Probabilistic design of systems with general distributions of parameters

Abstract

SUMMARYThis paper presents a new method for ÿnding optimal solutions of systems with design parameters which are random variables distributed with various general and possibly non-symmetrical distributions. A double-bounded density function is used to approximate the distributions. Speciÿcations may require tracking constraints in time domain and stability conditions in frequency domain. Using sensitivity information, the proposed method ÿrst ÿnds a linearized feasible region. Afterwards it attempts to place a tolerance box of the design parameters such that the region with higher yield lies in the feasible region. The yield is estimated by the joint cumulative density function over a portion of the tolerance box contained in the feasible region. Optimal designs are found for a fourth-order servomechanism and actual yields are evaluated by Monte-Carlo simulation.

Bayesian and E-Bayesian estimation for the Kumaraswamy distribution based on type-ii censoring

“…B ab  is the Bayes estimate  of given by (2-8), (2-11), (2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14) and (2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17). For more details see (Han [9,24]).…”

“…By the same way, we can derive the E-Bayes estimates of  relative to LLF based on 2 by using (2-11), (3)(4)(5) in (3-3) and (2)(3)(4)(5)(6)(7)(8)(9)(10)(11), (3)(4)(5)(6) …”

“…by using (2-17), (3)(4)(5) in (3-3) and (2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17), (3)(4)(5)(6) …”

“…EBQi i   Proposition 4: It follows from (3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16), (3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17) and (3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18) …”

“…The continuous part of Kumaraswamy distribution, denoted Kum ( , )  has probability density function (pdf) and the cumulative distribution function (cdf) specified by 11 ( ; , ) Where  and  are the shape parameters. Based on different values of  and  , Kumaraswamy [3] and Ponnambalam, et al [4], have referred to that the Kumaraswamy distribution can be used as an approximation for many distributions, such as uniform, triangular, and can also be reproduce result of beta distribution. Nadarajah [5] considered that the Kumaraswamy distribution is special case of the three parameter beta distribution.…”