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This paper develops a state space decomposition method for a class of discrete-time singular systems. The singular matrix in this system is allowed to vary not only in the value but also in the dimension. An orthogonal factorization is used to rewrite the original space as a direct sum of its two subspaces. Correspondingly, the singular system is replaced by two reduced-order systems, whose states are then calculated respectively. Based on this construction, the prediction of part of the state, optimal filtering result, an optimal one step smoothing result is proposed. Lastly, a numerical example is presented to demonstrate the effectiveness of the new method.

A filter based on characteristic functions is developed in this paper, to fit to a class of non-Gaussian dynamical systems, which state models and measurement models are all nonlinear and multi-dimensional. The new filter overcomes limitations and expands the application of this kind of filter, which is proved to just fit to one special kind of systems with multi-dimensional linear state models and one-dimensional nonlinear measurement models. Firstly, the filter using characteristic function is introduced and its limitation is analysed. Then, we design the new filter to fit to nonlinear states and multi-dimensional measurements. Thirdly, the matrix format of performance index is presented to match to the new filter gain, and the weighting function vector is given to ensure the uniform boundedness of such a performance index. Finally, the new filter gain can be obtained by minimizing this performance index, and the process of filtering design is accomplished. Simulation examples are given to illustrate the effectiveness of the proposed filter design scheme.

This paper designs a novel filter based on characteristic function for multidimensional observation systems, essentially extending the proposed filter, which just fitted to one-dimensional observations. For the dynamic model from grinding process cylindrical workpieces, this filter could result in enhanced and incremental productivity and quality control in manufacturing processes. In the processing of the filter design, a new form of filter will be given to adapt to the multidimensional observations, the matrix format of performance index will be designed to fit to matrix format of filter gain, the selecting range of the weighting function vector will be given to ensure the uniform boundedness of the designed performance index, and the filter gain can be obtained by minimizing the performance index. Finally, we illustrate the effectiveness of the proposed method by simulation examples in the field of the target tracking and the grinding process cylindrical workpieces.