| Title:
|
Rank-one LMI approach to robust stability of polynomial matrices (English) |
| Author:
|
Henrion, Didier |
| Author:
|
Sugimoto, Kenji |
| Author:
|
Šebek, Michael |
| Language:
|
English |
| Journal:
|
Kybernetika |
| ISSN:
|
0023-5954 |
| Volume:
|
38 |
| Issue:
|
5 |
| Year:
|
2002 |
| Pages:
|
[643]-656 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Necessary and sufficient conditions are formulated for checking robust stability of an uncertain polynomial matrix. Various stability regions and uncertainty models are handled in a unified way. The conditions, stemming from a general optimization methodology similar to the one used in $\mu $-analysis, are expressed as a rank-one LMI, a non-convex problem frequently arising in robust control. Convex relaxations of the problem yield tractable sufficient LMI conditions for robust stability of uncertain polynomial matrices. (English) |
| Keyword:
|
linear matrix inequality |
| Keyword:
|
stability |
| MSC:
|
15A39 |
| MSC:
|
93D09 |
| idZBL:
|
Zbl 1265.93194 |
| idMR:
|
MR1966952 |
| . |
| Date available:
|
2009-09-24T19:49:35Z |
| Last updated:
|
2015-03-25 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/135493 |
| . |
| Reference:
|
[1] Apkarian P., Tuan H. D.: A sequential SDP/Gauss–Newton algorithm for rank-constrained LMI problems.In: Proc. IEEE Conference on Decision and Control, Phoenix 1999, pp. 2328–2333 |
| Reference:
|
[2] Barmish B. R.: New Tools for Robustness of Linear Systems.Macmillan, New York 1994 Zbl 1094.93517 |
| Reference:
|
[3] Bhattacharyya S. P., Chapellat, H., Keel L. H.: Robust Control: The Parametric Approach.Prentice Hall, Upper Saddle River, N.J. 1995 Zbl 0838.93008 |
| Reference:
|
[4] Blondel V. D., Tsitsiklis J. N.: A survey of computational complexity results in systems and control.Automatica 36 (2000), 9, 1249–1274 Zbl 0989.93006, MR 1834719, 10.1016/S0005-1098(00)00050-9 |
| Reference:
|
[5] Boyd S., Ghaoui L. El, Feron, E., Balakrishnan V.: Linear Matrix Inequalities in System and Control Theory.SIAM Studies in Applied Mathematics, Philadelphia 1994 Zbl 0816.93004, MR 1284712 |
| Reference:
|
[6] Brixius N., Sheng, R., Potra F. A.: sdpHA: a Matlab implementation of homogeneous interior-point algorithms for semidefinite programming.Optimization Methods and Software 11/12 (1999), 583–596 Zbl 0973.90525, MR 1778430, 10.1080/10556789908805763 |
| Reference:
|
[7] Ghaoui L. El, Oustry, F., Rami M. Ait: A cone complementarity linearization algorithm for static output feedback and related problems.IEEE Trans. Automat. Control 42 (1997), 8, 1171–1176 MR 1469081 |
| Reference:
|
[8] Ghaoui L. El, Commeau J. L.: Lmitool 2.0 Package: An Interface to Solve LMI Problems. E-Letters on Systems, Control and Signal Processing, Issue 125, January 1999 |
| Reference:
|
[9] Ghaoui L. El, (eds.) S. I. Niculescu: Advances in Linear Matrix Inequality Methods in Control.SIAM Advances in Control and Design, Philadelphia, 1999 Zbl 0932.00034, MR 1736563 |
| Reference:
|
[10] Fan M., Tits, A., Doyle J.: Robustness in the presence of joint parametric uncertainty and unmodeled dynamics.IEEE Trans. Automat. Control 36 (1991), 1, 25–38 MR 1084243, 10.1109/9.62265 |
| Reference:
|
[11] Geromel J. C., Peres P. L. D., Bernussou J.: On a convex parameter space method for linear control design of uncertain systems.SIAM J. Control Optim. 29 (1991), 381–402 Zbl 0741.93020, MR 1092734, 10.1137/0329021 |
| Reference:
|
[12] Gupta S.: Robust stability analysis using LMI: Beyond small gain and passivity.Internat. J. Robust Nonlinear Control 6 (1996), 953–968 MR 1429435, 10.1002/(SICI)1099-1239(199611)6:9/10<953::AID-RNC261>3.0.CO;2-L |
| Reference:
|
[13] Haddad W. M., Bernstein D. S.: Explicit construction of quadratic Lyapunov functions for the small gain, positivity, circle and Popov theorems and their applications to robust stability.Part I: Continuous-time theory. Internat. J. Robust Nonlinear Control 3 (1993), 313–339 |
| Reference:
|
[14] Henrion D., Tarbouriech, S., Šebek M.: Rank-one LMI approach to simultaneous stabilization of linear systems.Systems Control Lett. 38 (1999), 2, 79–89 Zbl 1043.93545, MR 1751684, 10.1016/S0167-6911(99)00049-3 |
| Reference:
|
[15] Henrion D., Bachelier, O., Šebek M.: $\mathcal{D}$-stability of polynomial matrices.Internat. J. Control 74 (2001), 8, 845–856 MR 1832952 |
| Reference:
|
[16] Henrion D., Šebek, M., Bachelier O.: Rank-one LMI approach to stability of 2-D polynomial matrices.Multidimensional Systems and Signal Processing 12 (2001), 1, 33–48 Zbl 0976.93043, MR 1818911 |
| Reference:
|
[17] Henrion D., Arzelier D., Peaucelle, D., Šebek M.: An LMI condition for robust stability of polynomial matrix polytopes.Automatica 37 (2001), 3, 461–468 Zbl 0982.93057, MR 1843990, 10.1016/S0005-1098(00)00170-9 |
| Reference:
|
[18] Iwasaki T., Hara S.: Well-posedness of feedback systems: Insights into exact robustness analysis and approximate computations.IEEE Trans. Automat. Control 43 (1998), 5, 619–630 Zbl 0927.93038, MR 1618048, 10.1109/9.668829 |
| Reference:
|
[19] Karl W. C., Verghese G. C.: A sufficient condition for the stability of interval matrix polynomials.IEEE Trans. Automat. Control 38 (1993), 7, 1139–1143 Zbl 0800.93954, MR 1235240, 10.1109/9.231473 |
| Reference:
|
[20] Kučera V.: Discrete Linear Control: The Polynomial Approach.Wiley, Chichester 1979 MR 0573447 |
| Reference:
|
[21] Packard A., Doyle J.: The complex singular value.Automatica 29 (1993), 1, 71–109 MR 1200542, 10.1016/0005-1098(93)90175-S |
| Reference:
|
[22] Peaucelle D., Arzelier D.: New LMI-based conditions for robust ${\mathcal{H}}_2$ performance analysis: In Proc.American Control Conference, Chicago 2000, pp. 317–321 |
| Reference:
|
[23] Polyx, Inc.: Polynomial Toolbox for Matlab, Release 2.0.0, 1999. See the web page www.polyx.cz |
| Reference:
|
[24] Scherer C.: A full-block $\mathcal{S}$-procedure with applications: In: Proc.IEEE Conference on Decision and Control, San Diego 1997, pp. 2602–2607 |
| Reference:
|
[25] Scorletti G., Ghaoui L. El: Improved LMI conditions for gain scheduling and related control problems: Internat.J. Robust Nonlinear Control 8 (1998), 845–877 MR 1639959, 10.1002/(SICI)1099-1239(199808)8:10<845::AID-RNC350>3.0.CO;2-I |
| Reference:
|
[26] Shim D.: Quadratic stability in the circle theorem or positivity theorem.Internat. J. Robust Nonlinear Control 6 (1996), 781–788 Zbl 0864.93084, MR 1416914, 10.1002/(SICI)1099-1239(199610)6:8<781::AID-RNC189>3.0.CO;2-K |
| Reference:
|
[27] Vandenberghe L., Boyd S.: Semidefinite programming.SIAM Rev. 38 (1996), 49–95 Zbl 0845.65023, MR 1379041, 10.1137/1038003 |
| Reference:
|
[28] Willems J. C.: Paradigms and puzzles in the theory of dynamical systems.IEEE Trans. Automat. Control 36 (1991), 259–294 Zbl 0737.93004, MR 1092818, 10.1109/9.73561 |
| Reference:
|
[29] Zhou K., Doyle, J., Glover K.: Robust and Optimal Control.Prentice Hall, Upper Saddle River, N.J. 1996 Zbl 0999.49500 |
| . |
