Nonlinear Estimation with Perron-Frobenius Operator and Karhunen-Loève Expansion

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1 Nonlinear Estimation with Perron-Frobenius Operator and Karhunen-Loève Expansion Parikshit Dutta a, Abhishek Halder b, Raktim Bhattacharya b. a INRIA Rhône Alpes & Laboratoire Jean Kuntzmann, 655 avenue de l Europe, 3833 Montbonnot, France b Department of Aerospace Engineering, Texas A&M University, College Station, TX 77843, USA Abstract hal , version 1-14 Jan 13 In this paper, a novel methodology for state estimation of stochastic dynamical systems is proposed. In this formulation, finite-term Karhunen-Loève (KL) expansion is used to approximate the process noise, thus resulting in a non-autonomous deterministic approximation (with parametric uncertainty) of the original stochastic nonlinear system. It is proved that the solutions of the approximate dynamical system, asymptotically converge to the true solutions, in mean square sense. The evolution of uncertainty for the KL approximated system is predicted via Perron-Frobenius (PF) operator. Furthermore, a nonlinear estimation algorithm, using the proposed uncertainty propagation scheme, is developed. It is found that for finite dimensional linear and nonlinear filters, the evolving posterior densities obtained from the KLPF based estimator is closer, than particle filter, to the true posterior densities. The methodology is then applied to estimate states of a hypersonic reentry vehicle. It is observed that, the KLPF based estimator outperformed particle filter in terms of capturing localization of uncertainty through posterior densities, and reduction of uncertainty. Key words: Karhunen-Loève expansion; Perron-Frobenius operator; stochastic dynamical systems; state estimation. 1 Introduction Consider a stochastic nonlinear system given by the Itô stochastic differential equation (SDE) dx (t) = f (x (t), t, δ) dt + dw (ω, t), dy (t) = h (x (t), t, δ) dt + dv (ω, t), (1) () where x (t) R n and y (t) R m are the state and measurement vectors at time t; δ R p is the parameter vector, and W (ω, t) : Ω R + R n, V (ω, t) : Ω R + R m are mutually independent Wiener processes denoting process and measurement noise, respectively. Here Ω denotes the sample space. The functions f (.), and h (.) represent the dynamics and measurement models, respectively. If the uncertainty in initial conditions (x ) and parameters (δ) are prescribed through the joint probability density function (PDF) ϕ (z), supported over the extended state space z := x δ] T R n+p, then the propagation of This research was partially supported by NSF award #11699 with D. Helen Gill as the Program Manager. Some preliminary results of this work have been accepted as a regular paper in IEEE Multi-Conference in Systems and Control, 1, and is under review at American Control Conference, 13. addresses: parikshit.dutta@inria.fr (Parikshit Dutta), ahalder@tamu.edu (Abhishek Halder), raktim@tamu.edu (Raktim Bhattacharya). Preprint submitted to Automatica 11 January 13

2 uncertainty through (1) and () requires solving the Fokker-Planck equation (FPE) (or Kolmogorov forward equation) 5,37] ϕ n t = (ϕf i ) + x i n n j=1 x i x j (Q ij ϕ) (3) to compute ϕ (z, t) subject to ϕ (z, ) = ϕ (z), where Q is the process noise covariance matrix. In the nonlinear estimation problem, the prior PDF obtained by solving (3), is updated according to Bayes rule to result the posterior PDF. hal , version 1-14 Jan 13 Solving the parabolic partial differential equation (PDE) (3), second order in space and first order in time, is computationally hard 5,39]. Several methods 6,41,19] for approximate numerical solution of the FPE have been developed over the years. These algorithms, intend to solve the FPE using either grid based methods like FEM 6,61,49,1], or by using meshfree methods 4,57]. It is known 55,6] that for grid based methods, complexity in solving the problem increases exponentially with the increase in dimensions of the extended state space. Same is true 17,1,31] for Monte Carlo based methods 36,57] for solving FPE. On the other hand, function approximation techniques like 39] have been reported to work well for moderate (3 to 4) dimensions. However, finding the correct basis to get a sufficiently good approximation of the transient PDFs remains challenging, particularly when the problem is high dimensional 6]. Thus, most solution methods for FPE suffer from the curse of dimensionality 6]. State estimation, in such cases, is a difficult task when systems involved are highly nonlinear and measurement update is infrequent 17]. Hence, there is a pressing need to develop an uncertainty propagation method which will accurately predict uncertainty for high dimensional systems with process noise. State and parameter estimation for nonlinear systems are commonly done using sequential Monte Carlo (SMC) methods, particle filter being the most popular amongst them 3]. It is well known 17] that these methods require a large number of ensembles for convergence, leading to higher computational cost. This problem is tackled through resampling,48,9]. Particle filters with resampling technique, are commonly known as bootstrap filters 9]. It has been observed that bootstrap filters introduce other problems like loss of diversity amongst particles 53], if the resampling is not performed correctly. Recently developed techniques 44] have combined importance sampling and Markov Chain Monte Carlo (MCMC) methods to improve sampling quality, enabling better estimates of states and parameters. Several other methods, like regularized particle filter 51], and filters with MCMC move step 8], have been developed to enhance sample diversity. At the same time, even with resampling, due to the simulation based nature of these filters, the ensemble size scales exponentially with state dimension for large problems 58]. To circumvent this problem, particle filters based on Rao-Blackwellization 11] have been proposed to partially solve the estimation problem analytically. However, its application remains limited to systems where the required partition of the state space is possible. In this work, we have proposed an uncertainty propagation methodology by combining Karhunen-Loève (KL) expansion 35] with Perron-Frobenius (PF) operator theory 4]. Further, we have developed a nonlinear state estimation technique which uses the proposed method for forward propagation of uncertainty. Notice that, in the absence of process noise, (3) reduces to Liouville equation associated with the PF operator 4] that describes spatio-temporal drift of ϕ (z, t), accounting parametric and initial condition uncertainty. Being a first order quasi-linear PDE, Liouville equation can be solved 3] using the method of characteristics (MOC) 3] and unlike Monte Carlo, such a computation occurs in pointwise exact arithmetic 31,1]. In nonlinear estimation setting, it has already been shown 1,18] that for systems without process noise, PF operator based nonlinear filter outperforms particle filter and bootstrap filter. Hence, it is reasonable to hope that the proposed estimation technique based on KL expansion and PF operator (KLPF), will perform better than other SMC methods. The motivation behind investigating such mixed parametric-nonparametric approach stems from the fact that the presence of process noise necessitates solving a second order transport PDE, and consequently, MOC based exact arithmetic computation can not be achieved. This argument usually leads researchers to develop approximation algorithms to solve the second order FPE, which is an exact description of the problem. Here, instead, we derive an approximate ordinary differential equation (ODE) representation of the problem (using KL), and then solve this approximate problem in exact arithmetic (using PF). Hence our approach differs from those which strive to numerically solve the Fokker-Planck PDE, in the sense that we propose to approximate the problem while the latter strives to approximate the solution. Convincing numerical evidence in favor of this argument was given in our earlier work ]. Through a multivariate Kolmogorov-Smirnov test, that paper also showed that the KLPF solution satisfies at least weak distributional convergence to the true SDE solution, meaning our first KL, then PF algorithm is

3 asymptotically consistent in distribution. However, two issues remained unsettled there: (i) in exactly what sense the convergence of the solution occurs, and (ii) formal proof in support of this consistency result. This paper addresses both these issues. Further, a detailed numerical investigation is carried out about the performance of the KLPF filter. The rest of the paper is organized as follows. In Section II, KL expansion is introduced first, and used to approximate the process noise term. It is also shown why such an approximation yields a sufficiently good solution for the SDE. The KLPF methodology is presented next in Section III with some illustrative examples. Extension of this idea to other types of stochastic evolution equations is pointed out. Then the nonlinear estimation technique, based on KLPF, is proposed in Section IV. In Section V, we demonstrate that for finite dimensional linear and nonlinear filters with known solutions, KLPF filter outperforms particle filter. Then we apply this technique to estimate states of a six dimensional nonlinear system and compare its results with particle filter. Section VI concludes the paper. Karhunen-Loève Approximation of Stochastic Dynamical Systems KL expansion was derived independently by many researchers 38,35,34,45,47] to represent a stochastic process Y (ω, t) as a random linear combination of a set of orthonormal deterministic L functions {e i (t)}, i.e. Y (ω, t) = Z i (ω) e i (t), (4) hal , version 1-14 Jan 13 where Z i (ω) are random variables. The idea is similar to the Fourier series expansion, where a deterministic linear combination of orthonormal L functions is used. Further, if we write Z i (ω) = Λ i ζ i (ω), where Λ i R + and {ζ i (ω)} is a sequence of random variables to be determined, then {Λ i } and {e i (t)} can be interpreted as the eigenvalues and eigenfunctions of the covariance function 14] C (t 1, t ) := cov (Y (ω, t 1 ) E Y (ω, t 1 )], Y (ω, t ) E Y (ω, t )]), (5) that admits a spectral decomposition 3] of the form C (t 1, t ) = Λ i e i (t 1 ) e i (t ). Since the covariance function is bounded, symmetric and positive-definite, the eigenvalue problem can be cast as a homogeneous Fredholm integral equation of second kind, given by C (t 1, t ) e i (t 1 ) dt 1 = Λ i e i (t ). (6) D t Given the covariance function of a stochastic process, the eigenvalue-eigenfunction set can be found by solving (6), and the resulting expansion (4) converges to Y (ω, t) in mean-square sense 43]. The following results are useful for our formulation. Theorem 1 (KL expansion of Wiener process) For Wiener process W (ω, t) with variance σ, the eigenvalues and eigenfunctions {Λ i, e i (t)}, of its covariance function C (t 1, t ) = (t 1 t ), t 1, t, T ], T ], are given by Λ i = 4T (( π (i 1), e i (t) = T sin i 1 ) ) πt, (7) T and hence the KL expansion of W (ω, t), is of the form (( W (ω, t) m.s. sin i 1 ) ) πt T = ζ i (ω), (8) T ( i 1 ) π T where ζ i (ω) are i.i.d. samples drawn from N (, σ ). 3

4 Corollary (KL expansion of Gaussian white noise) Since dw (ω, t) = η (ω, t) dt, the KL expansion for Gaussian white noise η (ω, t) can be obtained by taking the derivative of (8) with respect to t, i.e. η (ω, t) m.s. = T (( ζ i (ω) cos i 1 ) ) πt, (9) T where the i.i.d. random variables ζ i (ω) N (, σ ). Proposition 1 (KL expansion applied to SDE) Consider the stochastic dynamical system (1). We will incorporate the sample of the Gaussian white noise ω Ω, in the argument of dw (ω, t). Important point to note is, Ω is independent of the sample space of X(t). Without loss of generality, let us consider only state (instead of extended state) uncertainty in the system. Therefore, (1) can be written in Langevin form ẋ (t) = f (x (t), t) + η (ω, t). (1) hal , version 1-14 Jan 13 We consider integrating up to a finite time T ; hence, η(ω, t) : Ω, T ] R n is a Gaussian white noise having autocorrelation Q. Utilizing corollary, where the expansion in (9) has been truncated to N-terms, (1) can be written as ẋ N (t) = f (x N (t), t) + T where x N (t) is the solution of the KL-approximated system..1 Convergence of Solution N (( ζ i (ω) cos i 1 ) ) πt T Due to finite term truncation of the white noise KL expansion, the solution of (11) is only an approximation of the SDE solution. Thus, it is imperative to quantify a notion of convergence, if any, between the solution of (11) and that of (1). In this section, we show that as N, x N (t) converges to x(t) in mean square sense, and then discuss about the rate of convergence..1.1 Asymptotic Convergence Theorem 3 Let x (ω, t) be the solution of the nonlinear Itô SDE where dw(ω, t) = η (ω, t) dt, and f : R n, T ] R n satisfies the following: (11) dx(t) = f(x, t)dt + dw(ω, t) (1a) dx (t) = f (x, t) + η(ω, t), dt (1b) (1) non-explosion condition: D, s.t. f (x, t) < D(1 + x ) where x R n, t, T ]; () Lipschitz condition: C, s.t. f (x, t) f ( x, t) < C x x, where x, x R n, t, T ]. Let x N (t) be solution of the ordinary differential equation (ODE), formed by N-term approximation of η(ω, t) in (1a), using orthonormal basis, which is given by, dx N (t) = f (x N, t) + η N (ω, t), dt ] T where η N (ω, t) is the N-term approximation and E η N (ω, t)dt <. Then, lim E x (t) x N (t) ] =, N (13) (14) iff x N (t) is the KL expansion of x (t). 4

5 Proof Throughout the proof, we will write x(t) as x(ω, t) where ω Ω, the sample space of the random process x(t). First we will prove that if (14) holds then x N (ω, t) is the KL expansion of x (ω, t). Let {φ m (t)} m=1 be any orthonormal basis. Then x (ω, t) can be written as a convergent series sum x (ω, t) = b m c m (ω)φ m (t). m=1 (15) Let the solution of (13) be a N term approximation of x (ω, t) which converge to true solution in mean square sense. Hence the truncation error e N is given by e N (ω, t) = b m c m (ω)φ m (t). m=n+1 (16) Projecting x (ω, t) in φ m results c m (ω) = 1 T x(ω, t)φ m (t)dt. b m (17) Hence hal , version 1-14 Jan 13 E e N (ω, t)] = = m=n+1 k=n+1 m=n+1 k=n+1 T T φ m (t)φ k (t) E x (ω, t 1 ) x (ω, t )] φ m (t 1 )φ k (t )dt 1 dt T T φ m (t)φ k (t) C xx (t 1, t )φ m (t 1 )φ k (t )dt 1 dt, (18) where C xx (t 1, t ) is the covariance function of x(ω, t). For convergence of error, the orthonormal basis φ m (t) should minimize T T E e N (ω, t)] dt. Taking the orthonormality of φ m (t) into account, we get E e N (ω, t)] dt = m=n+1 T T which is to be minimized over φ m (t), subject to the constraint T φ m (t)φ k (t)dt = δ mk m, k N. C xx (t 1, t )φ m (t 1 )φ m (t )dt 1 dt, (19) () Introducing Lagrange multipliers b m, our objective function J(.) becomes J(φ m (t)) = min φ m(t) m=n+1 b m T T C xx (t 1, t )φ m (t 1 )φ m (t )dt 1 dt ( ) T φ m (t)φ m (t)dt 1. (1) Differentiating (1) with respect to φ m (t) and setting the derivative to zero, we obtain T ] T C xx (t 1, t )φ m (t 1 )dt 1 b mφ m (t ) dt =, () T C xx (t 1, t )φ m (t 1 )dt 1 = b mφ m (t ), (3) 5

6 which is the Fredholm integral equation for the covariance function of random process x (ω, t). Hence φ m (t) and b m are the eigenfunction-eigenvalue pairs of C xx (t 1, t ). This completes our proof. Now we will prove that, if x N (ω, t) is the KL expansion of x(ω, t), then to ensure that solution of (13) converge to solution of (1a) in mean square sense, x N (ω, t) must be the solution of (13). To prove this, we recall the following uniqueness conditions on solution of (1a) and KL expansion of a random process. Proposition (5], Chap. 5) Given the non-explosion condition and the Lipschitz condition are satisfied for f (, ) in (1a). Let Z be a random variable, independent of the σ-algebra F, m generated by η(ω, t), t and E Z ]. Then the stochastic differential equation in (1a) where t, T ], X(ω, ) = Z, has a unique t- ] T continuous solution x(ω, t) adapted to the filtration Ft Z generated by Z, and E x(ω, t) dt <. Proposition 3 (7], Chap. ) The KL expansion of the random process x(ω, t), given by x(ω, t) = Λm ξ m (ω)φ m (t), is unique. m=1 hal , version 1-14 Jan 13 Let us assume that x N (ω, t) is the KL expansion of x(ω, t). Furthermore, assume that x N (ω, t) x N (ω, t), which is the solution of (13). Hence, we assume that although x N (ω, t) is the KL expansion of x(ω, t), it does not satisfy (13), whose solution converge to the solution of (1a) in mean square sense. Proposition and 3 tell us that the solution of SDE is unique and any random process has an unique KL expansion. Also (13) has unique solution as RHS of (13) satisfies Lipschitz condition. This can be proven as follows: for right hand side of (13) to satisfy Lipschitz condition, we must have, f(x, t) + η N (ω, t) f( x, t) η N (ω, t) C x x f(x, t) f( x, t) C x x, which is true since f(, ) itself satisfies Lipschitz condition. Hence (1a) has unique solution that admits a unique KL expansion. Also according to our assumption, the solution of (13) converge to solution of (1a) in mean square sense. This contradicts our assumption that x N (ω, t) x N (ω, t), which completes the proof. Theorem 3 states conditions upon the solutions of approximated and true systems for mean square convergence to hold, under certain assumptions on f(, ). However, no condition is imposed on the initial states, which we investigate next. Theorem 4 Given the stochastic dynamical system dx(t) = f(x, t)dt + dw (ω, t), (4) and its corresponding N-term KL approximation given by N dx N (t) = f(x N, t)dt + ξ i (ω) φ i (t)dt (5) where, lim E N N W t ξ i (ω)φ i (t) =. Then, lim E x(t) x N (t) ] =, if x() = x N (). N 6

7 hal , version 1-14 Jan 13 Proof Integrating (4) and (5) and taking the expected value of square of the difference, E x(t) x N (t) ] t t =E N (x() x N ()) + (f(x, s) f(x N, s))ds + d(w s ξ i (ω)φ i (s)) E (x() x N ()) ] t ] + E (f(x, s) f(x N, s))ds + t E N d(w s ξ i (ω)φ i (s)) E (x() x N ()) ] t ] + te f(x, s) f(x N, s) ds + t E N d(w s ξ i (ω)φ i (s)) (Using Chebyshev s integral inequality.) lim E x(t) x N (t) ] E (x() x N ()) ] t ] + lim te f(x, s) f(x N, s) ds + N N t lim E N d(w s ξ i (ω)φ i (s)) (6) N Using Lipschitz criterion and property of KL expansion, we get v(t) B + A v(t) (say) lim E x(t) x N (t) ] E (x() x N ()) ] N }{{}}{{} =:B (say) t t +tc lim E N x(s) x N (s) ] ds v(s)ds v(t) B exp(at) (Gronwall s inequality, where tc A, t (, T ]) lim E x(t) x N (t) ] =, since x() = x N (), as per our assumption. N Theorem 4 implies that, starting from the same initial condition, for sufficiently large N, solution of (5) converge to solution of (4) in mean square sense. Hence, it can be said, that if the process noise term in Langevin equation is well-approximated using KL expansion, it entails a good approximation in solution space. Notice that in presence of initial state uncertainty, Theorem 4, may not always be true. The following corollary quantifies when it is indeed true, for uncertain initial conditions. Corollary 5 If x N () x (), and if x N () is the generalized polynomial chaos (gpc) expansion of x(), then theorem 4 is true. Proof Starting from (6) in the proof of theorem 4, and taking limits on both sides, lim E x(t) x N (t) ] lim E (x() x N ()) ] t ] + lim te f(x, s) f(x N, s) ds + N N N t lim E N d(w s ξ i (ω)φ i (s)) (7) N Going through the subsequent steps as before, we arrive at the following final result, lim E x(t) x N (t) ] =, if lim E x() x N () ] = N N 7

8 If x N () is the gpc expansion of x(), then they converge in mean square sense 7]. x() x N () ] =, which implies that lim N E Hence lim N E inequality. This completes our proof. 4 x(t) x N (t) ] =, by using Gronwall s 3 1 x 1 3 hal , version 1-14 Jan Fig. 1. This plot illustrates the asymptotic convergence results developed in Section II.A.1 for Van der Pol oscillator given by (3). Starting from the same initial condition (1, 1), denoted by the filled circle, the dashed, and solid curves show the deterministic (zero noise), and stochastic (SDE sample path with zero-mean additive Gaussian noise of variance.5) trajectories, respectively. The dash-dotted curve is the trajectory of the KL-approximated system with N = 1 terms, starting from the same initial condition, with process noise same as that of the SDE path. As N increases, the dash-dotted curve converges to the solid curve in mean-square sense..1. Rate of Convergence It is known 5] that the rate-of-convergence of KL expansion is governed by the decay rate of the sum of the eigenvalues of the covariance function. Example 1 (Rate-of-convergence of KL expansion of Wiener process) Consider the stochastic process dx (ω, t) = dw (ω, t). Eigenvalues of the covariance function of x (ω, t) are given by Theorem 1. Thus, the truncation error due to N-term KL approximation of x (ω, t), is x 1 i=n+1 N Λ i = Λ i Λ i = 4T π π 8 1 ( (π ψ (1) N + 1 ))] = Tπ ( 8 ψ(1) N + 1 ), (8) which decays faster than O ( N 1), but slower than O ( N ). Here ψ (1) (.) denotes the trigamma function ]. Recently, it has been shown 56] that one can estimate the KL convergence rate by knowing the smoothness of the covariance function, even though the eigenvalues are not available analytically. The main result is that exponential convergence occurs when the covariance function is analytic. If the latter has Sobolev regularity, the KL expansion has algebraic decay. The following example on geometric Browinan motion (GBM) illustrates this. Example (Rate-of-convergence of KL expansion of GBM) Consider the GBM dx (ω, t) = ax (ω, t) dt + bx (ω, t) dw (ω, t), where a, b are deterministic ) constants. The covariance function can be computed from (5) as C (t 1, t ) = (x (ω)) e (e a(t1+t) b t 1. Since C (t 1, t ) is analytic, the convergence is exponential. 8

9 In the nonlinear estimation setting, the covariance function of the process being estimated, is not known beforehand. What is known here is the dynamics f (.). This motivates us to investigate the connection between the regularity of f (.) with the regularity of C (t 1, t ). However, such an analysis has not been done here and will be a topic of our future work. 3 KLPF Formulation In this section, we propose a methodology to predict evolution of uncertainty in KL-approximated system using the PF operator. Let us consider the KL approximated dynamical system given in (11) where x N (t) R n are the states, having initial PDF ϕ (x N (), ) = ϕ. We apply MOC to PF operator, using the formulation given in 3], and form an augmented dynamical system given by ẋ N (t) = f (x N (t)) + T N ϕ(x N (t)) = div (f (x N (t))) ϕ(x N (t)), (( ζ i (ω) cos i 1 ) ) πt, (9a) T (9b) hal , version 1-14 Jan 13 n f (x N (t)) where div (f (x N (t))) := x Ni (t), and x N 1, x N,..., x Nn ] are states of the KL approximated dynamical system. Equation (9) can be solved to get the value of ϕ (x N (t)) along the characteristic curves of PF operator equation. Detailed discussion of the solution methodology is omitted here for brevity, and the reader is referred to 1,1] for the same. 3.1 Illustrative Examples The proposed methodology is applied to a Vanderpol s oscillator, given by ẍ (t) = ( 1 x (t) ) ẋ (t) x (t) + η (ω, t), (3) and to a Duffing oscillator, with dynamics, ẍ (t) = 1x (t) 3x 3 (t) 1ẋ (t) + η (ω, t), (31) with η (ω, t) having autocorrelation πi. The initial state uncertainty, has a PDF ϕ N (, ] T, diag (1, 1) ) for both the systems in (3) and (31). Taking x 1 (t) = x (t) and x (t) = ẋ (t), the augmented dynamical system for the Vanderpol s oscillator, is given by ẋ 1 (t) = x (t), ẋ (t) = ( 1 x 1 (t) ) x (t) x 1 (t) + η (ω, t), ϕ (x (t)) = ( 1 x 1 (t) ) ϕ (x (t)), (3a) (3b) (3c) and for the Duffing oscillator, is given by ẋ 1 (t) = x (t), ẋ (t) = 1x 1 (t) 3x 3 1 (t) 1x (t) + η (ω, t), ϕ (x (t)) = 1ϕ (x (t)). (33a) (33b) (33c) Next, the initial PDF ϕ is sampled with sample size of ν = 5. For the Vanderpol s oscillator, final time T is taken to be 1s, for the Duffing oscillator T = 3s. Number of terms in the KL expansion is kept fixed as N = 7. Figure shows the evolution of probability densities with time for the two oscillators. It is observed that, for the Vanderpol s oscillator the probability mass accumulates along the limit cycle and for the Duffing oscillator, we get a bimodal PDF at final time. This is in agreement with the physical intuition for behavior of these systems. 9

10 hal , version 1-14 Jan 13 Fig.. Uncertainty Propagation for Vanderpol s oscillator (3) in top row, and Duffing oscillator (31) in bottom row. Darker regions have higher PDF value than lighter regions. 3. Extensions It is important to recognize that the main idea behind KLPF is to derive an approximate ODE initial value problem (IVP) from the original stochastic description, thereby facilitating an MOC based PF operator computation in exact arithmetic. Thus, it is possible to extend our KLPF formulation for more general class of stochastic systems, not necessarily described by an SDE IVP, as long as we can arrive at some ODE IVP approximation. In the SDE IVP case, this has been achieved through a two-step first KL, then PF algorithm (Figure 3). For general stochastic evolution equations, an intermediate approximation step is needed between the KL approximation and MOC based PF computation step. Fig. 3. Schematic of the generalized KLPF formulation. As the right-most flow-chart shows, an intermediate approximation step is needed for generic stochastic evolution, between KL approximation and PF operator based computation. 1

11 3..1 Example: Uncertainty quantification for stochastic delay differential equation (SDDE) Consider a dynamics described by nonlinear SDDE 46] dx (t) = f (t, x (t), x (t τ)) dt + dw (ω, t), t >, x (t) = ψ (t), t ( τ, ], (34) where τ denotes a fixed time-delay, ψ (t) is the history function, and W (ω, t) denotes the Wiener process. The drift vector field depends on both the current state x (t), and the delayed state x (t τ). Given (34), solving the delay Fokker-Planck equation 4] is computationally hard for general nonlinear drift. Instead, we can write the KL approximated dynamics corresponding to (34) as dx N = f (t, x N (t), x N (t τ)) + η N (ω, t), t >, dt x N (t) = ψ (t), t ( τ, ], (35) hal , version 1-14 Jan 13 which is a DDE boundary value problem (BVP), with η N (ω, t) being the N-term KL approximation of Gaussian white noise. We next use method-of-steps 1 (MOS) 5] to further reduce this infinite-dimensional BVP to a sequence of finite-dimensional ODE IVPs, as shown: ( ) ẋ (1) N (t) = f t, x (1) N (t), ψ (t τ) + η N (ω, t), t (, τ], ( ) ẋ () N (t) = f t, x () N (t), x(1) N (t τ) + η N (ω, t), t (τ, τ],. ( ) ẋ (m) N (t) = f t, x (m) N (t), x(m 1) N (t τ) + η N (ω, t), t ((m 1) τ, mτ], (36) where m Z +. Now we can apply the MOC based PF formulation to (36) by solving the following pair of ODEs in the j th interval ((j 1) τ, jτ]: ( ) ẋ (j) N (t) = f t, x (j) N (t), x(j 1) N (t τ) + η N (ω, t), (37a) ( ( )) ϕ (j) = div f t, x (j) N (t), x(j 1) N (t τ) ϕ (j), (37b) with the initial conditions ( being the) respective terminal values of the (j 1) th interval. Eqn. (37) computes the evolution of joint PDF ϕ (j) t, x (j) N (t) along the trajectory x (j) N (t). 4 Nonlinear Filtering Using KLPF Method Let us consider a nonlinear dynamical system with states x R n being measured by a discrete nonlinear measurement model with outputs ỹ k R m, modeled as ẋ(t) = f(x, t) + η(ω, t), ỹ k = h(x k, t k ) + ϑ k, (38a) (38b) where {t k } r k= are time instances when measurements are obtained, and {x k} r k= are the corresponding states. η(ω, t) and ϑ k are zero mean delta correlated Gaussian random processes with covariances Q and R, respectively. Let p k (.) and p+ k (.) denote the prior and posterior PDFs at time t k. The state estimation algorithm described next, is developed in similar manner as 1]. 1 MOS provides the intermediate approximation step from DDE BVP to ODE IVP, as shown in the third column of Figure 3. 11

12 4.1 Step 1: Initialization Step Sample ν particles from the domain of the initial random variable. Let the particles be {x,i } ν with corresponding PDF values {p,i } ν. The following steps are then performed sequentially starting from k = Step : Propagation Step Consider the N-term KL approximation of (38a) and the corresponding MOC based PF operator equation, given together by (9). With the initial states at (k 1) th step as x k 1,i, p k 1,i ], (9) is integrated over the interval t k 1, t k ] for each particle, to get the states x k,i and prior PDF p k,i at t k. 4.3 Step 3: Update Step First, the likelihood function is determined for each particle using Gaussian nature of measurement noise, which is given by 1 p k,i (ỹ k x k = x k,i ) = (π)m R exp 1 (ỹk h(x k,i, t k )) R 1 (ỹ k h(x k,i, t k )) ], where R denotes ther determinant of measurement noise covariance matrix. The posterior PDF value for the i th particle, is then obtained as hal , version 1-14 Jan 13 p + k,i = p k,i(ỹ k x k = x k,i )p k,i. ν p k,i (ỹ x k = x k,i )p k,i The actual posterior PDF is the limit of the above expression as ν. However, unlike particle filters, here we compute the prior density in exact arithmetic using PF operator. Hence, accounting the dynamics of the prior itself, makes (39) an accurate representation of the posterior PDF, as no histogram approximation is employed here. A comparative analysis of numerical performance for obtaining PDFs using MC histogram versus PF operator, can be found in 31]. 4.4 Step 4: Getting the State Estimate Computation of the state estimate for the k th step, depends on the desired optimality criterion involving posterior PDF. Most commonly used criteria are maximum likelihood criterion (MLC), minimum error criterion (MEC) and minimum variance criterion (MVC) 9]. For the examples, following this section, we have used MVC to obtain the ν ν state estimate, which is given by ˆx k = arg min x x x k,i p + k (x k,i) = x k,i p + k (x k,i). (39) One can use resampling techniques, often employed in particle filtering to avoid degeneracy and increase diversity amongst particles 53]. However, qualitatively, since histogram techniques are not used in determining density functions, this method is less sensitive to the issue of degeneracy. 5 Numerical Results 5.1 Comparison of Posterior Densities of Particle Filter and KLPF Estimator We compare the KLPF estimator to particle filters, by investigating the posterior PDFs obtained by the two methods after each filtering step. Our objective is to illustrate the performance improvement achieved by KLPF compared to particle filter, for cases where the estimation problems are known to admit finite dimensional sufficient statistics. In such cases, knowledge of the true posterior enables us to conclude the performance improvement accomplished by KLPF. 1

13 5.1.1 Case I: Kalman Filter For a linear system with Gaussian uncertainty, we know the exact evolution of PDFs through Kalman filter. We compute the Wasserstein metric 54] of the posteriors obtained by aforementioned methods, from the Kalman filter s posterior. The Wasserstein metric is a metric comparing the distance between the shapes of two distributions, as defined next. Definition 1 (Wasserstein distance) Let the pair (M, d) denote a metric space M endowed with metric d. For q 1, let P q (M) denote the collection of all probability measures µ on M with finite q th moment, i.e. for some x in M, M d(x, x) q dµ(x) <. Then the Wasserstein distance of order q, between two probability measures ς 1 and ς in P q (M), is defined as ( W q (ς 1, ς ) := inf µ M(ς 1,ς ) M M d(x, x) q dµ(x, x)) 1/q, (4) where, M(ς 1, ς ) is the set of all measures supported on the product space M M, such that their first marginal is ς 1 and second marginal is ς. hal , version 1-14 Jan 13 For the sake of brevity, detailed discussion about the Wasserstein metric has been omitted here and can be found in ref. 16,59]. If ς 1 N (µ 1, Σ 1 ) and ς N (µ, Σ ), and if we consider q = (requiring the nd moment to be finite), then the Wasserstein metric admits a closed form expression in terms of the moments of ς 1 and ς : W (ς 1, ς ) = µ 1 µ + tr(σ 1 ) + tr(σ ) tr( Σ 1 Σ Σ1 ) 1. (41) In general, the Wasserstein metric can be normalized by dividing W (ς 1, ς ) by the diameter of the product space. As a concrete numerical example, let us consider the linear system given by the following dynamical equations. ẋ(t) =.5I x(t) + 1 1] w(t), (4) where x(t) R are the states and I n R n n is the identity matrix. We consider a discrete measurement model given by ỹ k = 1 1] x k + v k, (43) where ỹ k are measurements coming at discrete times. w(t) and v k are assumed to be zero mean delta correlated Gaussian processes with covariance Q = 1/ 3 and R = 1/ respectively. We assume that the system has Gaussian initial condition uncertainty with mean and covariance as 1, 1] and diag(1, 1) respectively. We draw 1 samples, each of sample size 5 from the initial distribution, and plot the mean and standard deviation of the Wasserstein distances, as measured from the Kalman filter posterior, shown in Figure 4. It can be observed that the mean for the KLPF estimator is closer to zero than the particle filter. Moreover, the deviation from the mean is lower for the KLPF estimator than the particle filter. Hence it can be said that KLPF estimator is able to capture the evolution of densities more accurately than the particle filter Case II: Bene s Filter Bene s filter is one of the few known nonlinear filters which admit a finite-dimensional solution of the nonlinear estimation problem. Here, the nonlinear drift is assumed to satisfy a Riccati differential equation 7] and the measurement model is taken to be affine in states. For simplicity, we consider the continuous-time scalar Bene s filtering problem of the form: dx (t) = κex e x κe x dt + dw (ω, t), dy (t) = x (t) dt + dv (ω, t), (44) + e x 13

14 .5.45 Mean and Std. Dev. for W Dist time (s) Fig. 4. Plot of mean and standard deviation of the Wasserstein distances of the KLPF estimator (solid line) and the particle filter (hyphenated line) from the Kalman filter. The vertical lines about the means represent ±1σ limits. hal , version 1-14 Jan 13 with κ =.5 and deterministic initial condition x. The process and measurement noise densities are N (, Q) and N (, R) respectively, with Q = 1, R = 1. It can be shown 4] that the drift nonlinearity in (44) satisfy the necessary Riccati condition and the resulting solution 15] is given by the normalized posterior density ( coth(t) ϕ (x (t) Y t ) = π κe x + e x κe It(y(ω)) + e It(y(ω)) where Y t is the history (filtration) till time t, and ) exp ( 1 ) Γ (t), (45) I t (y (ω)) := sech(t) x + t ] sinh(s)dy s (ω), Γ (t) := tanh(t) + coth(t) (x I t (y (ω))). (46) Notice that for this nonlinear non-gaussian estimation problem, unlike Kalman filter case, we can not write the Wasserstein distance between the true posterior (45) and particle filter/klpf posterior, as an analytical expression in terms of the respective sufficient statistics. Thus, in order to compute the Wasserstein time history, we resort to the optimization problem (4), which can be cast 33] as a linear program (LP). At each time, we sample (45) using the Metropolis-Hastings MCMC technique 13], and solve the LP between the sampled true Bene s posterior and particle filter/klpf posterior, to result the normalized Wasserstein trajectories shown in Figure 5. Like the Kalman filter case, as time progresses, KLPF posterior gets closer, compared to particle filter, to true Bene s posterior. 14

15 .6.4 Mean and Std. Dev. for W Dist time (s) Fig. 5. Plot of mean and standard deviation of the Wasserstein distances of the KLPF estimator (solid line) and the particle filter (hyphenated line) for the Bene s filter. The vertical lines about the means represent ±1σ limits. 5. Application to Hypersonic Entry hal , version 1-14 Jan 13 The KLPF filtering technique is applied next to estimate states of a hypersonic spacecraft entering the atmosphere of Mars. The entry dynamics is given by the noisy version of Vinh s equation 1]: ṙ =v sin γ + η r, v cos γ cos ξ θ = + η θ, r cos λ v cos γ sin ξ λ = + η λ r v = ρv g sin γ Ω r cos λ(sin γ cos λ cos γ sin λ sin ξ) + η v B c v γ = r g cos(γ) + ρ L v D ξ = ρ B c (47a) (47b) (47c) (47d) «v cos σ + Ω cos λ cos ξ + Ω r cos λ(cos γ cos λ + sin γ sin λ sin ξ) + ηγ (47e) v B c «L v sin σ v cos γ cos ξ tan λ + Ω(tan γ cos λ sin ξ sin λ) Ω D r r v cos γ sin λ cos λ cos ξ + η ξ, where r is the distance of the vehicle from the planet s center, θ and λ are Mars-centric longitude and latitude respectively, v is the total velocity of the vehicle, γ is the flight path angle and ξ is the azimuth angle. The symbol g denotes the acceleration due to gravity, B c denotes the ballistic coefficient, L D stands for the lift-to-drag ratio of the the vehicle, ρ is the atmospheric density given by ρ = ρ e ( h h h ) 1, where ρ, h 1 and h are constants depending on the planet s atmospheric model, h = r R m is the height measured from the planet s surface and R m is the mean equatorial radius of the planet. η r, η θ, η λ, η v, η γ and η ξ are zero mean Gaussian process noises. Description of the constants in (47a) through (47f), used to simulate reentry in Martian atmosphere, are given in Table 1 1]. The system is assumed to have initial Gaussian state uncertainty with mean vector and covariance matrix given by, µ = R m + 54 Km, o, 3 o,.4 Km/s, 9 o,.573 o ] and Σ = diag (5.4 Km, 3 o, 3 o, 4 m/s,.9 o,.57 o ). (47f) The discrete measurement model consists of six measurements, and is given by ỹ = q, H, γ, θ, λ, ξ] + ϑ (48) where q = 1/ρv is the dynamic pressure and H = Kρ 1 v 3.15 is the heating rate, with K = being the scaled material heating coefficient 8]. ϑ R 6 is a vector of zero mean delta correlated discrete measurement noise. 15

16 Table 1 Parameter values for (47) Description of Constants Radius of Mars Value R m = m Acceleration due to gravity of Mars g = 3.71 m/s Ballistic coefficient of the vehicle B c = 7.8 kg/m L Lift-to-drag ratio of the vehicle =.3 D Density at the surface of mars ρ =.19 kg/m 3 Scale height for density computation Escape velocity of Mars Angular velocity of Mars Bank angle h 1 = 9.8 km h = km v c = 5.7 km/s Ω = rad/s σ = o Equation (47) and (48) are non-dimensionalized before applying the filtering algorithms. The characteristic mass, length and time, for this non-dimensionalization, are mass of the reentry vehicle (8 kg), R m and R m /v c = s, respectively. In scaled units the process noises are assumed to have covariance of Q = , and the measurement noise has covariance of R = I 6. hal , version 1-14 Jan 13 sqrt(σ θ CRLBθ ) (in deg) sqrt(σ r CRLBr ) (in km) sqrt(σ v CRLBv ) (in km/s) 5 1 time (s) sqrt(σ λ CRLBλ ) (in deg) time (s) sqrt(σ γ CRLBγ ) (in deg) 5 1 time (s) sqrt(σ ξ CRLBξ ) (in deg) 5 1 time (s) time (s) time (s) Fig. 6. Plots for σ CRLB for all the six states. The solid line represents KLPF filter (3 particles). The hyphenated, hyphen-dotted and solid-asterixed lines represent particle filters with 3, and 5 particles respectively. Figure 6 shows the plots for square root of the difference between variance and Cramer-Rao lower bounds (CRLB) for each state. The KLPF estimator with sample size 3 is compared with particle filters with 3, and 5 particles. We can see that the difference for KLPF estimator is lower than that of the particle filters for all the states. Hence, for the present case, the solution obtained from proposed estimation scheme is closer to the true minimum variance solution than that obtained from particle filters. Next, we plot the final posterior univariate and bivariate marginals obtained from the state estimation algorithms. The marginal PDFs were calculated using the algorithm given in ref. 3]. Figure 7 shows comparison of univariate marginals obtained from KLPF estimator and particle filter, both with 3 particles. Figure 8 plots bivariate marginal PDFs obtained from KLPF estimator and particle filter. It can be observed that the KLPF estimator is 16

17 r (km) λ (degrees N) θ (degrees E) v (km/s) γ (degrees) 3 1 ξ (degrees) hal , version 1-14 Jan 13 Fig. 7. Final posterior unidimensional marginal PDF for all states obtained from KLPF estimator (solid line) and particle filter (hyphenated line) with 3 particles. able to reduce variance and capture localization of uncertainty better than the particle filter with same number of particles. The KLPF estimator performs better than the particle filter with larger number of particles. Figure 9 and Fig. 1 show plots for the final posterior univariate and bivariate PDFs respectively for KLPF estimator with 3 particles and particle filter with 5 particles. It can be seen that, the KLPF estimator reduces variance and captures uncertainty localization better than the particle filter, although the particle filter has larger number of particles than the previous case. 6 Conclusion This work presents a method of approximating the noise term in stochastic dynamical systems using KL expansion. It is proved that the solution of the resulting approximate dynamical system converge to the true solution in mean square sense. A technique to propagate uncertainty in SDEs combining the above mentioned method and PF operator is proposed. A nonlinear estimation technique is developed using the KLPF uncertainty propagation methodology, which is then applied to estimate states of a nonlinear system and its performance is compared with particle filtering based estimation techniques. It is observed that the developed estimation technique is able to predict the evolution of posterior densities better than particle filters. Moreover, for the hypersonic reentry problem, the KLPF based estimator outperformed the sequential Monte-Carlo based methods in terms of reducing the variance and capturing the localization of uncertainty. We conclude that explicitly accounting prior dynamics via KLPF formulation, enables substantial filtering performance improvement over commonly used particle filtering technique. References 1] A. Halder and R. Bhattacharya. Beyond Monte Carlo: A computational framework for uncertainty propagation in planetary entry, descent and landing. In AIAA Guidance, Navigation and Control Conference, Toronto, ON, 1. ] M. Abramowitz and I.A. Stegun. Handbook of mathematical functions with formulas, graphs, and mathematical tables, volume 55. Dover publications,

18 hal , version 1-14 Jan 13 θ(degrees E) λ(degrees) (a) Top row- particle filter, bottom row- KLPF estimator. λ(degrees) r(degrees E) 3 λ(degrees) λ(degrees) (b) Top row- particle filter, bottom row- KLPF estimator. Fig. 8. Plots for the final posterior bivariate marginal PDFs obtained from KLPF estimator and particle filter with 3 particles. The darker regions represent lower PDF value and the lighter regions higher. 18

19 r (km) λ (degrees N) θ (degrees E) v (km/s).5 γ (degrees) 3 1 ξ (degrees) hal , version 1-14 Jan 13 Fig. 9. Final posterior unidimensional marginal PDF for all states obtained from KLPF estimator (solid line) with 3 particles and particle filter (hyphenated line) with 5 particles. 3] M.S. Arulampalam, S. Maskell, N. Gordon, and T. Clapp. A tutorial on particle filters for online nonlinear/non-gaussian Bayesian tracking. Signal Processing, IEEE Transactions on, 5(): ,. 4] A. Bain and D. Crişan. Fundamentals of stochastic filtering, volume 6. Springer Verlag, 8. 5] C.T.H. Baker, C.A.H. Paul, and DR Willé. Issues in the numerical solution of evolutionary delay differential equations. Advances in Computational Mathematics, 3(1): , ] R. Bellman. Dynamic Programming. Princeton, NJ: Princeton Univ. Press, ] VE Bene s. Exact finite-dimensional filters for certain diffusions with nonlinear drift. Stochastics: An International Journal of Probability and Stochastic Processes, 5(1-):65 9, ] K.P. Bollino, I.M. Ross, and D.D. Doman. Optimal nonlinear feedback guidance for reentry vehicles. In AIAA Guidance, Navigation and Control Conference, 6. 9] A.E. Bryson and Y.C. Ho. Applied optimal control: Optimization, estimation, and control. Hemisphere Pub, ] H.J. Bungartz and M. Griem. Sparse grids. Acta Numerica, 13(1):147 69, 4. 11] G. Casella and C.P. Robert. Rao-Blackwellisation of sampling schemes. Biometrika, 83(1):81 94, ] J.S. Chern and N.X. Vinh. Optimum reentry trajectories of a lifting vehicle. Technical Report NASA-CR-336, NASA, ] S. Chib and E. Greenberg. Understanding the Metropolis-Hastings algorithm. American Statistician, pages , ] N. Cressie. Statistics for spatial data. Terra Nova, 4(5): , ] D. Crisan. A direct computation of the Bene s filter conditional density. Stochastics: An International Journal of Probability and Stochastic Processes, 55(1-):47 54, ] J.A. Cuesta and C. Matran. Notes on the Wasserstein metric in Hilbert spaces. The Annals of Probability, pages , ] F. Daum, J. Huang, and R. Co. Curse of dimensionality and particle filters. In Proceedings of IEEE Aerospace Conference, pages , 3. 18] F. Daum and M. Krichman. Non-particle filters. In Proceedings of SPIE, volume 636, page 63614, 6. 19] M. Di Paola and A. Sofi. Approximate solution of the Fokker-Planck-Kolmogorov equation. Probabilistic Engineering Mechanics, 17(4): ,. ] A. Doucet, S.J. Godsill, and C. Andrieu. On sequential Monte Carlo sampling methods for Bayesian filtering. Statistics and Computing, 1(3):197 8,. 1] P. Dutta and R. Bhattacharya. Hypersonic state estimation using the Frobenius-Perron operator. Journal of Guidance, Control and Dynamics, 34():35 344, 11. ] Parikshit Dutta, Halder Abhishek, and Bhattacharya Raktim. Uncertainty quantification for stochastic nonlinear systems using Perron Frobenius operator and Karhunen Lo eve expansion. In IEEE Multiconference in Systems and Control, October, 1, Dubrovnik, Croatia, October 1. IEEE. 19

20 hal , version 1-14 Jan 13 θ(degrees E) λ(degrees) (a) Top row- particle filter, bottom row- KLPF estimator. λ(degrees) r(degrees E) 3 λ(degrees) λ(degrees) (b) Top row- particle filter, bottom row- KLPF estimator. Fig. 1. Plots for the final posterior bivariate marginal PDFs obtained from KLPF estimator with 3 particles and particle filter with 5 particles. The darker regions represent lower PDF value and the lighter regions higher.

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