Stabilization of exponentially unstable discrete-time linear systems by truncated predictor feedback

Predictor state feedback solves the problem of stabilizing a discrete-time linear system with input delay by predicting the future state with the solution of the state equation and thus rendering the closed-loop system free of delay. The solution of the state equation contains a term that is the convolution of the past control input with the state transition matrix. Thus, the implementation of the resulting predictor state feedback law involves iterative calculation of the control signal. A truncated predictor feedback law results when the convolution term in the state prediction is discarded. When the feedback gain is constructed from the solution of a certain parameterized Lyapunov equation, the truncated predictor feedback law has been shown to achieve asymptotic stabilization of a system that is not exponentially unstable in the presence of an arbitrarily large delay by tuning the value of the parameter small enough. In this paper, we extend this result to exponentially unstable systems. Stability analysis leads to a bound on the delay and a range of the values of the parameter for which the closed-loop system is asymptotically stable as long as the delay is within the bound. The corresponding output feedback result is also derived.