266 Y. Ben-Haim and S. Cogan Fig. 25.3 Robustness curve of Eq. (25.5). F/sD3. QK=s D1 m.h/ D F QKCsh xc ) b h.xc/ D 1 s F xc Q K (25.5) or zero if this is negative which occurs if the performance requirement, xc, is too large to be obtained with the nominal system. The robustness function in Eq. (25.5) demonstrates two fundamental properties that hold for all info-gap robustness functions: trade off and zeroing, illustrated in Fig. 25.3. The performance requirement is that the displacement, x, be no less than the critical value xc. This requirement becomes more demanding as xc increases. We see from Eq. (25.5) and Fig. 25.3 that the robustness decreases as the requirement becomes more demanding. That is, robustness trades off against performance: the robustness can be increased only by relaxing the performance requirement. The negative slope in Fig. 25.3 represents the trade off between robustness and performance: strict performance requirements, demanding very good outcome, are less robust against uncertainty than lax requirements. This trade off quantifies the intuition of any healthy pessimist: more demanding requirements are more vulnerable to surprise and uncertainty than more modest requirements. The second property illustrated in Fig. 25.3 is zeroing. Our best estimate of the stiffness is QK, so the predicted displacement is x DF=QK. Eq. (25.5) shows that the robustness becomes zero precisely at the value of the critical displacement, xc, that is predicted by the nominal model: xc DF=QK, which equals 3 for the parameter values in Fig. 25.3. Stated differently, the zeroing property asserts that best-model predictions have no robustness against error in the model. Like trade off, this is true for all info-gap robustness functions. Models reflect our best understanding of the system and its environment. Nonetheless, the zeroing property means that model predictions are not a good basis for design or planning decisions, because those predictions have no robustness against errors in the models. Recall that we’re discussing situations with large info-gaps. If your models are correct (no info-gaps), then you don’t need robustness against uncertainty. The zeroing property asserts that the predicted outcome is not a reliable characterization of the design. The trade off property quantifies how much the performance requirement must be reduced in order to gain robustness against uncertainty. The slope of the robustness curve reflects the cost of robustness: what decrement in performance “buys” a specified increment in robustness. Outcome-quality can be exchanged for robustness, and the slope quantifies the cost of this exchange. In Fig. 25.3 we see that the cost of robustness is very large at large values of xc, and decreases as the performance requirement is relaxed. The robustness function is useful as a response to the pernicious side of uncertainty. In contrast, the opportuneness function is useful in exploiting the potential for propitious surprise. We now discuss the info-gap opportuneness function. Info-gap opportuneness is based on three components: a system model, an info-gap model of uncertainty, and a performance aspiration. The performance aspiration expresses a desire for better-than-anticipated outcome resulting from propitious surprise. This differs from the performance requirement for robustness that expresses an essential or critical outcome without which the result would be considered a failure. We will illustrate the opportuneness function by continuing the same example, with positiveFandx. The robust-satisficing decision maker requires that the displacement be no less that xc. The opportune windfalling decision maker recognizes that a larger displacement would be better, especially if it exceeds the anticipated displacement, F=QK. For the opportune windfaller, the displacement would be wonderful if it is as large as xw, which exceeds the anticipated displacement. The windfaller’s aspiration is not a performance requirement, but it would be great if it occurred.
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