Abstract
An interval matrix can be represented in terms of a “center” matrix and a nonnegative error matrix, specifying maximum elementwise perturbations from the center matrix. A commonly proposed robust stability (regularity) characterization for an interval matrix with a stable (nonsingular) center matrix identifies the minimum scaling of this error matrix for which instability (singularity) is achieved. In this paper it is shown that approximating this minimum scaling is a MAX-SNP-hard problem. This implies that in the general case, unless the class of deterministic polynomial-time decision problems, P, equals the class of nondeterministic polynomial-time decision problems, NP, thought to be highly unlikely, this minimum scaling cannot be approximated with a ratio arbitrarily close to unity in polynomial time.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
S. Arora, C. Lund, R. Motwani, M. Sudan, and M. Szegedy, Proof verification and hardness of approximation problems,Proceedings of the 33rd Annual IEEE Conference on Foundations of Computer Science, Pittsburgh, PA, October, 1992, pp. 14–23.
R. Braatz, P. Young, J. Doyle, and M. Morari, Computational complexity of mu calculation,IEEE Transactions on Automatic Control,39 (1994), 1000–1002.
G. Coxson and C. DeMarco, Computing the real structured singular value is NP-hard, Report ECE-92-4, Department of Electrical and Computer Engineering, University of Wisconsin-Madison, June, 1992.
J. Demmel, The componentwise distance to the nearest singular matrix,SI AM Journal of Matrix Analysis and Applications, 13 (1992), 10–19.
J. Doyle, Analysis of feedback systems with structured uncertainty,IEE Proceedings, D,129 (1982), 242–250.
R. Fagin, Generalized first-order spectra and polynomial-time recognizable sets, in Complexity of Computation, SIAM-AMS Proceedings, vol. 7, American Mathematical Society, Providence, RI, 1974.
F. Gantmacher,The Theory of Matrices, Chelsea, New York, 1959.
M. Garey, D. Johnson, and L. Stockmeyer, Some simplified NP-complete graph problems,Theoretical Computer Science,1 (1976), 237–267.
M. Goemans, and D. Williamson,. 878-approximation algorithms for MAX CUT and MAX 2SAT,Proceedings of the 26th ACM Symposium on the Theory of Computing, New York, pp. 422–431.
A. Nemirovskii, Several NP-hard problems arising in robust stability analysis,Mathematics of Control, Signals, and Systems,6 (1993), 99–106.
A. Neumaier,Interval Methods for Systems of Equations, Cambridge University Press, Cambridge, 1990.
C. Papadimitriou, and M. Yannakakis, Optimization, approximation and complexity classes,Proceedings of the 20th ACM Symposium on the Theory of Computing, 1988, pp. 229–234.
S. Poljak and J. Rohn, Checking robust nonsingularity is NP-hard,Mathematics of Control, Signals, and Systems,6 (1993), 1–9.
M. Safonov, Stability margins of diagonally perturbed multivariable feedback systems,Proceedings of the 20th IEEE Conference on Decision and Control, San Diego, CA, December, 1981, pp. 1472–1478.
L. Saydy, A. Tits, and E. Abed, Robust stability of linear systems relative to guarded domains,Proceedings of the 27th IEEE Conference on Decision and Control, Austin, TX, December, 1988, pp. 544–551.
P. Young, M. Newlin, and J. Doyle, Let's get real, presented at the SIAM Conference on Control and Its Applications, Minneapolis, MN, Sept. 1992.
Author information
Authors and Affiliations
Additional information
This research was supported by the National Science Foundation under Grant ECS-8857019.
Rights and permissions
About this article
Cite this article
Coxson, G.E., DeMarco, C.L. The computational complexity of approximating the minimal perturbation scaling to achieve instability in an interval matrix. Math. Control Signal Systems 7, 279–291 (1994). https://doi.org/10.1007/BF01211520
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01211520