---
tag: K22P72HMT1
kind: concept
name: Discrete-time algebraic Riccati equation
msc: [15A24, 93C55, 49N10]
specializes: [GGZV75ZZMV]
sources:
  - url: https://en.wikipedia.org/wiki/Algebraic_Riccati_equation
---

## Summary

The 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 $X$.
It is the discrete-time form of the [[algebraic Riccati equation|GGZV75ZZMV]].
Its stabilising solution gives the optimal state feedback for the linear-quadratic regulator of the system $x_{k+1} = A x_k + B u_k$.
A 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.

## Explanations

### L1 human

A 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.
The best policy multiplies the current state by a fixed gain matrix, and the gain is computed from the solution $X$ of this equation.
The matrix $X$ measures the total future cost that starts from each state.
For one variable with $A = B = Q = R = 1$, the equation becomes $X^2 - X - 1 = 0$, and the useful solution is the golden ratio $(1 + \sqrt{5})/2$.