Converse results on existence of sum of squares Lyapunov functions
Despite the pervasiveness of sum of squares (sos) techniques in Lyapunov analysis of dynamical systems, the converse question of whether sos Lyapunov functions exist whenever polynomial Lyapunov functions exist has remained elusive. In this paper, we first show via an explicit counterexample that if...
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| Format: | Article |
| Language: | en_US |
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Institute of Electrical and Electronics Engineers (IEEE)
2012
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| Online Access: | http://hdl.handle.net/1721.1/72483 https://orcid.org/0000-0003-1132-8477 |
| _version_ | 1826211793973805056 |
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| author | Ahmadi, Amir Ali Parrilo, Pablo A. |
| author2 | Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science |
| author_facet | Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science Ahmadi, Amir Ali Parrilo, Pablo A. |
| author_sort | Ahmadi, Amir Ali |
| collection | MIT |
| description | Despite the pervasiveness of sum of squares (sos) techniques in Lyapunov analysis of dynamical systems, the converse question of whether sos Lyapunov functions exist whenever polynomial Lyapunov functions exist has remained elusive. In this paper, we first show via an explicit counterexample that if the degree of the polynomial Lyapunov function is fixed, then sos programming can fail to find a valid Lyapunov function even though one exists. On the other hand, if the degree is allowed to increase, we prove that existence of a polynomial Lyapunov function for a homogeneous polynomial vector field implies existence of a polynomial Lyapunov function that is sos and that the negative of its derivative is also sos. The latter result is extended to develop a converse sos Lyapunov theorem for robust stability of switched linear systems. |
| first_indexed | 2024-09-23T15:11:29Z |
| format | Article |
| id | mit-1721.1/72483 |
| institution | Massachusetts Institute of Technology |
| language | en_US |
| last_indexed | 2024-09-23T15:11:29Z |
| publishDate | 2012 |
| publisher | Institute of Electrical and Electronics Engineers (IEEE) |
| record_format | dspace |
| spelling | mit-1721.1/724832022-09-29T13:16:14Z Converse results on existence of sum of squares Lyapunov functions Ahmadi, Amir Ali Parrilo, Pablo A. Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science Massachusetts Institute of Technology. Laboratory for Information and Decision Systems Parrilo, Pablo A. Ahmadi, Amir Ali Parrilo, Pablo A. Despite the pervasiveness of sum of squares (sos) techniques in Lyapunov analysis of dynamical systems, the converse question of whether sos Lyapunov functions exist whenever polynomial Lyapunov functions exist has remained elusive. In this paper, we first show via an explicit counterexample that if the degree of the polynomial Lyapunov function is fixed, then sos programming can fail to find a valid Lyapunov function even though one exists. On the other hand, if the degree is allowed to increase, we prove that existence of a polynomial Lyapunov function for a homogeneous polynomial vector field implies existence of a polynomial Lyapunov function that is sos and that the negative of its derivative is also sos. The latter result is extended to develop a converse sos Lyapunov theorem for robust stability of switched linear systems. 2012-08-30T18:43:00Z 2012-08-30T18:43:00Z 2012-03 2011-12 Article http://purl.org/eprint/type/ConferencePaper 978-1-61284-799-3 978-1-61284-800-6 http://hdl.handle.net/1721.1/72483 Ahmadi, Amir Ali, and Pablo A. Parrilo. “Converse Results on Existence of Sum of Squares Lyapunov Functions.” 50th IEEE Conference on Decision and Control and European Control Conference 2011 (CDC-ECC). 6516–6521. https://orcid.org/0000-0003-1132-8477 en_US http://dx.doi.org/10.1109/CDC.2011.6161493 50th IEEE Conference on Decision and Control and European Control Conference 2011 (CDC-ECC) Creative Commons Attribution-Noncommercial-Share Alike 3.0 http://creativecommons.org/licenses/by-nc-sa/3.0/ application/pdf Institute of Electrical and Electronics Engineers (IEEE) MIT web domain |
| spellingShingle | Ahmadi, Amir Ali Parrilo, Pablo A. Converse results on existence of sum of squares Lyapunov functions |
| title | Converse results on existence of sum of squares Lyapunov functions |
| title_full | Converse results on existence of sum of squares Lyapunov functions |
| title_fullStr | Converse results on existence of sum of squares Lyapunov functions |
| title_full_unstemmed | Converse results on existence of sum of squares Lyapunov functions |
| title_short | Converse results on existence of sum of squares Lyapunov functions |
| title_sort | converse results on existence of sum of squares lyapunov functions |
| url | http://hdl.handle.net/1721.1/72483 https://orcid.org/0000-0003-1132-8477 |
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