Cubic Hermite Interpolation and Lobatto Collocation for Nonlinear Sampled-Data Control

Abstract

Nonlinear controls designed in continuous time suffer from an increasing model mismatch and performance degradation when implemented in sampled-data control loops at relatively low sampling rates. The model mismatch is reduced when higher order numerical integration is used as modeling basis to describe the open-loop and the desired closed-loop dynamics. We recently presented a modular and minimally invasive way to implement continuously designed nonlinear controls with high accuracy in discrete time based on Gauss-Legendre collocation. In this paper, we show how the main drawback of the latter approach, highly discontinuous control signals, is removed by using piecewise cubic spline interpolation, or equivalently, Lobatto collocation. We present the rationale behind our approach, set up the nonlinear systems of equations for the required one-step predictions of the target dynamics, and discuss several relations between Hermite, Lobatto and Gauss collocation. On a benchmark nonlinear simulation example, we illustrate and discuss the performance of the approach with increasing sampling times.