Flight controllers for micro-air UAVs are generally designed using proportional-integral-derivative (PID) methods,where the tuning of gains is difficult and time-consuming,and performance is not guaranteed.In this paper,we develop a rigorous method based on the sliding mode analysis and nonlinear backstepping to design a PID controller with guaranteed performance.This technique provides the structure and gains for the PID controller,such that a robust and fast response of the UAV (unmanned aerial vehicle) for trajectory tracking is achieved.First,the second-order sliding variable errors are used in a rigorous nonlinear backstepping design to obtain guaranteed performance for the nonlinear UAV dynamics.Then,using a small angle approximation and rigorous geometric manipulations,this nonlinear design is converted into a PID controller whose structure is naturally determined through the backstepping procedure.PID gains that guarantee robust UAV performance are finally computed from the sliding mode gains and from stabilizing gains for tracking error dynamics.We prove that the desired Euler angles of the inner attitude controller loop are related to the dynamics of the outer backstepping tracker loop by inverse kinematics,which provides a seamless connection with existing built-in UAV attitude controllers.We implement the proposed method on actual UAV,and experimental flight tests prove the validity of these algorithms.It is seen that our PID design procedure yields tighter UAV performance than an existing popular PID control technique.