The new nonprobabilistic criterion of failure for dynamical systems based on convex models

By a counter example, we show that there seem to be some problems in Ben-Haim's theory of robust reliability of dynamical systems based on convex models. We still point out that the property of the expansion of convex models is just the addition of a convex model and a real vector, and the property of the translation of convex models is just the scalar multiplication convex models. By means of the partial-order relation of the superscribed hyperrectangle or interval vectors of convex models, we present a correct criterion of reliability of the dynamical system with bounded uncertainty. Based on them, we propose the expansion function which is different from the one of Ben-Haim. Following Ben-Haim's thoughts, based on the new expansion function, we again define the input, failure, and overall reliability indices. By Ben-Haim's example, we obtain some results different from his. The conclusion and results may be thought of as to the further development of Ben-Haim's robust reliability.