# Mid-State Kalman Filter for Nonlinear Problems
###### Abstract  
When tracking very long-range targets, wide-band radars capable of measuring targets with high precision at ranges have severe measurement nonlinearities. The existing nonlinear filtering technology, such as the extended Kalman filter and untracked Kalman filter, will have significant consistency problems and loss in tracking accuracy. A novel mid-state Kalman filter is proposed to avoid loss and preserve the filtering consistency. The observed state and its first-order state derivative are selected as the mid-state vector. The update process is transformed into the measurement space to ensure the Gaussian measurement distribution and the linearization of the measurement equation. In order to verify the filter performance in comparison, an iterative formulation of Cramer-Rao Low Bound for the nonlinear system is further derived and given in this paper. Simulation results show that the proposed method has excellent performance of high filtering accuracy and fast convergence by comparing the filter state estimation accuracy and consistency.  
osniscrete; Kalman filter; nonlinear systems; radar target tracking 20222222130222222130  
through the transformation of the point set [9; 10]. The UKF has less computation and better approximation performance than the EKF. The CKF is based on the spherical-radial criterion and uses a group of cubature points with equal weight to calculate the mean and covariance of state variables [11]. [PERSON] conducts a comparative analysis on the UKF and CKF for low-dimensional and high-dimensional models under nonlinear conditions. The simulation results show that the CKF has the optimal numerical stability and filtering accuracy under high dimensional conditions [12; 13]. A Particle Filter (PF) is not limited by the linearization error or gaussian noise assumption and approximates the probability density function corresponding to the nonlinear function. However, the considerable amount of calculation is unbearable for a real-time target tracking system [14].  
Due to the complexity and uncertainty of very long-range target tracking, the Gaussian noise distribution in the measurement space will become a severe non-Gaussian distribution when it is converted into the state space in the update process. This phenomenon is often encountered in wide-band radar systems with high-range accuracy [15]. Due to the nonlinear distribution of the measurement area, existing nonlinear filters have specific problems in the accuracy of state estimation and the consistency of the filter and even divergence of the filter may occur in the EKF [16]. Because of this, a mid-state Kalman filter (MSKF) is proposed in this paper. A MSKF takes the observed state and its first-order state derivative as the mid-state vector and converts the predicted state covariance into the covariance of the mid-state vector according to the third-order spherical-radial criterion. Then the KF update process is converted to the measurement space and realizes the linearization of the measurement equation. Therefore, information loss in updating will be reduced to a minimum and the MSKF will have a higher accuracy in state estimation. The MSKF was applied to very long-range tracking problems and the simulation results prove the superiority and applicability of the algorithm.