Input delay compensation of linear systems with both state and input delays by adding integrators

This paper studies stabilization of linear systems with both state and input delays. A dynamic input-delay compensator obtained by adding integrators is established to compensate the input delays that can be arbitrarily large. With the input delay compensator, the original stabilization problem reduces to the problem of stabilizing an augmented linear time-delay system without input delay. Three methods are also proposed to design stabilizing controllers for the augmented linear time-delay system. The first method is based on linear matrix inequalities (LMIs) and the second method is based on model reduction. The third method is based on pole placement and is built for the particular case that the original time-delay system has only a pure delayed state vector on its right hand side. For this method, the optimal gain such that the decay rate of the closed-loop system is maximized is also proposed. The effectiveness of the proposed approaches is illustrated by three linear time-delay systems that are open-loop unstable.