched subject for the past several decades. The Kalman filter [1] and later, the extended Kalman filter (EKF) have been the most widely used tools for this problem. The former is the optimal state estimator for systems governed by linear dynamics and linear measurement models. The latter extends the Kalman filter to nonlinear systems by considering the first order approximation of the actual nonlinear dynamics and measurement model. In other words, the Kalman filter and EKF deal with linear and linearized dynamical systems respectively. The primary drawback of EKF, namely linearization, is also its main appeal among practitioners because it greatly simplifies its implementation. However, several instances have been encountered over the years in which they fail to perform well or even diverge. Despite interest in nonlinear filtering methods since the 1980 s (e.g. see [2]), they have become practicable only in the past two decades with the sudden explosion in computing capability. This paper considers the problem of nonlinear filtering for continuous dynamical systems and discrete measurement updates. The key objective of filtering, also known as state estimation, is to determine the probability density function (pdf) of the system states conditioned on all available measurements. This leads us to the key research issue: a general probability density function is fully characterized only when all its moments, which are infinite in number, are taken into account. Therefore, any general nonlinear filter is equivalent to a set of propagation and update rules in the infinite dimensional space of moments. An alternate but equivalent description of the nonlinear filtering problem This work was not supported by any organization *M. Kumar is a Postdoctoral Research Assistant in the Department of Aerospace Engineering at the Texas A&M University, College Station, TX USA. mrinal@tamu.edu **S. Chakravorty is with the Faculty of Aerospace Engineering at the Texas A&M University, College Station, TX 77843, USA. schakrav@aero.tamu.edu can be given in functional space. Here, the Fokker-Planck equation (FPE) describes the propagation phase of the pdf and the Bayes rule describes the measurement update step. Unfortunately, analytical solutions of FPE, which is a partial differential equation, are not known to exist except for the case of linear dynamical systems (Ref. [3]). Additionally, numerical attempts have consistently been foiled by several difficult issues, most notable being the curse of dimensionality (Refs. [4] [6]). Furthermore, the update rule also suffers dimensionality issues which prevents its real-time implementation, as is required in the filtering application. As a result, the nonlinear filtering literature has branched into three broad directions (Ref. [7]). The first category of nonlinear filters can be classified as closure-filters. Here, the state uncertainty is characterized by moments of the pdf truncated up to a pre-determined order. The EKF is a special case which considers only the first two moments, leading to Gaussian closure. In the same spirit as EKF, propagation and update rules can be determined for higher order moments (higher order closures), thus capturing desired nonlinear effects (see Refs. [8], [9]). As with EKF, the higher order filters are subject to similar issues of poor performance/divergence depending on the degree of nonlinearity and/or intensity of noise in the underlying system. The second class of nonlinear filters, known as particlefilters treat the problem differently. They characterize the state uncertainty in terms of the statistics of a sample of points called particles. Each particle carries a weight and is individually propagated forward through the stochastic dynamics of the system (Refs. [10] [16]). The particle weights are updated as new measurements come in and the desired moments of the pdf are computed by studying the statistics of the weighted particles. Since the true pdf of the state is not known a-priori, the particles are sampled from a proposal density function, one that is known and relatively easy to sample from. Particle filters face two key research issues, first of which is determining how many points are sufficient to confidently describe the statistics of the uncertain state. The second difficulty faced by particle filters is called particle depletion - a scenario in which particles residing in regions of high probability mass migrate to the tail regions of the pdf. This causes thinning of the particle density in regions where the bulk of the conditional pdf resides. This requires re-sampling every few time steps so that particles are replenished in the regions of high probability density and removed from the tail regions. An important technique in this context is that of sampling/importantance resampling (SIR) (Refs. [10], [11], [17]). Recently, numerous particle filters

2 based on the Markov chain Monte Carlo sampling technique have been designed, e.g. Ref [18]. The third category of nonlinear filters, also the subject of this paper, is called projection filters. In general, this class of filters assumes the state conditional pdf to belong to a family of functions and determines the parameters associated with the said functions. Ref. [19] assumes the pdf to reside in the family of exponential pdfs. Xu et al. [20] represented the state conditional pdf in terms of Dirac delta functions and employed a direct quadrature method of moments to solve the associated FPE. Similarly, Gaussian sum filters (Refs. [21] [23]) assume the state conditional pdf to be composed of a mixture of Gaussians. Beard [24] proposed a projection filter in which the state pdf was expanded in terms of global shape functions. The FPE was solved in weak form via the Galerkin projection technique and the update was implemented as a weak form of the Bayes rule. Due to numerical issues associated with solving FPE, the application of this filter was limited to low dimensional state spaces. In addition, the implementation of Bayes rule in variational form is not suitable for the filtering application because it requires the computation of a large measurementstiffness matrix every time a new measurement is obtained. In Kumar et al. [25], a similar projection filter was proposed where the FPE was solved in real time using a meshless particle-partition of unity finite element technique. However, this filter also solved the Bayes rule in variational form, thereby restricting its application to cases with low-frequency updates. The current paper is an extension of the work in Ref. [25]. It utilizes the same technique as in Refs. [5], [6] and [25] for the propagation phase, namely, a meshless semianalytical method for solving FPE in near real-time. For the measurement update however, the current paper utilizes a Markov chain Monte Carlo sampling based technique to speed up the process, thus making the nonlinear filter more amenable to high-frequency update scenarios. The semianalytical FPE solver is a robust tool based on the meshless particle-partition of unity finite element method (ppufem) for dealing with nonlinear uncertainty propagation. ppufem naturally extends FPE discretization to high dimensional systems due to its meshless nature ( node based as opposed to element based). Equipped with h and p refinements, it is capable of generating highly accurate solutions of FPE using a small number of local shape functions. In Refs. [5] and [6], ppufem was combined with modal analysis to generate transient FPE response in real time for nonlinear dynamical systems, thus making it ideally suited for filtering applications. Markov chain Monte Carlo, or MCMC is a well established technique [26] for sampling non-standard distributions. The Metropolis-Hastings (MH) algorithm can be used to quickly generate a small sample from (a possibly unnormalized) pdf. Following the propagation step, the current paper utilizes the MH algorithm to generate a sample from the un-normalized posterior obtained from the Bayes rule. A function approximation algorithm (least-squares) is then implemented on this sample to approximate the posterior. This can be done online because the mass matrix associated with least-squares can be pre-computed. The outcome is that the MCMC based approach considerably speeds up the measurement update step. The remainder of this paper is organized as follows: Section II states the problem of nonlinear filtering with FPE and Bayes rule. Sections II-A - II-D describe each step of the proposed filter, including pdf approximation (Sec.II- A), filter initialization (Sec.II-D), propagation (Sec.II-B) and measurement update (Sec.II-C). Results are shown in Sec.III and conclusions drawn in Sec.IV. II. THE BAYES NONLINEAR FILTERING PROBLEM As mentioned earlier, we shall consider the case of continuous time dynamical systems and discrete measurement updates. The Bayes nonlinear filtering problem involves two steps implemented recursively - uncertainty propagation ( prediction) and measurement update ( filtering). Following filter initialization, the true description of state uncertainty propagation is given by the Fokker Planck equation. Measurements obtained are incorporated into state uncertainty via the Bayes rule. The filter starts again with propagation until the next batch of measurements are obtained. To set up the mathematical framework, consider a stochastic dynamical system governed by the following Itô stochastic differential equation (SDE): dx = f(t, x)dt + g(t, x)db(t), E[x(t 0 )] = x 0 (1) where, B represents a M-dimensional zero mean Brownian motion process with correlation function Qδ(t 1 t 2 ), and x 0 represents the mean initial state. Vector functions f(t, x) : [0, ) R N R N and g(t, x) : [0, ) R N R N M are measurable functions. The initial probability density of the state is known to be W(0, x) = W 0 (x), which captures the state uncertainty at time t = 0. Incoming measurements are captured by the following discrete nonlinear measurement model: y k = h(x k ) + v k (2) where, y R l, v is a l-dimensional Brownian motion process (measurement noise), h(x) : R n R l is measurable and k denotes the time instant of measurement. The growth of information with successive measurements is denoted by the filtration Y k = {y 0, y 1,..., y k }. Then, the nonlinear Bayes filtering problem is to determine the state probability density function conditioned on the filtration Y k, i.e., W(x k Y k ) utilizing the following recursive equations: W(x k Y k 1 ) = W(x k x k 1 )W(x k 1 Y k 1 )dx k 1 (3) W(x k Y k W(y k x k )W(x k Y k 1 ) ) = Ω W(y (4) k ξ)w(ξ Y k 1 )dξ Eq.3 represents the propagation part between two measurement updates. The left hand side of this equation is the prior state pdf, which can be obtained by integrating the associated Fokker-Planck equation between time labels

3 t k 1 and t k. The posterior state pdf is obtained from the Bayes rule in Eq.4. In this equation, W(y k x k ) represents the likelihood probability density function, given by: 1 W(y k x k ) = exp( 1 (2π det R) m 2 [y k h(x k )] T R 1 [y k h(x k )]), where, R k is the measurement error covariance matrix. As mentioned above, in order to obtain the prior pdf, i.e. W(x k Y k 1 ), it is required to solve the FPE associated with equation 1. FPE is a linear, parabolic partial differential equation given by: t W(t, x Y) = L FPW(t, x Y) (5) where, N N L FP = D (1) 2 i (, ) + D (2) ij (, ) (6) x i x i x j i,j=1 D (1) (t, x) = f(t, x) g(t, x) Qg(t, x) (7) x D (2) (t, x) = 1 2 g(t, x)qgt (t, x) (8) where, L FP is the Fokker-Planck operator, D (1) is known as the drift coefficient vector and D (2) is the diffusion coefficient matrix, both understood in the Stratonovich sense. There are numerous problems associated with solving FPE for general nonlinear high dimensional systems. Recently, Kumar et al. ( [5], [6]) have developed a robust meshless algorithm coupled with modal analysis to obtain near realtime transient response to FPE. This algorithm is used in the current paper for the propagation phase. We shall next consider the individual steps of the proposed nonlinear filter. A. State pdf Characterization Note that the Bayes nonlinear filtering problem is equivalent to an infinite dimensional system of equations involving all the moments of the unknown state pdf conditioned on available measurements. True characterization of the state pdf should therefore include all these moments. Unfortunately, a numerical method cannot do so directly because they are infinite in number. The Kalman filter/ekf truncates the pdf characterization at the first two moments, namely mean and covariance. By thus ignoring all higher moments, they implicity assume that the state pdf remains Gaussian at all times. The current filter does not characterize the pdf directly in terms of moments, but locally supported shape functions in the following manner: Ŵ(t, x Y) = P Q i a ij (t)ψ ij (x) (9) j=1 where, Ŵ(t, x Y) is an approximation of the true conditional state pdf, W(t, x Y). A similar annotation is used for all other approximated variables. The above characterization has been derived from a local, meshless approximation of W(t, x Y). The most important feature of a meshless approximation is that it is constructed in terms of nodes k distributed over the domain of solution. This is a clear paradigm shift from the traditional element-based, meshbased finite-element method (FEM). A mesh is nearly impossible to construct in dimensions greater than 3, thus severely restricting the utility of FEM for FPE. On the other hand, a meshless method such as the particle-partition of unity finite element method (ppufem) (Refs. [5], [6], [27], [28]) approximates the unknown purely in terms of nodes distributed over the domain of solution. Such a method is not concerned with how the nodes are connected to each other, thereby eliminating the mesh. Each node carries individually chosen local shape functions which best locally approximate the unknown. The local approximations of individual nodes are automatically blended together using weight functions called pasting functions. More detail about this technique can be obtained from Refs. [6], [27] and [28]. Due to space constraints, it is only important to consider Eq.9 in the present context. The outer summation (i = 1,... P ) enumerates the number of nodes in the approximation. The inner summation, (j = 1,... Q i ) counts the local shape functions assigned to the i th node. Therefore, Ψ ij (x) denotes the j th shape function assigned to the i th node in the approximation. The size of the approximation clearly is D = P Q i. As a result, the propagation and undate rules of the proposed nonlinear filter need to be determined for the associated parameters, a ij (t), totaling D in number. We will assume that the initial state conditional pdf is a posterior, i.e. the first step in the filter recursion is propagation. B. Filter Propagation For reasons which will soon be clear, filter initialization will be described last in this paper, following propagation and update steps. As noted before, the propagation step is given by Eq.3. This equation is equivalent to solving the associated FPE given in Eq.5. Substitution of Eq.9 in Eq.5 results in a residual. When the residual is minimized following the standard Galerkin projection technique, we obtain a D dimensional discretized representation of the FP operator: Mȧ(t) = Ka(t) + l. Here, M is the mass matrix, K is the stiffness matrix and l is the load vector. In Ref. [5], this discretization step was combined with modal analysis and spurious mode rejection to obtain near real-time transient FPE response. Modal analysis involves solving the generalized eigenvalue problem associated with the discretized system, i.e. Kφ = λmφ. The resulting set of eigenvectors {φ i } D is scrutinized for admissibility and physically irrelevant modes are blocked from participation in the approximation. Details of this process, which is fully automatic (does not require user input) can be obtained from Refs. [5], [6]. Spurious mode rejection leaves behind a reduced set (A) of admissible eigenfunctions that can be used to approximate the solution of the transient FPE as follows: Ŵ(t, x Y) = card(a) á i (t)φ i (x), φ i (x) A (10)

4 where, φ i (x) A are admissible eigenfunctions of the discretized FP operator and á i (t) are undetermined coefficients in modal space obtained from the transformation: á = V 1 a. Here, V is a matrix whose columns are eigenvectors of the system (K, M). Before we proceed, we note here that the parameters of the state pdf have now changed from a i (t) to á i (t), and their number reduced from D to card(a). The pdf is still characterized by shape functions, but now, by a special set of shape functions, namely the admissible eigenfunctions of the FP operator (Eq.10). This is also the reason why the filter initialization was left for the end because we actually deal with a transformed set of parameters, i.e. modal coefficients, á i (t), instead of the regular Fourier coefficients a i (t). A key point is that computation of admissible eigenfunctions is a pre-processing step that does not depend on initial state uncertainty. Therefore, once the eigenfunctions of the discretized FP operator are computed and stored away, the propagation step can be completed in real time. By virtue of working in the modal space, we obtain the following analytically solvable decoupled form: á(t) = Λá(t) + ĺ. In this relationship, Λ is a diagonal matrix containing the generalized eigenvalues of (K, M), and ĺ = V 1 M 1 l is the load vector in modal coordinates. The time history of these admissible modal amplitudes can now be written as: ( ) á i (t) = á i (t 0 ) + ĺi λ i exp(λ i t) ĺi λ i. (11) Accuracy of Approximation: Given the admissible set of eigenfunctions, it is possible to state the following result about the nature of computed transient FPE behavior: [4] Theorem 2.1: Let A = {φ i : Real(λ i ) < 0, ε i = L FP (φ i ) λ i φ i L2(Ω) < δ} be the set of stable admissible eigenfunctions. Define equation error in FPE as e(t) = tŵ(t, x Y) LFP(Ŵ(t, x) Y). If the initial equation-error is within a specified tolerance, i.e. e(t 0 ) < L2(Ω) ɛ, then the equation-error at all subsequent times is bounded by the initial equation-error, i.e. e(t) < ɛ. The proof of the above theorem is fairly straightforward and is excluded from the current paper (see Ref. [5]). The above theorem implies that equation error in FPE resulting from the modal basis actually reduces with time. In fact, a stronger result can be stated that the equation error is bounded above by an exponentially decaying envelope [5]. C. Filter Measurement Update The measurement update in a nonlinear Bayes filter is a computationally expensive step that requires implementation of the Bayes update rule. In most existing papers on projection-type nonlinear filters (e.g., see Beard [24]), this is achieved by solving the Bayes rule in weak form. This involves reducing Eq.4 to a discretized finite dimensional algebraic form via variational formulation: Ka = f, where K and f are the measurement stiffness matrix and load vector respectively. Unfortunately, neither K nor f can be computed off-line because they require input from the incoming new measurements. Because of the large size of K, this step cannot be completed fast enough for filtering applications. The current paper presents an accelerated method of implementing the measurement update step. Using the functional representation of the prior pdf from the propagation step (Eq.11) and the functional form of the likelihood function, the numerator of the RHS of Eq.4 is constructed. This is nothing but the un-normalized posterior state pdf. Next, a Markov chain Monte Carlo (MCMC) routine, such as the Metropolis-Hastings algorithm is run to generate a small sized sample from this un-normalized posterior. The motivation behind this step is to obtain a small-sized representation of the posterior density function. Let us denote this sample and the associated un-normalized probabilities by (ξ, W(ξ Y k )). Examples of such samples in 1 and 2 dimensions are shown in Fig.1. The sample size is typically very small, e.g. of the order of a few hundred points. Next, a least-squares function approximator is executed on this sample using the admissible eigenfunctions of the FP operator ({φ i } card(a) ) as basis functions. This results in the following normal equations: card(a) á + i (t k)φ i (x), φ j (x) = W(ξ Y k ), φ j (ξ) (12) where, á + i (t k) represent undetermined coefficients approximating the un-normalized posterior density function. The above system is equivalent to: Má = f, where the mass matrix M can be pre-computed. Only the load vector, f, in the RHS is required to be computed online as new measurements come in. This is much faster than the variational formulation of the Bayes update rule, which requires online computation of a measurement stiffness matrix. No such matrix is involved in the above formulation. We are now left with the task of computing the normalization factor, i.e. the denominator of Eq.4. Note that it is not possible to determine the normalization factor of Ŵ (x Y k) from its generated MCMC sample. Therefore, looking closer at the denominator of Eq.4, we treat it as the integral of the prior, weighted by the likelihood function. In the MCMC framework, this requires us to generate a sample of the likelihood function, which is a trivial task because it is Gaussian in nature. Thus the normalization constant can be computed as a weighted sum of the prior evaluated at points sampled from the likelihood function: W(ξ Y k 1 )W(y k ξ)dξ 1 N W(ζ i Y k 1 ); Ω N where, {ζ i } N W(y k x) (13) D. Filter Initialization From the propagation part, we know that the state conditional pdf is characterized in terms of the admissible eigenfunctions (φ A) of the FP operator. Therefore in filter initialization, we are required to find the modal coefficients at time t 0, i.e. á i (t 0 ). In the current filter, this is done by solving a function approximation problem using the pre-

5 + Actual pdf (un-normalized) MCMC Sample + MCMC Sample (a) MCMC samples for a bimodal pdf in 1D (b) MCMC samples for a bimodal pdf in 2D Fig. 1. MCMC sampling of probability density functions (Metropolis-Hastings algorithm) computed admissible eigenfunctions of the FP operator as basis functions. The function being approximated is the given initial state uncertainty distribution, W( x 0 ). As in the measurement update step, this can also be achieved via MCMC sampling using the same pre-computed mass matrix. We are thus led to the initial uncertainty estimate: Ŵ(x 0 Y 0 ) = card(a) á + i (t 0)φ (x). The superscript + signifies that the above pdf is a posterior, in accordance with the assumption made earlier that the filter is set into motion by propagation rather than update. In other words, if there is a measurement update involved at t = 0, it is assumed to be built into the function W( x 0 ). III. RESULTS Consider first a two-state nonlinear oscillator modeled by the Duffing equation with state-multiplied noise: ẍ + 2ηẋ x + ɛx 3 = xg 1 (t) + G 2 (t). The two independent components of noise, G 1 and G 1 have intensities D 11 and D 22. We consider the case of D 11 = 0.0 for easy comparison with the extended Kalman filter. Note that the current method is fully equipped to deal with state multiplied noise. Values of other parameters used are: η = 0.1, ɛ = 0.5 and D 22 = 0.4. An artificial measurement model measuring system energy is considered: h(x, ẋ) = x 2 +ηẋ 2 +ɛx 4. Measurement noise is assumed to be R = 2 and measurements are assumed to arrive every 9 seconds. The initial state distribution is given by the following Gaussian pdf: N({5, 5}, 0.5I 2 2 ). Filter-errors are plotted in Figs.2(a) and 2(b) along with 3-σ confidence bounds. Fig. 2(c) shows conditional densities obtained from the nonlinear filter at four time instants. It is instructive to consider Fig.2(c) first: since the measurement model does not provide unique information about the state (model is symmteric about the x and ẋ axis), the measurement update is unable to convert the bimodal prior into a unimodal function. Due to the nature of the measurement model, the posterior is bimodal, i.e., the update step only makes the two modes sharper (see Fig.2(c)) rather than eliminating one of the modes. On the other hand, the EKF estimate of state x tends to drift away from the truth, and despite tuning efforts, leads to inconsistent behavior. This is clearly apparent in Fig.2(a), wherein the filter error in x breaks the EKF 3 σ confidence boundary. This behavior is due to the ambiguity introduced by the measurement model in conjunction with long propagation time and high process noise. The key result of this paper is the time of computation for the measurement update step. The results presented in this section were obtained using modest computing resources: 1.86 GHz Pentium M processor with 1 GB RAM. For this machine, the time required for the measurement step for the current method was about 2 seconds. Note that with the time between measurements being 9 s, this filter is feasible in real time. However, the variational form of the Bayes rule for this system as described in Refs. [24], [25] requires about 400 s on the same computing platform. Therefore, the current approach for measurement update is clearly superior to variational implementation of Bayes rule and represents a significant step towards real-time feasibility of a FPE based nonlinear filter. System 2: Filtering in 4D: (Coupled Vibration Isolation Suspension) We next consider a coupled two-degree-offreedom (i.e. four-state) nonlinear vibration isolating suspension model studied by Ariaratnam [29]: x 1 = x 3, x 2 = x 4 x 3 = αx 3 1 M x 4 = βx 4 1 I V + ζ 1 x 1 V + ζ 2 (14) x 2 In this example, we consider a 2 dimensional measurement model in which the state-rates, x 1 and x 2, i.e. x 3 and x 4 are measured. The measurements are assumed to arrive every 4 seconds with an error covariance matrix of 5I 2 2. Error estimates in the various states are shown in Figs.3(a) and 3(b). The behavior of EKF for this system was observed to be unpredictable because different noise samples led to different steady state behaviors of x 2. This is a result of high process noise coupled with relatively long propagation times, causing the EKF errors to border on inconsistency (see error estimates of states x 2 and x 4 in Figs.3(a),3(b)). On the

6 EKF FPE/MCMC Based EKF FPE/MCMC Based (a) Comparative error estimates for x prior pdfs (b) Comparative error estimates for ẋ posterior pdfs t = 54 s t = 18 s t = 9 s t = 0 s (c) Full state pdf t = 0s to t = 54s Fig. 2. State estimates and conditional pdfs for the Duffing oscillator. EKF FPE Based EKF FPE Based (a) Comparative error estimates for x 1 and x 2. (b) Comparative error estimates for x 3 and x 4. Fig. 3. Error estimates for system 2. other hand, with only the information available about the states x 3 and x 4, the FPE based filter is unable to decide between the two modes of the system. For this system, the measurement update step was about 500 times faster using the MCMC sampling based technique than the variational form technique. IV. CONCLUSIONS In this paper, a nonlinear filter based on Fokker-Planck equation for propagation and MCMC based Bayes rule for measurement update was presented. FPE for the propagation phase was solved in real time using a robust semianalytical meshless technique developed in previous work. The measurement update was implemented as a function approximation problem on a small MCMC sample of the unnormalized posterior density obtained from the Bayes rule. Considerable speed-up in the update step (as compared with weak formulation of the Bayes rule) was reported. Examples were presented in 2 and 4 dimensional state space systems to demonstrate the effectiveness of this filter. Significant improvement in the time required for enforcing measurement updates are obtained. This is an important step towards the realization of a real time FPE based nonlinear filter. Work is currently under progress to implement this filter on more complex problems.

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