Numerical Methods for Control

Feedback controls that are designed based on continuous-time models, can be implemented in sampled control loops with high sampling frequency without relevant loss of accuracy and stability margin (emulation or quasi-continuous controller). For lower sampling frequencies (higher sampling times) the increasing mismatch between the underlying model and the neglected sample and hold mechanism leads to deterioration of the closed-loop behavior up to destabilization. A remedy is the use of higher order integration schemes in combination with appropriate input signal shaping to represent the system model and the target closed-loop dynamics. Using the implicit midpoint is the simplest way to re-use continuous-time control laws with a minimally invasive modification of the control structure - a block that generates the half step prediction of the closed-loop dynamics (Kotyczka, Thoma, 2021). The approach can be easily extended to the higher order, multi-stage setting with Gauss-Legendre collocation (Kotyczka, Martens, Lefèvre, 2021). The videos below show the improvement in the application to the trajectory control of a two-link flexible-joint robot at DLR, Oberpfaffenhofen. The emulation controller with high controlled stiffness produces unsatisfactory behavior at a sampling time of 8 ms (left video). The improvement when using the control implementation based on the implicit midpoint rule (right video) is remarkable. See (Thoma, Wu, Dietrich, Kotyczka, 2021) for details.

A lecture given at CIRM in April 2022 on the definition of discrete-time port-Hamiltonian systems using symplectic integration (Kotyczka and Lefèvre, 2019) and its use for control is available here: