## 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.

### MSC:

 93B35 Sensitivity (robustness) 93C41 Control/observation systems with incomplete information 93D99 Stability of control systems
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### References:

 [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
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