Unmanned Aerial Vehicles (UAVs), particularly quadcopters, have seen an explosion in use, yet their nonlinear, coupled, and underactuated dynamics make control a challenging problem. This thesis presents a comprehensive simulation-based platform developed in MATLAB to model quadcopter flight dynamics and evaluate control strategies without the risks of hardware testing. A physically accurate nonlinear mathematical model was derived, incorporating rigid-body dynamics, motor response, and environmental disturbances. Two control architectures, a cascaded PID controller and a Linear Quadratic Regulator (LQR), were designed and subjected to comparative analysis under nominal and disturbed conditions. Results demonstrate that while PID provides baseline stability, the LQR controller offers superior trajectory tracking, robustness, and disturbance rejection. This work establishes a validated framework for future advanced control development and hardware integration.