---
tag: 9PW3GX1579
kind: definition
name: Discrete-time algebraic Riccati equation
msc: [15A24, 93C55, 49N10]
defines: K22P72HMT1
uses: [0VD9ZSWWJG, 0VR671PHE5, 01MMVW8CKC, 0KX66CZMBB, 09DXWYRD43, 0EB9N0S33C, GGZV75ZZMV, FM9Y4Y6D8F]
sources:
  - url: https://en.wikipedia.org/wiki/Algebraic_Riccati_equation
---

## Definition

Let $n, m \ge 1$.
Let $A$ be a real $n \times n$ [[matrix|0VD9ZSWWJG]] and let $B$ be a real $n \times m$ matrix.
Let $Q$ be a real [[symmetric|0VR671PHE5]] [[positive semidefinite|01MMVW8CKC]] $n \times n$ matrix, and let $R$ be a real symmetric [[positive definite|0KX66CZMBB]] $m \times m$ matrix.
Write $M^T$ for the [[transpose|09DXWYRD43]] of a matrix $M$.
The *discrete-time algebraic Riccati equation* is
$$X = A^T X A - A^T X B \left(R + B^T X B\right)^{-1} B^T X A + Q$$
for an unknown real symmetric positive semidefinite $n \times n$ matrix $X$.

## Remarks

For such $X$ the matrix $R + B^T X B$ is positive definite, hence [[invertible|0EB9N0S33C]], so the equation makes sense.
A solution $X$ is *stabilising* if every eigenvalue of $A - BK$ has absolute value less than $1$, where $K = (R + B^T X B)^{-1} B^T X A$.
If the pair $(A, B)$ is stabilisable and the pair $(Q^{1/2}, A)$ is detectable, the equation has a unique positive semidefinite solution, and it is stabilising.
The optimal control of the infinite-horizon linear-quadratic regulator is then $u_k = -K x_k$.
Example: for $n = m = 1$ and $A = B = Q = R = 1$, the equation reads $X = X - X^2/(1 + X) + 1$, that is, $X^2 - X - 1 = 0$.
Its nonnegative solution is $X = (1 + \sqrt{5})/2$, and then $K = X/(1 + X) = 1/X$ and $A - BK = 1 - 1/X \approx 0.38$, so it is stabilising.
Non-example: the other root $X = (1 - \sqrt{5})/2$ satisfies the matrix identity but is negative, so it is not an admissible solution.
For $B = 0$ the equation becomes the linear [[Stein equation|FM9Y4Y6D8F]] $X - A^T X A = Q$.