UC San Diego

The length of time a jet aircraft is capable of remaining airborne can be maximized by flying at the speed that produces the least amount of drag. This speed may be predicted based on wind tunnel models, but the optimal speed for any aircraft in service differs somewhat from the calculated speed. Identifying the optimal speed has the potential to realize fuel savings and improve endurance. Extremum seeking is a non-model based form of real time nonlinear optimization that is suitable for problems such as this; however, traditional extremum seeking involves adding a small periodic perturbation to the control input. In this application, this would mean perturbing the throttle, which could erase the fuel savings otherwise achieved by the optimization process. To address this problem, a modified form of extremum seeking is developed that uses atmospheric turbulence in place of throttle perturbations. Using stochastic averaging, it is proven analytically that the extremum-seeking controller stabilizes the speed of the aircraft to the minimum-drag speed, with an average offset proportional to the third derivative of the drag curve and the variance of the airspeed. Brief simulation results illustrate the performance of the basic algorithm. Next, a new form of extremum seeking is introduced that extends a recent development in extremum seeking (called Newton method extremum seeking) to systems using stochastic perturbations. This work is parallel to the work on endurance optimization, but is relevant because the gradient estimator developed herein correctly estimates a two-dimensional gradient with perturbations of different amplitudes in the two dimensions. This is used in a refinement of the basic endurance optimization algorithm that involves a two-dimensional dependence; lift and drag are treated as functions of not only angle of attack (as implicitly assumed to this point) but also Mach number. Optimization proceeds along a line of constant lift in this two-dimensional plane. Analysis proves similar convergence properties for the refined algorithm, and the algorithm is tested in a high fidelity simulation lent by local industry. Simulation results show improvement over the nominal loiter speed