A neat linear equation often sits behind LQR. The moving pole on screen follows nonlinear dynamics. To interpret the gap, first ask where the linear model was built and which physical limits the run encountered.
Start at the equilibrium, not at an arbitrary angle
The upright cart-pole model is linearized around a specified upright condition. It uses deviations from that condition and derivatives of the nonlinear equations there. Small-angle approximations such as sin θ≈θ work only to the accuracy you need within a neighborhood; they are not global identities.
There is no universal cutoff angle. A model useful for predicting the direction of motion might be too rough for a tight settling-time specification. The controller, force limit and rail length also affect what can be recovered.
Separate three reasons a run can differ
First, neglected higher-order terms grow as the trajectory moves away from upright. Second, the actuator may saturate: the calculated force is not the applied force. Third, the cart may hit the finite rail before the pole is recovered. These mechanisms can occur together, but they are not the same error.
EIGENROOM currently shows the nonlinear run and controller signals, not a time-aligned trajectory from the linearized cart-pole model. Therefore a comparison of two starting angles reveals the behavior of the nonlinear closed loop, not a plotted linearization residual.
A fair comparison in the Lab
Select LQR and pin a run with a small initial angle. Increase only that angle while keeping Q/R, disturbance, seed and run duration fixed. Look at pole angle, cart position, applied force and rail status. If the larger run fails, report the changed initial condition and which limit appeared first.
To claim a numerical range of linearization accuracy, a future feature would have to run the linear and nonlinear models from matching deviation states and inputs, then compare trajectories over a stated time window. That is a separate, validated experiment.
When a local controller struggles, check approximation range, actuator saturation and rail limits separately. Keep the conclusion tied to the conditions actually tested.
CHECK YOUR UNDERSTANDING: A larger initial angle makes an LQR run fail. Does that prove the linearization error caused the failure?
No. Check force saturation and rail limits too. The current Lab does not plot a matched linear-model trajectory, so the error cannot be isolated from this run.