LQG: control what you estimate
Combine LQR state feedback with a Kalman state estimate.
First, picture it
LQR knows what to do if it knows all four cart-pole states. The observer supplies estimates when only position and angle sensors are available. The controller then acts on those estimates.
What the model says
A linear-quadratic-Gaussian controller combines an LQR gain with a Kalman filter for a stated linear model, quadratic cost and Gaussian-noise assumptions. Under those conditions, the separation principle lets the two parts be designed independently. The lab applies the resulting gain to a nonlinear, force-limited cart-pole with bounded injected noise, so the textbook optimality guarantee does not apply to its runs. Compare LQR, estimator-only and LQG under the same initial angle and kick.
uₖ=−K_LQR x̂ₖFull-state LQR uses true x; LQG closes the loop with estimated x̂.
Make it concrete
Pin an LQG run, switch to LQR without changing the kick, and compare position recovery, angle and force. Any difference comes from state estimation in this setup.
Open the LQG experimentLQG is not a guaranteed cure for saturation or model error; a mathematically optimal linear gain can fail against a rail limit.
Check your understanding+
Which signal drives the LQG feedback law?
The estimated state from the observer, rather than the inaccessible true state.