An n–th Order Generalization of the Activity Measure for Continuous Systems

¹ Department of Computer Science, FCEN, Universidad de Buenos Aires and CONICET, Argentina
² Department of Environmental Systems Sciences, ETH Zurich, Switzerland
³ Department of Control, FCEIA, Universidad Nacional de Rosario and CIFASIS-CONICET, Argentina

^a e-mail: rcastro@dc.uba.ar
^b e-mail: kofman@cifasis-conicet.gov.ar

Abstract

In this work we present an analytical expression that generalizes the definition of activity measure in continuous time signals. We define the activity of order n and show that it allows to estimate the number of sections of polinomials up to order n that are needed to represent that signal with certain accuracy. We apply this concept to obtain a lower bound for the number of steps performed by quantization–based integration algorithms in the simulation of ordinary differential equations. We performed a practical analysis over a first order example system, computing the activity of order n and comparing it with the number of steps required integration methods of different orders. We corroborated the theoretical predictions, which indicate that the activity measure can be used as a reference for assessing the suitability of different algorithms depending on how close they perform in comparison with the theoretical lower bound. Finally, a discussion is provided which indicates that further research is needed in order to test the results presented in this work in the context of stiff systems.