Extremum seeking control for UAV close formation flight via improved pigeon-inspired optimization

This paper proposes a comprehensive design scheme for the extremum seeking control (ESC) of the unmanned aerial vehicle (UAV) close formation flight. The proposed design scheme combines a Newton-Raphson method with an extended Kalman filter (EKF) to dynamically estimate the optimal position of the following UAV relative to the leading UAV. To reflect the wake vortex effects reliably, the drag coefficient induced by the wake vortex is considered as a performance function. Then, the performance function is parameterized by the first-order and second-order terms of its Taylor series expansion. Given the excellent performance of nonlinear estimation, the EKF is used to estimate the gradient and the Hessian matrix of the parameterized performance function. The output feedback of the proposed scheme is determined by iterative calculation of the Newton-Raphson method. Compared with the traditional ESC and the classic ESC, the proposed design scheme avoids the slow continuous time integration of the gradient. This allows a faster convergence of relative position extremum. Furthermore, the proposed method can provide a smoother command during the seeking process as the second-order term of the performance function is taken into account. The convergence analysis of the proposed design scheme is accomplished by showing that the output feedback is a supermartingale sequence. To improve estimation performance of the EKF, a improved pigeon-inspired optimization (IPIO) is proposed to automatically tune the noise covariance matrix. Monte Carlo simulations for a three-UAV close formation show that the proposed design scheme is robust to the initial position of the following UAV.