A verified method for solving piecewise smooth initial value problems

Abstract

In many applications, there is a need to choose mathematical models that depend on non-smooth functions. The task of simulation becomes especially difficult if such functions appear on the right-hand side of an initial value problem. Moreover, solution processes from usual numerics are sensitive to roundoff errors so that verified analysis might be more useful if a guarantee of correctness is required or if the system model is influenced by uncertainty. In this paper, we provide a short overview of possibilities to formulate non-smooth problems and point out connections between the traditional non-smooth theory and interval analysis. Moreover, we summarize already existing verified methods for solving initial value problems with non-smooth (in fact, even not absolutely continuous) right-hand sides and propose a way of handling a certain practically relevant subclass of such systems. We implement the approach for the solver VALENCIA-IVP by introducing into it a specialized template for enclosing the first-order derivatives of non-smooth functions. We demonstrate the applicability of our technique using a mechanical system model with friction and hysteresis. We conclude the paper by giving a perspective on future research directions in this area.

Set-membership identifiability of nonlinear models and related parameter estimation properties

“…In the last years, there has been quite a lot of emphasis on bounded-error models, as opposed to stochastic ones, for achieving several tasks, e.g., fault diagnosis and fault tolerant control (Puig, 2010;Seybold et al, 2015), robust robot localization (Kieffer et al, 2000), reachability analysis (Auer et al, 2013;Maiga et al, 2016). This has been stressed by the success of operational estimation methods aiming at computing sets guaranteed to contain the feasible parameter/state set, i.e., the set of all the parameter/state vectors consistent Interval analysis has brought a set of tools that indifferently apply to linear and nonlinear systems as opposed to ellipsoidal and zonotope-based estimation methods.…”