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  "body": "\n## Summary\n\nAn algebraic Riccati equation is a quadratic matrix equation for an unknown symmetric matrix $X$, such as $A^T X + XA - XBR^{-1}B^T X + Q = 0$.\nIt is the steady state of the matrix Riccati differential equation.\nIts stabilising solution gives the optimal feedback of the linear-quadratic regulator and the gain of the steady-state Kalman filter.\nWithout the quadratic term it reduces to a [[Lyapunov equation|92GVPZH9FT]].\n\n## Explanations\n\n### L1 human\n\nA controller wants to steer a linear system to rest while keeping both the error and the effort small.\nThe best policy feeds back the state through a matrix gain, and that gain comes from a matrix $X$ solving a Riccati equation.\nThe equation is quadratic in $X$, like $ax^2 + bx + c = 0$ for numbers, so it can have several solutions.\nFor one variable with $A = 0$ and $B = Q = R = 1$, it reads $1 - x^2 = 0$, and the useful solution is $x = 1$.\n",
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