Exponential Enclosure Techniques for Initial Value Problems with Multiple Conjugate Complex Eigenvalues
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.
KeywordsOrdinary differential equations Initial value problems Complex interval arithmetic ValEncIA-IVP
- 1.Auer, E., Rauh, A., Hofer, E.P., Luther, W.: Validated modeling of mechanical systems with SmartMOBILE: improvement of performance by ValEncIA-IVP. In: Hertling, P., Hoffmann, C.M., Luther, W., Revol, N. (eds.) Real Number Algorithms. LNCS, vol. 5045, pp. 1–27. Springer, Heidelberg (2008)CrossRefGoogle Scholar
- 5.Lin, Y., Stadtherr, M.A.: Validated solution of initial value problems for odes with interval parameters. In: NSF Workshop Proceeding on Reliable Engineering Computing. Savannah GA, February 22–24 2006Google Scholar
- 6.Nedialkov, N.S.: Computing Rigorous Bounds on the Solution of an Initial Value Problem for an Ordinary Differential Equation. Ph.D. thesis, Graduate Department of Computer Science, University of Toronto (1999)Google Scholar
- 9.Rauh, A., Auer, E., Hofer, E.P.: ValEncIA-IVP: a comparison with other initial value problem solvers. In: CD-Proceedings of 12th GAMM-IMACS Intenational Symposium on Scientific Computing, Computer Arithmetic, and Validated Numerics SCAN 2006. IEEE Computer Society, Duisburg, Germany (2007)Google Scholar
- 11.Rauh, A., Westphal, R., Aschemann, H.: Verified simulation of control systems with interval parameters using an exponential state enclosure technique. In: CD-Proceedings of IEEE International Conference on Methods and Models in Automation and Robotics MMAR. Miedzyzdroje, Poland (2013)Google Scholar
- 12.Rauh, A., Westphal, R., Auer, E., Aschemann, H.: Exponential enclosure techniques for the computation of guaranteed state enclosures in ValEncIA-IVP. In: Proceedings of 15th GAMM-IMACS International Symposium on Scientific Computing, Computer Arithmetic, and Validated Numerics SCAN 2012, vol. 19(1), pp. 66–90. Novosibirsk, Russia, Special Issue of Reliable Computing (2013)Google Scholar
Open Access This chapter is licensed under the terms of the Creative Commons Attribution-NonCommercial 2.5 International License (http://creativecommons.org/licenses/by-nc/2.5/), which permits any noncommercial use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.
The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.