Multiple-segment fourier-based approach for linear quadratic optimal control

A Fourier-based state parameterization approach for determining the near optimal trajectories of linear time-invariant dynamic systems with quadratic performance indices has been developed. The necessary condition of optimality is derived as a system of linear algebraic equations in terms of free boundary values and Fourier coefficients. In contrast to earlier work in which the state trajectory was represented by a single segment Fourier-type approximation, here the use of multiple segment approximations is developed. Simulation results show that the single segment Fourier-based approach is faster than standard transition-matrix and Riccati-based approaches. The multiple segment Fourier-based approach is numerically more robust than the transition-matrix approach and computationally more efficient than Riccati-based approaches in solving linear quadratic optimal control problems. Furthermore, compared to the single segment Fourier-based approach, the multiple segment implementation improves the accuracy of the near optimal solution for highly responsive dynamic systems.