270 Y. Ben-Haim and S. Cogan The system fails if x<xc where xDF/K and F is a known positive constant. x is now a random variable (because K is random) so the performance requirement is that the probability of failure not exceed a critical value Pc. We will explore the robustness function. We consider the special case that F/xc Cıs, meaning that the failure threshold for Klies outside the interval in which the pdf of Kis known. The probability of failure is: Pf.p/ DProb.x <xc/ DProb.K>F=xc/ DZ 1 F=xc p.K/ dK (25.13) For the estimated pdf, Qp.K/, one finds the following expression for the estimated probability of failure: Pf .Qp/ D1 ˆ .F=xc/ (25.14) where ˆ( ) is the cumulative probability distribution function of the standard normal variate. The robustness function, b h.Pc/, is the greatest horizon of uncertainty h up to which all pdf’s p(K) in the uncertainty set U(h) do not have failure probability Pf(p) in excess of the critical value Pc: b h.Pc/ Dmax h W max p2U.h/ Pf.p/ Pc (25.15) After some algebra one finds the following expression for the robustness: b h.Pc/ D 8ˆ< ˆ: 0 if 0 Pc <Pf .Qp/ Pc Pf.Qp/ 1 if Pf .Qp/ Pc 2Pf .Qp/ 1 otherwise (25.16) The robustness curve in Eq. (25.16) is illustrated in Fig. 25.7 for [(F/xc) ]/ D3, meaning that the failure threshold is three standard deviations above the mean. Hence the estimated probability of failure is Pf .Qp/ D0:00135. The trade off property is evident in this figure: lower (better) required probability of failure, Pc, entails lower (worse) robustness b h.Pc/. We see the discontinuous jump of robustness to infinity at Pc D2Pf .Qp/. This is because the actual probability of failure, Pf(p), cannot exceed more than twice the estimated value, Pf .Qp/. This results from the constraints on the pdf’s in the info-gap model of Eq. (25.12). The zeroing property is expressed by robustness equaling zero when the performance requirement, Pc, equals the estimated value Pf .Qp/. Fig. 25.7 Robustness curve for Eq. (25.16)
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