An optimal state and fault estimation scheme is proposed for two-dimensional discrete systems subject to either deterministic disturbances or stochastic disturbances (noises). A direct solution to the deterministic estimation problem is obtained first, based on a well-designed regularized least squares problem with a dynamic constraint of a two-dimensional singular system, augmented from the original system state and unknown disturbance. After proving the solution equivalence between the deterministic scenario and the stochastic one in the sense of optimal state and fault estimation, a unified solution, based on a Riccati-like equation recursion, can be established by weighting parameterization for two-dimensional systems in deterministic and stochastic cases. The unified solution also works as the optimal state observer and generalized Kalman filter for two-dimensional singular systems. Generalization discussions concerning different system descriptions with respect to fault as well as the implementations of the proposed estimator are also presented. Simulation illustrates the effectiveness of the proposed method.

针对具有确定性干扰或随机性干扰（噪声）的二维离散系统，提出了一种最优的状态和故障估计方案。首先，基于精心设计的具有二维奇异系统动态约束的正则化最小二乘问题，直接获得对确定性估计问题的直接解决方案，该问题由原始系统状态和未知扰动增加。从最优状态和故障估计的角度证明确定性方案与随机方案之间的等价解后，可以通过对Riccati式方程递归进行加权，确定和确定二维系统的参数，从而建立统一的解决方案。随机情况。统一的解决方案还可以用作二维奇异系统的最佳状态观测器和广义卡尔曼滤波器。还介绍了有关故障的不同系统描述的一般化讨论以及所提出的估计器的实现。仿真表明了该方法的有效性。