Geodesic
Rank-one LMI approach to robust stability of polynomial matrices
Kybernetika, Tome 38 (2002) no. 5, pp. 643-656 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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.
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.
Classification : 15A39, 93D09
Keywords: linear matrix inequality; stability 
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     title = {Rank-one {LMI} approach to robust stability of polynomial matrices},
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Henrion, Didier; Sugimoto, Kenji; Šebek, Michael. Rank-one LMI approach to robust stability of polynomial matrices. Kybernetika, Tome 38 (2002) no. 5, pp. 643-656. http://geodesic.mathdoc.fr/item/KYB_2002_38_5_a11/

[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

[2] Barmish B. R.: New Tools for Robustness of Linear Systems. Macmillan, New York 1994 | Zbl

[3] Bhattacharyya S. P., Chapellat, H., Keel L. H.: Robust Control: The Parametric Approach. Prentice Hall, Upper Saddle River, N.J. 1995 | Zbl

[4] Blondel V. D., Tsitsiklis J. N.: A survey of computational complexity results in systems and control. Automatica 36 (2000), 9, 1249–1274 | DOI | MR | Zbl

[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 | MR | Zbl

[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 | DOI | MR | Zbl

[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

[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

[9] Ghaoui L. El, (eds.) S. I. Niculescu: Advances in Linear Matrix Inequality Methods in Control. SIAM Advances in Control and Design, Philadelphia, 1999 | MR | Zbl

[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 | DOI | MR

[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 | DOI | MR | Zbl

[12] Gupta S.: Robust stability analysis using LMI: Beyond small gain and passivity. Internat. J. Robust Nonlinear Control 6 (1996), 953–968 | DOI | MR

[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

[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 | DOI | MR | Zbl

[15] Henrion D., Bachelier, O., Šebek M.: $\mathcal{D}$-stability of polynomial matrices. Internat. J. Control 74 (2001), 8, 845–856 | MR

[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 | MR | Zbl

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[20] Kučera V.: Discrete Linear Control: The Polynomial Approach. Wiley, Chichester 1979 | MR

[21] Packard A., Doyle J.: The complex singular value. Automatica 29 (1993), 1, 71–109 | DOI | MR

[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

[23] Polyx, Inc.: Polynomial Toolbox for Matlab, Release 2. 0.0, 1999. See the web page www.polyx.cz

[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

[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 | DOI | MR

[26] Shim D.: Quadratic stability in the circle theorem or positivity theorem. Internat. J. Robust Nonlinear Control 6 (1996), 781–788 | DOI | MR | Zbl

[27] Vandenberghe L., Boyd S.: Semidefinite programming. SIAM Rev. 38 (1996), 49–95 | DOI | MR | Zbl

[28] Willems J. C.: Paradigms and puzzles in the theory of dynamical systems. IEEE Trans. Automat. Control 36 (1991), 259–294 | DOI | MR | Zbl

[29] Zhou K., Doyle, J., Glover K.: Robust and Optimal Control. Prentice Hall, Upper Saddle River, N.J. 1996 | Zbl