A Memoryless Delay-Adaptive Feedback Law for the Regulation of Discrete-Time Linear Systems

It is challenging to regulate a linear system with input delay when the delay information is absent. A solution is proposed in this paper to regulate the class of discrete-time linear systems whose open loop poles are located inside or on the unit circle. With the delay below an established upper bound, a memoryless delay-adaptive feedback design achieves the convergence of the system state and input to zero as time approaches infinity. In the event that the system has all its open loop poles strictly inside the unit circle or at $z=1$, regulation is achieved for any, arbitrarily large, bounded delay without resorting to any knowledge of the delay. The proposed delay-adaptive control design utilizes only the state at the current time instant for feedback, resulting in a memoryless feedback law convenient for implementation.