Maximum delay bounds of linear systems under delay independent truncated predictor feedback

Abstract

In a predictor feedback law for a linear system with input delay, the future state is predicted as the state solution of the linear system. The zero input solution contains the transition matrix. The zero state solution gives rise to the distributed nature of the feedback law. In a 2007 IEEE TAC paper, it is established that, when the system is not exponentially unstable, low gain feedback can be designed such that the predictor feedback law, with the distributed term truncated, still achieves stabilization for an arbitrarily large delay. Furthermore, in the absence of purely imaginary poles, the transition matrix in the truncated predictor feedback (TPF) can be safely dropped, resulting in a delay independent TPF law, which is simply a delay independent linear state feedback. In this paper, we first construct an example to show that, in the presence of purely imaginary poles, the linear delay independent TPF in general cannot stabilize the system for an arbitrarily large delay. By using the Lyapunov–Krasovskii Stability Theorem, we derive a bound on the delay under which the delay independent truncated predictor feedback law achieves stabilization for a general system that may be exponentially unstable.