Computation of Viability Kernels on Grid Computers for Aircraft Control in Windshear

Abstract

This paper is devoted to the analysis of aircraft dynamics in the cruise flight phase under windshear conditions. The study is conducted with reference to a point-mass aircraft model restricted to move in a vertical plane. We formulate the problem as a differential game against the wind disturbances: The first player, autopilot, manages, via additional smoothing filters, the aircraft's angle of attack and power setting. The second player, wind, produces disturbances that are transferred, also via smoothing filters, into most dangerous wind gusts. The state variables of the game are subject to state constraints representing aircraft safety conditions related, for example, to the altitude, path inclination and velocity. Viability theory is used to compute the so-called viability kernels, the maximal subsets of state constraints where an appropriate feedback strategy of the first player can keep aircraft trajectories arbitrary long for all admissible disturbances generated by the second player. A grid method is utilized, and challenging computations in seven dimensions are conducted on a supercomputer system.

Dynamic Models with Sigmoid Corrections to Generation of an Achievable 4D-Trajectory for a UAV and Estimating Wind Disturbances

“…In this paper, we use the sigma functions directly in the feedback loop of the tracking differentiator as corrective action. The scaling factors are assigned at the design stage based on equations that consider the "worst case" ( 24), ( 26) disturbances (23). During the control process in the virtual system (22), the sigmoid corrective actions converge to the disturbances in a finite time and repeat their shape with a specified accuracy.…”

“…For their implementation, the wind velocity field is assumed to be known, but it is difficult to obtain accurate information about the changing wind velocity field. Within the framework of differential game theory, researchers use the grid method [23], but its implementation requires complex calculations in seven dimensions on a supercomputer system.…”