Observer-Based Control of Linear Systems with Mismatched Input and Output Delays

This paper investigates the stabilization of linear systems subject to simultaneous, mismatched time delays in both the control input and system output vectors. The proposed control framework is developed in two primary stages. First, an asymptotically stabilizing delayed state-feedback controller is synthesized by leveraging recent advancements in Linear Matrix Inequality (LMI) techniques. Second, this controller is realized using novel time-delay compensators \cite{trinhnam26}. This architecture successfully accommodates an output measurement delay $\tau_y$ that is independent of the input delay $\tau_u$, enabling direct estimation of the delayed state-feedback control law. The proposed methodology is then extended to target output controllers to account for simultaneous, mismatched time delays in both the control input and system output vectors.