Uncertainty Propagation for Nonlinear Dynamical Systems using Gaussian Mixture Models

Transcription

1 Uncertainty Propagation for Nonlinear Dynamical Systems using Gaussian Mixture Models Gabriel Terejanu, Puneet Singla, Tarunraj Singh, Peter D. Scott A Gaussian mixture model approach is proposed for accurate uncertainty propagation through a general nonlinear system. The transition probability density function, is approximated by a finite sum of Gaussian density functions whose parameters (mean and covariance) are propagated using linear propagation theory. Two different approaches are introduced to update the weights of different components of a Gaussian mixture model for uncertainty propagation through nonlinear system. The first method updates the weights such that they minimize the integral square difference between the true forecast probability density function and its Gaussian sum approximation. The second method uses Fokker-Planck equation error as a feedback to adapt for the amplitude of different Gaussian components while solving a quadratic programming problem. The proposed methods are applied to a variety of problems in the open literature, and argued to be an excellent candidate for higher dimensional uncertainty propagation problems. Introduction The state of a stochastic dynamic system, x, is characterized by its time dependent probability density function (pdf), p(t, x). The random nature of the state may be due to the uncertain initial conditions and/or due to the random excitation that is driving the dynamic system. The knowledge about the time evolution of the pdf of a state of a dynamic system is important to quantify the uncertainty in the state at a future time. Numerous fields of science and engineering present the problem of pdf evolution through nonlinear dynamic systems with stochastic excitation and uncertain initial conditions. 1, 2 This is a difficult problem and has received much attention over the past century. One may be interested in the determination of the response of engineering structures such as beams, plates, entire buildings beams under random excitation (in structure mechanics 3 ), or the propagation of initial condition uncertainty of an asteroid for the determination of its probability of collision with a planet (in astrodynamics 4 ), or the motion of particles under the influence of stochastic force fields (in particle physics 5 ), or simply the computation of the prediction step in the design of a Bayes filter (in filtering PhD Student, Department of Computer Science & Engineering, University at Buffalo, Buffalo, NY-1426, terejanu@buffalo.edu. Assistant Professor, AAS Member, Department of Mechanical & Aerospace Engineering, University at Buffalo, Buffalo, NY-1426, psingla@buffalo.edu. Professor, Department of Mechanical & Aerospace Engineering, University at Buffalo, Buffalo, NY-1426, tsingh@buffalo.edu. Professor, Department of Computer Science & Engineering, University at Buffalo, Buffalo, NY-1426, peter@buffalo.edu. 1 of 24

2 theory 6 ). All these applications require the study of the time evolution of the pdf, p(t, x), corresponding to the state, x, of the relevant dynamic system. For nonlinear systems, the exact description of the transition pdf is provided by a linear partial differential equation (pde) known as the Fokker Planck Kolmogorov Equation (FPKE), or simply the Fokker Planck Equation (FPE). 2 Analytical solutions exists only for stationary pdf and are restricted to a limited class of dynamical systems. 1, 2 Thus researchers are looking actively at numerical approximations to solve the Fokker-Planck equation, 7 1 generally using the variational formulation of the problem. The Finite Difference Method (FDM) and the Finite Element Method (FEM) have been used successfully for two and even three dimensional systems. However, these methods are severely handicapped for higher dimensions because the generation of meshes for spaces beyond three dimensions is still impractical. Furthermore, since these techniques rely on the FPE can only be applied to continuous-time dynamical systems. For discrete-time dynamical systems there is no equation to describe the evolution of the actual transition pdf, therefore we can only rely on approximations in this case. Several other approximate techniques exist in the literature to attack the pdf evolution, the most popular being Monte Carlo methods, 11 Gaussian closure 12 (or higher order moment closure), Equivalent Linearization, 13 and Stochastic Averaging. 14, 15 Monte Carlo methods require extensive computational resources and effort, and becomes increasing infeasible for high-dimensional dynamic systems. The latter are similar in several respects, and they are suitable only for linear or moderately nonlinear systems, because the effect of higher order terms can lead to significant errors. Furthermore, all these approaches provide only an approximate description of the uncertainty propagation problem by restricting the solution to a small number of parameters - for instance, the first N moments of the sought pdf. The present paper is concerned with improving the approximation to the forecast density function using a finite Gaussian mixture. With a sufficient number of Gaussian components, any pdf may be approximated as closely as desired. Many algorithms 16, 17 have been discussed in literature for time evolution of the initial state pdf through a nonlinear dynamical system in discrete-time or continuous-time. In conventional methods, the weights of different components of a Gaussian mixture are kept constant while propagating the uncertainty through a nonlinear system and are updated only in the presence of measurement data. 16, 17 This assumption is valid if the underlying dynamics is linear or the system is marginally nonlinear. The same is not true for the general nonlinear case and new estimates of weights are required for accurate propagation of the state pdf. However, the existing literature provides no means for adaptation of the weights of different Gaussian components in the mixture model during the propagation of state pdf. The lack of adaptive algorithms for weights of Gaussian mixture are felt to be a serious disadvantages of existing algorithms and provide the motivation for this paper. 2 of 24

3 In this paper, two different schemes are introduced to update the weights corresponding to different components of Gaussian mixture model during pure propagation. The first method updates the forecast weights by minimizing the integral square difference between the true forecast pdf and its Gaussian sum approximation. The second method updates the weights by constraining the Gaussian sum approximation to satisfy the Fokker-Planck equation for continuous-time dynamical systems. Both the methods leads to a convex quadratic programming problem and hence, are computationally efficient. The structure of paper is as follows: First, an introduction to conventional Gaussian sum approximation is presented followed by two new schemes for updating the weights of different components of Gaussian mixture. Finally, several numerical examples are considered to illustrate the efficacy of the proposed methods. Update Scheme for Discrete-Time Dynamic Systems Problem Statement Consider the following nonlinear discrete-time dynamical system driven by white noise, x k+1 = f(k, x k ) + η k (1) η k N (, Q k ) (2) with the probability density function of the initial conditions given by the following Gaussian sum, where p(t, x ) = N wkn i (x k µ i k, P i k) (3) N (x µ, P) = 2πP 1/2 exp [ 1 ] 2 (x µ)t P 1 (x µ) (4) We are interested in approximating the probability density function p(t k+1, x k+1 ) as a Gaussian mixture. A linear mapping, will transform a Gaussian mixture into another Gaussian mixture where the parameters of the mixands are easily computed. But the outcome of a Gaussian that undergoes a nonlinear transformation is generally non-gaussian. By linearizing the nonlinear transformation, a Gaussian sum approximation to the forecast density function p(t k+1, x k+1 ) is obtained, N ˆp(t k+1, x k+1 ) = wk+1n i (x k+1 µ i k+1, P i k+1) (5) where the reference values of the parameters of the Gaussian components are given by the time update step 3 of 24

4 of the Kalman filter: w i k+1 = w i k (6) µ i k+1 = f(k, µ i k) (7) P i k+1 = A k (µ i k)p i ka T k (µ i k) + Q k (8) where A k (x k ) = f(k, x k) x k (9) Other evolution schemes other than linearization using Taylor series, such as statistical linearization, may be used for obtaining the moments of the Gaussian components. Since they imply linearizations, such approximations are computational convenient and may be easily used in linear applications which have been intensively studied. The reason that the weights are not changed is because it is assumed that the covariances are small enough 16 such that the linearizations become representative for the dynamics around the means. To better understand this assumption, consider that the dynamical system is not driven by noise, the nonlinear mapping is invertible, and we want to propagate our Gaussian Sum for just one time step. Then the unnormalized density function 18 after one iteration, is given by: p(t k+1, x k+1 ) = p(t k, x k ) deta k (x ) 1 (1) N = wkn i (x k µ i k, P i k) deta k (x k ) 1 (11) Assuming that all covariances are small enough such that the linearization around a mean is representative for the dynamics in the vicinity of the respective mean, then N p(t k+1, x k+1 ) wkn i (x k µ i k, P i k) detak (µ i k) 1 (12) N = wkn i (x k+1 µ i k+1, P i k+1) (13) In practice this assumption may be easily violated resulting in a poor approximation of the forecast pdf. Practically, the dynamic system may exhibit strong nonlinearities and the total number of Gaussian components, needed to represent the pdf, may be restricted due to computational requirements. The existing literature provides no means for adaption of the weights of different Gaussian components in the mixture model during the propagation of the state pdf. The lack of adaptive means for updating the weights of Gaussian mixture are felt to be a serious disadvantage of existing algorithms and provides the motivation for this paper. 4 of 24

5 Weight Update I A better approximation of the forecast pdf may be obtained by updating the forecast weights. The new weights may be obtained in the least square sense to minimize the following integral square difference between the true density and its approximation: min w i k+1 s.t. 1 p(tk+1, x k+1) ˆp(t k+1, x k+1) 2 dx k+1 (14) 2 N wk+1 i = 1 w i k+1, i = 1,, N Here, no a priori structure is used for the true density function p(t k+1, x k+1 ), it is completely unknown. The expression (1) is only used to motivate the introduction of an update scheme for the forecast weights. By substituting (5) in (14) and expanding and grouping the terms in the cost function w.r.t. the weights, the new cost function is given by, J = 1 2 wt k+1mw k+1 w T k+1y (15) where w k+1 = [w 1 k+1 w2 k+1... wn k+1 ]T, and M N N is a symmetric matrix given by: M N N = M(x k+1 )M T (x k+1 )dx k+1 (16) where M is a N 1 vector that contains all the Gaussian components at time k + 1: M(x k+1 ) = T [ N (x k+1 µ 1 k+1, P 1 k+1) N (x k+1 µ 2 k+1, P 2 k+1)... N (x k+1 µ N k+1, Pk+1)] N (17) Thus, the components of matrix M are easily given by the product rule of two Gaussian density functions which yields another Gaussian density function. By integrating the product we are left only with the 5 of 24

6 normalization constant: m ij = N (x k+1 µ i k+1, P i k+1)n (x k+1 µ j k+1, Pj k+1 ) dx k+1, i j (18) = N (µ i k+1 µ j k+1, Pi k+1 + P j k+1 ) ( ) N x k+1 P ij k+1[ (P i k+1 ) 1 µ i k+1 + (P j k+1 ) 1 µ j ] k+1, P ij k+1 dx k+1 = N (µ i k+1 µ j k+1, Pi k+1 + P j k+1 ) [ = 2π(P i k+1 + P j k+1 ) 1/2 exp 1 ] 2 (µi k+1 µ j k+1 )T (P i k+1 + P j k+1 ) 1 (µ i k+1 µ j k+1 ) (19) m ii = N (µ i k+1 µ i k+1, P i k+1 + P i k+1) = 4πP i k+1 1/2 where P ij k+1 = [ (P i k+1) 1 + (P j k+1 ) 1] 1 (2) The components of the vector y N 1 are given by: y i = p(t k+1, x k+1 )N (x k+1 µ i k+1, P i k+1) dx k+1, i = 1... N (21) = p(t k, x k )N (f(k, x k ) µ i k+1, P i k+1) dx k (22) ˆp(t k, x k )N (f(k, x k ) µ i k+1, P i k+1) dx k (23) N = N (x k µ j k, Pj k )N (f(k, x k) µ i k+1, P i k+1) dx k (24) w j k j=1 The transition from (21) to (22) is given by the Law of the Unconscious Statistician, avoiding this way to deal with the unknown true forecast pdf, at a cost of computing the expectation of a composite function (22). We assume that we have a good Gaussian sum approximation at time t k such that ˆp(t k, x k ) p(t k, x k ). If this is not the case, we can use the Law of the Unconscious Statistician (LOTUS) 19 until we reach a good approximation at an earlier time t k p. y i = = = p(t k, x k )N (f(k, x k ) µ i k+1, P i k+1) dx k (25) p(t k 1, x k 1 )N (f(k, f(k 1, x k 1 ) µ i k+1, P i k+1) dx k 1 (26) p(t k p, x k p )N (f f... f(k p, x k p ) µ i k+1, P i k+1) dx k p (27) 6 of 24

7 Using (24), the cost function may be rewritten given the new y as, J = 1 2 wt k+1mw k+1 w T k+1nw k (28) where w k = [w 1 k w2 k... wn k ]T is the prior weight vector, and the matrix N N N has the following components: n ij = N (f(k, x k ) µ i k+1, P i k+1)n (x k µ j k, Pj k ) dx k (29) = E N (xk µ j k,pj k ) [ N (f(k, xk ) µ i k+1, P i k+1) ] (3) The expectations (3) may be computed using Gaussian Quadrature, Monte Carlo integration or Unscented Transformation. 2 While in lower dimensions the unscented transformation is mostly equivalent with Gaussian quadrature, in higher dimensions the unscented transformation is computational more appealing in evaluating integrals since the number of points only grows linearly with the number of dimensions. However, this comes with a loss in accuracy, 21 hence the need for a larger number of points 22 to capture additional information. In the case of linear transformation M = N, hence no change will occur to the weights. However, the approximations made in computing the expectations (3) may not give a perfect match between the two matrices, thus deteriorating the overall results in this case. Hence, such an update scheme may be recommended only for nonlinear systems. The final formulation of the optimization (14) may be posed in the quadratic programming framework and solved using readily available solvers: min J = 1 w k+1 2 wt k+1mw k+1 w T k+1nw k (31) s.t. 1 T N 1w k+1 = 1 w k+1 N 1 Here 1 N 1 is a vector of ones and N 1 is a vector of zeros. Now, if we show that the matrix M is a positive semi-definite and the cost function J is lower bounded, then the aforementioned optimization problem can be posed as a convex optimization problem and we are guaranteed to have a unique solution. 23 Notice that the integrand M(x i k+1 )MT (x i k+1 ) is a rank one symmetric positive semi-definite matrix with non-zero eigenvalue equal to M T (x i k+1 )M(xi k+1 ). Further, we write the integral (16) as a infinite sum as follows: M = i M(x i k+1)m T (x i k+1) (32) 7 of 24

8 Making use of the fact that summation of two positive semi-definite matrix is also a positive semi-definite matrix, we conclude that matrix M is a positive semi-definite matrix since M is a sum of positive semidefinite matrices. Further, notice that the linear term w T k+1 Nw k in cost function J is lower bounded since N and both w k+1 and w k have to lie between and 1. Hence, we can conclude that the cost function J is always bounded below and we are guaranteed to have a unique solution. Update Scheme for Continuous-Time Dynamic Systems Problem Statement Consider a general n-dimensional noise driven nonlinear dynamic system with uncertain initial conditions, given by the equation: ẋ = f(t, x) + g(t, x)γ(t) (33) where, Γ(t) represents a Gaussian white noise process with the correlation function Qδ(t 1 t 2 ), and the initial state uncertainty is captured by the pdf p(t, x). Then, the time evolution of p(t, x) is described by the following FPE, which is a second order pde in p(t, x): t p(t, x) = L FPp(t, x) (34) where, n L FP = D (1) i (.) + x i n j=1 D (1) (t, x) = f(t, x) n 2 D (2) ij x i x (.) (35) j g(t, x) Qg(t, x) (36) x D (2) (t, x) = 1 2 g(t, x)qgt (t, x) (37) issues: Despite its innocuous appearance, the FPE is a formidable problem to solve, because of the following 1. Positivity of the pdf: p(t, x), t & x 2. Normalization constraint of the pdf: p(t, x)dv = No fixed Solution Domain: how to impose boundary conditions in a finite region and restrict numerical computation to regions where p > of 24

9 Let us assume that underlying pdf can be approximated by a finite sum of Gaussian pdfs N ˆp(t, x) = w i p gi where p gi = N (x(t) µ i, P i ) (38) where µ i and P i represent the mean and covariance of the i th component of the Gaussian pdf, respectively and w i denotes the amplitude of i th Gaussian in the mixture. The positivity and normalization constraint on the mixture pdf, ˆp(t, x), leads to following constraints on the amplitude vector: N w i = 1, w i (39) Since all the components of the mixture pdf (38) are Gaussian and thus, only estimates of their mean and covariance need to be maintained in order to obtain the optimal state estimates which can be propagated using the celebrated Kalman filter time update equations: µ i = f(µ i ) (4) Ṗ i = A T i P i + P i A i + g(t, µ i )Qg T (t, µ i ) (41) f(t, x) where A i = x (42) x=µi Notice that the weights w i corresponding to each Gaussian component are unknown. In conventional Gaussian sum filter the weights are initialized such that initial mixture pdf approximates the given initial pdf and it is assumed that the weights do not change over time. This assumption is valid if the underlying dynamics is linear or the system is marginally nonlinear. The same is not true for the general nonlinear case and new estimates of weights are required for accurate propagation of the state pdf. Weight Update II In this section, a novel method is described to update the weights of different components of the Gaussian mixture (38). The main idea is that the mixture pdf (38), ˆp(t, x), should satisfies the Fokker-Planck equation (35) and the Fokker-Planck equation error can be used as a feedback to update the weights of different Gaussian components in the mixture pdf. 24 In other words, we seek to minimize the Fokker-Planck equation error under the assumption (38), (4) and (41). 9 of 24

10 The substitution of (38) in (35) leads to e(t, x) = = ˆp(t, x) t ˆp(t, x) t L FP (ˆp(t, x)) (43) n D (1) i (t, x) n n 2 D (2) ij (t, x) + ˆp(t, x) (44) x i x i x j j=1 where L FP ( ) is the so called Fokker-Planck operator, and, ˆp(t, x) t = N w i p g i µ i T µ i + n n p gi P ijk P j=1 ijk k=1 (45) where P ijk is the jk th element of the i th covariance matrix P i. Further, substitution of (36) and (37) along with (45) in (44) leads to N e(t, x) = w i L i (t, x) = L T w (46) where w is a N 1 vector of Gaussian weights, and L i is given by L i (t, x) = p g i µ i T f(µ i ) + n n p gi P j=1 ijk k=1 d (1) j (t, x)p gi 1 x j 2 n k=1 P ijk + n j=1 2 d (2) jk (t, x)p g i x j x k ( f j (t, x) p g i f j (t, x) + p gi x j x j )] (47) d (1) (t, x) and d (2) (t, x) are given as: d (1) (t, x) = 1 2 g(t, x) Qg(t, x) (48) x d (2) (t, x) = 1 2 g(t, x)qgt (t, x) (49) 1 of 24

11 Further, different derivatives in the above equation can be computed using the following analytical formulas: p gi µ i = P 1 i (x µ i ) p gi (5) p gi = p g i P i p g i P i 2 P i P i 2 p gi = p g i 2 P 1 i p g i 2 = p g i 2 P 1 i + p g i (x µ i ) T P 1 i (x µ i ) P i (51) (x µ i ) T P 1 i (x µ i ) P i (52) 2 P 1 i (x µ i ) (x µ i ) T P 1 i (53) x = P 1 i (x µ i ) p gi (54) [ ] 2 p gi xx T = P 1 i I + (x µ i ) p T g i p gi (55) x Now, at a given time instant, after propagating the mean, µ i and the covariance, P i, of individual Gaussian elements using (4) and (41), we seek to update weights by minimizing the FPE equation error over some volume of interest V : 1 min e 2 (t, x)dv (56) w i 2 s.t V N w i = 1 w i, i = 1,, N Since, the Fokker-Planck equation error (46) is linear in the Gaussian weights, w i, hence, the aforementioned problem can be written as a quadratic programming problem. min w 1 2 wt Lw (57) s.t 1 T N 1w = 1 w N 1 where 1 N 1 is a vector of ones, N 1 is a vector of zeros and L is given by L = V L(x)L T (x)dv = L 1 L 1 dv V L 1 L 2 dv V. L 1 L N dv V L 2 L 1 dv L N L 1 dv V L 2 L 2 dv V V L N L 2 dv.... L 2 L N dv L N L N dv V V V (58) 11 of 24

12 Now, if we show that the matrix L is a positive semi-definite, then the aforementioned optimization problem can be posed as a convex optimization problem and we are guaranteed to have a unique solution. Notice that the integrand L(x)L T (x) is a rank one symmetric positive semi-definite matrix with non-zero eigenvalue equal to L T (x)l(x). Further, the integral in the definition of matrix L can be approximated by an infinite sum of positive semi-definite matrix: V L(x)L T (x)dv = i L(x i )L T (x i ) (59) Making use of the fact that summation of two positive semi-definite matrix is also a positive semi-definite matrix, we conclude that matrix L is a positive semi-definite matrix and thus the optimization problem (57) has a unique solution. Notice, to carry out this minimization, we need to evaluate integrals involving Gaussian pdfs over volume V which can be computed exactly for polynomial nonlinearity and in general can be approximated by the Gaussian quadrature method. Numerical Results The two update schemes are compared with the usual procedure of not updating the weights on several dynamical systems where either a closed form solution for the stationary pdf is available or a Monte Carlo simulation is easily carried. For the examples where we have an analytical solution available, the following integral of the absolute error has been used to compare the methods: E = p(t, x) ˆp(t, x) dx (6) The expectation integrals needed by the first approach have been computed using either Unscented Transformation 2 or Continuous Square-Root Unscented Transformation 25 with a 6n+1 sigma point selection scheme. 22 For the second method, the expectations have been computed using Gaussian quadrature method. Example 1 The first update method has been applied on the following nonlinear discrete-time dynamical system with uncertain initial condition given by (62): x k x k+1 = 1 2 x k x 2 k + η k where η k N (, 1) (61) x.1n (.5,.1) +.9N (.5, 1) (62) 12 of 24

13 The moments of the two Gaussian components are propagated for 2 time steps using (7) and (8). Since there is no analytical solution for the forecast pdf, we run a Monte Carlo simulation using 1 samples and compute the histogram of the samples at each time step, Fig. 1(a). Fig. 1(b) shows the pdf approximation without updating the weights and in Fig. 1(c) is plotted the pdf approximation with updated weights. By updating the weights, we are able to better capture the two modes presented in Fig. 1(a). In addition to this, log probability of the monte carlo sample points was computed according to the following relationship: M N L = log w i N (x j µ i, P i ) (63) j=1 Here, M is the total number of samples used in the Monte Carlo approximation. The higher value of L represents a better pdf approximation. Fig. 1(d) shows the plot of log probability of the samples with and without updating the weights of the Gaussian mixture. As expected, updating the weights of the Gaussian mixture leads to higher log probability of the samples. Hence, we conclude that the adaptation of the weights during propagation leads to more accurate pdf approximation than without weight update. Example 2 For the second example we consider the following continuous-time dynamical system with uncertain initial condition given by (65): ẋ = sin(x) + G(t) where Q = 1 (64) x.1n (.5,.1) +.9N (.2, 1) (65) The moments of the two Gaussian components are propagated for 15 sec using (4) and (41). Again, since there is no analytical solution for the transition pdf, we run a Monte Carlo simulation using 1 samples and compute the histogram of the samples at each time step ( t = 1 sec.), Fig. 2(a). Fig. 2(b) shows the pdf approximation without updating the weights and Figs. 2(c) and 2(d) show the plot of approximated pdf using the two methods introduced in this paper. It is clear from these plots that we are able to better capture the transition pdf with the incorporation of weight update scheme. Also, the shape of approximated pdf is in accordance with the histograms generated by Monte-Carlo simulations and is consistent with the behavior of the dynamical system, which has two attractors at π and π. 13 of 24

14 5.4 #points p(x) time step x time step x 5 1 (a) Histogram - approximate true pdf (b) Transition pdf (without Weight Update 2 x p(x) log.prob.data time step x no update weight update time step (c) Transition pdf (without Weight Update) (d) Log probability of the samples Figure 1. Numerical Results Example 1 14 of 24

15 .8 p(x) time x (b) Transition pdf (without Weight Update p(x) p(x) (a) Histogram - approximate true pdf time time x (c) Transition pdf (without Weight Update I) 5 x (d) Transition pdf (without Weight Update II) Figure 2. Numerical Results Example 2 15 of 24

16 Example 3 We consider the following 2-D nonlinear dynamical system for analysis: ẍ + ηẋ + αx + βx 3 = g(t)g(t) (66) Eq. (66) represents a noise driven duffing oscillator with a soft spring (αβ <, η > ), and included damping (to ensure the presence of a stationary solution - a bimodal pdf). For simulation purposes, we use g(t) = 1, Q = 1, η = 1, α = 1, β = 3 and the stationary probability density function (67) is plotted in Fig. 3(b). ( p(x, ẋ) exp 2 η ( α g 2 Q 2 x2 + β 4 x4 + 2ẋ2 1 )) (67) Notice that the stationary pdf is an exponential function of the steady state system Hamiltonian, scaled by the parameter 2 η g 2 Q.1 To approximate the stationary pdf given by (67), we assume the initial pdf to be a mixture of 1 Gaussian pdfs with centers uniformly distributed between ( 5, 5) and (5, 5). The covariance matrix for each Gaussian component is chosen to be P = 1 I 2 2, where I is the 2 2 identity matrix and the initial weights of the Gaussian components are randomly assigned a number between and 1. The weights have been randomly generated in 2 different runs (see Fig. 3(i) for weights/run), and the initial pdf for the last run. is plotted in Fig. 3(a). The center and corresponding covariance matrices of the Gaussian components are linearly propagated for 1 seconds. Fig. 3(c) shows the plot of the approximated pdf without updating the weights of Gaussian mixture and Fig. 3(f) shows the corresponding approximation error plot for the weights of the last run. As expected, the final pdf is biased towards one of the mode of the true pdf due to the fact that initial weights of those components were large. It is clear that although the approximated pdf did capture some of non-gaussian behavior, it fails to capture both the modes accurately. Further, Figs. 3(d) and 3(h) show the plot of the approximated stationary pdfs after updating the weights of different components of the Gaussian mixture using weight update schemes I and II, respectively. It is clear that both methods (Update I and Update II) yield approximately the same pdf with corresponding approximation error given in Figs. 3(g) and 3(h). The integral absolute error (6) for the updated Gaussian sum is lower than the one without weight update. This result is consistent in all 2 runs as shown in Fig. 3(j), where for different initial weights both methods are able to yield approximately same approximation error and considerably lower than without updating the weights. The first method has been applied recursively every second to update the weights and the second method 16 of 24

17 has been applied only once at the end of the simulation. The results of the 2 runs have been averaged and tabulated at the end of the section, Table 1. As expected, with the adaptation of the weights, the approximation errors have been reduced considerably. Example 4 For the fourth example we have considered the following the 2D noise driven quintic oscillator (68): ẍ + ηẋ + x(ɛ 1 + ɛ 2 x 2 + ɛ 3 x 4 ) = g(t)g(t) (68) For simulation purposes, we choose the following parameters: g(t) = 1, Q = 1, η = 1, ɛ 1 = 1, ɛ 2 = 3.65, ɛ 3 = The true stationary pdf (69), plotted in Fig. 4(b), is trimodal in this case with each of the mode centered at three equilibrium points of the oscillator. ( p(x, ẋ) exp 2 η ( ɛ1 g 2 Q 2 x2 + ɛ 2 4 x4 + ɛ 3 6 x6 + 1 )) 2ẋ2 (69) To approximate the stationary pdf, we assume the initial pdf to be a mixture of 11 Gaussian pdfs with centers uniformly distributed between ( 5, 5) and (5, 5). The covariance matrix for each Gaussian component is chosen to be P =.33 I 2 2, and initially all Gaussian components are assumed to weigh equally as shown in Fig. 4(a). The center and corresponding covariance matrices of the Gaussian components are propagated for 1 seconds in accordance with (4) and (41). Fig. 4(c) shows the plot of the approximated pdf without updating the weights of the Gaussian mixture and Fig. 4(f) shows the corresponding approximation error plot. As expected, the final pdf is not able to capture the mode centered at unstable equilibrium point (, ). It is clear that although the approximated pdf did capture some of non-gaussian behavior, it fails to capture all three modes accurately. Figs. 4(d) and 4(e) show the plot of the approximated stationary pdf after updating the weights of different components of the Gaussian mixture model using the first and the second method respectively. Figs. 4(g) and 4(h) show the approximation error of the two methods. Their integral absolute errors of Eq. (6), tabulated at the end of the section, shows that with the adaptation of the weights, the approximation errors have been reduced considerably. Both methods have been applied once at the end of the simulation. 17 of 24

18 (a) Initial pdf (b) True pdf (c) Final pdf (without Weight Update - run #2) (d) Final pdf (with Weight Update I - run #2) (e) Final pdf (with Weight Update II - run #2) (f) Error pdf (without Weight Update - run #2) (g) Error pdf (with Weight Update I - run #2) (h) Error pdf (with Weight Update II - run #2) No Update Update I Update II random weights.15.1 int.abs.err # random run # random run (i) Random Weights Run (j) Integral Absolute Error Run Figure 3. Numerical Results Example 3 18 of 24

19 (a) Initial pdf (b) True pdf (c) Final pdf (without Weight Update) (d) Final pdf (with Weight Update I) (e) Final pdf (with Weight Update II) (f) Error pdf (without Weight Update) (g) Error pdf (with Weight Update I) (h) Error pdf (with Weight Update II) Figure 4. Numerical Results Example 4 19 of 24

20 Example 5 The last example involves the state pdf propagation through the noise driven energy dependent damping oscillator given by the following equation: ẍ + βẋ + x + α(x 2 + ẋ 2 )ẋ = g(t)g(t) (7) For simulation purposes, we choose the following parameters: g(t) = 1, Q = 1/π, α =.125, β =.5, and the stationary probability density function is given by: ( p(x, ẋ) exp η ( 2g 2 β(x 2 + ẋ 2 ) + α 2 (x2 + ẋ 2 ) 2)) (71) Fig. 5(b) shows the plot of true stationary pdf with most of the probability mass centered at the boundary of stable limit cycle in this case. To approximate the stationary pdf, we assume the initial pdf to be a mixture of 1 Gaussian pdfs with centers uniformly distributed between ( 2, 2) and (2, 2). The covariance matrix for each Gaussian component is chosen to P =.1347 I 2 2. Initially the weight of the first Gaussian component is w 1 =.5 and the rest of the 99 Gaussian components have equal weights, w =.51. The center and corresponding covariance matrices of the Gaussian components are propagated for 1 seconds using (4) and (41). Fig. 5(c) shows the plot of the approximated pdf without updating the weights of the Gaussian mixture and Fig. 5(f) shows the corresponding approximation error plot. As expected, the final pdf is not able to capture the non-gaussian behavior in this case. Figs. 5(d) and 5(e) show the plot of the approximated stationary pdf after updating the weights of different components of the Gaussian mixture model using the first and the second method respectively. Figs. 5(g) and 5(h) show the approximation error of the two methods. Their integral absolute errors (6), tabulated at the end of the section, shows that with the adaptation of the weights, the approximation errors have been reduced considerably. The first method has been applied recursively every 1 seconds to update the weights and the second method has been applied only once at the end of the simulation. Finally, we mention that the numerical results presented in this section reiterates our observation and support the conclusion that the improved performance of proposed algorithms can be attributed to the adaptation of amplitude corresponding to different components of Gaussian mixture. These dramatic advantages from five different problems provide compelling evidence for the merits of proposed algorithms. 2 of 24

21 (a) Initial pdf (b) True pdf (c) Final pdf (without Weight Update) (d) Final pdf (with Weight Update I) (e) Final pdf (with Weight Update II) (f) Error pdf (without Weight Update) (g) Error pdf (with Weight Update I) (h) Error pdf (with Weight Update II) Figure 5. Numerical Results Example 5 21 of 24

22 Table 1. Numerical approximation of the integral of the absolute error No Update Update I Update II Example 3 (avg. 2 runs) Example 4 (1 run) Example 5 (1 run) Conclusions Two update schemes for the forecast weights are presented in order to obtain a better Gaussian sum approximation to the forecast pdf. The difference between the two methods comes from the particularities of their derivations. The first method updates the weights such that they minimize the integral square difference between the true probability density function and its approximation. The derivation of the first method indicates that it is more appropriate for discrete time systems. When dealing with stochastic differential equations the expectations (3) are computed using Continuous Square-Root Unscented Transformation, 25 and the decision to update the weights recursively is made in order for the sigma points to capture the effect of the process noise on the evolution of the pdf. The second method is derived such that the Gaussian sum is satisfying the Fokker-Planck equation, which indicate that it is more appropriate for continuous time systems. The weights are updated such that they minimize the Fokker-Planck equation error. Both the method leads to two different convex quadratic minimization problems which are guaranteed to have a unique solution. Several benchmark problems have been provided to compare the two methods with the usual procedure of not updating the weights in pure propagation settings. In all of these diverse test problems, the proposed two algorithms are found to produce considerably smaller errors as compared to existing methods. The results presented in this paper serve to illustrate the usefulness of adaptation of weights corresponding to different components of Gaussian sum model. In filtering settings, where observations are available, the study of the impact of such an update scheme for Gaussian Sum Filtering is set as future work. Finally, we fully appreciate the truth that results from any test are difficult to extrapolate, however, testing the new algorithm on five benchmark problems does provide compelling evidence and a basis for optimism. 22 of 24

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