Nonlinear Gaussian Filtering via Radial Basis Function Approximation

Transcription

1 51st IEEE Conference on Decision and Contro December Maui Hawaii USA Noninear Gaussian Fitering via Radia Basis Function Approximation Huazhen Fang Jia Wang and Raymond A de Caafon Abstract This paper presents a nove type of Gaussian fiter the radia basis Gaussian fiter (RB-GF) for noninear state estimation In the RB-GF we propose to use radia basis functions (RBFs) to approximate the noninear process and measurement functions of a system considering the superior approximation performance of RBFs Optima determination of the approximators is achieved by RBF neura network (RBFNN) earning Using the RBF based function approximation the chaenging probem of integra evauation in Gaussian fitering can be we soved guaranteeing the fitering performance of the RB-GF The proposed fiter is studied through numerica simuation in which a comparison with other existing methods vaidates its effectiveness I INTRODUCTION Fiter design for noninear state estimation has been a ongstanding chaenge in many fieds incuding contro systems signa processing navigation and guidance etc with a arge amount of research effort having been dedicated to this topic during the past decades 1] The extended Kaman fiter (EKF) is arguaby the most popuar technique ] However the performance of the EKF is often unsatisfactory in terms of convergence speed and robustness to serious noninearities A number of other KF variants have thus been proposed for improvements eg unscented KF (UKF) and ensembe KF (EnKF) Much attention in recent years has been directed towards Gaussian fitering for the purpose of obtaining anaytic or cosed-form noninear fiters Gaussian fiters buid on Bayesian state estimation and assumed density fitering (ADF) A Bayesian estimator sequentiay updates the conditiona probabiity density functions (pdf s) of unknown state variabes given the output measurements 3] The ADF assumes a particuar form of density which are often mathematicay tractabe to dea with for the pdf s invoved in the Bayesian estimator and then computes the state estimates 4] If the assumed densities are Gaussian the ADF wi ead to the Gaussian fiters A crucia probem in Gaussian fitering is to evauate a number of integras The integrand of each integra is the mutipication of a noninear function (stemming from the noninear system equations) and a Gaussian function To address the probem two main approaches have been proposed One of them is based on Monte Caro samping that H Fang is with the Department of Mechanica & Aerospace Engineering University of Caifornia San Diego CA 9093 USA J Wang is with the Schoo of Contro Science & Engineering Daian University of Technoogy Daian China and is currenty visiting the Department of Mechanica & Aerospace Engineering University of Caifornia San Diego CA 9093 USA RA de Caafon is with the Department of Mechanica & Aerospace Engineering University of Caifornia San Diego CA 9093 USA caafon@ucsdedu approximates pdf s by a set of random sampes In fitering a set of state estimates (ie sampes) are generated in ight of the a priori conditiona pdf The KF computation is impemented to each of them and then the individua estimation resuts wi be aggregated to yied the fina state estimate The aforementioned integras are impicity approximated during the process Two typica methods that fa into this category are the EnKF 5] and the UKF 6] the difference between which is that the atter uses deterministic samping The other approach is direct numerica integration Making use of the Gauss-Hermite quadrature rue the Gauss-Hermite fiter (GHF) is capabe of giving amost accurate evauation of the integras arising in Gaussian fitering through a weighted summation of the noninear function evauated at some fixed points 7; 8] In 9] the cubature KF (CKF) is proposed which adopts a spherica-radia cubature rue for numerica computation of these integras The objective of this paper is to deveop a new Gaussian fiter that is reaized by a radia basis function (RBF) based approach Widey used in pattern cassification and curve fitting probems 10] RBFs aso find important appications in the deveopment of neura network based contro systems eg 11; 1; 13; 14] We propose that integra evauation in Gaussian fitering can be reduced to function approximation the construction of which can be achieved using a set of RBFs In this paper we wi use the Gaussian RBF (GRBF) because Gaussian-type functions which frequenty appears in Gaussian fitering are easy to manipuate to derive integration in cosed form With an equivaent structure to a RBF Neura Network (RBFNN) the RBF based function approximator can be estabished through training the RBFNN The obtained fiter which we refer to as the radia basis Gaussian fiter (RB-GF) has high estimation performance and satisfactory computationa efficiency Essentiay a Gaussian fiter with RBFNNs empoyed to assist in deaing with integra evauation the proposed RB-GF differs significanty from existing RBFNN based noninear fitering schemes eg 15] in which RBFNNs are used to mode unknown or uncertain system dynamics In addition instead of doing fixed-point quadrature or cubature approximation ike in 8; 9] it is a customized fiter incorporating construction of approximators for different noninear functions in different systems The remainder of the paper is as foows Section II presents the Gaussian fitering technique the derivation of which from Bayesian estimation theory and ADF is introduced We then deveop the RB-GF in Section III showing the nove appication of the RBFs to Gaussian fitering A simuation study is iustrated in Section IV to show /1/$ IEEE 604

2 the effectiveness of the RB-GF Finay some concuding remarks are offered in Section V II NONLINEAR GAUSSIAN FILTERING Let us consider the foowing noninear discrete-time system: { xk+1 = f(x k ) + w k (1) y k = h(x k ) + v k where x k R nx is the unknown system state and y k R ny is the output The process noise w k and the measurement noise v k are mutuay independent zeromean white Gaussian sequences with covariances Q k and R k respectivey For a Gaussian random vector x R n we use the notation N (x a A) := (π) n A 1 exp 1 (x a)t A 1 (x a) ] where x a R n and A denotes the determinant of A R n n Then we have p(w k ) = N (w k 0 Q k ) and p(v k ) = N (v k 0 R k ) The noninear mappings f : R nx R nx and h : R nx R ny represent the process dynamics and the measurement mode respectivey Define the measurement set Y k := {y 1 y y k } At time k 1 a statistica description of x k from Y k 1 is given as p(x k Y k 1 ) When the new measurement y k containing further information about x k arrives p(x k Y k 1 ) wi be updated to p(x k Y k ) Appying the Bayes rue to this process we have the foowing two-step Bayesian estimation paradigm that sequentiay computes p(x k Y k 1 ) and p(x k Y k ): Prediction p(x k Y k 1 ) = p(x k x k 1 )p(x k 1 Y k 1 )dx k 1 Update () p(x k Y k ) = p(y k x k )p(x k Y k 1 ) (3) p(y k Y k 1 ) A key assumption to be made throughout the paper is that p(x k Y k 1 ) and p(y k Y k 1 ) are Gaussian In this circumstance p(x k Y k ) is ensured to be Gaussian as we and furthermore the Bayesian fiter in ()-(3) propagates forward in a Gaussian manner Accordingy the Gaussian fitering equations can be obtained by determining the means and covariances of p(x k Y k 1 ) and p(x k Y k ) The prediction x k given Y k 1 denoted as ˆx is given by ˆx = x k p(x k Y k 1 )dx k ] = x k p(x k x k 1 )dx k p(x k 1 y k 1 )dx k 1 = f(x k 1 ) N (x k 1 ˆx k 1 k 1 P x k 1 k 1 )dx k 1 (4) The associated prediction error covariance is P x = (x k ˆx )(x k ˆx ) T p(x k Y k 1 )dx k = x k x T k p(x k Y k 1 )dx k ˆx ˆx T = f(x k 1 )f(x k 1 ) T N (x k 1 ˆx k 1 P x k 1 k 1 )dx k 1 ˆx ˆx T + Q k 1 (5) The derivation of (4)-(5) uses (A1)-(A) in the Appendix When y k is avaiabe et us consider the joint conditiona pdf of x k and y k given Y k 1 which according to the assumption is Gaussian: p(x k y k Y k 1 ) = ] ] N xk ˆx Px P xy ( ) T y k P y ŷ P xy (6) Here ŷ is the prediction of y k given Y k 1 given by ŷ = y k p(y k Y k 1 )dy k (7) It is noted that p(y k Y k 1 ) = p(x k y k Y k 1 )dx k = p(y k x k )p(x k Y k 1 )dx k Inserting the above equation into (7) yieds ] ŷ = y k p(y k x k )dy k p(x k Y k 1 )dx k = h(x k )p(x k Y k 1 )dx k = h(x k ) N (x k ˆx P x )dx k (8) The associated covariance is P y = (y k ŷ )(y k ŷ ) T p(y k Y k 1 )dy k = h(x k )h T (x k ) N (x k ˆx P x )dx k ŷ ŷ T + R k (9) and the cross-covariance is P xy = (x k ˆx )(y k ŷ ) T p(x k y k Y k 1 )dx k dy k = x k h(x k ) T N (x k ˆx P x )dx k ˆx ŷ T (10) It foows from (6) and (A3) that p(x k Y k ) = N (x k ˆx k k P x k k ) 6043

3 where ˆx k k = ˆx + P xy P x k k = Px Pxy ( P y ) 1 (yk ŷ ) (11) ( ) 1 ( T P y P) xy (1) The Gaussian fiter is summarized in Agorithm 1 It is noteworthy however that the Gaussian fiter ony deineates a conceptua framework for this type of fitering methods To make it truy appicabe in practice it is needed to deveop methods for evauation of the integras in (4)-(5) and (8)- (10) Initiaize: k = 0 ˆx 0 0 = E(x 0 ) P x 0 0 = p 0I where p 0 > 0 repeat k k + 1 Prediction: State prediction via (4) Computation of prediction error covariance via (5) Update: Measurement prediction via (8) with the associated covariance via (9) Computation of the cross-covariance via (10) State estimation via (11) Computation of the estimation error covariance via (1) unti no more measurements arrive Agorithm 1: The Gaussian fiter for noninear state estimation III RADIAL BASIS GAUSSIAN FILTERING As is observed the integras in (4)-(5) and (8)-(10) take one of the foowing forms: Ω 1 = g(x) N (x µ Σ)dx (13) Ω = xg T (x) N (x µ Σ)dx (14) Ω 3 = g(x)g T (x) N (x µ Σ)dx (15) where x R n and g is assumed without oss of generaity to be a mapping over R n R m where n and m are an arbitrary positive integers In this section we introduce the notion and reaization of the RBF approximation of g(x) and then continue to show how to construct the RB-GF based on the proposed function approximation A Fitering Integra Evauation For g i (x) that is the i-th eement of g(x) we consider a nonineary parameterized approximator ĝ i (x) = N w ij s j (x) (16) where s j (x) for j = 1 N is a set of N RBFs and w ij s are the weighting factors A wide variety of RBFs such as muti-quadratics inverse muti-quadratics and Gaussian functions have been studied in the iterature We propose to use the Gaussian RBFs (GRBFs) which wi faciitate addressing the probem of Gaussian fitering Then s j (x) is given by ] s j (x) = exp (x c j) T (x c j ) σj = α j N ( x c j σj I ) where α j = (πσ j ) n c j and σ j are the center and width of the RBF respectivey For simpicity we assume that c j s and σ j s are fixed and known The assumption does not imit the extension of the ensuing derivation to the case when both of them are unknown and need to be determined It foows from (A4) that the mutipication of two Gaussian functions is another unnormaized Gaussian function Hence we consider where β j N (x µ j Σ j ) = N ( x c j σ j I ) N (x µ Σ) γ j N (x µ j Σ j ) = N ( x c j σ j I ) N ( x c σ I ) Σ j = ( σ j I + Σ 1) 1 µ j = Σ j ( σ j c j + Σ 1 µ ) σ N (x µ Σ) β j = (π) n σ n j Σ 1 Σj 1 exp 1 ( σ j c T j c j + µ T Σ 1 µ µ T j ( ) 1 1 Σ j = I + Σ j µ j = Σ j (σ 1 c + Σ j µ j ) Σ 1 j µ j ) ] γ j = β j (π) n σ n Σ j 1 Σj 1 exp 1 ( σ c T c + µ T 1 j Σ j µ j µ T j Σ 1 j j) ] µ Note that if µ and Σ are variabes Σj µ j β j Σ j µ j and γ j are a functions of µ and Σ We have the foowing integration formuae before proceeding further: s j (x) N (x µ Σ)dx = α j β j (µ Σ) xs j (x) N (x µ Σ)dx = α j β j (µ Σ) µ j (µ Σ) s j (x)s (x) N (x µ Σ)dx = α j α γ j (µ Σ) 6044

4 R m j = 1 M} find the optima W to minimize the convex cost function J(W) defined as Fig 1: The architecture of a RBFNN Define the foowing matrices and vectors: W = w ij s(x) = s j (x) ]T β(µ Σ) = α j β j (µ Σ) Ψ(µ Σ) = α j β j (µ Σ) µ j (µ Σ) ] Γ(µ Σ) = α j α γ j (µ Σ) Here W R m N s R N β R N Ψ R N m and Γ R N N We then have ĝ(x) = W s(x) and Ω 1 ĝ(x) N (x µ Σ)dx = W β(µ Σ) (17) Ω xĝ T (x) N (x µ Σ)dx = Ψ(µ Σ) W T (18) Ω 3 ĝ(x)ĝ T (x) N (x µ Σ)dx = W Γ(µ Σ) W T (19) We see that (17)-(19) construct a computationa foundation on which the Gaussian fitering integras in (4)-(5) and (8)-(10) can be evauated easiy The foundation is based on RBF approximation If the approximation is accurate (17)-(19) wi be a cosed-form soution to Gaussian fitering B RBF Approximation via Neura Network Learning Prior to integra evauation in (17)-(19) the weight matrix W must be determined optimay in the sense that approximation error between g(x) and ĝ(x) is minimized A formuation of this probem is: Given a data set containing M eements D = {(d j z j ) z j = g(d j ) d j R n z j J(W) = 1 = 1 = 1 z j ĝ(d j ) z j W s(d j ) i=1 m z ji w i s(d j ) (0) where z ji is the i-th eement of z j and w i is the i-th row vector of W It is noteworthy that each w i can be determined separatey by minimizing J(w i ) = z ji w i s(d j ) The above weight determination probem is equivaent to training a RBF Neura Network (RBFNN) A RBFNN usuay performs curve fitting in a high dimensiona space or more specificay to find a hypersurface that provides a best fit for the high-dimensiona training data 10] For g i (x) the schematic diagram of a RBFNN is shown in Fig 1 It has three ayers The first one is the input ayer which has n nodes corresponding to each eement of the input vector d The second ayer is a hidden ayer with N units to each of which a nodes in the first ayer are connected The activation functions of the j-th unit is the GRBF s j (x) indicating that this is indeed a GRBFNN Each s j (x) in the hidden ayer is connected through the weight w ij to the output ayer that has ony a singe unit This unit computes a weighted sum of the outputs of the hidden units as the output of the network It is noted that the RBFNN transates the function approximation under consideration into neura network earning which appies earning strategies to the training data set D to determine the weights of the output ayer A few different types of earning strategies have been proposed in the iterature A straightforward approach is to use the pseudoinverse method to derive the east squares soution to (0) However it is computationay inefficient especiay whenever new data become avaiabe and aso poory scaabe to arge data sets To remedy this situation most other approaches for RBFNN earning carry out recursive updating Among them we highight the one based on gradient descent 10] Consider ĝ i in (16) which can be rewritten as ĝ i (x) = w i s(x) The recursive earning procedure of w i is expressed as φ() = s T (d j ) (z ji w i () s(d j )) (1) w i ( + 1) = w i () η φ() () where φ = wi J(w i ) is the gradient η is the earning coefficient and denotes the recursion step 6045

5 Approximation properties of the RBFNN is of much significance in practica impementation The Universa Approximation Theorem states that if g(x) is continuous then there is a RBFNN such that the function ĝ(x) reaized by the RBFNN is cose to g(x) in the L p norm for p 1 ] 10] Furthermore it is pointed out in 1] that ĝ(x) can approximate the continuous g(x) to an arbitrary accuracy over a compact set Thus for the considered Gaussian fitering probem if the function approximators are we designed via RBFNNs high-accuracy approximation can be achieved thus ensuring the fitering performance C The RB-GF Putting together the formuae in the Gaussian fiter function approximation and integra evauation yieds the RB-GF as described in Agorithm Initiaize: k = 0 ˆx 0 0 = E(x 0 ) P x 0 0 = p 0I where p 0 > 0 typicay a arge positive vaue Function approximation Construction of the RBF based approximators for f and h using RBFNN via (1)-() repeat k k + 1 Prediction: State prediction via (4) and (17) Computation of prediction error covariance via (5) and (19) Update: Measurement prediction via (8) and (17) with the associated covariance via (9) and (19) Computation of the cross-covariance via (10) and (18) State estimation via (11) Computation of the estimation error covariance via (1) unti no more measurements arrive Agorithm : The Radia Basis Gaussian Fiter (RB-GF) IV SIMULATION EXAMPLE In this section we present a numerica exampe to evauate the performance of the proposed RB-GF Consider the onedimensiona noninear system { xk+1 = f(x k ) + w k (3) y k = h(x k ) + v k where f(x k ) = sin(x k ) + 005x k (1 x k) h(x k ) = 001(x k 005) The noises w k and v k are zero-mean white Gaussian with Q = 0001 and R = 0001 respectivey We then appy the RB-GF agorithm to data generated from the above system Another three types of Gaussian fiters UKF GHF CKF are aso impemented for an overa comparison In simuation x vs ˆx x ˆx (a) k True UKF GHF CKF RB-GF (b) k UKF GHF CKF RB-GF Fig : (a) True states and estimated ones; (b) estimation errors the initia condition x 0 = 0 ˆx 0 0 = 06 and P 0 0 = 3 The functions f(x) and h(x) are approximated over ] by 10 GRBFs with the centers eveny distributed and widths equa to 1 The number of Sigma points in the UKF is 11 and the number of quadrature points in the GHF is 10 Fig shows the state estimation resuts yieded by the four fiters From Fig and numerous simuation runs we consistenty observe comparabe performance between the UKF GHF and the proposed RB-GF with the RB-GF usuay performing sighty better Athough the CKF can achieve the same-eve accuracy ony when a sufficient number of measurements arrive it is the most computationay efficient From the simuation we gain an insight that the RB-GF can be enhanced in two ways First instead of designing the approximators by RBFNN earning once and for a dynamicay approximating the functions over the neighborhood of the current state estimate wi ead to better approximation performance Second the notion deveoped 6046

6 in this paper can be extended to mixed Gaussian fiter which empoys a inear combination of Gaussian densities to represent any type of probabiity densities Despite additiona computationa compexity both methods are foreseeabe to boost the estimation performance significanty V CONCLUSIONS We have investigated the noninear state estimation probem and proposed a new Gaussian fiter based on RBF approximation A distinct advantage of Gaussian fiters is the cosed-form description However their practica impementation requires evauation of certain forms of integras To dea with this chaenge we have proposed to buid apprximators composed of a weighed sum of RBFs for noninear system functions Determination of the approximators ie optima seection of the weights can be addressed by RBFNN earning It has been shown that with the RBF based function approximators the integras in Gaussian fitering can be deicatey evauated giving rise to the RB-GF agorithm We have demonstrated the effectiveness of the RB-GF through a comparison with other fiters in numerica simuation Future work wi be devoted to fitering performance enhancement by deveoping dynamic function approximation and the mixed Gaussian fiter with RBF based function approximators APPENDIX For the reader s convenience we provide some Gaussian identities the proofs of which are widey avaiabe in textbooks and thus omitted A the vectors and matrices invoved beow are assumed to have compatibe dimensions If p(x) = N (x a A) then (Mx + m)p(x)dx = Ma + m (A1) (x m)(x m) T p(x)dx = (a m)(a m) T + A If ( ] ] ]) x a A C p(x y) = N y b C T B (A) then the margina distributions are p(x) = N (x a A) and p(y) = N (y b B) and the conditiona distributions is p(x y) = N ( x a + CB 1 (y b) A CB 1 C T) (A3) Given two gaussian functions their mutipication eads to another gaussian function ie where N (x a A) N (x b B) = λ N (x c C) C = ( A 1 + B 1) 1 c = C(A 1 a + B 1 b) (A4) λ = (π) n A 1 B 1 1 C exp 1 ( a T A 1 a + b T B 1 b c T C 1 c )] REFERENCES 1] B D O Anderson and J B Moore Optima fitering Prentice-Ha Engewood Ciffs 1979 ] D Simon Optima State Estimation: Kaman H and Noninear Approaches Wiey-Interscience 006 3] J V Candy Bayesian Signa Processing: Cassica Modern and Partice Fitering Methods New York NY USA: Wiey-Interscience 009 4] P S Maybeck Stochastic modes estimation and contro Voume II A Press Ed ] G Evensen Sequentia data assimiation with a noninear quasi-geostrophic mode using monte caro methods to forecast error statistics Journa of Geophysica Research vo 99 pp ] S Juier and J Uhmann Unscented fitering and noninear estimation Proceedings of the IEEE vo 9 no 3 pp ] H Kushner and A Budhiraja A noninear fitering agorithm based on an approximation of the conditiona distribution IEEE Transactions on Automatic Contro vo 45 no 3 pp ] K Ito and K Xiong Gaussian fiters for noninear fitering probems IEEE Transactions on Automatic Contro vo 45 no 5 pp ] I Arasaratnam and S Haykin Cubature Kaman fiters IEEE Transactions on Automatic Contro vo 54 no 6 pp ] S Haykin Neura Networks: A Comprehensive Foundation nd ed Upper Sadde River NJ USA: Prentice Ha PTR ] S S Ge T H Lee and C J Harris Adaptive Neura Network Contro for Robotic Manipuators London UK: Word Scientific ] R Sanner and J-J Sotine Gaussian networks for direct adaptive contro IEEE Transactions on Neura Networks vo 3 no 6 pp ] B Ren S Ge T H Lee and C-Y Su Adaptive neura contro for a cass of noninear systems with uncertain hysteresis inputs and time-varying state deays IEEE Transactions on Neura Networks vo 0 no 7 pp ] B Ren S S Ge K P Tee and T H Lee Adaptive neura contro for output feedback noninear systems using a barrier Lyapunov function IEEE Transactions on Neura Networks vo 1 no 8 pp ] S Eanayar VT and Y Shin Radia basis function neura network for approximation and estimation of noninear stochastic dynamic systems IEEE Transactions on Neura Networks vo 5 no 4 pp