Exponential enclosure techniques for initial value problems with multiple conjugate complex eigenvalues

Abstract

The computation of guaranteed state enclosures has a large variety of applications in engineering if initial value problems for sets of ordinary differential equations are concerned. One possible scenario is the use of such state enclosures in the design and verification of linear and nonlinear feedback controllers as well as in predictive control procedures. In many of these applications, system models are characterized by a dominant linear part (commonly after a suitable coordinate transformation) and by a not fully negligible nonlinear part. To compute guaranteed state enclosures for such systems, general purpose approaches relying on a Taylor series expansion of the solution can be employed. However, they do not exploit knowledge about the specific system structure. The exponential state enclosure technique makes use of this structure, allowing users to compute tight enclosures that contract over time for asymptotically stable dynamics. This paper firstly gives an overview of exponential enclosure techniques, implemented in ValEncIA-IVP, and secondly focuses on extensions to dynamic systems with single and multiple conjugate complex eigenvalues.