Stabilising an Inverted Pendulum on a Cart: LQR versus Angle-Only PD Control in Nonlinear Simulation

The inverted pendulum on a cart is the standard benchmark for unstable cyber-physical systems. This paper compares linear-quadratic-regulator (LQR) full-state feedback with three weight sets against an angle-only proportional-derivative (PD) controller in a nonlinear simulation with a force limit of 20 N. For an initial tilt of 0.2 rad, the angle-only PD controller stabilised the pendulum fastest (0.42 s to within 2 % of the initial angle) but left the cart drifting to 1.10 m, because the cart position is not controlled. The LQR controllers returned the cart to its origin, with settling times of 2.2 to 4.2 s depending on the weights. Heavier state weights shortened the settling time and reduced the cart travel (from 0.35 m to 0.19 m) at the cost of larger forces (peak 4.2 N to 7.7 N) and a smaller region of attraction (largest recovered initial tilt 1.10 to 0.85 rad). The model parameters are assumed and the study has no hardware validation. This is a simulation study; all results are outputs of the deposited Python code with assumed parameters. Source code is attached.