Algebraic Riccati equation

Summary

An algebraic Riccati equation is a quadratic matrix equation for an unknown symmetric matrix $X$, such as $A^T X + XA - XBR^{-1}B^T X + Q = 0$. It is the steady state of the matrix Riccati differential equation. Its stabilising solution gives the optimal feedback of the linear-quadratic regulator and the gain of the steady-state Kalman filter. Without the quadratic term it reduces to a Lyapunov equation.