Algebraic Riccati equation

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The algebraic Riccati equation is either of the following matrix equations:

the continuous time algebraic Riccati equation (CARE):

A^T X + X A - X B B^T X + Q = 0 \,

or the discrete time algebraic Riccati equation (DARE):

X = A^T X A -(A^T X B)(R + B^T X B)^{-1}(B^T X A) + Q.\,

X is the unknown n by n symmetric matrix and A, B, Q, R are known real coefficient matrices.

The name Riccati is given to the CARE equation by analogy to the Riccati differential equation: the unknown appears linearly and in a quadratic term (but no higher-order terms). The DARE arises in place of the CARE when studying discrete time systems; it is not obviously related to the differential equation studied by Riccati.

The algebraic Riccati equation determines the solution of the infinite horizon time-invariant Linear-Quadratic Regulator problem (LQR) as well as that of the infinite horizon time-invariant Linear-Quadratic-Gaussian control problem (LQG). These are two of the most fundamental problems in control.

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