Analytical and Veri fied Numerical Results Concerning Interval Continuous-time Algebraic Riccati Equations

Authors

1 Department of Applied Mathematics, Faculty of Mathematics and Computer, Shahid Bahonar University of Kerman, Kerman, Iran.

2 Department of Applied Mathematics, Faculty of Mathematics and Computer, Shahid Bahonar University of Kerman, Kerman, Iran

3 Department of Applied Mathematics, Faculty of Mathematics and Computer, Shahid Bahonar University of Kerman, Kerman, Iran. Member of Young Researchers Society of Shahid Bahonar University of Kerman, Kerman, Iran.

Abstract

This paper focuses on studying the interval continuous-time algebraic Riccati equation A∗X + XA + Q − XGX = 0, both from the theoretical aspects and the computational ones. In theoretical parts, we show that Shary’s results for interval linear systems can only be partially generalized to this interval Riccati matrix equation. We then derive an efficient technique for enclosing the united stable solution set based on a modified variant of the Krawczyk method which enables us to reduce the computational complexity, significantly. Various numerical experiments are also given to show the efficiency of proposed scheme.

Keywords

References

1. Karl J. A strom and P.R. Kumar. Control. Automatica, 50(1):3-43, January 2014. Available from: http://dx.doi.org/10.1016/j.automatica.2013.10.012, doi:10.1016/j. automatica.2013.10.012.
2. H. Abou-Kandil, G. Freiling, V. Ionescu, and G. Jank. Matrix Riccati Equations in Control and Systems Theory. Systems & Control: Foundations & Applications. Birkhauser Basel, 2003. Available from: https://books.google.com/booksid=DmyfOTn_D9kC.
3. E Adams and U Kulisch. Scientific computing with automatic result verification. i. language and programming support for verified scientific computation, ii. enclosure methods and algorithms with automatic result verification, iii. applications in the engineering sciences. Academic Press, San Diego, 1993.
4. Gotz Alefeld and Jurgen Herzberger. Introduction to interval computations. Computer Science and Applied Mathematics. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1983. Translated from the German by Jon Rokne.
5. Gtz Alefeld and Gnter Mayer. Interval analysis: theory and applications. Journal of Computational and Applied Mathematics, 121(12):421 - 464, 2000. Available from: http://www.sciencedirect.com/science/article/pii/S0377042700003423, doi:http://dx.doi.org/10.1016/S0377-0427(00)00342-3.
6. P. Benner, A. Laub, and V. Mehrmann. A collection of benchmark examples for the numerical solution of algebraic Riccati equations I: the continuous-time case. Technical Report SPC 95-22, Forschergruppe `Scientific Parallel Computing', Fakultat furMathematik, TU Chemnitz Zwickau, 1995. Version dated February 28, 1996.
7. D.A. Bini, B. Iannazzo, and B. Meini. Numerical Solution of Algebraic Riccati Equations. Fundamentals of Algorithms. Society for Industrial and Applied Mathematics, 2012. Available from: https://books.google.ro/booksid=Gi1jZAz3pncC.
8. S. Bittanti, A.J. Laub, and J.C. Willems. The Riccati Equation. Communications and Control Engineering. Springer Berlin Heidelberg, 2012. Available from: https: //books.google.com/booksid=6SzvCAAAQBAJ.
9. Marzieh Dehghani-Madiseh and Milan Hladik. Efficient approaches for enclosing the united solution set of the interval generalized sylvester matrix equation. CoRR, abs/1510.04853, 2015. Available from: http://arxiv.org/abs/1510.04853.
10. Andreas Frommer and Behnam Hashemi. Verified computation of square roots of a matrix. SIAM J. Matrix Anal. Appl., 31(3):1279{1302, 2009. Available from: http://dx.doi.org/10.1137/090757058, doi:10.1137/090757058.
11. Alexandre Goldsztejn. A right-preconditioning process for the formal-algebraic approach to inner and outer estimation of ae-solution sets. Reliable Computing, 11(6):443-478, 2005. Available from: http://dblp.uni-trier.de/db/journals/rc/rc11.html#Goldsztejn05.
 
Global Analysis and Discrete Mathematics
Volume 2, Issue 1
May 2017
Pages 55-74

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History

  • Receive Date: 23 September 2016
  • Revise Date: 04 July 2017
  • Accept Date: 08 March 2017

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