Praktische methodes voor benaderend robuuste periodische optimalecontrole van niet-lineare mechanische systemen

This thesis investigates efficient formulations and methods to solve robust periodic optimal control problems. We assume that systems are essentially nonlinear over the broad range of state space spanned by a limit cycle, requiring nonlinear programming techniques, yet behave as time varying linear systems along that limit cycle. The existing Lyapunov framework is taken as a given. This framework introduces the periodic Lyapunov differential equations into the optimal control problem. Using an interpretation of these continuous equations as covariance propagation, the framework allows one to robustify path constraints in a first-order approximation with respect to Gaussian disturbances. The framework was improved in this thesis in terms of formulation, discretisation accuracy, computational complexity, and structure exploitation as to allow larger scale applications. The main formulation proposed in this work is based on a separate treatment of Lyapunov states such that the discrete periodic Lyapunov equations (DPLE) arise, which come with a guarantee on preservation of positive definiteness. The computational complexity of the resulting nonlinear program is reduced by eliminating the Lyapunov states and using a dedicated DPLE solver based on the periodic Schur decomposition. This solver is implemented as infinitely-differentiable component in the modular open-source optimal control framework CasADi: it is embeddable into a symbolic computational graph. This formulation is found to be highly beneficial for large scale applications with limited time-horizon. In addition to open-loop trajectory/control design, time-variant linear feedback control design is demonstrated with the method. Applications include an automated process to select the safest modes of operation of a carousel device for launching an airborne wind-energy system, and a benchmark application for time-optimal periodic quadcopter flight.nrpages: 276status: publishe