Bayesian Unscented Kalman Filter for State Estimation of Nonlinear and Non-Gaussian Systems

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1 Bayesian Unscented Kaman Fiter for State Estimation of Noninear and Non-aussian Systems Zhong Liu, Shing-Chow Chan, Ho-Chun Wu and iafei Wu Department of Eectrica and Eectronic Engineering, he University of Hong Kong, Hong Kong, P. R. China {iuzhong, scchan, andrewhcwu, Abstract his paper proposes a Bayesian unscented Kaman fiter with simpified aussian mixtures (BUKF-SM for dynamic state space estimation of noninear and non-aussian systems. n the BUKF-SM, the state and noise densities are approximated as finite aussian mixtures, in which the mean and covariance for each component are recursivey estimated using the UKF. o avoid the exponentia growth of mixture components, a aussian mixture simpification agorithm is empoyed to reduce the number of mixture components, which eads to ower compexity in comparing with conventiona resamping and custering techniques. Experimenta resuts show that the proposed BUKF-SM can achieve better performance compared with the partice fiter (PF-based agorithms. his provides an attractive aternative for noninear state estimation probem. Keywords Bayesian unscented Kaman fiter; dynamic state estimation; noninear and non-aussian system; aussian mixture; partice fiter.. NRODUCON State estimation of stochastic dynamic systems from a sequence of noisy observations pays a crucia roe in many practica appications such as target tracing, faut detection, signa processing and automatic contro probems. he dynamic system under consideration is usuay modeed by the state-space approach, where difference or differentia equations are used to mode the evoution of the system over time. When the mode is inear with aussian distributed noises, the ceebrated Kaman fiter (KF [] is an optima state estimator which can provide the mean and covariance sequentiay. For more compicated systems with noninear modes and non-aussian noises, there is no cosed form anaytic expression for the posteriori densities. herefore, the state and noise densities have to be approximated by a variety of methods such as histogram or aussian mixtures equipped with noninear fiters say the exted Kaman fiter (EKF and unscented Kaman fiter (UKF []. he histogram approach represents the density as a set of partices and it gives rise to the partice fitering technique [3], which approximates the non-aussian densities with a set of weighted partices. However the computationa compexity of such methods grows exponentiay with the dimension of the states and partices. n addition, due to the probems of degeneracy and samping impoverishment [4], the standard PF may be rather inefficient in samping. o aeviate the probems, many partice fiter (PF-based agorithms [5-7] have been proposed. hey empoy re-samping and custering techniques to simpify the density to achieve reduction in compexity. For instance, in [6], a greedy expectation maximization (EM-based agorithm for mode order reduction using Kubac-Leiber divergence (KLD [8] has been empoyed to simpify the density after re-samping. Simiar concept has been exted to a noninear setting as the aussian sum partice fiter (SPF [5], in which the state density for mixture components is approximated using weighted partices. Recenty, for inear non-aussian systems, a Bayesian Kaman fiter (BKF has been reported in our previous wor [9], which is abe to approximate the non-aussian noise as simpified aussian mixtures with reativey ow compexity. he effectiveness of the BKF agorithm has been demonstrated in video object tracing and other appications []. Since state estimation of noninear systems is frequenty encountered in many practica appications, it wi be highy desirabe if these nice properties can be exted to the noninear settings. t is the aim of this paper to further ext the BKF concept to noninear state estimation using the unscented transformation technique. n particuar, we propose a new Bayesian unscented Kaman fiter with simpified aussian mixtures (BUKF-SM for state estimation in non-inear and non-aussian systems. Among different noninear fitering methods, we have chosen the unscented transform as it is renowned for its we performance in approximating the mean and covariance of the state using a minima set of sampe points, rather than inearizing the noninear function as in the EKF. Whie the noninearity is captured using the unscented transform, the state density in the proposed approach is modeed using an efficient M simpification procedure in BKF, which directy simpifies the Ms by minimizing an upper bound of the approximation error between the origina and the simpified modes. his avoids the high compexity of the re-samping and custering encountered in [5-7], whie /6/$3. 6 EEE 443

2 aowing more genera state and measurement noises to be modeed resuting in better tracing of the states. Lie its BKF counterpart, the arithmetic compexity of the new BUKF- SM agorithm is ony a poynomia, instead of exponentia, function of the state dimension. he proposed BKF-SM is appied to sove a noninear and non-aussian time series estimation probem and the simuation resuts show that it is abe to achieve a better tracing accuracy than the PF-based agorithms. t therefore serves an attractive aternative to PF for noninear state estimation, especiay for systems with more number of states. he rest of the paper is organized as foows: Section describes the detais of the proposed BUKF-SM agorithm. n Section, the performance of the BUKF-SM is evauated and compared with the PF-based agorithms. he concusion is drawn in Section V.. BUKF-SM FOR SAE ESMAON OF NONLNEAR AND NON-AUSSAN SYSEMS For simpicity, we consider the foowing autonomous discrete-time noninear state-space mode: x = f ( x + w, ( z = h + v ( x, ( where x and z denote respectivey the state and observation vectors at time instant, and f ( x and h ( x are respectivey nown noninear state and measurement functions. w and v are random vectors of given distributions, which are assumed to be mutuay indepent and indepent of x.. Unscented Kaman Fiter (UKF n the UKF framewor, the probabiity density functions (pdfs of the state noise w and observation noise v are usuay assumed to be zero mean aussian distributed, i.e.: w = N( w, w, Q, (3 v = N( v, v, R, (4 with w = and v =, and N( u, u, C denotes a aussian distribution with mean u and covariance C. Rather than inearizing the noninear functions as in the EKF, UKF utiizes an unscented transform to approximate the non- aussian state density arising from the noninearity by a aussian distribution through samping at a set of sigma points. o be specific, given an L-dimensiona state vector x with mean μ and covariance P, the UKF agorithm for state estimation can be summarized as foows: χ ( with Construct the foowing sigma points { } corresponding weights { w } from the aussian distribution with mean μ and covariance P : κ χ ( = μ, w =, (5 L + κ χ ( = μ + ( ( L + κ P, w =, (6 ( L + κ χ μ L + κ w = w, (7 ( ( L ( L+ ( = P, + where =,..., L, κ is the scaing parameter and + κ P represents the -th coumn of the matrix ( ( L ( ( L + κ P. Propagate each sigma point through the noninear system state function: χ ( = f ( χ (,. (8 3 Approximate the predicted density after the noninear state function by a aussian distribution by using the ensembe mean and covariance estimates as foows: μ = = w χ (, (9 P = w ( χ μ ( χ μ + Q. ( = ( ( 4 Construct a simiar set of sigma points from the approximatey predicted aussian distribution with mean μ and covariance P : κ χ ( = μ, w =, ( L + κ χ ( = μ ( ( + L + κ P, w =, ( ( L + κ ( ( L + wl w χ κ, (3 ( L+ ( = μ P, + = where =,..., L. 5 Propagate the sigma points through the noninear system measurement function and cacuate the predicted measurement mean: z ( = h ( χ (, (4 z w z. (5 = = ( 6 Estimate the measurement covariance and crosscovariance: P z, = = w ( z ( z ( z ( z + R, (6 P xz = w χ μ ( z z. (7, = ( ( ( 7 Cacuate the posteriori estimation of the state and covariance matrix from the ensembe averages obtained from the sigma points: K = P xz, Pz,, (8 μ = μ + K ( z z, (9 P = P K Pz, K. ( n the above UKF, steps to 3 (from eq. (5 to eq. ( are nown as the time update and the remaining steps are referred to as the measurement update. Whie the UKF assumes the state noise w and measurement noise v to be aussian, we consider a new BUKF-M, which aows the noises to be non-aussian and they can be approximated using Ms.. Bayesian UKF with aussian Mixture (BUKF-M n the proposed BUKF-M, the pdfs of the state noise w and observation noise v are characterized by finite Ms: w = i = α i N( w, wi, Qi, ( 444

3 v = j = β j N( v, v j, R j, ( where ( is the number of components of the corresponding noise, α i ( β j is the probabiity of the corresponding component with i = α = ( j = β j =. Now suppose further that the previous posteriori pdf of the state is modeed by the foowing M with components: p x z = g ( N( x, μ g ( (, (3 ( g = where g ( is the weight of the corresponding component. hen the predictive a priori density can be obtained as: x z = x z x x dx = N( x, μg ( (, (4 g = g ( where =, ( = g ( α i g = i = g ( α i, and μ g ( and g ( g P are respectivey the mean and covariance of corresponding aussian component, which can be predicted using the time update of UKF from eq. (5 to eq. (. Furthermore, given the current observation at time instant, the corrected a posteriori density can be derived as foows: p ( x z = c z x x z = c j= β j N( z, h ( x, R j g = g ( N( x, μg ( ( ""= N( x, μg, P g, "g = g (5 where c = x z z x dx represents a normaizing constant, = = is the updated component number, ( ( = β j p j z x g = j= ( β j p j ( z x, and "g g g μ g "and P g "are respectivey the mean and covariance of the corresponding aussian component, which can be estimated using the measurement update of UKF from eq. ( to eq. (. t can be noticed that the number of mixture components wi grow from to in the predictive step and from to in the subsequent measurement correction step. Hence, the number of mixture components and hence the compexity wi grow exponentiay with time, which compicates onine appications. We now address this issue by means of an order reduction approach, which hep to maintain the number of components over time..3 BUKF with Simpified M (BUKF-SM Consider the foowing M mode with n components: f ( x = n j = α jφ j ( x, (6 where φ j ( x = N( x u j, H j is the j-th components and α j n are the mixing coefficients such that j = α j =. n the BUKF-SM, our goa is to approximate f (x as a simpified mixture mode with fewer components g( x = m i = w i g i ( x, ~ where gi ( x = N( x ti, H i with m < n. w i, t i and H ~ i are respectivey the weight, center and covariance matrix of the i- th component g i (x. w i s shoud be summed up to one. ABLE BUKF-SM ALORHM nitiaization p ( x = N ( x, μ, P g = g g g for =,,... BUKF-SM Prediction: for i = : for g = : g ' = g + ( i Estimate μ and P g ( with μ g ( g ( ( and Q i (UKF ime Update α g = ( g ( i = ( ( ( = = g g g i g( αi BUKF-SM Correction: for j = : for g '= : g " = g' + ( j Estimate g"μ and g"p with μ, P and g ( g ( R j (UKF Measurement Update β p ( z x ="g g ( " = "( β g g j g = j= g ( BUKF-SM Order Reduction: Compute: { g }, { μ g } and { P } j g iven a distance metric D ( f ( x, g( x between functions f (x and g (x, the error of approximating f (x with g (x is D ( f ( x, g( x = ( ( f ( x g( x dx. 5. Different from the conventiona re-samping and custering techniques, we adopted the two-step M simpification procedure deveoped in []. More precisey, at the -th iteration, the component ( ( ( mixture is partitioned into m groups { S, S,..., S m }. Step (Mean update: he representative component ( ( C i ( x for S i that minimizes the oca quantization error ( is C ( arg min ( ( ( ( i x =. ( j S j i α C x φ j x dx nterested C x readers are referred to [] and references therein for soving this probem using coordinate descent. ( ( m Step (Custering: iven C = { Ci ( x } i =, one reassign φ j (x to the nearest C i ( x based on the distortion ( ( ( measure Di, j = D( Ci ( x, φ j ( x, and then update S i. he above process is repeated unti either the change in ( ( m tota distortion or C = { Ci ( x } i = is ess than a certain threshod, or a maximum number of iteration is reached. Since the unscented transformation heps us to approximate the noninear density for each component after j p ( z j x 445

4 the noninear measurement and state transformation, a set of aussian components are obtained at the of each iteration, which can be simpified using the M simpification procedure with fewer components so that the number of components after each iteration can be maintained at a constant eve. his eads to a great reduction in compexity in comparing with conventiona re-samping approaches. For instance, if the component number and the dimension of the state is respectivey n and d, the compexity of conventiona approach such as the KLD-based mode order reduction method in [7] is O (n d, whie the compexity of the function approximation-based method is O ( d 3n( L + m, where and L are the numbers of iterations which are typicay sma []. abe summarizes the proposed BUKF- SM agorithm.. SMULAON RESULS We consider the foowing noninear and non-aussian time series estimation probem [7], which is commony used in the iterature. he process mode is given by: x = + sin( ωπ ( + φx + w, (7 where w has a amma pdf, w = a(3,.5, and ω =.4 and φ =. 5 are scaar parameters. Data sampes are assumed to be taen from = to 6. he underying state distribution can be heavy taied and asymmetric due to the amma distributed process noise and the noninearity of the state function. he non-stationary observation mode is defined as: φx + v, 3, z = (8 φ 3x + v, > 3, where φ =. and φ 3 =. 5. he measurement noise is a 5 zero-mean aussian distribution, i.e., v ~ N( v,,. We compare the performance of the proposed BUKF- SM agorithm with the standard PF, SPF [5], SPPF [] and MSPPF [7] in terms of the foowing root mean squared error (RMSE measure, which is defined as: M RMSE = m = ( x ( m xˆ ( m M, (9 where x (m and xˆ ( m denote respectivey the true and fitered state at the m-th Monte Caro (MC run. he settings are: For the standard PF, the number of partices is and re-samping is performed with a threshod vaue of.. For SPF, the initia state pdf is modeed by five Ms with 5 equa weights, i.e., x = g =.N ( x,, and the amma pdf of the process noise is approximated by two Ms using the EM agorithm. he number of partices for SPF is the same as the standard PF. 3 For SPPF, MSPPF and BUKF-SM, the initia state pdf is identica to that in the SPF whie the scaing parameter κ for sigma point generation is set to the recommed vaue with κ = 3 L =. 4 he number of partices for SPF, SPPF and MSPPF is the same as the standard PF. Experiments were conducted on a computer with nte(r Core(M i7-6k CPU and 89MB RAM. he RMSE PF SPF SPPF MSPPF BUKF-SM Fig.. RMSE of the estimated states for different agorithms are compared. An enarged figure for the comparison between MSPPF and BUKF-SM from time instant to 3 is aso potted. Average RMSE Average ime (s ABLE AVERAE RMSE/ME COMPARSON SPF SPPF MSPPF PF [5] [] [7] BUKF-SM averaged over Monte Caro trias for different agorithms is compared in Fig.. he RMSE vaues in Fig. are further averaged over time and are shown in abe. he resuts demonstrate that the proposed BUKF-SM agorithm is abe to obtain better estimation accuracy in comparing with the other agorithms. he performance of PF is simiar with [7]. he reason for its bad performance is mainy due to the highy peaed ieihood function of the observations (arising from the sma observation noise variance fused with the spurious jumps in the state introduced by the heavy taied process noise, which causes degeneracy and samping impoverishment for the PF. SPF, SPPF and MSPPF can hande the case propery. However their accuracy mainy reies on the number of samping partices and as we mentioned before the computationa compexity wi grow rapidy when the dimension of the states or the number of partices becomes arger. Compared with these PF-based agorithms, the overa performance of the proposed BUKF-SM is better and its computationa time averaged over MC trias is aso reativey ower as shown in tabe. V. CONCLUSON A nove BUKF agorithm with simpified aussian mixtures (BUKF-SM has been presented for dynamic state space estimation in noninear and non-aussian systems. Simuations show that the proposed BUKF-SM agorithm can achieve better performance than the PF-based agorithms, which provides an attractive aternative to conventiona noninear state estimation agorithms. REFERENCE [] R. E. Kaman, A new approach to inear fitering and prediction probems, rans. ASME,. Basic Eng., vo. 8D, pp.35-45, Mar. 96. []. Uhmann et a, A new method for the non inear transformation of means and covariances in fiters and estimations, EEE rans. on Automatic Conto, vo. 45,. [3]. S. Liu and R. Chen, Sequentia Monte Caro methods for dynamic systems,. of Amer. Stat. Assoc.,vo. 93, no. 443, pp

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