Abstract—We analyze a probabilistic variation on the classic robustness radius. Instead of measuring the distance to the closest destabilizing state, we look at a set of probability distributions around our initial state. The probabilistic radius is the radius of the largest sphere, for which none of the distributions supported within it takes the expected values of system parameters outside a prescribed set. We show that when the set of acceptable parameter values is closed and convex, the radius for the family of distributions which are nondecreasing with respect to distance reduces to the same radius for uniform distributions. We generalize this result for distributions of higher order convexity or concavity with respect to distance, obtaining an equal radius using a simple family of polynomials.
Index Terms—Robustness radii, distributional robustness, decision analysis, control theory, aircraft control.
I. Kangasniemi and J. P. Nikkarila are with the University of Helsinki, They are also with Finnish Defence Forces Research Agency, P.O.Box 10, FI-11311 Riihimäki, Finland (e-mail: ilmari.kangasniemi@helsinki.fi, juha-pekka.nikkarila@mil.fi).
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Cite: Ilmari Kangasniemi and Juha-Pekka Nikkarila, "Probabilistic Robustness Radii with n :th Order Convex Distributions," International Journal of Modeling and Optimization vol. 6, no. 4, pp. 211-218, 2016.
