| Title:
|
On some relaxations commonly used in the study of linear systems (English) |
| Author:
|
Bachelier, Olivier |
| Author:
|
Mehdi, Driss |
| Language:
|
English |
| Journal:
|
Kybernetika |
| ISSN:
|
0023-5954 (print) |
| ISSN:
|
1805-949X (online) |
| Volume:
|
51 |
| Issue:
|
5 |
| Year:
|
2015 |
| Pages:
|
830-855 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
This note proposes a quite general mathematical proposition which can be a starting point to prove many well-known results encountered while studying the theory of linear systems through matrix inequalities, including the S-procedure, the projection lemma and few others. Moreover, the problem of robustness with respect to several parameter uncertainties is revisited owing to this new theorem, leading to LMI (Linear Matrix Inequality)-based conditions for robust stability or performance analysis with respect to ILFR (Implicit Linear Fractional Representation)-based parametric uncertainty. These conditions, though conservative, are computationally very tractable and make a good compromise between conservatism and engineering applicability. (English) |
| Keyword:
|
LMI relaxations |
| Keyword:
|
robust analysis |
| Keyword:
|
parametric uncertainty |
| MSC:
|
93C05 |
| MSC:
|
93C35 |
| MSC:
|
93D09 |
| idZBL:
|
Zbl 06537783 |
| idMR:
|
MR3445987 |
| DOI:
|
10.14736/kyb-2015-5-0830 |
| . |
| Date available:
|
2015-12-16T19:05:09Z |
| Last updated:
|
2018-01-10 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/144746 |
| . |
| Reference:
|
[1] Apkarian, P., Gahinet, P.: A linear matrix inequality approach to ${H}_{\infty}$ control..Int. J. Robust Nonlinear Control 4 (1994), 421-448. MR 1286148, 10.1002/rnc.4590040403 |
| Reference:
|
[2] Arzelier, D., Peaucelle, D., Sahli, S.: Robust static output feedback stabilization for polytopic uncertain systems..In: Robust Control Design, ROCOND, Milan 2003. |
| Reference:
|
[3] Bachelier, O., Henrion, D., Pradin, B., Mehdi, D.: Robust matrix root-clustering of a matrix in intersections or unions of subregions..SIAM J. Control Optim. 43 (2004), 3, 1078-1093. MR 2114390, 10.1137/s0363012903432365 |
| Reference:
|
[4] Bachelier, O., Mehdi, D.: Robust matrix root-clustering through extended KYP Lemma..SIAM J. Control Optim. 45 (2006), 1, 368-381. MR 2225310, 10.1137/s036301290444349x |
| Reference:
|
[5] Bosche, J., Bachelier, O., Mehdi, D.: An approach for robust matrix root-clustering analysis in a union of regions..IMA J. Math. Control Inform. 22 (2005), 227-239. Zbl 1108.93063, MR 2160204, 10.1093/imamci/dni007 |
| Reference:
|
[6] Boyd, S., Ghaoui, L. El, Féron, E., Balakrishnan, V.: Linear matrix inequalities in system and control zheory..SIAM Studies in Applied Mathematics 15 (1994). MR 1284712, 10.1137/1.9781611970777 |
| Reference:
|
[7] Chesi, G., Garulli, A., Tesi, A., Vicino, A.: Polynomially parameter-dependent Lyapunov functions for robust stability of polytopic systems: An LMI approach..IEEE Trans. Automat. Control 50 (2005), 3, 365-370. MR 2123097, 10.1109/tac.2005.843848 |
| Reference:
|
[8] Chilali, M., Gahinet, P.: ${H}_{\infty}$ design with pole placement constraints: An LMI approach..IEEE Trans. Automat. Control 41 (1996), 3, 358-367. MR 1382985, 10.1109/9.486637 |
| Reference:
|
[9] Oliveira, M. C. de, Bernussou, J., Geromel, J. C.: A new discrete-time robust stability condition..Systems Control Lett. 37 (1999), 4, 261-265 Zbl 0948.93058, MR 1751256, 10.1016/S0167-6911(99)00035-3 |
| Reference:
|
[10] Oliveira, M. C. de, Geromel, J. C., Bernussou, J.: Robust filtering of discrete-time linear systems with parameter dependent Lyapunov functions..SIAM J. Control Optim. 41 (2002), 3, 700-711. Zbl 1022.93048, MR 1939867, 10.1137/s0363012999366308 |
| Reference:
|
[11] Dinh, M., Scorletti, G., Fromion, V., Magarotto, E.: Parameter dependent {H}$_{\infty}$ control by finite dimensional LMI optimization: Application to trade-off dependent control..Int. J. Robust and Nonlinear Control 15 (2005), 383-406. Zbl 1100.93016, MR 2139464, 10.1002/rnc.994 |
| Reference:
|
[12] Ebihara, Y., Peaucelle, D., Arzelier, D.: S-variable Approach to LMI-based Robust Control..Communications and Control Engineering Series, Springer, London 2014. Zbl 1314.93005, MR 3244313, 10.1007/978-1-4471-6606-1 |
| Reference:
|
[13] Ebihara, Y., Hagiwara, T.: A dilated LMI approach to robust performances analysis of linear time-invariant uncertain systems..Automatica 41 (2005), 11, 1933-1941. MR 2168657, 10.1016/j.automatica.2005.05.023 |
| Reference:
|
[14] Ebihara, Y., Hagiwara, T.: On the degree of polynomial parameter-dependent Lyapunov function for the robust stability of single parameter-dependent LTI systems: A counter-example to Barmish's conjecture..Automatica 42 (2006), 1599-1603. MR 2246853, 10.1016/j.automatica.2006.04.011 |
| Reference:
|
[15] Feng, Y., Yagoubi, M., Chevrel, P.: Dilated LMI characterizations for linear time-invariant singular systems..Int. J. Control 83 (2010), 11, 2276-2284. MR 2747292, 10.1080/00207179.2010.514947 |
| Reference:
|
[16] Finsler, P.: Über das Vorkommen definiter und semidefiniter Formen in Scharen quadratischer Formen: Comment..Math. Helv. 9 (1937), 188-192. MR 1509554, 10.1007/bf01258188 |
| Reference:
|
[17] Geromel, J. C., Oliveira, M. C. de, Hsu, L.: LMI characterization of structural and robust stability..Linear Algebra Appl. 285 (1998), 69-80. Zbl 0949.93063, MR 1653487, 10.1016/s0024-3795(98)10123-4 |
| Reference:
|
[18] Graham, M. R, Oliveira, M. C. de: Robust analysis with respect to real parameter uncertainty..In: Proc. 47th IEEE Conference on Decision and Control (CDC), Cancun 2008. 10.1109/cdc.2008.4739449 |
| Reference:
|
[19] Graham, M. R., Oliveira, M. C. de: Linear matrix inequality tests for frequency domain inequalities with affine multipliers..Automatica 46 (2010), 897-901. Zbl 1191.93106, MR 2877163, 10.1016/j.automatica.2010.02.009 |
| Reference:
|
[20] Graham, M. R, Oliveira, M. C. de, Callafon, R. A.: Frequency domain conditions via linear matrix inequalities..In: Proc. 46th IEEE Conference on Decision and Control, New Orleans 2007. 10.1109/cdc.2007.4434854 |
| Reference:
|
[21] Graham, M. R., Oliveira, M. C. de, Callafon, R. A.: An alternative Kalman-Yakubovich-Popov lemma and some extensions..Automatica 45 (2009), 6, 1489-1496. Zbl 1166.93007, MR 2879454, 10.1016/j.automatica.2009.02.006 |
| Reference:
|
[22] Gusev, S. V., Likhtarnikov, A. L.: Kalman-Popov-Yakubovich lemma and the S-procedure: A historical essay..Automation Remote Control 67 (2006), 11, 1768-1810. Zbl 1195.93002, MR 2275456, 10.1134/s000511790611004x |
| Reference:
|
[23] Hecker, S., Varga, A.: Generalized LFT-based representation of parametric models..Europ. J. Control, 10 (2004), 4, 326-337. MR 2120424, 10.3166/ejc.10.326-337 |
| Reference:
|
[24] Hill, R. D.: Eigenvalue location using certain matrix functions and geometric curves..Linear Algebra Appl. 16 (1977), 83-91. Zbl 0448.15009, MR 0466177, 10.1016/0024-3795(77)90022-2 |
| Reference:
|
[25] Iwasaki, T., Hara, S.: Generalized KYP lemma: Unified frequency domain inequalities with design applications..IEEE Trans. Automat. Control 50 (2005), 1, 41-59. MR 2110808, 10.1109/tac.2004.840475 |
| Reference:
|
[26] Iwasaki, T., Meinsma, G., Fu, M.: Generalized S-procedure and finite frequency KYP lemma..Math. Problems Engrg. 6 (2000), 305-320. Zbl 1056.93596, MR 2055916, 10.1155/s1024123x00001368 |
| Reference:
|
[27] Lasserre, J.-B.: Moments, Positive Polynomials and Their Applications..Imperial College, 2009. Zbl 1211.90007, MR 2589247, 10.1142/p665 |
| Reference:
|
[28] Lyapunov, A.: Problème général de la stabilité du mouvement..Annales de la Faculté de Sciences de Toulouse 1907, Translated into French from the original Russian text, Kharkov 1892. 10.5802/afst.246 |
| Reference:
|
[29] Manceaux-Cumer, C., Chrétien, J.-P.: Minimal LFT form of a spacecraft built up from two bodies..In: Proc. AIAA Guidance, Navigation, and Control Conference, Montréal 2001. 10.2514/6.2001-4350 |
| Reference:
|
[30] Ostrowski, A., Schneider, H.: Some theorems on the inertia of general matrices..J. Math. Anal. Appl. 4 (1962), 72-84. Zbl 0112.01401, MR 0142555, 10.1016/0022-247x(62)90030-6 |
| Reference:
|
[31] Peaucelle, D.: Quadratic separation for uncertain descriptor system analysis, strict LMI conditions..In: Proc. 46th IEEE Conference on Decision and Control (CDC), New Orleans 2007. 10.1109/cdc.2007.4434238 |
| Reference:
|
[32] Peaucelle, D., Arzelier, D., Bachelier, O., Bernussou, J.: A new robust D-stability condition for real convex polytopic uncertainty..Systems Control Lett. 40 (2000), 1, 21-30. Zbl 0977.93067, MR 1829071, 10.1016/s0167-6911(99)00119-x |
| Reference:
|
[33] Peaucelle, D., Arzelier, D., Henrion, D., Gouaisbault, F.: Quadratic separation for feedback connection of an uncertain matrix and an implicit linear transformation..Automatica 43 (2007), 796-804. MR 2306727, 10.1016/j.automatica.2006.11.005 |
| Reference:
|
[34] Pipeleers, G., Demeulenaere, B., Swevers, J., Vandenberghe, L.: Extended LMI characterizations for stability and performance of linear systems..Systems Control Lett. 58 (2009), 7, 510-518. Zbl 1166.93014, MR 2536426, 10.1016/j.sysconle.2009.03.001 |
| Reference:
|
[35] Rantzer, A.: On the Kalman-Yakubovich-Popov lemma..Systems Control Lett. 28 (1996), 7-10. Zbl 0866.93052, MR 1393670, 10.1016/0167-6911(95)00063-1 |
| Reference:
|
[36] Sari, B., Bachelier, O., Mehdi, D.: Improved robust $D_U$-stability measures via S-procedure..In: Proc. American Control Conference (ACC), Seattle 2008. 10.1109/acc.2008.4587253 |
| Reference:
|
[37] Scherer, C. W.: A full block S-procedure with applications..In: Proc. 36th Conference on Decision Control (CDC), San Diego 1997. 10.1109/cdc.1997.657686 |
| Reference:
|
[38] Scherer, C. W.: LPV control and full block multipliers..Automatica 37 (2001), 361-375. Zbl 0982.93060, MR 1844000, 10.1016/s0005-1098(00)00176-x |
| Reference:
|
[39] Skelton, R. E., Iwasaki, T., Grigoriadis, K.: A Unified Approach to Linear Control Design..Taylor and Francis series in Systems and Control, 1997. |
| Reference:
|
[40] Stein, P.: Some general theorems on iterants..J. Res. National Bureau of Standards 48 (1952), 82-83. MR 0047001, 10.6028/jres.048.010 |
| Reference:
|
[41] Yakubovich, V. A.: S-procedure in nonlinear control theory..Vestnik Leningrad Univ. 1 (1971), 62-77. Zbl 0232.93010, MR 0297429 |
| . |
