Models developed from first principles often contain coefficients that are nonlinear functions of design parameters, which themselves are generally only known within some tolerance. Analysis and design of controllers for such models (plants) are often expedited through the use of an overbounding interval plant representation of the original plant. However, such representations are known to introduce conservativeness. As an alternative, we provide an algorithm for synthesizing a frequency-dependent convex hull approximation of an uncertain plant with nonlinear coefficients, which we refer to as a minimized plant. We develop several theorems that demonstrate reduced conservativeness through the use of minimized plants. In particular, the use of these plants to determine robust stability margins results in an improvement over the use of overbounding interval plants. We also illustrate in an example how a constant coefficient compensator and minimized plant meet a given set of design specifications, but the same compensator fails to meet specifications when analysed using an interval plant. As reduced conservativeness simplifies the design process, the convex hull synthesis algorithm and associated theorems provided in this paper facilitate robust controller analysis and design.