Optimal solution of the two-stage Kalman estimator

The two-stage Kalman estimator was originally proposed to reduce the computational complexity of the augmented state Kalman filter. Recently, it was also applied to the tracking of maneuvering targets by treating the target acceleration as a bias term. Except in certain restrictive conditions, the conventional two-stage estimators are suboptimal in the sense that they are not equivalent to the augmented state filter. In this paper, the authors propose a new two-stage Kalman estimator, i.e., new structure, which is an extension of Friedland’s estimator and is optimal in general conditions. In addition, we provide some analytic results to demonstrate the computational advantages of two-stage estimators over augmented ones. Index Terms—Augmented state Kalman filter, bias-free filter, dynamical bias, optimal filter, two-stage Kalman estimator. I. INTRODUCTION Consider the problem of estimating the state of a dynamic system in the presence of a dynamical bias. It is common to treat the bias as part of the system state and then estimate the bias as well as the system state. This leads to an augmented state Kalman filter (ASKF) whose implementation can be computationally intensive. To reduce the computational cost, Friedland [1] proposed to employ the two-stage Kalman estimator to decouple the augmented filter into two parallel reduced-order filters. In recent years, the computational efficiency of the two-stage estimator is also appreciated when it is used to address the maneuvering target tracking problem, in which the target acceleration is treated as a random bias [14]. While Friedland’s decomposition is optimal for the case of a constant bias, it is suboptimal for a random/dynamical bias unless an algebraic constraint on the statistics of the bias process is satisfied [10], [12]. Since this Manuscript received November 8, 1995; revised June 10, 1996, October 28, 1996, and April 28, 1997. This work was supported in part by the National Science Council of the Republic of China under Grant NSC 83-0404-E-009085. The authors are with the Department of Electrical and Control Engineering, National Chiao Tung University, Hsinchu, Taiwan, R.O.C. (e-mail: [email protected]). Publisher Item Identifier S 0018-9286(99)00667-4. 0018–9286/99$10.00  1999 IEEE IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 44, NO. 1, JANUARY 1999 195 algebraic constraint is seldom satisfied for practical systems, the twostage Kalman estimator cannot exactly implement the ASKF. The motivation for our work is generalization of the two-stage structure to recover the optimal performance when the bias is a random process. Here we review some previous works. After [1], many researchers have also contributed in this area, e.g., Tacker et al. [2], Tanaka [4], Mendel et al. [6], and Ignagni [7]. Recently, Ignagni [8] considered the case of a bias driven by a white noise which is uncorrelated with the system noise. However, the result he obtained is suboptimal. In [12], Alouani et al. considered a random bias in which the bias noise is correlated with the system noise. It was proved that under an algebraic constraint on the correlation between the system noise and the bias noise, the proposed two-stage Kalman estimator is optimal. Since almost all practical systems will not satisfy this algebraic constraint, they also concluded that all two-stage Kalman estimators are suboptimal. In [10], Alouani et al. extended the result of [12] to color noises. The two-stage Kalman estimator is also applied to the maneuvering target tracking problems (e.g., [9], [11], and [14]) and the nonlinear estimation problems (e.g., [3], [5], and [13]). The objectives of this paper are to propose an optimal twostage Kalman estimator (OTSKE) to evaluate its performance and to describe its applications. As shown in [12], the conventional twostage Kalman estimator (CTSKE) is suboptimal unless an algebraic constraint is satisfied. Using the matrix transformation technique, we generalize the CTSKE to obtain the OTSKE, in which the algebraic constraint [12] is removed and the optimal performance is guaranteed. The OTSKE is optimal in the minimum mean square error (MMSE) sense, and this is verified in a theorem by proving that it is equivalent to the ASKF. This paper is organized as follows. In Section II, we state the problem of interest. In Section III, the OTSKE is derived for state estimation in the presence of a dynamical bias without any constraint. Performance and applications of the proposed OTSKE filter are given in Sections IV and V, respectively. Section VI is the conclusion. A detailed proof is provided in the Appendix. II. STATEMENT OF THE PROBLEM The problem of interest is described by the discretized equation set Xk+1 = AkXk +Bk k +W x k (1) k+1 = Ck k +W k (2) Yk = HkXk +Dk k + k (3) where Xk 2 R is the system state, k 2 R is the dynamical bias, and Yk 2 R is the measurement vector. Matrices Ak; Bk; Ck; Dk; and Hk are of appropriate dimensions with the assumption that Ck is nonsingular. The process noises W x k ;W k and the measurement noise k are zero-mean white Gaussian sequences with the following covariances: E[W x k (W x l ) 0] = Q k kl; E[W k (W l ) 0] = Q k kl; E[W x k (W l ) 0] = Q k kl; E[ k( l) 0] = Rk kl; E[W x k ( l) 0] = 0; and E[W k ( l) 0] = 0, where 0 denotes transpose. The initial states X0 and 0 are assumed to be uncorrelated with the white noise sequences W x k ; W k ; and k. The initial conditions are assumed to be Gaussian random variables with E[X0] = X0; CovfX0g = P x 0 ; E[ 0] = 0; Covf 0g = P 0 > 0; and CovfX0 0 0g = P x 0 . Treating Xk and k as the augmented system state, the ASKF is described by X a kjk 1 = Ak 1X a k 1jk 1 (4) X a kjk = X a kjk 1 +Kk Yk HkX a kjk 1 (5) Pkjk 1 = Ak 1Pk 1jk 1 A 0 k 1 +Qk 1 (6) Kk = Pkjk 1 H 0 kf HkPkjk 1 H 0 k +Rkg 1 (7) Pkjk = (I Kk Hk)Pkjk 1 (8) where X a ( ) = [ X( ) ( ) ]; Kk = [ K k K k ] P( ) = CovfX a ( )g = [ P x ( ) (P ( ) ) 0 P x ( ) P ( ) ] Ak=[ Ak 0 Bk Ck ]; Hk = [Hk Dk]; Qk=[ Q k (Q k ) 0 Q x k Q k ]: The computational cost of the ASKF increases with the augmented state dimension. Hence, the filter model (4)–(8) may be impractical to implement. The reason for this computational complexity is the extra computation of P x ( ) terms. Therefore, if these P x ( ) terms can be eliminated, we can reduce the complexity from implementational point of view. In the next section, we propose an optimal two-stage implementation of the above filter without explicitly calculating these P x ( ) terms. III. DERIVATION OF THE OPTIMAL TWO-STAGE KALMAN ESTIMATOR The design of a new two-stage estimator is described as follows. First, form a modified bias-free filter by ignoring the bias term and by adding an external bias-compensating input. Second, take the bias into account and derive a bias filter to compensate the modified biasfree filter in order to reconstruct the original filter. These two filters are used to build a new algorithm which is equivalent to the ASKF. This new algorithm is named the OTSKE. If the bias term is ignored ( = 0), the bias-free filter is just a Kalman filter based on the model (1) and (3). Hence, the bias-free filter is given by Xkjk 1 = Ak 1 Xk 1jk 1 (9) Xkjk = Xkjk 1 + K x k (Yk Hk Xkjk 1) (10) P x kjk 1 = Ak 1 P x k 1jk 1A 0 k 1 +Q x k 1 (11) K k = P x kjk 1H 0 k Hk P x kjk 1H 0 k +Rk 1 (12) P x kjk = I K x kHk P x kjk 1 (13) where Xkjk represents the estimate of the state process when the bias is ignored and P x kjk is the error covariance of Xkjk. Accounting for the bias noise effect, we modify the bias-free filter by changing the predicted state and covariance equations, i.e., (9) and (11), into Xkjk 1 = Ak 1 Xk 1jk 1 + uk 1 (14) P x kjk 1 = Ak 1 P x k 1jk 1A 0 k 1 + Q x k 1 (15) where uk, a new external input, and Q k , a new statistic for W x k , are yet to be determined. To distinguish this modified filter from Friedland’s bias-free filter, the new filter [(14), (10), (15), (12), and (13)] is called the modified bias-free filter. The modified bias-free filter X can be corrected by adding a bias filter, denoted by f ; K ; P g, to reconstruct the original filter. This creates the OTSKE filter which will later be presented as a linear combination of the estimates of the modified bias-free filter and the bias filter. The bias filter is derived in the following. First, we propose the following two-stage U -V transformation: X a kjk 1 = T (Uk) X a kjk 1 (16) X a kjk = T (Vk) X a kjk (17) Pkjk 1 = T (Uk) Pkjk 1T (Uk) (18) Kk = T (Vk) Kk (19) Pkjk = T (Vk) PkjkT (Vk) (20) 196 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 44, NO. 1, JANUARY 1999