Fast Construction of Robustness Degradation Function

Abstract

We develop a fast algorithm to construct the robustness degradation function, which describes quantitatively the relationship between the proportion of systems guaranteeing the robustness requirement and the radius of the uncertainty set. This function can be applied to predict whether a controller design based on an inexact mathematical model will perform satisfactorily when implemented on the true system.

A new statistical approach for the analysis of uncertain systems

“…It can be verified that r i r i+1 = 1 − 1 m−1 λ−1 +i for i = 1, 2, · · · , m − 1. Let n k be the total number of simulations on the direction associated with directional sample U k , k = 1, 2, · · · , N. Applying Theorem 1 of [5] and Theorem 5 in this paper to a sample reuse process conditioned upon a direction with grid points r 1 , r 2 , · · · , r m and sample size N = 1,…”

“…From another perspective, it also shows that the number of grid points and computational complexity tend to infinity as the tolerance tends to zero. The computational complexity can be reduced by the sample reuse techniques of [5]. It is recently shown in [13] that the equivalent number of grid points is bounded by 1 + d ln λ (see Appendix C for a proof).…”

“…This theorem concerns the sampling complexity of the Sample Reuse Algorithm proposed in page 1963 of [5].…”

“…Aimed at overcoming the computational complexity and conservatism of the classical deterministic worst-cast approach, there are growing interests in developing probabilistic methods and randomized algorithms (see, [1][2][3][4][5][6][7][8][9][10][11] and the references therein). Specially, a probabilistic robustness measure, referred to as the confidence degradation function or robustness function is proposed in [3].…”

On the binomial confidence interval and probabilistic robust control

“…Significant contributions have been made by a number of researchers (see, e.g., [1,2,3,4,5], [10,11,12,13,14,15,16,17] and the reference therein). A fundamental problem in the area of probabilistic robustness analysis is to estimate the probability that a certain robustness requirement is guaranteed for an uncertain dynamic system.…”