Reach-Set Estimation for DAE Systems under Uncertainty and Disturbances Using Trajectory Sensitivity and Logarithmic Norm

Abstract

Trajectory sensitivity analysis is useful for analyzing the dynamic behaviour of differential-algebraic equation (DAE) systems under uncertain initial conditions and/or parameters. However, the approximate trajectories obtained using trajectory sensitivities are not accompanied by explicit error bounds. In this paper, we provide an efficient method to obtain a numerical error bound for the first-order trajectory approximation. This approach uses second-order trajectory sensitivities. A theoretical result quantifying the excursion of trajectories induced by uncertain initial conditions and external disturbances is derived based on the logarithmic norm, and is extended to DAE systems. Although this result itself provides a guaranteed over-approximation of the reach-set of nonlinear DAE systems, by combining this result with the efficient bound obtained from trajectory sensitivities, we are able to provide a much less conservative reach-set estimate for systems under uncertain initial conditions and/or parameters, and external disturbances.