Regulators for optimal periodic processes

A feedback controller called a periodic regulator is developed for implementing an optimal periodic process to a physical system under state deviations and constant parameter variation. The objective of the periodic regulator is to bring the perturbed plant back to a neighboring optimal periodic path in such a manner that the infinite-time second-variation of the cost is minimized. The transition matrix associated with the variational problem evaluated over one period is symplectic and generically possesses two unity eigenvalues coupled in the same Jordan box; the primary eigenvector associated with the unity eigenvalue corresponds to the orbital velocity direction. It is shown that there exists a locally stable subspace in the neighborhood of the periodic orbit. Moreover, a simple condition is given for defining the state variation which guarantees that it is contained in this stable subspace. A controller including feedback of the parameter variation is derived which provides first-order convergence to a new optimal periodic trajectory. Finally, The periodic regulator developed is applied to a simple two-state periodic optimal control problem to demonstrate its convergence properties