Numerical Methods For Optimal Control Of Switched Systems

The challenges in real-life applications, like e.g., managing systems with load fluctuations, start-up, and shut-down, represent complex mathematical problems. This complexity stems from strong nonlinearities (especially in transients), mixed-integer decision variables and controls (e.g., for coupling components), state-dependent discontinuities (from phase transitions or controllers), and the large system dimension. While industry often relies on decoupling and recipe-based controls, these prove insufficient for such intricate, coupled systems, highlighting a need for innovative nonlinear optimization methods. For processes under uncertainty, static open-loop controls are inadequate; optimal feedback control laws, dependent on estimated states, are preferred. The presently most popular approach for general nonlinear optimal control problems with state and control constraints is Nonlinear Model Predictive Control (NMPC). The main idea is to estimate the present state from measured data on a finite "moving" time horizon of the past and to optimize the control on a "moving" time horizon in an open-loop. The first instant of the control is then applied during a sampling time interval, during which the next re-optimization is computed. This dissertation develops numerical methods for computing open-loop and feedback controls in certain classes of mixed-integer optimal control problems with switched ODEs (SwOCP), which exhibit important applications to characterize the complex properties of dry friction problems. We follow Filippov's rule, according to which the SwOCP is reformulated to an optimal control problem with mixed integer controls and special mixed control-state constraints. We investigate the relaxed formulation of this optimal control problem and derive necessary optimality conditions from the Pontryagin maximum principle (PMP), where the regularity property of the mixed constraints is carefully considered. Numerical methods for the relaxed problem based on the multiple shooting approach and an appropriate "rounding scheme" to handle implicit switching are investigated. In order to compute optimal feedback control laws, we generalize the "NMPC" approach to the general SwOCP class above. We develop a direct approach to derive feedback control laws. It is based on the PMP approach to computing "neighbouring feedback" controls to find out the explicit switching of integer controls. The numerical methods are illustrated with benchmark problems via the MUSCOD-II tool software with PGPLOT or MATLAB.