Abstract
This paper considers the problem of determining the minimum euclidean distance of a point from a polynomial surface in R n. It is well known that this problem is in general non-convex. The main purpose of the paper is to investigate to what extent Linear Matrix Inequality (LMI) techniques can be exploited for solving this problem. The first result of the paper shows that a lower bound to the global minimum can be achieved via the solution of a one-parameter family of LMIs. Each LMI problem consists in the minimization of the maximum eigen value of a symmetric matrix. It is also pointed out that for some classes of problems the solution of a single LMI provides the lower bound. The second result concerns the tightness of the bound. It is shown that optimality of the lower bound can be readily checked via the solution of a system of linear equations. In addition, it is pointed out that lower bound tightness is strictly related to some properties concerning real homogeneous forms. Finally, an application example is developed throughout the paper to show the features of the approach.
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References
J.C. Doyle, J.E. Wall, G. Stein, “Performance and robustness analysis for structured uncertainty,” Proc. 21st IEEE Conf. on Decision and Control, Orlando, Florida, pp. 629–636, 1982.
J.C. Doyle, “Analysis of feedback systems with structured uncertainties,” IEE Proceedings-D Control Theory and Applications, vol. 129, pp. 242–250, 1982.
S.P. Bhattacharyya, H. Chapellat and L. H. Keel, Robust Control: The Parametric Approach, Prentice Hall PTR, NJ, 1995.
R. Genesio, M. Tartaglia, and A. Vicino, “On the estimation of asymptotic stability region. State of the art and new proposals”, IEEE Transactions on Automatic Control, vol. 30, pp. 747–755, 1985.
H. D. Chiang and J. S. Thorp, “Stability regions of nonlinear dynamical systems: a constructive methodology”, IEEE Transactions on Automatic Control, vol. 34, pp. 1229–1241, 1989.
S. Boyd, L. El Ghaoui, E. Feron and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory, Siam, Philadelphia, 1994.
D. Hershkowitz, “Recent directions in matrix stability,” Linear Algebra and its Applications, vol. 171, pp. 161–186, 1992.
H. K. Khalil, Nonlinear Systems, McMillan Publishing Company, New York, 1992.
A.J. van der Schaft, “L 2-gain analysis of nonlinear state feedback H ∞-control,” IEEE Transactions on Automatic Control, vol. 37, pp. 770–784, 1992.
G. Chesi, A. Tesi, A. Vicino, R. Genesio, “On convexification of some minimum distance problems,” Research Report DSI-19/98, Dipartimento di Sistemi e Informatica, Firenze, 1998.
Yu. Nesterov and A. Nemirovsky, Interior Point Polynomial Methods in Convex Programming: Theory and Applications, Siam, Philadelphia, 1993.
G. Hardy, J.E. Littlewood, G. Pólya, Inequalities: Second edition, Cambridge University Press, Cambridge, 1988.
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© 1999 Springer-Verlag London Limited
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Chesi, G., Tesi, A., Vicino, A., Genesio, R. (1999). A convex approach to a class of minimum norm problems. In: Garulli, A., Tesi, A. (eds) Robustness in identification and control. Lecture Notes in Control and Information Sciences, vol 245. Springer, London. https://doi.org/10.1007/BFb0109880
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DOI: https://doi.org/10.1007/BFb0109880
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