Linear Quadratic Optimal Control of Nonlinear Dynamic Systems

Abstract

Despite the progress made in the field of control engineering, there are still significant challenges involved in the control of nonlinear dynamic systems. This thesis presents a novel approach to addressing these challenges by leveraging the simplicity of linear optimal quadratic controllers to achieve efficient control of nonlinear dynamic systems that are otherwise considered too difficult to control. In this thesis, three nonlinear dynamic applications are presented: the 3-DOF helicopter, the 6-DOF aircraft landing gear, and the loudspeaker system. These systems face various challenges, including unstable dynamics, landing vibrations, intermodulation distortions, and under-actuation. As such, the Linear Quadratic Regulator (LQR) controller is proposed to control the 3-DOF Helicopter, the Linear Quadratic Gaussian (LQG) controller is proposed for the control of the 6-DOF aircraft landing gear, and both the LQR and the LQG controllers are proposed to control the loudspeaker system.The LQR controller computes the control signals of the systems while the LQG controller acts as a state estimator. The state-space model of each nonlinear dynamic system is derived, and then, the mathematical models of the optimal control strategies are calculated. The control strategies are also tested under various conditions and compared with an equally simple control strategy, the PID controller. To better evaluate the execution and the performance of the LQR and LQG control strategies, two quantitative tracking performance metrics are presented; i) the integral of the tracking errors, and ii) the integral of the control signals of the system. The results obtained affirm the robustness and competence of the proposed control strategies.