Piece-wise analytic trajectory computation for polytopic switching between stable affine systems

Abstract

Our problem is to compute trajectories of a hybrid system that switches between stable affine ODEs, with switching triggered by hyperplane crossings. Instead of integrating over relatively short time steps, we propose to analytically calculate the affine ODE trajectories between switching times. Our algorithm computes the switching times themselves by Chebyshev interpolation of the analytic trajectory pieces, and polynomial root finding. We shrink the interpolation time intervals using bounds on the times needed by the affine ODE trajectories to enter certain Lyapunov sub-level sets. Based on the Chebfun package, we give a MATLAB implementation of our algorithm. We find that this implementation simulates Relay feedback systems as accurately and sometimes faster than conventional algorithms.