A new vertex result for robustness problems with interval matrix uncertainty. (English) Zbl 1154.93023

Summary: This paper addresses a family of robustness problems in which the system under consideration is affected by interval matrix uncertainty. The main contribution of the paper is a new vertex result that drastically reduces the number of extreme realizations required to check robust feasibility. This vertex result allows one to solve, in a deterministic way and without introducing conservatism, the corresponding robustness problem for small and medium size problems. For example, consider quadratic stability of an autonomous \(n_x\) dimensional system. In this case, instead of checking \(2^{n_x^2}\) vertices, we show that it suffices to check \(2^{2n_x}\) specially constructed systems. This solution is still exponential, but this is not surprising because the problem is NP-hard. Finally, vertex extensions to multiaffine interval families and some sufficient conditions (in LMI form) for robust feasibility are presented. Some illustrative examples are also given.


93B35 Sensitivity (robustness)
93C41 Control/observation systems with incomplete information
93D99 Stability of control systems
Full Text: DOI Link


[1] Barmish, B.R., New tools for robustness of linear systems, (1994), MacMillan New York · Zbl 1094.93517
[2] Ben-Tal, A.; Nemirovski, A., Lectures on modern convex optimization, () · Zbl 0977.90052
[3] Ben-Tal, A.; Nemirovski, A., On tractable approximations of uncertain linear matrix inequalities affected by interval uncertainty, SIAM journal on optimization, 12, 811-833, (2002) · Zbl 1008.90034
[4] Boyd, S.; El Ghaoui, L.; Feron, E.; Balakrishnan, V., ()
[5] Calafiore, G.; Polyak, B.T., Stochastic algorithms for exact and approximate feasibility of robust lmis, IEEE transactions on automatic control, 46, 1755-1759, (2001) · Zbl 1007.93080
[6] Garofalo, F.; Celentano, G.; Glielmo, L., Stability robustness of interval matrices via Lyapunov quadratic forms, IEEE transactions on automatic control, 38, 281-284, (1993) · Zbl 0774.93061
[7] Hertz, D., The extreme eigenvalues and stability of real symmetric interval matrices, IEEE transactions on automatic control, 37, 532-535, (1992)
[8] Horisberger, H.P.; Belanger, P.R., Regulators for linear, time invariant plants with uncertain parameters, IEEE transactions on automatic control, 21, 705-708, (1976) · Zbl 0339.93013
[9] Kharitonov, V.L., Asymptotic stability of an equilibrium position of a family of systems of linear differential equations, Differential’nye uravneniya, 14, 1483-1485, (1978) · Zbl 0409.34043
[10] Kothare, M.V.; Balakrishnan, V.; Morari, M., Robust constrained model predictive control using linear matrix inequalities, Automatica, 32, 1361-1379, (1996) · Zbl 0897.93023
[11] Liberzon, D.; Tempo, R., Common Lyapunov functions and gradient algorithms, IEEE transactions on automatic control, 49, 990-994, (2004) · Zbl 1365.93469
[12] Nemirovskii, A., Several NP-hard problems arising in robust stability analysis, Mathematics of control, signal and systems, 6, 99-105, (1993) · Zbl 0792.93100
[13] Poljak, S.; Rohn, J., Checking robust nonsingularity is NP-hard, Mathematics of control, signal and systems, 6, 1-9, (1993) · Zbl 0780.93027
[14] Polyak, B.T.; Tempo, R., Probabilistic robust design with linear quadratic regulators, Systems and control letters, 43, 343-353, (2001) · Zbl 0974.93070
[15] Rohn, J., Systems of linear interval equations, Linear algebra and its applications, 126, 39-78, (1989) · Zbl 0712.65029
[16] Seif, N.P.; Hussein, S.A.; Deif, A.S., The interval Sylvester equation, Computing, 52, 233-244, (1994) · Zbl 0807.65046
[17] Tempo, R.; Calafiore, G.; Dabbene, F., Randomized algorithms for analysis and control of uncertain systems, (2005), Springer-Verlag London · Zbl 1079.93002
[18] Young, P.M., Robustness analysis with full-structured uncertainties, Automatica, 33, 2131-2145, (1997) · Zbl 0903.93007
[19] Young, P.M.; Doyle, J.C., Properties of the mixed \(\mu\) problem and its bounds, IEEE transactions on automatic control, 41, 155-159, (1996) · Zbl 0845.93023
[20] Zhou, K.; Doyle, J.C.; Glover, K., Robust and optimal control, (1996), Prentice-Hall Upper Saddle River · Zbl 0999.49500
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.