Linear quadratic regulator

The linear quadratic regulator (LQR) is the optimal state-feedback controller for a linear system with a quadratic cost function. Given a state-space representation and a cost that penalizes both state deviations and control effort, LQR produces a feedback matrix \(K\) such that the control law \(u = -Kx\) minimizes the total cost over an infinite horizon. The solution is computed by solving the algebraic Riccati equation, a nonlinear matrix equation whose solution encodes the trade-off between performance and control authority.

LQR is the workhorse of modern control design because it converts an infinite-dimensional optimization problem — choosing a control trajectory over all future time — into a finite-dimensional algebraic problem: find the right \(K\). The resulting closed-loop system has guaranteed stability margins — at least 60 degrees phase margin and 6 dB gain margin — that are remarkable given that the design process never explicitly considers stability margins. They emerge as a side effect of optimality, a gift from the mathematics that engineers did not ask for but gratefully accept.

The limitation is that LQR assumes full state measurement, which is rarely true in practice. When only outputs are measured, the Kalman filter provides state estimates, and the combination of LQR with Kalman filtering produces the linear quadratic Gaussian (LQG) controller. The separation principle guarantees that this combination is optimal — but only for the model. In reality, the separation between estimation and control breaks down when the model is imperfect, and LQG controllers can have arbitrarily poor robustness properties.