Abstract
In this paper, we consider several robust control analysis and design problems. As has become well known over the last few years, most of these problems are NP hard. We show that if instead of worst-case guaranteed conclusions, one is willing to draw conclusions with a high degree of confidence, then the computational complexity decreases dramatically.
This work was supported in part by Airforce Office of Scientific Research under contract no. F-49620-93-1-0246DEF and Army Research Office under grant no. DAAH04-93-G-0012
This is a preview of subscription content, log in via an institution to check access.
Preview
Unable to display preview. Download preview PDF.
References
G. Balas, J.C. Doyle, K. Glover, A. Packard, and R. Smith, μ-Analysis and Synthesis Toolbox User's Guide, MUSYN Inc., 1994.
B. R. Barmish and C.M. Lagoa, “The uniform distribution: a rigorous justification for its use in robustness analysis,” Technical Report ECE-95-12, Dept. of ECE, Univ. of Wisconsin-Madison, 1995.
R. Braatz, P. Young, J. Doyle, and M. Morari, “Computational complexity of μ calculation,” Proc. of the Automatic Control Conference, pp: 1682–1683, San Francisco, 1993.
M. Deng and Y.-C. Ho, “Sampling-selection method for stochastic optimization problems,” Preprint, Harvard University, 1995.
J. C. Doyle, B. A. Francis, and A. R. Tannenbaum, Feedback Control Theory, Macmillan Publishing Company, New York, 1992.
M. K. H. Fan, A. L. Tits, and J. C. Doyle, “Robustness in the presence of joint parametric uncertainty and unmodeled dynamics,” Proc. of the American Control Conference, pp. 1195–1200, 1988.
W. Feller, An Introduction to Probability Theory and its Applications, Volume 2, Wiley & Sons, New York, 1965.
J. Friedman, P. Kabamba, and P. Khargonekar, “Worst-case and average H 2 performance analysis against real constant parametric uncertainty,” Automatica, vol. 31, pp 649–657, 1995.
M. Fu and B. R. Barmish, “Maximal unidirectional perturbation bounds for stability of polynomials and matrices,” Systems & Control Letters, 11 (1988) pp 173–179.
D. E. Goldberg, Genetic Algorithms in Search, Optimization, and Machine Learning, Addison-Wesley Publishing Co., Reading, Mass., 1989.
S. Hall, D. MacMartin, and D. Bernstein, “Covariance averaging in the analysis of uncertain systems,” IEEE Trans. Auto. Contr., vol. 38, pp 1858–1862, 1993.
Y.-C. Ho, “Heuristics, rules of thumb, and the 80/20 proposition,” IEEE Trans. Auto. Contr., vol. 39, pp. 1025–1027, 1994.
W. Hoeffding, “Probability inequalities for sums of bounded random variables,” Amer. Stat. Assoc. J., pp. 13–30, 1963.
V. L. Kharitonov, “Asymptotic stability of an equilibrium position of a family of systems of linear differential equations,” Differencial'nye Uravnenija 14 (11) (1978) pp. 2086–2088.
T.W.E. Lau and Y.-C. Ho, “Universal alignment probabilities and subset selection for ordinal optimization,” Preprint, Harvard University, 1996.
L. Lovasz and M. Simonovits, “Random walks in a convex body and an improved volume algorithm,” Random Structures and Algorithms, vol. 4, pp. 359–412, 1993.
MATH WORKS. MATLAB Reference Guide. The MATH WORKS, Inc., 1992.
C. Marrison and R. Stengel, “The use of random search and genetic algorithms to optimize stochastic robustness functions,” Proc. of the American Control Conference, pp. 1484–1489, Baltimore, Maryland, 1994.
A. Nemirovskii, “Several NP-hard problems arising in robust stability analysis,” Math. of Control, Signals, and Systems, vol. 6, pp 99–105, 1993.
A. Packard and J.C. Doyle, “The complex structured singular value,” Automatica, vol. 29, pp. 71–110, 1993.
S. Poljak and J. Rohn, “Checking robust nonsingularity is NP-hard,” Math. of Control, Signals, and Systems, vol. 6, pp 1–9, 1993.
L. R. Ray and R. F. Stengel, “Stochastic robustness of linear time-invariant control systems,” IEEE Trans. Auto. Contr., vol. 36, pp. 82–87, 1991.
H. E. Romeijn, Global Optimization by Random Walk Sampling Methods, book nr. 32 of Tinbergen Institue Research Series, Thesis Publishers, Amsterdam, The Netherlands, 1992.
R.Y. Rubinstein, Monte Carlo Optimization, Simulation and Sensitivity of Queueing Networks, Wiley and Sons, New York, 1986.
R. Tempo and E. W. Bai, “Robustness analysis with nonlinear parametric uncertainty: A probabilistic approach,” CENS-CNR Report 95/11, Poiltecnico di Torino, Torino, Italy, 1995.
P. Young and J. Doyle, “Computation of μ with real and complex uncertainties,” Proc. of 29th Conf. on Decision and Control, pp. 1230–1235, Honolulu, Hawaii, 1990.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1999 Springer-Verlag London Limited
About this paper
Cite this paper
Khargonekar, P.P., Tikku, A. (1999). Probabilistic search algorithms for robust stability analysis and their complexity properties. In: Yamamoto, Y., Hara, S. (eds) Learning, control and hybrid systems. Lecture Notes in Control and Information Sciences, vol 241. Springer, London. https://doi.org/10.1007/BFb0109719
Download citation
DOI: https://doi.org/10.1007/BFb0109719
Published:
Publisher Name: Springer, London
Print ISBN: 978-1-85233-076-7
Online ISBN: 978-1-84628-533-2
eBook Packages: Springer Book Archive