Application of Unscented Transformation for Nonlinear State Smoothing

Application of Unscented Transformation for Nonlinear State Smoothing WANG Xiao-Xu 1 PAN Quan 1 LIANG Yan 1 ZHAO Chun-Hui 1 Abstract Motivated by the well-known fact that the state estimate of a smoother is more accurate than that of the corresponding filter, this paper is concerned with the state smoothing problem for a class of nonlinear stochastic discrete systems. Firstly, a novel type of optimal smoother, which provides a unified theoretical framework for the solution of state smoothing problem no matter that system is linear or nonlinear, is derived on the basis of minimum mean squared error (MMSE) estimation theory. Further, in the case that the dynamic model and measurement functions are all nonlinear, a new suboptimal smoother is developed by applying the unscented transformation for approximately computing the smoothing gain in the optimal smoothing framework. Finally, the superior performance of the proposed smoother to the existing extended Kalman smoother (EKS) is demonstrated through a simulation example. Key words Nonlinear, optimal smothering framework, minimum mean squared error (MMSE), unscented transformation, extended Kalman smoother (EKS) Citation Wang Xiao-Xu, Pan Quan, Liang Yan, Zhao Chun-Hui. Application of unscented transformation for nonlinear state smoothing. Acta Automatica Sinica, 2012, 38(7): 1107 1112 DOI 10.1016/S1874-1029(11)60284-X Discrete-time filtering for nonlinear dynamic system is an important research area and has attracted considerable interest. Recently, a kind of optimal Gaussian recursive filter for nonlinear systems has been developed in 1] to achieve the whole posterior description (mean and covariance) of nonlinear state, which provided a unified theoretical framework for solving nonlinear filtering problem. However, such a complete description calls for infinite parameters and complex calculation, which results in that this optimal Gaussian filter is hard to be applied in practice. So, a large number of suboptimal schemes 2 4] to approximate the posterior distribution of nonlinear state are developed. As is well-known, the most widely used filter for nonlinear estimation problem is the extended Kalman filter (EKF) 2]. It is derived from Kalman filter based on the successive linearization of the nonlinear state-space model. Recently, a novel nonlinear filtering algorithm named unscented Kalman filter (UKF) 5] is proposed as an improvement to EKF. It utilizes the unscented transformation (UT) technique to approximate the posterior mean and covariance of the nonlinear state with a second order accuracy, while EKF uses the linearization technique to achieve only first order accuracy. Moreover, unlike EKF, UKF does not ask for computing Jacobian matrices of the nonlinear functions, hence it is easier in implementation than EKF. Under the multi-dimensional Taylor series, Julier et al. 6] gave the theoretical analysis of performance about UKF, e.g., the approximation of the mean and the covariance and the general rule of selecting sigma points. van der Merwe et al. 7] designed the square-root unscented Kalman filter (SR-UKF) to strengthen the filtering numerical stability. Cho et al. 8], through introducing the interacting multiple model algorithm 9] to the aided design of the sigma points Kalman filters, presented an adaptive IIR/FIR fusion filter, which showed robustness against model uncertainty, temporary disturbing noise, large initial estimation error, Manuscript received January 20, 2011; accepted January 10, 2012 Supported by National Natural Science Foundation of China (61135001, 61075029, 61074179, 61074155) and the Postdoctoral Science Foundation of China (20110491692) Recommended by Associate Editor FANG Hai-Tao 1. Institute of Control and Information, School of Automation, Northwestern Polytechnical University, Xi an 710072, P. R. China etc, compared with the standard UKF. Simandl et al. 10] derived the new relations among the Stirling sinterpolation and the unscented transformation, whose impact on the covariance of the state estimation was analyzed. Xiong et al. 11] showed that the design of noise covariance matrix played an important role in improving the stability of the UKF and proved that under certain condition of appropriate choice of the extra positive matrix, the estimation error of the UKF remained bounded without the requirement of small initial estimation error. Leven et al. 12] applied the UKF for multiple target tracking with symmetric measurement equations, and showed that the UKF algorithm had better performance and was simpler in implementation than the previous EKF. Hermoso-Carazo et al. 13] considered the filtering problem for a class of nonlinear discrete-time systems using measurements which can be randomly delayed by two sampling time; in this situation, a generalization of EKF and UKF algorithms based on twostep randomly delayed measurements has been proposed and a comparison made between the two filters. Bruno et al. 14] developed constrained unscented filtering algorithms to address the nonlinear state estimation problem in the case that prior knowledge is available in the form of interval constraints on the states. The similar problem was investigated in 15]. Saleh et al. 16] presented a modification on delay estimation and tracking using the unscented filter for an asynchronous CDMA system, and the simulation results showed that the modified UKF tracks the delay variations with less errors than the standard UKF. Meanwhile, in addition to target tracking 12, 17 18], UKF has also widespread applications including navigation 19 21], chemical process 22], communication 23], signal processing 24] and so on. Despite UKF and its improved algorithms 13 17] have been proved to be superior in different areas 18 24],itis necessary to emphasize that the unscented transformation is only applied for dealing with the state filtering problem of nonlinear systems while there are scarcely attempts made for using UT to solve the nonlinear state smoothing problem. In addition, up to the present, the solution for nonlinear state smoothing issue is seldom reported publicly. Furthermore, smoother can obtain better estimation

1108 ACTA AUTOMATICA SINICA Vol. 38 precision than filter, because smoother which is based on the measured data from step k = 0 to future can use more measurement information than filter which is only from k = 0 to present. Therefore, there is an urgent need to carry on this theoretical and practical research and to design a new nonlinear state smoother. The goal of this paper is the derivation of the nonlinear state smoother based on UT. Firstly, the minimum mean squared error (MMSE) estimation theory is introduced. Secondly, the optimal smoother for a class of discrete stochastic dynamic systems described by density functions on the basis of MMSE estimation is derived. Thirdly, we consider a general nonlinear stochastic discrete system and obtain nonlinear suboptimal smoother based on UT. Finally, the performance of the new smoother through a simulation example is demonstrated. 1 Problem formulation Consider state-space model of discrete stochastic dynamic system as the following two density functions: xk p(x k x k 1 ) (1) y k p(y k x k ) where x k R n and y k R m are the state and observation vectors at time k, respectively, p(x k x k 1 ) is the transition probability density function of the dynamic model, p(y k x k ) is the likelihood probability density function of the measurements. The purpose is to find the approximation to the smoothing distributions p(x k y 1:T ) for all k 0, 1,,T} based on the MMSE estimation and the measurements sequence y 1:T = y 1,y 2,,y k,,y T }, the smoothing distributions at time steps k and k + 1 are all assumed to be Gaussian, i.e. p(x k y 1:T )=N(m s k,pk s ) (2) p(x k+1 y 1:T )=N(m s k+1,p s k+1) (3) where the mean m s k+1 and the covariance P s k+1 are known. 2 Optimal smoother 2.1 MMSE estimation For deriving the optimal smoother for the state model (1), some standard results about the MMSE estimation 25] are recalled. Lemma 1. Assume that x and z are random vector variables with the second order moment, so the linear MMSE estimation of x based on z, which is marked as ˆx = Ê(x z), has the following form: ˆx = Ê(x z) =E(x)+cov(x, z)var(z)] 1 z E(z)] (4) where var(z) =Ez E(z)]z E(z)] T }, E denotes the mean. Lemma 2. Assume that x, y, andz are random variables with the second order moment, if A and B are nonrandom matrices, then Ê(Ax + By) z] =AÊ(x z)+bê(y z) (5) Lemma 3. Assume that x, z a,andz b are random variables with the second order moment, if z can be expressed as a set containing z a and z b, i.e., z = z a,z b }, then Ê(x z) =Ê(x za)+ê( x z b)= 1 Ê(x z a)+e x z b T ] E( z b z b )] T zb (6) where x = x Ê(x za), z b = z b Ê(z b z) (7) It has been proved that if the conditional distribution of x based on z is assumed to be Gaussian, the linear MMSE estimator of x based on z is equivalent to the conditional mean 26 27], i.e. ˆx = Ê(x z) =E(x z) = xp(x z)dx (8) 2.2 Optimal smoother based on the linear MMSE estimation It has been shown in (2) and (3) that the smoothing distributions are assumed to be Gaussian, then under the condition of Gaussian distributions, on the basis of equivalence between the linear MMSE estimator and the conditional mean, we can obtain m s k =E(x k y 1:T )=Ê(x k y 1:T ) Pk s =E m s k( m s k) T] =E (x k m s k)(x k m s k) T] = (x k m s k)(x k m s k) T p(x k y 1:T )dx k (9) m s k+1 =E(x k+1 y 1:T )=Ê(x k+1 y 1:T ) Pk+1 s =E m s k+1( m s k+1) T] = E (x k+1 m s k+1)(x k+1 m s k+1) T] = (x k+1 m s k+1)(x k+1 m s k+1) T p(x k+1 y 1:T )dx k+1 (10) where m s k+1 and P s k+1 are known while m s k and P s k are unknown, the goal of this paper is to derive the expression of m s k and P s k on the conditions of the known m s k+1 and P s k+1. In addition, assume that the filtering and prediction distributions are also Gaussian, i.e. p(x k y 1:k )=N(m k,p k ) (11) p(x k+1 y 1:k )=N(m k+1,p k+1) (12) Further, the joint distribution of x k and x k+1 can be written as (( ) ) (( ) ( )) xk mk Pk C k+1 p y x 1:k N k+1 m, k+1 Ck+1 T P k+1 (13) Therefore m k =E(x k y 1:k )=Ê(x k y 1:k ) P k =E m k ( m k ) T] =E (x k m k )(x k m k ) T] = (x k m k )(x k m k ) T p(x k y 1:k )dx k (14) m k+1 =E(x k+1 y 1:k )=Ê(x k+1 y 1:k ) P k+1 =E m k+1( m k+1) T] = E (x k+1 m k+1)(x k+1 m k+1) T] = (x k+1 m k+1)(x k+1 m k+1) T p(x k+1 y 1:k )dx k+1 (15)

No. 7 WANG Xiao-Xu et al.: Application of Unscented Transformation for Nonlinear State Smoothing 1109 C k+1 =E m k ( m k+1) T] = Obviously, on the basis of (19), it can be noticed that E (x k m k )(x k+1 m k+1) T] = E(x k x k+1,y 1:k )= x k p(x k x k+1,y 1:k )dx k = (x k m k )(x k+1 m k+1) T p(x k,x k+1 y 1:k )dx k dx k+1 (16) x k p(x k x k+1,y 1:T )dx k =E(x k x k+1,y 1:T ) Then, subtracting (17) from (21) gives (22) The derivation of optimal smoother for the state model (1) is shown as follows. m s k = m k + D k+1 (m s k+1 m k+1) (23) Step 1. Considering Gaussian properties in (2), (3), and Step 2. The following is the derivation of Pk s. From (13), on the basis of Lemma 3 and (8), it can be obtained (23), we have that m s k = x k m s k = x k m k D k+1 (m s k+1 m k+1) = E(x k x k+1,y 1:k )= x k p(x k x k+1,y 1:k )dx k = m k D k+1 m k+1 + D k+1 m s k+1 (24) Ê(x k x k+1,y 1:k )=Ê(x k y 1:k )+Ê( m k m k+1) = m k +E m k ( m k+1) T] E m k+1( m k+1) T]} 1 m k+1 = m k + C k+1 Pk+1] 1 (x k+1 m k+1) = m k + D k+1 (x k+1 m k+1) (17) E(x k x k+1,y 1:T )= x k p(x k x k+1,y 1:T )dx k = Ê(x k x k+1,y 1:T )=Ê(x k y 1:T )+Ê( ms k m s k+1) = m s k +E m s k( m s k+1) T] E m s k+1( m s k+1) T]} 1 m s k+1 = m s k +E m s k( m s k+1) T] Pk+1] s 1 (x k+1 m s k+1) (18) The following is the derivation of E m s k( m s k+1) T]. Due to the Markov properties 28] of the state-space model described by (1), we have p(x k x k+1,y 1:T )=p(x k x k+1,y 1:k ) (19) Using (19) and (17) yields E m s k( m s k+1) T] = m s k( m s k+1) T p(x k,x k+1 y 1:T )dx k dx k+1 = m s k( m s k+1) T p(x k x k+1,y 1:T )p(x k+1 y 1:T )dx k dx k+1 = ] m s kp(x k x k+1,y 1:T )dx k ( m s k+1) T p(x k+1 y 1:T )dx k+1 = E(x k x k+1,y 1:k ) m s k]( m s k+1) T p(x k+1 y 1:T )dx k+1 = mk m s k + D k+1 (m s k+1 m k+1)+d k+1 m s k+1] ( m s k+1) T p(x k+1 y 1:T )dx k+1 = D k+1 m s k+1( m s k+1) T p(x k+1 y 1:T )dx k+1 = D k+1 P s k+1 (20) Substituting (20) into (18), the equation becomes E(x k x k+1,y 1:T )=m s k + D k+1 (x k+1 m s k+1) (21) E m s k( m s k+1) T] = E ( m k D k+1 m k+1 + D k+1 m s k+1)( m s k+1) T] = E ( m k D k+1 m k+1)( m s k+1) T] + D k+1 Pk+1 s (25) According to (20) and (25), it can be derived that the first item of the right-hand side in (25) is zero, i.e. E ( m k D k+1 m k+1)( m s k+1) T] = 0 (26) On the basis of (26) and (24), the smoothing covariance matrix Pk s is computed as follows: Pk s =E m s k( m s k) T] = E ( m k D k+1 m k+1 + D k+1 m s k+1) ( m k D k+1 m k+1 + D k+1 m s k+1) T] = E ( m k D k+1 m k+1)( m k D k+1 m k+1) T] + D k+1 E m s k+1( m s k+1) T] Dk+1 T = P k + D k+1 P s k+1 P k+1]d T k+1 (27) From (23) and (27), we can conclude that the optimal smoother can be written as D k+1 = C k+1 P k+1] 1 (28) m s k = m k + D k+1 m s k+1 m k+1] (29) P s k = P k + D k+1 P s k+1 P k+1]d T k+1 (30) where D k (k>0) is the smoothing gain. It is not difficult to find that (29) and (30) give a unified theoretical framework for state smoothing, no matter the system (1) is linear or nonlinear. When the system (1) is linear, the smoothing gain D k can be computed accurately by linear transformation, and the well known smoother for linear system is Kalman smoother 29]. When the system (1) is nonlinear, its smoothing recursive formula will be discussed in the next section. 3 Nonlinear suboptimal smoother based on unscented transformation In this section, we shall consider the general nonlinear stochastic discrete-time system model: xk = f k 1 (x k 1 )+w k 1 y k = h k (x k )+v k (31)

1110 ACTA AUTOMATICA SINICA Vol. 38 where x k R n is the state vector, y k R m is the measurement vector, f k 1 ( ) is the dynamic model function, h k ( ) is the measurement function, the process noise w k N(q k,q k ) and the measurement noise v k N(r k,r k ) are uncorrelated Gaussian white noises respectively. Obviously, the model (31) is a special case of the model (1). With respect to the system model (31), if f k 1 ( ) and h k ( ) are both linear functions, D k is accurately computed by simple linear transformation. However, when f k 1 ( ) and h k ( ) are both nonlinear functions, obtaining a precise value of D k is very hard in most cases. This is because that if we want to obtain the accurate solution of D k,it has to achieve the whole posterior probability description of nonlinear state. But this whole description needs to calculate the complex multi-dimensional integral of the nonlinear function 1], which calls for infinite parameters and large calculation, so it is almost impossible to get the analytical solution of D k in most cases. For the above reasons, D k has to be calculated approximately through some suboptimal methods, for example the linearization of f k 1 ( ) and h k ( ), which yields extended Kalman smoother (EKS) 29], and the unscented transformation 3] which yields nonlinear suboptimal smoother based on UT, etc. The specific steps of the nonlinear suboptimal smoother based on the UT are given as follows. Step 1. Initialization At time step 0, there is no measurement, except the prior distribution x 0 N(m 0,P 0). Step 2. Sigma points calculation and propagation The n-dimensional random vector x k with the mean m k and the covariance P k is approximated by sigma points selected by using the following equations: ξ 0,k = m k, W 0 = κ n + κ ( ) 1 ξ i,k = m k + (n + κ)pk, W i = i 2(n + κ) ( ) 1 ξ i+n,k = m k (n + κ)pk, W i+n = i 2(n + κ) ( (32) (n ) where i =1, 2,,n, + κ)pk denotes the i-th column of the square-root of the matrix (n + κ)p k, κ is the scaling parameter which is usually set as κ =3 n for the Gaussian distribution, W i} is a set of scalar weights. Then, propagate the sigma points through the dynamic model function γ i,k+1 k = f k (ξ i,k )+q k, i =0, 1, 2,, 2n (33) Step 3. Computing the smoothing gain D k Firstly, the predicted mean m k+1, the predicted covariance P k+1, and the cross-covariance C k+1 can be computed as follows: m k+1 = P k+1 = C k+1 = W iγ i,k+1 k + q k = i W if k (ξ i,k )+q k (34) W i γ i,k+1 k m k+1]γ i,k+1 k m k+1] T} + Q k (35) W i ξ i,k m k ]γ i,k+1 k m k+1] T} (36) Then, the smoothing gain D k can be computed through (28). Substituting D k into (29) and (30) can obtain the nonlinear state smoother based on UT. The aforementioned procedure is a recursion, which can be used for computing the smoothing distribution of step k from the smoothing distribution of time step k+1. Because the smoothing distribution and filtering distribution of the last time step T are the same, we have m s T = m T and PT s = P T, and thus, the recursion can be used for computing the smoothing distributions of all time steps by starting from the last step k = T and proceeding backwards to the initial step k =0. In addition, for implementing the smoother (29) and (30), the approximate mean m k and the covariance P k of the filtering distribution in (14), as well as m k+1 and P k+1 of the predictive distribution in (15) must be known. For the model (31). m k, P k, m k+1,andp k+1 are assumed to have been computed by the standard UKF 3]. 4 Simulation analysis Consider a numerical example described as follows: x k+1 = x 1,k+1 x 2,k+1 = x 3,k+1 3sin(2x 2,k ) x 1,k +e 0.05x 3,k +10 + 1 1 w k 0.2x 1,k (x 2,k + x 3,k ) 1 y k = x 1,k + x 2,k x 3,k + v k (37) where the process noise w k and the measurement noise v k are independent of each other and all Gaussian white noises have the following statistics: wk N(0.3, 0.7) v k N(0.5, 1.0) (38) The initial state x 0 is taken to be a Gaussian random variable, which has the following mean and variance: ˆx 0 = 0.7 1 1] T P 0 = I (39) We select κ = 0.0 and time step k = 50 for filtering and smoothing estimation. Firstly, we use the standard EKF and UKF to estimate the state x 1 in (37) for the purpose of comparing with the existing EKS and the proposed nonlinear state smoother based on UT. The corresponding simulation results are given in Figs. 1 and 2, respectively. Moreover, the root mean square error (RMSE) curves of the estimated results using EKF and UKF are given in Fig. 3. As it can be seen from Figs. 1 3, despite the estimation of state x 1 with UKF and EKF fails to track the theoretical value, UKF is still superior to EKF in estimation precision, which can be proved by the fact that the estimation and the corresponding RMSE of UKF are bounded while those of EKF are divergent. This is because the unscented transformation for UKF can approximate the posterior mean and covariance of the nonlinear state with a second order accuracy; on contrary, the linearization technique applied for EKF can only achieve first order accuracy.

No. 7 WANG Xiao-Xu et al.: Application of Unscented Transformation for Nonlinear State Smoothing 1111 Fig. 1 Estimation of the state in UKF and nonlinear smoother basedonut Fig. 2 Fig. 3 Estimation of the state in EKF and EKS RMSE of the state in UKF and EKF Secondly, in order to improve estimation precision of the state x 1, EKS and the proposed nonlinear smoother based on UT in Section 3 are applied to estimating x 1. With a duplicate choice of κ =0.0 andk = 50, the estimation results of the state x 1 smoothed by using EKS and the proposed nonlinear smoother based on UT are shown in Figs. 1 and 2, respectively. Moreover, Fig. 4 gives the RMSE estimation of the state in UKF, EKS, and nonlinear smoother based on UT. Fig. 4 RMSE of the state in UKF, EKS, and nonlinear smoother based on UT In Figs. 1 and 2, the smoothing results follow the theoretical curve of the state x 1 while EKF and UKF are failures. Accordingly, it is obviously seen from Fig. 4 that despite the RMSE curves in three algorithms are all bounded and convergent, the steady errors of EKS and nonlinear smoother based on UT are all far less than that of UKF, which implies that they outperform UKF in precision of estimation. In addition, the steady RMSE of the proposed smoother is close to zero and less than that of EKS, because UT has at least the second order accuracy in capturing the posterior mean and covariance of the nonlinear state while the linearization applied for EKS can only achieve firstorder accuracy. This indicates that the proposed nonlinear smoother in the paper is feasible and effective in dealing with nonlinear state smoothing issue and has higher estimation accuracy than EKS and UKF. 5 Conclusion In this paper, the unified optimal framework for description of the linear or nonlinear state smoothers is derived according to the MMSE estimation theory. The proposed framework is only formal in the sense that it rarely can be directly used in practice since the model nonlinearity results in the intractability and infeasibility of analytically computing the smoothing gain. A novel type of suboptimal nonlinear state smoother is developed through applying the unscented transformation to compute approximately the smoothing gain appearing in the proposed optimal smoothing framework. The performance of the developed nonlinear smoother is demonstrated by a numerical example, and the simulation results show that its estimation accuracy is superior to that of the standard UKF and EKS. References 1 Ito K, Xiong K Q. Gaussian filters for nonlinear filtering problems. IEEE Transactions on Automatic Control, 2000, 45(5): 910 927 2 Sorenson H W. Kalman Filtering: Theory and Application. New York: IEEE Press, 1985 3 Julier S, Uhlmann J, Durrant-Whyte H F. A new method for the nonlinear transformation of means and covariances in filters and estimators. IEEE Transactions on Automatic Control, 2000, 45(3): 477 482 4 Nørgarrd M, Poulsen N K, Ravn O. New developments in state estimation for nonlinear systems. Automatica, 2000, 36(11): 1627 1638 5 Julier S J, Uhlmann J K, Durrant-Whyte H F. A new approach for filtering nonlinear systems. In: Proceedings of American Control Conference. Soallle, Wuhlnglon, USA: IEEE, 1995. 1628 1632 6 Julier S, Uhlmann Jeffrey K. A General Method for Approximating Nonlinear Transformations of Probability Distributions. Technical Report OX1 PJ, Robotics Research Group, Department of Engineering Science, University of Oxford, UK, 1996 7 van der Merwe R, Wan Eric A, Julier S J. Sigma-point Kalman filters for nonlinear estimation and sensor fusion: applications to integrated navigation. In: Proceedings of the AIAA Guidance Navigation and Control Conference. Providence, RI, USA: AIAA, 2004. 1735 1764 8 Cho S Y, Kim B D. Adaptive IIR/FIR fusion filter and its application to the INS/GPS integrated system. Automatica, 2008, 44(8): 2040 2047 9 Blom H A P, Bar-Shalom Y. The interacting multiple model algorithm for systems with Markovian switching coefficients. IEEE Transactions on Automatic Control, 1988, 33(8): 780 783 10 Šimandl M, Duník J. Derivative-free estimation methods: new results and performance analysis. Automatica, 2009, 45(7): 1749 1757 11 Xiong K, Zhang H Y, Chan C W. Performance evaluation of UKF-based nonlinear filtering. Automatica, 2006, 42(2): 261 270

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