Ellipsoidal bounds for uncertain linear equations and dynamical systems

Automatica

Volumn 40, Issue 5, 2004, Pages 773-787

  Calafiore, Giuseppe ^a   El Ghaoui, Laurent ^b  

^a POLITECNICO DI TORINO   (Italy)

^b UNIVERSITY OF CALIFORNIA   (United States)

Author keywords

Bounding ellipsoids; Interior point methods; Linear fractional transformation; Linear matrix inequalities (LMI); Set valued filtering; Structured and parametric uncertainty; Uncertain linear equations

Indexed keywords

COMPUTER SIMULATION; CONTROL SYSTEM ANALYSIS; LINEAR EQUATIONS; MATHEMATICAL MODELS; MATHEMATICAL TRANSFORMATIONS; MATRIX ALGEBRA; OPTIMIZATION; PARAMETER ESTIMATION; STATE ESTIMATION; VECTORS;

BOUNDING ELLIPSOIDS; INTERIOR-POINT METHODS; LINEAR FRACTIONAL TRANSFORMATIONS (LFT); LINEAR MATRIC INEQUALITY (LMI); SET-VALUED FILTERING; UNCERTAIN LINEAR EQUATIONS;

UNCERTAIN SYSTEMS;

EID: 1642367258     PISSN: 00051098     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.automatica.2004.01.001     Document Type: Article

References (41)

1. Ben-Tal A., Nemirovski A. Lectures on modern convex optimization. 2001;SIAM, Philadelphia, PA.
2. Ben-Tal A., Nemirovski A. On tractable approximations of uncertain linear matrix inequalities affected by interval uncertainty. SIAM Journal on Optimization. 12(3):2002;811-833.
3. Bertsekas D.P., Rhodes I.B. Recursive state estimation with a set-membership description of the uncertainty. IEEE Transactions on Automatc Control. 16:1971;117-128.
4. Boyd, S., El Ghaoui, L., Feron, E., & Balakrishnan, V. (1994). Linear matrix inequalities in system and control theory. Vol. 15, Studies in applied mathematics. Philadelphia, PA: SIAM.
5. Chernousko F.L. State estimation of dynamic systems. 1993;CRC Press, Boca Raton, FL.
6. Durieu C., Walter E., Polyak B. Multi-input multi-output ellipsoidal state bounding. Journal of Optimization Theory and Applications. 111(2):2001;273-303.
7. El Ghaoui, L. (2002). Inversion error, condition number, and approximate inverses of uncertain matrices. Linear Algebra and its Applications, 342 (1-3).
8. El Ghaoui, L., & Calafiore, G., (1999a). Confidence ellipsoids for uncertain linear equations with structure. In Proceedings of the IEEE conference on decision and control, Vol. 2, Phoenix, AZ, December 1999. (pp. 1922-1927).
9. El Ghaoui, L., & Calafiore, G. (1999b). Worst-case simulation of uncertain systems, Vol. 245, Lecture notes in control and information sciences. (pp. 134-146). London: Springer.
10. El Ghaoui L., Calafiore G. Robust filtering for discrete-time systems with bounded noise and parametric uncertainty. IEEE Transactions on Automatic Control. 46(7):2001;1084-1089.
11. Fan M.K.H., Tits A.L., Doyle J.C. Robustness in the presence of mixed parametric uncertainty and unmodeled dynamics. IEEE Transactions on Automatic Control. 36:1991;25-38.
12. Fogel E., Huang Y.F. On the value of information in system identification - bounded noise case. Automatica. 18(2):1982;229-238.
13. El Ghaoui L., Lebret H. Robust solutions to uncertain semidefinite programs. SIAM Journal of Optimization. 9(1):1998;33-52.
14. Goemans, M. X., & Williamson, D. P. (1994). 878-approximation for MAX CUT and MAX 2SAT. Proceedings of the 26th ACM symposium on theoretical computing. (pp. 422-431).
15. Hansen E.R. Bounding the solution of interval linear equations. SIAM Journal on Numerical Analysis. 29:1992;1493-1503.
16. Henrion D., Tarbouriech S., Arzelier D. LMI approximations for the radius of the intersection of ellipsoids. Journal of Optimization Theory and Applications. 108(1):2001;1-28.
17. Horn R.A., Johnson C.R. Matrix analysis. 1985;Cambridge University Press, Cambridge, UK.
18. Van Huffel S., Vandewalle J. The total least squares problem: Computational aspects and analysis. 1991;SIAM, Philadelphia.
19. Köyluoglu, U., Cakmak, A. S., & Nielsen, S. R. K. (1995). Interval algebra to deal with pattern loading and structural uncertainty. Journal of Engineering Mechanics, November, 1149-1157.
20. Kurzhanski A.B., Valyi I. Ellipsoidal calculus for estimation and control. 1996;Birkhauser, Boston.
21. Lasserre J.B. Global optimization with polynomials and the problem of moments. SIAM Journal on Optimization. 11(3):2001;796-817.
22. Maskarov D.G., Norton J.P. State bounding with ellipsoidal set description of the uncertainty. International Journal of Control. 65(5):1996;847-866.
23. Muhanna R.L., Mullen R.L. Uncertainty in mechanics problems - interval-based approach. ASCE Journal of Engineering Mechanics. 127(6):2001;557-566.
24. Nemirovski A., Roos C., Terlaky T. On maximization of quadratic form over intersection of ellipsoids with common center. Mathematical Programming. 86(3):1999;463-473.
25. Nesterov Y., Nemirovski A. Interior point polynomial methods in convex programming: Theory and applications. 1994;SIAM, Philadelphia.
26. Nesterov Yu. Semidefinite relaxation and non-convex quadratic optimization. Optimization Methods and Software. 12:1997;1-20.
27. Neumaier A. Interval methods for systems of equations. 1990;Cambridge University Press, Cambridge, UK.
28. Norton J.P., Veres S.M. Developments in parameter bounding. Gerencser L., Caines P.E. Topics in stochastic systems: Modelling, estimation and adaptive control. 1991;1137-1158 Springer, Berlin.
29. Ovseevich A.I. On equations of ellipsoids approximating attainable sets. Journal of Optimization Theory and Applications. 95(3):1997;659-676.
30. Parrilo, P. A. (2000). Structured semidefinite programs and semialgebraic geometry methods in robustness and optimization. Ph.D. Thesis, California Institute of Technology.
31. Polyak B.T. Convexity of quadratic transformations and its use in control and optimization. Journal of Optimization Theory and Applications. 99(3):1998;553-583.
32. Polyak, B. T., Nazin, S. A., Durieu, C., & Walter, E. (2002). Ellipsoidal estimation under model uncertainty. In Proceedings of IFAC 2002 world congress, Barcelona, Spain.
33. Rohn J. Systems of linear interval equations. Linear Algebra and its Applications. 126:1989;39-78.
34. Rohn J. Overestimations in bounding solutions of perturbed linear equations. Linear Algebra and its Applications. 262:1997;55-66.
35. Schweppe F.C. Recursive state estimation: unknown but bounded errors and system inputs. IEEE Transactions on Automatic Control. 13(1):1968;22-28.
36. Schweppe F.C. Uncertain dynamic systems. 1973;Prentice-Hall, New York.
37. Shamma J.S., Tu K.-Y. Set-valued observers and optimal disturbance rejection. IEEE Transactions on Automatic Control. 44(2):1999;253-264.
38. Skelton R.E., Iwasaki T., Grigoriadis K. A unified algebraic approach to linear control design. 1998;Taylor & Francis, London.
39. Vandenberghe L., Boyd S. Semidefinite programming. SIAM Review. 38(1):1996;49-95.
40. Vandenberghe L., Boyd S., Wu S.-P. Determinant maximization with linear matrix inequality constraints. SIAM Journal on Matrix Analysis and Applications. 19(2):1998;499-533.
41. Zhou K., Doyle J.C., Glover K. Robust and optimal control. 1996;Prentice-Hall, Upper Saddle River