For an implicitly defined discrete system, a new algorithm for Kalman filtering is developed and an efficient numerical implementation scheme is proposed. Unlike the traditional explicit approach, the implicit filter can be readily applied to ill-conditioned systems and allows for generalization to descriptor systems. The implementation of the implicit filter depends on the solution of the congruence matrix equation (A1)(Px)(AT1) = Py. We develop a general iterative method for the solution of this equation, and prove necessary and sufficient conditions for convergence. It is shown that when the system matrices of an implicit system are sparse, the implicit Kalman filter requires significantly less computer time and storage to implement as compared to the traditional explicit Kalman filter. Simulation results are presented to illustrate and substantiate the theoretical developments.

中文翻译：

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隐式卡尔曼滤波

针对隐式定义的离散系统，开发了一种新的卡尔曼滤波算法，并提出了一种有效的数值实现方案。与传统的显式方法不同，隐式过滤器可以很容易地应用于病态系统，并允许泛化到描述符系统。隐式滤波器的实现取决于同余矩阵方程 (A1)(Px)(AT1) = Py 的解。我们开发了求解该方程的通用迭代方法，并证明了收敛的充分必要条件。结果表明，当隐式系统的系统矩阵稀疏时，与传统的显式卡尔曼滤波器相比，隐式卡尔曼滤波器需要明显更少的计算机时间和存储来实现。