Ensemble Consider Kalman Filtering* Tai-shan Lou, Nan-hua Chen, Hua Xiong, Ya-xi Li, and Lei Wang Abstract—For the nonlinear systems, the ensemble Kalman filtering[7], desensitized Kalman filtering[8], set-valued filter can avoid using the Jacobian matrices and reduce the estimation[9] and consider Kalman filtering (also called computational complexity. However, the state estimates still Schmidt-Kalman filtering)[6, 10, 11]. A consider method is suffer greatly negative effects from uncertain parameters of the proposed by Schmidt to account for the parameter dynamic and measurement models. To mitigate the negative uncertainties by incorporating the covariance of the effects, an ensemble consider Kalman filter (EnCKF) is designed parameters into the Kalman filtering formulations. by using the “consider” approach and resampling the ensemble members in each step to incorporate the statistics of the To overcome the drawbacks of the EnKF coming from the uncertain parameters into the state estimation formulations. The unknown parameters, this paper proposes an ensemble effectiveness of the proposed EnCKF is verified by two consider Kalman filter (EnCKF) for the nonlinear dynamic numerical simulations. systems with uncertain parameters. The covariance matrices between the uncertain parameters and the states and the I. INTRODUCTION measurements are computed and propagated by using the Nonlinear Kalman filtering plays an important role in ensemble integration in the filtering algorithm. The formula information and communication systems, control systems, and of the EnCKF are derived by using augmented-state many other areas, such as target tracking, navigation of methods[12] in Section Ⅱ. Two numerical simulations are aerospace vehicles[1, 2], fault diagnosis[3], chemical plant shown in Section Ⅲ. control, signal processing and fusion of multi-sensor data[4, 5]. A lot of Kalman filtering algorithms have been proposed to II. ENSEMBLE CONSIDER KALMAN FILTERING different engineering problems, such as extended Kalman For the nonlinear dynamic system model, the EnKF gives filter, central differential Kalman filter, unscented Kalman a suboptimal solution of the Fokker-Planck equation by using filter, cubature Kalman filter and ensemble Kalman filter an ensemble integration to approximate the error statistics. (EnKF)[6]. The unscented Kalman filter and cubature Kalman Here, based on the consider method and the EnKF, the EnCKF filter belong to the deterministic sampling filter algorithm, in method is presented to consider the uncertain parameters in the which the sigma points are generated deterministically on the dynamic models. state and covariance matrix[6]. The EnKF belongs to a general class of known particle algorithm, in which an ensemble is Consider a nonlinear discrete dynamic system model with used to represent for the probability distribution functions uncertain parameters and additive noises, in which its state (PDFs), the time-update PDFs and the posterior PDF of the equation is given by measurements are modeled by the stochastic models of the x f x, b w , (1) ensemble integration, respectively. The EnKF is widely used k k11 k in the nonlinear models with the extremely high order, high and measurement equation is given by uncertainty of the initial states and a large number of observations[7]. zk h x k, b v k (2) For the above nonlinear Kalman filtering, an underlying n p assumption is that the dynamic and measurement equations where xk is the state vector at time step k , and zk can be accurately modeled without any unknown parameters is the measurement vector, functions f and h are respectively or biases. However, in practice, it is always difficulty to obtain the nonlinear dynamic and the measurement equations, b l the accurately parameter values, and sometimes the parameters is referred to as the uncertain parameter vector, and wv, are are time-varying[8]. Neglecting the uncertainties of the kk parameters may have unexpected state estimate errors and assumed to be zero-mean Gaussian white noise with even lead to diverge. Many methods have been proposed to covariance matrices Qk and Rk , respectively. Moreover, solve these uncertain model parameters, such as H wvkk, are assumed to be uncorrelated. *Resrach supported by the National Nature Science Foundation of China Nan-hua Chen is with School of Electrical and Information Engineering, under grant #61603346, the Key Science and Technology Program of Henan Zhengzhou University of Light Industry, Zhengzhou, 450002 China (e-mail: Province under grant #182102110014, the Fundamental Research Programs [email protected]). for the Provincial Universities under grant #15KYYWF01, and the Key Hua Xiong is with Beijing Institute of Electronic System Engineering, research projects of Henan higher education institutions under grant Beijing 100854 China (e-mail: [email protected]). #18A413003. Ya-xi Li is with School of Electrical and Information Engineering, Tai-shan Lou is with School of Electrical and Information Engineering, Zhengzhou University of Light Industry, Zhengzhou, 450002 China (e-mail: Zhengzhou University of Light Industry, Zhengzhou, 450002 China [email protected]). (corresponding author: 86-370-6355-6790; e-mail: [email protected], Lei Wang is with School of Automation Science and Electrical [email protected]). Engineering, Beihang University, Beijing, 100083 China. (e-mail: lwang @buaa.edu.cn). In this work, the uncertain parameters are modeled as a x n m ensemble error matrix Mk is the difference between constant vector with a priori known statistic. Here, the the ensemble members and the ensemble mean, and the reference values and the covariance are assumed to be b and b q m ensemble Mk is the difference between ensemble Qb . i ˆ error matrix of bk and bk , which is replaced by the constant Following the consider approach, the estimated state x k b of the uncertain parameter in consider approach to reduce and the uncertain parameters b are augmented into a state the computation . X nl , which is described by k Estimating the predicted covariance matrix PXX, k , which T Xk [,] x k b k (3) is given by the estimated and consider components, and is defined as and its covariance are defined by PP PPxx xb xx,, k xb k P (4) PXX, k XX PP PPbx bb bx,, k bb k To propagate the error distribution in the predictive step, 1 XXT MMkk m forecasted state estimates with sample random errors are m 1 x generated as in an ensemble at time step k 1. The ensemble TT (13) 1 Mk xb f nm b MMkk χk 1 R is defined as m 1 Mk χ fm XXX12,,, (5) x xTT x b k1 k 1 k 1 k 1 1 MMMMk k k k TT where the superscript i denote the i-th forecast ensemble m 1 MMMMb x b b k k k k member, the augmented ensemble member is i i i T from which it can be seen that Xk1[ x k 1 , b k 1 ] ,im 1,2, , . T The propagated ensemble members are 1 xx PMMxx, k k k m 1 i i f ()X k 1 w i k 1 xbT X k (6) PMM (14) ˆi 0 xb, k k k bk 1 m 1 T 1 bb Estimating the priori state Xˆ by PMMbb, k k k k m 1 m i xˆk 1 The measurement ensemble members Z (im 1,2, , ) XXˆ i (7) k kkˆ are computed from the augmented state ensemble members b m i1 k i Xk (im 1,2, , ) : and the ensemble error matrix M X () n l m , which is the k i i i i i Zk h()(,) X k h x k b k (15) difference between the true state Xk (im 1,2, , ) and the ˆ Estimating the priori measurement zˆ by ensemble mean X k , is defined by k x m 1 i XmMk 1 ˆ ˆ ˆ (8) zZkk (16) MXXXXkb k k,, k k m i1 Mk z p m From Eqs. (7) and (8), it can be seen that and the measurement error matrix Mk R is defined by m 1 zm1 ˆ i M[,,] Z zˆ Z zˆ (17) xxkk (9) k k k k k m i1 The innovations covariance Pzz, k and the cross-covariance m ˆ 1 i P of the augmented measurement and state are computed bbkk (10) Xz, k m i1 by using the state and measurement ensemble members xm1 M x xˆ ,, x xˆ (11) T k k k k k 1 zz PMMzz, k k k (18) m 1 Mbm b1 bˆ,, b bˆ (12) k k k k k 1 T PMM Xz (19) n Xz, k k k where xˆk is the ensemble mean of m state xk , m 1 ˆ l bk is the ensemble mean of m parameter bk . The The measurement and state cross covariance P in Eq. Xz, k PPbb,, k bb k Qb (34) (19) is defined by the estimated and considered terms where the last formulation of Eq. (30) is obtained by the x P 1 M T substituting Eq.(25) into the second formulation of the Eq.(27). PMxz, k k z (20) Xz, kP b k bz, k m 1Mk Lastly, the posteriori estimate and the posteriori parameter can be given by it can be seen that 1 m ˆ ˆ i 1 xzT xxkk (35) PMMxz, k k k (21) m i1 m 1 ˆˆ bbkk (36) 1 bzT PMMbz, k k k (22) m 1 Note that the standard EnKF gain matrix can be recovered by setting Q 0 . Then, the augmented Kalman gain is defined as: b Following above formulations, the ensemble consider K P x,, k11 xz k Kalman filter is summarized by the following equations, Kk PPP Xz,, k zz k zz (23) Kb,, k P bz k which include two parts: it can be seen that Time update: 1 i i i Kx,,, k PP xz k zz k (24) xk f (,) xˆ k11 b k 1 m 1 ˆ i Kb,,, k PP bz k zz k (25) xxkk m i1 (37) Then, the augmented posteriori estimate is 1 xxT PMMxx, k k k ˆ iˆ i i i m 1 Xk X k K k() z k Z k (26) 1 xbT PMMxb, k k k and the augmented posteriori covariance matrix is m 1 PPxx,, k xb k Measurement update: P XX, k PP bx,, k bb k m (27) 1 i TT zZˆ kk PPPPxxk, xbk ,KKKK xk , zz, k xk , xkzzkbk , , , m i1 PPPP KKKKTT bxk,,,,,,,, bbk bkzzk xk bkzzk bk 1 zzT PMMzz, k k k i m 1 where the perturbed measurements zk are given by (38) 1 xzT ii PMMxz, k k k zk h(,) x k b v k (28) m 1 T i 1 bz in which the perturbation variable vkkNR(0, ) .