Optimal Control Design and Simulation of a Second-Order Inverted Pendulum System Based on Linear Quadratic Regulator (LQR)

Inverted pendulum systems (IPSs) are typically nonlinear, highly coupled, multivariable underactuated systems. Due to their complex control characteristics, they have become a research focus in control engineering. This paper mainly studies the second-order inverted pendulum system. Aiming at the control challenges brought by its highly nonlinear, double-arm coupling and underactuated characteristics, an optimal control method based on a linear quadratic regulator (LQR) is proposed. Firstly, the nonlinear dynamic equations of the system are derived through VISIO modeling and mechanical analysis and then linearized at specific operating points to obtain an approximate linear model. On this basis, an optimal state feedback controller is designed to achieve stable control of the second-order inverted pendulum system. Subsequently, MATLAB simulation is used to verify the performance of the designed LQR control, and a comparative analysis is performed with the proportional-integral-differential (PID) controller. The research results show that LQR control can improve the robustness and control performance of the system while optimizing the cost function, providing an effective solution for controlling complex, underactuated systems.