Application of Pattern Search in Worst-Case Flutter Solution

ROBUST flutter solution based on analysis was first proposed by Lind and Brenner [1] and Lind [2], who used both the flighttest data and nominal aeroelastic model in the method. One of the main objectives of -method is to generate worst-case flutter solutions with respect to potential modeling errors.When computing the robust flutter speed by the method, is taken as “an exact measure of robust stability for systems with structured uncertainty, and the value of determines the allowable size of uncertainty matrices for which the plant is robustly stable as demonstrated in Theorem 3.3. 2” [1]. Mathematically, the norm of the smallest structured uncertainty matrix that causes instability is exactly 1= . After thework of Lind and Brenner [1] and Lind [2], Borglund [3] and Borglund and Ringertz [4] developed the -k method which is based on traditional frequency-domain flutter analysis thus allowing for detailed aerodynamic uncertainty modeling. The fairly recent -pmethod proposed byBorglund [5] generalizes the -kmethod to the Laplace domain and computes possible variation of subcritical damping and frequency of a particular aeroelastic mode with respect to deterministic uncertainty, which is quite useful in flight flutter testing. The -pmethod is successfully implemented in robustflutter analysis with respect to complex valued aerodynamic perturbations [6,7] andmixed structural/aerodynamic uncertainty [8]. However, as noted in [8], “the structured singular value has to be estimated by a computable upper bound, which can lead to some amount of conservativeness in the analysis (in particular formixed real/complex uncertainty). ” It may become crucial when computing with respect to purely real valued parametric uncertainty, because discontinuity may arise to [9,10] and lead to unreasonable results. The computation of itself is just an optimization problem taking as the objective function [11]. In addition to the method, there still exist other approaches such as optimization methods in worstcase flutter solution. Kuttenkeuler and Ringertz [12] and Becus [13] searched the most critical flutter configuration by optimizing the damping or the flutter speed in the uncertainty parameter space. However, this type of optimization problem is nonconvex in general, thus the efficient local method developed by Becus [13] can not always get the global optimal point, while the globalmethod [13] that sweeps the parameter space with a varying step would become timeconsuming subject to increasing number of uncertainty parameters. Moreover, these approaches lack a structured framework as noted in [8]. Fortunately, global optimization techniques have evolved a lot in recent years especially for the pattern search algorithm [14,15], which was proved to be with guaranteed global convergence if the function to be minimized is continuously differentiable [14]. In this Note, the aeroelastic model with purely real valued uncertainties is reviewed, and the promising pattern search algorithm is introduced and applied to solve this type of optimization problem by taking the worstor best-case flutter speed or aeroelastic damping or frequency as the objective function and taking the structured uncertainty operators as design variables. Sample test cases are studied with this approach and the results are validated by the exhaustive method. A special case with purely real uncertainties shows that the method can not give reasonable worst-case flutter solution while the present approach can.