On the Formulation of Minimum-State Approximation as a Nonlinear Optimization Problem

Abstract

The solution of the minimum-state (MS) approximation of the unsteady aerodynamic forces is brought in this work into the form of a nonlinear optimization problem. This new formulation takes as design variables all of the aerodynamic lag terms (known also as aerodynamic roots) as well as the two matrices that directly operate on these lag terms. This new formulation enables the explicit determination of the remaining matrices that form the MS approximation, and it does not require enforcing any equality constraints. Furthermore, it also permits the derivation of simple analytical expressions for the gradients of the least-square (LS)-type objective function. This combination of explicit expressions for both the gradients and some of the unknown matrices leads to a dramatic reduction in computational labor. It is also shown that by appropriately scaling the tabulated aerodynamic matrix a significantly accelerated rate of convergence is obtained during the process of optimization, whereas a general weighting scheme might considerably slow down this convergence. It is also shown that the preceding scaling of the tabulated aerodynamic matrix can also significantly reduce the computational labor required by current methods of solution that are based on iterative LS analysis. The new formulation presented in this work leads to better results (i.e., lower values for the objective function) than those obtained using current iterative LS-based methods that use preassumed values for the aerodynamic lag terms. At the same time, the computer CPU time required by these two methods of solution is of the same order, with only slightly higher CPU values needed for the new formulation. On the other hand, current methods based on iterative LS analysis that attempt to optimize the aerodynamic roots rather than use preassumed values need extensive added computational labor to the extent that makes them practically unattractive.