Optimization of a class of linear time-periodic systems: a new approach via transformation to a canonical form

We address optimization of commutative time-periodic linear systems steered by a single input. In this paper, the time-periodic dynamic equations are transformed to a canonical form, referred to as the higher-order form, such that the states and control inputs can be written as higher derivatives of a single variable. This higher-order form is then used to eliminate the dynamic equations explicitly from the optimization problem. It is shown that for a system with n states and a single input, the optimal solution satisfies a 2n order differential equation in a single variable along with 2n boundary conditions on higher derivatives of this variable split between the two end time. This differential equation can be solved numerically in an efficient way using weighted residual methods. It is also shown here that for a set of problems using this method, closed-form solution is possible. It is impossible to achieve these closed form solutions using conventional methods.