Optimal Tracking Control for Underactuated Airship

Background: A non-linear mathematical model of underactuated airship is derived in this paper based on Euler-Newton approach. The model is linearized with small disturbance theory, producing a linear time varying (LTV)model. The LTV model is utilized to design a linear quadratic tracking (LQT) controller. Two scenarios of LQT are presented in this work according to assumed costates transformations to compute the LQT control law. Results: : The LTV model is verified by comparing its output response with the result of the nonlinear model for a given input signal. It shows an acceptable error margin. The verified LTV model is used in designing the LQT controller. The controller is designed to minimize the error between the output and required states response with acceptable control signals using a weighted cost function. Two LQT controllers are presented in this work based on two different transformations used in solving the differential Riccati equation (DRE). These controllers are tested by a sample trajectory to deduce the characteristics of each assumption. Finally, a hybrid LQT controller is used and tested on circular, helical, and bowed trajectories. Conclusion: The first assumption of costates transformation has a good tracking performance, but it is sensitive to the change of trajectory profile. Whereas, the second one overcomes this problem due to considering the trajectory dynamics. Therefore, the first assumption is performed across the whole trajectory tracking except for parts of trajectory profile changes where the second assumption is applied.