The classical recursive three-step filter can be used to estimate the state and unknown input when the system is affected by unknown input, but the recursive three-step filter cannot be applied when the unknown input distribution matrix is not of full column rank. In order to solve the above problem, this paper proposes two novel filters according to the linear minimum-variance unbiased estimation criterion. Firstly, while the unknown input distribution matrix in the output equation is not of full column rank, a novel recursive three-step filter with direct feedthrough was proposed. Then, a novel recursive three-step filter was developed when the unknown input distribution matrix in the system equation is not of full column rank. Finally, the specific recursive steps of the corresponding filters are summarized. And the simulation results show that the proposed filters can effectively estimate the system state and unknown input.

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By constructing Green's function, we give the natural formulae of solutions for the following nonlinear impulsive fractional differential equation with generalized periodic boundary value conditions:where 1 < q < 2 is a real number, c D q t is the standard Caputo differentiation. We present the properties of Green's function. Some sufficient conditions for the existence of single or multiple positive solutions of the above nonlinear fractional differential equation are established. Our analysis relies on a nonlinear alternative of the Schauder and Guo-Krasnosel'skii fixed point theorem on cones. As applications, some interesting examples are provided to illustrate the main results.

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