Optimization and pseudospectra, with applications to robust stability

SIAM Journal on Matrix Analysis and Applications

Volumn 25, Issue 1, 2004, Pages 80-104

  Burke, J V ^a   Lewis, A S ^b   Overton, M L ^c  

^a UNIVERSITY OF WASHINGTON   (United States)

^b SIMON FRASER UNIVERSITY   (Canada)

^c NEW YORK UNIVERSITY   (United States)

Author keywords

Distance to instability; Eigenvalue optimization; H norm; Nonsmooth analysis; Pseudospectrum; Robust control; Robust optimization; Spectral abscissa; Stability radius; Subdifferential regularity

Indexed keywords

EID: 1342310021     PISSN: 08954798     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0895479802402818     Document Type: Article

References (45)

1. V. I. ARNOLD, On matrices depending on parameters, Russian Math. Surveys, 26 (1971), pp. 29-43.
2. V. BALAKRISHNAN, S. BOYD, AND S. BALEMI, Branch and bound algorithm for computing the minimum stability degree of parameter-dependent linear systems, Internat. J. Robust Nonlinear Control, 1 (1991), pp. 295-317.
3. H. BAUMGÄRTEL, Analytic Perturbation Theory for Matrices and Operators, Birkhäuser-Verlag, Basel, 1985.
4. A. BEN-TAL AND A. NEMIROVSKI, Lectures on Modern Convex Optimization: Analysis, Algorithms, and Engineering Applications, SIAM, Philadelphia, 2001.
5. V. BLONDEL AND J. N. TSITSIKLIS, NP-hardness of some linear control design problems, SIAM J. Control Optim., 35 (1997), pp. 2118-2127.
6. J. F. BONNANS AND A. SHAPIRO, Perturbation Analysis of Optimization Problems, Springer, New York, 2000.
7. ∞-norm, Systems Control Lett., 15 (1990), pp. 1-7.
8. ∞ norm of a transfer matrix and related problems, Math. Control Signals Systems, 2 (1989), pp. 207-219.
9. S. BOYD, L. EL GHAOUI, E. FERON, AND V. BALAKRISHNAN, Linear Matrix Inequalities in System and Control Theory, SIAM, Philadelphia, 1994.
10. M. BRÜHL, A curve tracing algorithm for computing the pseudospectrum, BIT, 36 (1996), pp. 441-454.
11. ∞-norm of a transfer function matrix, Systems Control Lett., 14 (1990), pp. 287-293.
12. J. V. BURKE, A. S. LEWIS, AND M. L. OVERTON, Optimal stability and eigenvalue multiplicity, Foundations Comput. Math., 1 (2001), pp. 205-225.
13. J. V. BURKE, A. S. LEWIS, AND M. L. OVERTON, Robust stability and a criss-cross algorithm for pseudospectra, IMA J. Numer. Anal., to appear.
14. J. V. BURKE, A. S. LEWIS, AND M. L. OVERTON, Two numerical methods for optimizing matrix stability, Linear Algebra Appl., 351/352 (2002), pp. 117-145.
15. J. V. BURKE AND M. L. OVERTON, Variational analysis of non-Lipschitz spectral functions, Math. Programming, 90 (2001), pp. 317-352.
16. R. BYERS, A bisection method for measuring the distance of a stable matrix to the unstable matrices, SIAM J. Sci. Stat. Comput., 9 (1988), pp. 875-881.
17. F. H. CLARKE, Optimization and Nonsmooth Analysis, Wiley, New York, 1983. Republished as Classics in Applied Math. 5, SIAM, Philadelphia, 1990.
18. F. H. CLARKE, Optimization and Nonsmooth Analysis, Wiley, New York, 1983. Republished as Classics in Applied Math. 5, SIAM, Philadelphia, 1990.
19. ∞ norms and related questions, in Proceedings MTNS-98, G. Picci and D. S. Gilliam, eds., Birkhäuser-Verlag, Basel, 1998.
20. C. HE AND G. A. WATSON, An algorithm for computing the distance to instability, SIAM J. Matrix Anal. Appl., 20 (1998), pp. 101-116.
21. Pseudospectra Gateway, http://web.comlab.ox.ac.uk/projects/pseudospectra.
22. D. HINRICHSEN AND A. J. PRITCHARD, Stability radii of linear systems, Systems Control Lett., 7 (1986), pp. 1-10.
23. R. A. HORN AND C. R. JOHNSON, Topics in Matrix Analysis, Cambridge University Press, Cambridge, UK, 1991.
24. T. KATO, A Short Introduction to Perturbation Theory for Linear Operators, Springer, New York, 1982.
25. C. LAWRENCE, A. L. TITS, AND P. VAN DOOREN, A fast algorithm for the computation of an upper bound on the μ-norm, Automatica, 36 (2000), pp. 449-456.
26. A. S. LEWIS, Convex analysis on the Hermitian matrices, SIAM J. Optim., 6 (1996), pp. 164-177.
27. A. S. LEWIS, Robust regularization, Math. Program., submitted.
28. V. B. LIDSKII, Perturbation theory of non-conjugate operators, USSR Comput. Math. and Math. Phys., 1 (1965), pp. 73-85.
29. C. F. VAN LOAN, How near is a stable matrix to an unstable matrix? Contemp. Math., 47 (1985), pp. 465-477.
30. J. W. MILNOR, Singular Points of Complex Hypersurfaces, Princeton University Press, Princeton, NJ, 1968.
31. J. MORO, J. V. BURKE, AND M. L. OVERTON, On the Lidskii-Vishik-Lyusternik perturbation theory for the eigenvalues of matrices with arbitrary Jordan structure, SIAM J. Matrix Anal. Appl., 18 (1997), pp. 793-817.
32. Y. NESTEROV AND A. NEMIROVSKII, Interior-Point Polynomial Algorithms in Convex Programming, SIAM, Philadelphia, 1994.
33. B. T. POLYAK, Convexity of nonlinear image of a small ball with applications to optimization, Set-Valued Anal., 9 (2001), pp. 159-168.
34. X. RAO, K. GALLIVAN, AND P. VAN DOOREN, Stabilization of large scale dynamical systems, in Proceedings MTNS 2000, S. L. Campbell, ed., Int. J. Appl. Math. Comput. Sci. II, University of Zielona Góra, Zielona, Góra, Poland, 2001.
35. R. T. ROCKAFELLAR AND R. J.-B. WETS, Variational Analysis, Springer, Berlin, 1998.
36. R. A. SILVERMAN, Introductory Complex Analysis, Prentice-Hall, Englewood Cliffs, NJ, 1967.
37. J. SREEDHAR, P. VAN DOOREN, AND A. L. TITS, A fast algorithm to compute the real structured stability radius, in Stability Theory: Proceedings of Hurwitz Centenary Conference, R. Jeltsch and M. Mansour, eds., Birkháuser-Verlag, Basel, 1996, pp. 219-230.
38. G. W. STEWART AND J. G. SUN, Matritc Perturbation Theory, Academic Press, Boston, 1990.
39. M. J. TODD, Semidefinite optimization, Acta Numer., 10 (2001), pp. 515-560.
40. L. N. TREFETHEN, Pseudospectra of linear operators, SIAM Rev., 39 (1997), pp. 383-406.
41. L. N. TREFETHEN, Computation of pseudospectra, Acta Numer., 8 (1999), pp. 247-295.
42. L. N. TREFETHEN, private communication, 2002.
43. ∞ norm minimization problems, in Proceedings of the 40th IEEE Conference on Decision and Control (Orlando, FL), IEEE, Piscataway, NJ, 2001.
44. M. VIDYASAGAR, Control Systems Synthesis: A Factorization Approach, MIT Press, Cambridge, MA, 1985.
45. G. ZAMES AND B. A. FRANCIS, Feedback, minimax sensitivity, and optimal robustness, IEEE Trans. Automat. Control, 28 (1983), pp. 585-601.