Fixed-Time Stabilization of Linear Delay Systems by Smooth Periodic Delayed Feedback

This article studies fixed-time stabilization (FxTS) of a general controllable linear system with an input delay
$\tau$
. It is shown that such a problem is not solvable if the prescribed convergence time
$T_{\tau }$
is smaller than
$2\tau$
. For
$T_{\tau }\geq 3\tau$
, a solution based on linear periodic delayed feedback (PDF) without any distributed delay is established. For
$T_{\tau }>2\tau$
, a solution based on linear predictor-based PDF containing a distributed delay is proposed. For both cases, the gains of the PDF can be chosen as continuous, continuously differentiable, and even smooth, in the sense of infinitely many times differentiable. If only an output signal is available for feedback, two classes of linear observers with periodic coefficients are designed, so that their states converge to the current and future states of the system at a prescribed finite time, respectively. With the observed current and future states, FxTS can also be achieved by using, respectively, the PDF and observer-based PDF. A linear periodic feedback (without delay) is also established to solve the FxTS problem of linear systems with both instantaneous and delayed controls, which cannot be stabilized by any constant instantaneous feedback in certain cases. Two numerical examples verify the effectiveness of the proposed approaches.