Discrete-time 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. 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.