control//feedback control//LQG

Optimal control of a linear system from noisy sensors, built from an optimal regulator and an optimal estimator. **LQR**, the linear quadratic regulator, is state feedback \(u=-Kx\) whose gain minimizes a quadratic cost, the sum over time of \(x^{\mathsf T}Qx+u^{\mathsf T}Ru\): how far the state is from where it should be, against how much actuation it costs to bring it there. Heavy weight on the state makes it aggressive; heavy weight on the input makes it gentle. It assumes the state is known exactly, which it never is.

Optimal control of a linear system from noisy sensors, built from an optimal regulator and an optimal estimator. LQR, the linear quadratic regulator, is state feedback u=−Kxu=-Kxu=−Kx whose gain minimizes a quadratic cost, the sum over time of xTQx+uTRux^{\mathsf T}Qx+u^{\mathsf T}RuxTQx+uTRu: how far the state is from where it should be, against how much actuation it costs to bring it there. Heavy weight on the state makes it aggressive; heavy weight on the input makes it gentle. It assumes the state is known exactly, which it never is.

LQG, linear quadratic Gaussian, closes that gap. A Kalman filter estimates the state from noisy sensors, and the LQR acts on the estimate instead of the state. For linear systems with Gaussian noise the separation principle holds: the best controller and the best estimator can be designed separately, each with its own Riccati equation, and put together they are optimal. The estimator does not need to know what the controller wants, and the controller treats the estimate as if it were the truth.

The two problems are mirror images. The regulator's Riccati equation and the filter's have the same structure with the roles of input and measurement exchanged, which is why they are called dual: controllability is to the one what observability is to the other (controllability and observability).

The QQQ and RRR of the regulator are design weights, chosen; the QQQ and RRR of the filter are noise covariances, measured or tuned (process noise, measurement noise). Same letters, different questions.

Optimal is not robust. LQG has no guaranteed stability margins: an LQR alone tolerates large gain errors, but with the filter in the loop a small mismatch between model and plant can erode the margins, which is why practical designs check robustness separately (feedback control).