Variational integrator based optimal feedback control for constrained mechanical systems

摘要

Today, a lot of mechanical systems have to operate with an improved performance compared to equal constructions decades ago. To stay competitive, engineers need to optimise all aspects of a mechanical system including its feedback control. An approach to minimize the control costs and ensuring a stable deviation control is the Riccati controller. To describe the discrete dynamics, a constrained variational integrator is used, which is a variant of a structure-preserving integration scheme. The desired optimal trajectory and according control input is determined solving the discrete mechanics and optimal control (DMOC) algorithm based on the variational integrator. Then, during time stepping of the perturbed system, the discrete Riccati equations yield the optimal deviation control input. Adding the optimal control input to the deviation control value causes a structure preserving trajectory as both DMOC and Riccati equations are based on the same variational integrator. In this work, criteria for the choice of quadrature rules required to derive the variational integrator are investigated to close a gap of previous works in this field. This procedure is applied to three different coordinate choices, minimal, redundant and nullspace coordinates. Simulation examples show that a stable handling of highlynonlinear systems is assured.With applying the formulas derived in this work, a transformation between different coordinate parametrisations and the corresponding cost matrices leads to the same deviation control and thus to the same system behaviour. Nevertheless, it is in the responsibility of the application engineer to choose appropriate cost-matrices for the occurring perturbations.