Algebraic Riccati equation

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.

External links

* [http://www.mathworks.com/access/helpdesk/help/toolbox/control/ref/care.html CARE solver help of MATLAB Control toolbox.]
* [http://www.mathworks.com/access/helpdesk/help/toolbox/control/ref/dare.html DARE solver help of MATLAB Control toolbox.]