Chebyshev Pseudospectral Method for Nonlinear Stabilization using Control Contraction Metrics

Real-time implementation of the control contraction metric (CCM) method for nonlinear stabilization involves computation of a shortest path (a geodesic) between pairs of states. In this paper we propose the use of a direct numerical method, namely a Chebyshev pseudospectral method, to compute a geodesic. We investigate the influence of various parameters and tolerances and provide practical recommendations. We also compare the proposed method to a multipleshooting algorithm (ACADO), and show that the proposed method is fast and accurate. An adaptive algorithm for finding a suitable degree and number of nodes is given. With an example nonlinear system, a controller is constructed using control contraction metrics and is compared against a linear controller and a controller calculated from nonlinear model predictive control, again using ACADO.