Neighbouring optimal feedback law for linear time-delayed dynamical systems

Abstract

One way to obtain a neighbouring feedback law for a time-delayed optimal control problem is to first transform it into a ‘standard’ optimisation problem. That is one without terms having a time-delayed argument. To this end, I use a Padé approximation to determine a differential relation for y(t), an augmented state that represents x(t−τ). The time-delayed optimisation problem can then be rewritten in terms of an augmented state vector consisting of both the physical state x(t) and the delayed state y(t). Once reformulated, one may use to advantage existing well-developed techniques such as the backward sweep method to obtain the neighbouring feedback law. Results obtained from two examples show good agreement between the exact results and those predicted by the feedback law for small variations in both the initial condition and a system parameter.