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  "body": "\n## Summary\n\nThe discrete-time algebraic Riccati equation is the nonlinear matrix equation $X = A^T X A - A^T X B (R + B^T X B)^{-1} B^T X A + Q$ for an unknown symmetric matrix $X$.\nIt is the discrete-time form of the [[algebraic Riccati equation|GGZV75ZZMV]].\nIts stabilising solution gives the optimal state feedback for the linear-quadratic regulator of the system $x_{k+1} = A x_k + B u_k$.\nA dual form of it, with $A$ and $B$ replaced by $A^T$ and $C^T$, gives the steady-state error covariance of the Kalman filter.\n\n## Explanations\n\n### L1 human\n\nA controller acts on a system once per time step and wants to keep both the state and its own effort small over all future steps.\nThe best policy multiplies the current state by a fixed gain matrix, and the gain is computed from the solution $X$ of this equation.\nThe matrix $X$ measures the total future cost that starts from each state.\nFor one variable with $A = B = Q = R = 1$, the equation becomes $X^2 - X - 1 = 0$, and the useful solution is the golden ratio $(1 + \\sqrt{5})/2$.\n",
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