On the Delay Bounds of Discrete-time Linear Systems under Delay Independent Truncated Predictor Feedback

Predictor state feedback law achieves stabilization for a discrete-time linear system with input delay by making the closed-loop system free of delay. It is known that the predictor feedback law disposed of the term corresponding to the zero state solution in the state predictor, which is referred to as truncated predictor feedback (TPF), can be designed using low gain feedback technique to stabilize a system that is not exponentially unstable for an arbitrarily large delay. Moreover, when the transition matrix in the truncated predictor feedback is removed, a delay independent TPF law is formulated, and has the ability to stabilize a system for an arbitrarily large delay if all open loop poles of the system are inside the unit circle or at z = 1. This paper first proposes an example to show that a delay independent TPF law is practically not able to achieve stabilization for a sufficiently large delay if the system is not exponentially unstable but has at least one pole on the unit circle at z not equal 1. We then determine a delay bound and the values of the low gain parameter that guarantee the asymptotic stability of the closed-loop system. Generalization of the results to the exponentially unstable system is also considered.