Linear Quadratic Gaussian Control

When the state is only partially measured and noisy, first estimate it with a Kalman filter, then compute control with LQR.

LQG is a classic optimal-control problem: the system is linear, both the process and the measurements carry Gaussian white noise, the state can't be fully measured directly, and the goal is to minimize the expected value of a quadratic cost. The solution splits into two pieces: a Kalman filter estimates the current state x̂ from noisy sensor readings, and LQR then computes the control from that estimate, u = −K·x̂ (K the LQR gain). The two pieces can be designed separately, each optimal on its own, and the combination is still overall optimal — this is the separation principle. The difference from plain LQR is that LQR assumes the full state is known exactly, while LQG faces the real-world case of partial, noisy measurement. It's worth noting that LQR comes with good stability margins by default, while LQG offers no such guarantee — a point Doyle's 1978 paper specifically raised — so robustness needs to be checked separately in engineering practice. LQG is a foundational model for understanding how robots split ‘state estimation’ from ‘control.’

ExampleA cart-pole balancing system with only encoders measuring cart position and pole angle, where velocity has to be estimated and the readings are noisy: a Kalman filter estimates all four states — position, velocity, angle, and angular velocity — and multiplying by the LQR gain gives the motor command, together forming an LQG controller.

Also called

LQG

Related

Linear Quadratic Regulator · Kalman Filter · State Estimation · Optimal Control · Robust Control · State Observer

Sources

Linear–quadratic–Gaussian control - Wikipedia