We propose formulating preliminary-stage aircraft design problems as geometric programs (GPs), which are a specific type of convex optimization problem. Recent advances in convex optimization offer significant advantages over the general nonlinear optimization methods typically used in MDO. Modern GP solvers are extremely fast even on large problems, require no initial guesses or tuning of solver parameters, and guarantee globally optimal solutions. These benefits come at a price: all objective functions and constraints -the mathematical models that describe aircraft design -must be expressed within the restricted functional forms of GP. Perhaps surprisingly, this restricted set of functional forms appears again and again in prevailing physics-based models for aircraft design tradeoffs. Moreover, for various models that cannot be manipulated algebraically into the forms required by GP, we can often fit compact GP models which accurately approximate the original models. The speed of GP solution methods makes their application to large MDO problems a promising approach.
Motivated by practical applications in engineering, this article considers the problem of approximating a set of data with a function that is compatible with geometric programming (GP). Starting with well-established methods for fitting max-affine functions, it is shown that improved fits can be obtained using an extended function class based on the softmax of a set of affine functions. The softmax is generalized in two steps, with the most expressive function class using an implicit representation that allows fitting algorithms to locally tune softness. Each of the proposed function classes is directly compatible with the posynomial constraint forms in geometric programming. Maxmonomial fitting and posynomial fitting are shown to correspond to fitting special cases of the proposed implicit softmax function class. The fitting problem is formulated as a nonlinear least squares regression, solved locally using a Levenberg-Marquardt algorithm. Practical implementation considerations are discussed. The article concludes with numerical examples from aerospace engineering and electrical engineering.
For a UAV to perch on a wire, aircraft control systems which operate far outside typical operating envelopes must be developed. The relevant transient aerodynamics at high angle of attack are not addressed today by control-accessible aerodynamic models. In this work, we present a set of physically-inspired basis functions which have enabled system identification of a nonlinear aerodynamics model along perching trajectories. Data is collected using a motion capture system which, critically, allows free-flight data from real system trajectories to be gathered. When simulated forward, the identified model accurately predicts the observed perching trajectories, making it an indispensable tool for designing feedback controllers that stabilize perching trajectories.
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