Human balancing tasks are modelled by differential equations and are compared to experimental observations. First, the classical inverted pendulum model is revisited with respect to stabilizability. Namely, the relation between the reaction time delay and the shortest pendulum length (critical length) of the stick is derived and is demonstrated experimentally. Conclusions are drawn related to human tests, such as stick balancing on the fingertip, balancing a linearly driven inverted pendulum and virtual stick balancing.Second, the ball and beam balancing is considered, where the task is to stabilize a rolling ball at the mid-point of a beam by manipulating the angular position of the beam. Assuming a delayed proportional-derivative feedback mechanism, the governing equation is delay-differential equation. Performance of the control system is analyzed in terms of overshoot and settling time. Experiments over 5-days trials shows that control parameters are tuned to the optimal point associated with minimal overshoot and the shortest settling time. Finally, some further balancing tasks are briefly demonstrated and discussed.
Strong stability preserving (SSP) time integrators have been developed to preserve nonlinear stability properties (e.g., monotonicity, boundedness) of the numerical solution in arbitrary norms or convex functionals, when coupled with suitable spatial discretizations. Currently, all existing general linear methods (including Runge-Kutta and linear multistep methods) either attain small step sizes for nonlinear stability, or they are only first order accurate. In order to obtain larger step sizes discretizations of PDEs that contain both upwind- and downwind-biased operators have been employed.
In this talk, we review SSP Runge-Kutta methods that use upwind- and downwind-biased discretizations in the framework of perturbed Runge-Kutta methods. We show how downwinding improves the SSP properties of time-stepping methods and breaks some order barriers. In particular, we focus on implicit perturbed SSP Runge-Kutta methods that have unbounded SSP coefficient. We present a second- and third-order one-parameter family of perturbed Runge-Kutta methods, for which the CFL-like step-size restriction can be chosen arbitrarily large. The stability of this family of methods is analyzed, and we demonstrate that the desired order of accuracy is obtained for large CFL numbers. Finally, we investigate the computational challenges of the implicit problem and propose ideas that lead to an efficient implementation of Newton's method.