A model-based framework for robust design

In this paper we develop an as-yet-missing theoretical framework as well as a general methodology for model-based robust design. At the outset, a distinction is made between three sets: the set of design variables, grouped in the n-dimensional vector x, which are to be assigned values as an outcome of the design job; the set of design-environment parameters (DEP), grouped in the v-dimensional vector p, over which the designer has no control; and the set of performance functions, arrayed in the m-dimensional vector f, representing the functional relations among performance, design variables and DER Resorting to the mathematical model available for the object under design, an in x v design performance matrix F, mapping the space of relative variations of p into that of relative variations of f, is derived. Moreover, two pertinent concepts are introduced: the design sensitivity matrix, which plays a major role in the transmission of the variations of p into variations of f, and its associated bandwidth, defined as the logarithm of the square root of the ratio between the maximum to the minimum singular values of the design performance matrix, measured in decades. A result stating the relation between the bandwidth of a matrix and its inverse is shown. Consequently, the aforementioned bandwidth represents an index for evaluating the robustness of a design. To demonstrate our approach, case studies are included.