An approach to stabilize linear systems with state and input delay

A sliding mode control scheme has been designed to ensure the asymptotic stability of a linear system with delay in both the input and state variables. A key step is the use of a transformation that eliminates the input delay. The resulting state-delay system is controlled through a sliding mode approach, where the equivalent control term uses state feedback and the discontinuous control term involves the sign and value of the switching function. It is shown that the sliding mode is reached in finite time. Stability conditions are derived using a Lyapunov technique combined with a Razumikhin approach to bound the difference between the original and the transformed states. The result is a set of sufficient conditions for asymptotic stability. The theoretical developments are illustrated via an example.