Linear-Quadratic-Gaussian Control (LQG)
LQG combines an LQR optimal feedback gain with a Kalman filter estimator to control a noisy system using only its measured outputs.
Optimal Control With Noisy Measurements
The linear-quadratic regulator assumes perfect knowledge of the full state. Real systems provide only noisy, partial measurements. Linear-quadratic-Gaussian control removes that assumption by pairing an LQR gain with a Kalman filter that estimates the state from the available outputs.
The two halves
- Estimation: a Kalman filter produces the optimal state estimate x-hat from the noisy measurements, accounting for process and sensor noise.
- Control: the LQR gain is applied to the estimate rather than the true state, giving u = -K*x-hat.
- The process noise and measurement noise covariances tune the filter; the Q and R matrices tune the regulator.
The separation principle
LQG works because of the separation principle: the optimal estimator and the optimal regulator can be designed independently and then joined, and the combination is still optimal. The controller gain K is computed as if the state were known, the filter gain L is computed as if control were separate, and using x-hat in place of x loses no optimality for the linear-Gaussian problem.
Structure
The resulting controller is itself a dynamic system of the same order as the plant: it runs a model of the plant, corrects that model with each measurement, and feeds the corrected estimate through the LQR gain. This is the canonical model-based output-feedback controller.
A caution on robustness
LQR alone has excellent guaranteed margins, but those guarantees do not carry over to LQG. Inserting the Kalman filter can erode the robustness margins substantially. The technique of loop transfer recovery adjusts the filter design to recover much of the lost margin, and where robustness is critical, H-infinity methods are preferred.
Despite that caveat, LQG is a foundational design: it shows how estimation and control combine cleanly, and it is the direct ancestor of modern robust and predictive controllers that also run an internal model of the plant.