Linear Quadratic Regulator

The optimal feedback controller for a linear system: trading off state error against control effort, solving for a fixed gain u = −Kx.

LQR is the most fundamental optimal-control method, with its theory laid down by Kálmán and others around 1960. It assumes the system is linear: ẋ = Ax + Bu, where x is the state, u the control input, and A, B matrices describing the system; the cost is quadratic, the time integral of xᵀQx + uᵀRu. Q reflects how much you care about state deviation, R how much you care about control effort, and both are tuned by the designer. Solving a Riccati equation gives the gain matrix K, and the optimal control is u = −Kx — at runtime this is just a single matrix multiplication. Compared to PID, LQR handles multiple-input, multiple-output systems naturally and directly trades off ‘error’ against ‘effort’; compared to MPC, it can't handle constraints like torque limits. Robots are nonlinear, so a common approach is to linearize around an equilibrium point and apply LQR there, valid only near that point; linearizing at every point along a trajectory instead gives a time-varying LQR, which is also the basis of iLQR.

ExampleLinearizing an inverted pendulum or an Acrobot (a two-link gymnast robot) around its upright equilibrium and computing an LQR gain lets it be balanced steadily upright; the balance controller on a two-wheeled self-balancing vehicle is often built the same way.

Also called

LQR

Related

Linear Quadratic Gaussian Control · Iterative Linear Quadratic Regulator · Optimal Control · Model Predictive Control · Proportional-Integral-Derivative Control · Inverted Pendulum Model (IPM)

Sources

Underactuated Robotics (Russ Tedrake), Ch. Linear Quadratic Regulators
Linear–quadratic regulator - Wikipedia