Numerical solution of nonlinear periodic optimal control problems using a Fourier integral pseudospectral method

Many real-world systems exhibit cyclical behavior and nonlinear dynamics. Optimal control theory provides a framework for determining the best periodic control strategies for such systems. These strategies achieve the desired goals while minimizing the costs, energy use, or other relevant metrics. This study addresses this challenge by introducing the Fourier integral pseudospectral (FIPS) method. This method is applicable to a general class of nonlinear periodic process control problems with equality and/or inequality constraints, assuming sufficiently smooth solutions. The FIPS method performs collocation of the problem's integral form at an equidistant set of nodes. Furthermore, it utilizes highly accurate Fourier integration matrices (FIMs) to approximate all necessary integrals. This approach transforms the original problem into a nonlinear programming problem (NLP) with algebraic constraints. We employed a direct numerical optimization method to solve this NLP effectively. This study establishes rigorous convergence properties and derives error estimates for the Fourier series, interpolants, and quadratures employed within the context of process control applications, focusing on smooth and continuous periodic functions. Finally, the accuracy and efficiency of the FIPS method are demonstrated through two illustrative nonlinear process-control problems.