25 Innovations and Info-Gaps: An Overview 265 Fig. 25.2 Mechanically linearized gap-closing electrostatic actuator. The figure is reproduced here with the permission of Prof. David Elata where K is a constant stiffness coefficient. The degree of linearity depends on the shapes of the cams and on the degree of mechanical and structural uniformity of the pair of beams. We will explore the robustness to uncertainty in the stiffness coefficient of the linearized beam. We will also explore robustness to uncertainty in a probabilistic model. Finally we will consider opportuneness. We consider Fand x to be positive. In our first approach to this problem we suppose that our knowledge of the stiffness coefficient, K, is quite limited. We know an estimated value, QK, and we have an estimate of the error, s, but the most we can confidently assert is that the true stiffness, K, deviates from the estimate by ˙s or more. We do not know a worst case or maximum error, and we have no probabilistic information about K. There are many types of info-gap models of uncertainty [1]. A fractional-error info-gap model is suitable to this state of knowledge: U.h/ D(KW K>0; ˇ ˇ ˇ ˇ ˇ K- QK s ˇ ˇ ˇ ˇ ˇ h) ; h 0 (25.3) The info-gap model of uncertainty in Eq. (25.3) is an unbounded family of sets of possible values of the uncertain entity, which is the stiffness coefficient Kin the present case. For any non-negative value of h, the set U(h) is an interval of Kvalues. Like all info-gap models, this one has two properties: nesting and contraction. ‘Nesting’ means that the set, U(h), becomes more inclusive as h increases. ‘Contraction’ means that U(h) is a singleton set containing only of the known nominal value, QK, when hD0. These properties endowh with its meaning as an ‘horizon of uncertainty’. Info-gap robustness is based on three components: a system model, an info-gap uncertainty model, and one or more performance requirements. In this present case, Eq. (25.2) is the system model and Eq. (25.3) is the uncertainty model. Our performance requirement is that the displacement, x, be no less than the critical value xc. The info-gap robustness is the greatest horizon of uncertainty, h, up to which the system model obeys the performance requirement: b h.xc/ Dmax h W min K2U.h/ x xc (25.4) Reading this equation from left to right we see that the robustness b h is the maximum horizon of uncertainty, h, up to which all realizations of the uncertain stiffness Kin the uncertainty set U(h) result in displacement x no less than the critical value xc. Robustness is a useful decision support tool because more robustness against uncertainty is better than less. Given two options that are more or less equivalent in other respects but one is more robust than the other, the robust-satisficing decision maker will prefer the more robust option. In short, “bigger is better” when prioritizing decision options in terms of robustness. Derivation of the robustness function is particularly simple in this case. From the system model we know that x D F K. Let m(h) denote the inner minimum in Eq. (25.4), and note that this minimum occurs, at horizon of uncertainty h, when KD QKCsh. The robustness is the greatest value of h up towhich m(h) is no less than xc:
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