Minimal-order specified-time unknown input observers

This paper presents a framework on minimal-order specified-time unknown input observers for linear systems based on a pairwise observer structure. A minimal-order specified-time observer is first proposed for the linear system without the unknown input, which can estimate the state exactly at the preset time by seeking for the unique solution of a system of linear equations. To further release the computational burden, another form of the specified-time observer is designed. In the presence of the unknown input, it is shown that the existing reduced-order specified-time unknown input observer is not the minimal-order one. By introducing a singular transformation to decouple the effect of the unknown input, the minimal-order specified-time unknown input observer is presented, with the strong* observability capturing the sufficient and necessary condition. The robustness of the proposed observers is also discussed.