High Order Discrete-Time Control Based on Gauss-Legendre Collocation

摘要

We present a framework for high order discrete-time implementation of nonlinear control laws, which is based on Gauss-Legendre integration. High order refers to the accuracy of the numerical approximations of the sampled control system model and the desired closed-loop (target) dynamics. The family of symplectic Gauss-Legendre s-stage integration schemes with the highest possible order 2s, which in numerical simulation go along with the conservation of a modified energy, appears as a natural vehicle to translate energy-based control designs like IDA-PBC to discrete time. The work extends our results on the implicit midpoint rule in conjunction with zero order hold towards the multi-stage case, where the input signal is shaped via Lagrange interpolation polynomials. Our method is modular in the sense that the structure of the continuous-time control law remains unaltered, and only predicted stage values must be replaced to generate stage input values for the s - 1 order hold element. Simulations and experiments with nonlinear IDA-PBC control for a magnetic levitation system illustrate the functioning of the approach at low sampling rates and give hints for future developments.