Adaptive Lyapunov redesign for a class of linear systems with delays in the state

A new adaptive control strategy based on barrier functions for a class of time-delay systems is proposed. The adaptive algorithm is at the joint of Lyapunov redesign and Sliding Mode Control: it requires the knowledge of a Lyapunov functional, and the construction of a sliding manifold. Through the adaptation of unit control gain, the proposed strategy does not require any a priori knowledge on the upper bound of perturbation and allows assigning an ultimate bound of the closed-loop solution despite the delay. This is achieved by driving the systems solutions within a predefined neighborhood of the sliding set in a predefined time accounting for the delay, and then ensuring a stable motion within the same neighborhood where the unit control gain is following the norm of the perturbation.