Exact State Estimation for Linear Systems with Unbounded Disturbances

A finite-time observer is designed for linear invariant systems in the presence of unknown inputs or disturbances with unavailable upper bound. The main condition for designing the observer is strongly detectability. Geometric control theory is used to decompose the given system into two parts: strongly observable subsystem and strongly detectable subsystem. Through a series of transformations, the former can be partitioned into two parts: affected and unaffected by unknown inputs (UI-free and UI-dependent), and then the states can be exactly estimated via time-delayed observer in pre-defined time. The nonstrongly observable subsystem can be observed asymptotically. Without the upper bound of disturbances, the observer ensures that the convergence time can be set arbitrarily. A numerical example illustrates the effective of the proposed estimation schemes.