Abstract
This paper presents a comprehensive Linear Quadratic Gaussian (LQG) control design for a self-balancing two wheeled robot. The nonlinear dynamics are derived using Lagrange equations and linearized around the upright equilibrium to obtain a state space model. The LQG controller integrates an optimal Linear Quadratic Regulator (LQR) for stabilization with a Kalman filter for state estimation under sensor noise. The entire design is implemented in MATLAB using a transparent, script based approach, ensuring full reproducibility and mathematical traceability. An optimal LQR is designed for stabilization, while a Kalman filter estimates the unmeasured states under sensor noise and disturbances. The integrated LQG controller is implemented entirely in MATLAB using a script based approach, ensuring transparency and reproducibility. Simulation results demonstrate rapid disturbance rejection, smooth transient responses, and bounded control effort. Extensive analyses including time domain, frequency domain, and pole zero assessments confirm the closed loop system’s stability, robustness, and noise resilience. The proposed framework validates LQG as a reliable and practical control strategy for inherently unstable robotic systems.