Simultaneous input and state estimation with multi-step delay for linear stochastic systems based on infinity filtering and smoothing

Abstract

This paper presents a novel method for simultaneous unknown input and state estimation (UISE) with multi-step delay for linear stochastic systems with rank-deficient direct feed-through matrix. The augmented Kalman filter and smoothing are established with an input model, where the input signal follows a specific probability distribution with finite covariance. Then, in light of the established properties of matrix limiting, the infinity filtering is deduced as the covariance matrix of the input signal tends to infinity. Moreover, the infinity smoothing algorithm is also derived to obtain the multi-step delayed unknown input and state estimation. Furthermore, the infinity filtering is strictly proved to coincide with the existing five-step recursive filter (FSRF). It is shown that some previous UISE algorithms, such as the unbiased minimum-variance (UMV) filter and recursive three-step filter (RTSF), are special cases of the proposed infinity filtering algorithm. This work reveals that the existing UMV, RTSF, and FSRF algorithms are essentially the particular limit of augmented Kalman filter and smoothing, which gives a new insight into the area of UISE.