Article

 Title: Robust decentralized $H_2$ control of multi-channel descriptor systems with norm-bounded parametric uncertainties (English) Author: Gui, Weihua Author: Chen, Ning Author: Zhai, Guisheng Language: English Journal: Kybernetika ISSN: 0023-5954 Volume: 45 Issue: 1 Year: 2009 Pages: 49-62 Summary lang: English . Category: math . Summary: This paper considers a robust decentralized $H_2$ control problem for multi-channel descriptor systems. The uncertainties are assumed to be time-invariant, norm-bounded, and exist in both the system and control input matrices. Our interest is focused on dynamic output feedback. A necessary and sufficient condition for an uncertain multi-channel descriptor system to be robustly stabilizable with a specified $H_2$ norm is derived in terms of a strict nonlinear matrix inequality (NMI), that is, an NMI with no equality constraint. A two-stage homotopy method is proposed to solve the NMI iteratively. First, a decentralized controller for the nominal descriptor system is computed by imposing block-diagonal constraints on the coefficient matrices of the controller gradually. Then, the decentralized controller is modified, again gradually, to cope with the uncertainties. On each stage, groups of variables are fixed alternately at the iterations to reduce the NMI to linear matrix inequalities (LMIs). An example is given to show the efficiency of this method. (English) Keyword: decentralized control Keyword: descriptor systems Keyword: parametric uncertainty Keyword: homotopy method Keyword: nonlinear matrix inequality MSC: 93A14 MSC: 93B40 MSC: 93B51 MSC: 93B52 MSC: 93C15 idZBL: Zbl 1158.93304 idMR: MR2489580 . Date available: 2010-06-02T18:18:44Z Last updated: 2012-06-06 Stable URL: http://hdl.handle.net/10338.dmlcz/140005 . Reference: [1] J. H. Alva: The feasibility of continuation methods for nonlinear equations.SIAM J. Numer. Anal. 11 (1974), 1, 102–120. MR 0418448 Reference: [2] T. N. Chang and E. J. Davison: Decentralized control of descriptor systems.IEEE Trans. Automat. Control 46 (2001), 1589–1595. MR 1858060 Reference: [3] N. Chen, M. Ikeda, and W. Gui: Robust decentralized $H_\infty$ control of interconnected systems: A design method using homotopy.In: Preprints 10th IFAC/IFORS/IMACS/IFIP Symposium on Large Scale Systems: Theory and Applications, Osaka 2004, pp. 330–336. Reference: [4] L. Dai: Singular Control Systems.Springer-Verlag, Berlin 1989. Zbl 0669.93034, MR 0986970 Reference: [5] M. Ikeda, T. W. Lee, and E. Uezato: A strict LMI condition for $H_2$ control of descriptor systems.In: Proc. 39th IEEE Conference on Decision and Control 2000, pp. 601–604. Reference: [6] M. Ikeda, G. Zhai, and E. Uezato: Centralized design of decentralized stabilizing controllers for large-scale descriptor systems.DCDIS Journal, Series B: Algorithms and Applications 11 (2004), 509–520. MR 2071224 Reference: [7] I. Masubuchi, Y. Kamitane, A. Ohara, and N. Suda: $H_\infty$ control for descriptor systems: A matrix inequalities approach.Automatica 33 (1997), 669–673. MR 1448959 Reference: [8] I. R. Petersen: A stabilization algorithm for a class of uncertain linear systems.Systems Control Lett. 8 (1987), 351–357. Zbl 0648.93042, MR 0884885 Reference: [9] K. Takaba: $H_2$ output feedback control for descriptor systems.Automatica 34 (1998), 7, 841–850. Zbl 0961.93012, MR 1635088 Reference: [10] D. Wang and P. Bao: Robust impulse of uncertain singular systems by decentralized output feedback.IEEE Trans. Automat. Control 45 (2000), 500–505. MR 1762865 Reference: [11] D. Wang and C. B. Soh: On regularizing singular systems by decentralized output feedback.IEEE Trans. Automat. Control 44 (1999), 148–152. MR 1666924 Reference: [12] S. L. Wo and Y. Zou: Decentralized robust stabilization for singular large scale systems with parameter uncertainty.Control and Decision 19 (2004), 8, 931–934 (in Chinese). MR 2079743 Reference: [13] H. S. Yang, Q. L. Zhang, and C. M. Shang: $H_2$ optimal control for descriptor systems.Acta Automat. Sinica 28 (2002), 6, 920–926 (in Chinese). Reference: [14] : .R. Yu and D. Wang. (2003). On impulsive modes of linear singular systems subject to decentralized output feedback. IEEE Trans. Automat. Control 48 (2003), 1804–1809. MR 2021433 Reference: [15] G. Zhai, M. Ikeda, and Y. Fujisaki: Decentralized $H_\infty$ controller design: A matrix inequality approach using a homotopy method.Automatica 37 (2001), 4, 565–573. MR 1832530 Reference: [16] G. Zhai, M. Yoshida, J. Imae, and T. Kobayashi: Decentralized $H_2$ controller design for descriptor systems: An LMI approach.Nonlinear Dynamics and Systems Theory 6 (2006), 1, 98–108. MR 2222782 .

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