Linear quadratic regulator

Balance tracking error against effort#

LQR starts with linear dynamics and a cost that penalizes state error and control input through weighting matrices. The solution is a feedback law of the form u = -Kx. MIT's Underactuated Robotics notes derive the gain from the algebraic Riccati equation for the continuous, infinite-horizon case.

The state weights express which errors matter most, while the input weights penalize aggressive actuation. On a robot, the state might include joint-angle and velocity errors around a standing pose. A larger penalty on body tilt can make the controller correct tilt more strongly, subject to the chosen actuation cost.

Local feedback for nonlinear robots#

Robot dynamics are usually nonlinear. A common use is to linearize them around an equilibrium or a nominal trajectory and apply LQR to deviations from that reference. Time-varying LQR changes the gain along a trajectory and is often paired with a motion produced by trajectory optimization.

LQR differs from model predictive control. A basic LQR law is computed from a linear model and fixed quadratic objective, then applied directly. MPC repeatedly solves a finite-horizon optimization and can include explicit state and input constraints.

Optimal only for the stated model and cost#

The optimality claim is conditional. It applies to the specified linear dynamics, quadratic cost, horizon, and mathematical assumptions. Poor state or input weights can yield a controller that is optimal for the formula but unsuitable for the physical task.

Large deviations can invalidate the local linear model. Actuator saturation, contact changes, delays, and incomplete state estimates also break assumptions of the basic design. Engineers therefore test the region of attraction and combine LQR with constraint handling, gain scheduling, or mode-specific controllers when the robot operates across wider regimes.