Algebraic Riccati equation
Encyclopedia
The algebraic Riccati equation is either of the following matrix equations:
the continuous time algebraic Riccati equation (CARE):
or the discrete time algebraic Riccati equation (DARE):
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 theory
.
A solution to the algebraic Riccati equation can be obtained by matrix factorizations or by iterating on the Riccati equation.
the continuous time algebraic Riccati equation (CARE):
or the discrete time algebraic Riccati equation (DARE):
X is the unknown n by n symmetric matrix and A, B, Q, R are known real
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
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
Differential equation
A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...
studied by Riccati
Jacopo Riccati
Jacopo Francesco Riccati was an Italian mathematician, born in Venice. He is now remembered for the Riccati equation. He died in Treviso in 1754.-Education:...
.
The algebraic Riccati equation determines the solution of the infinite horizon time-invariant Linear-Quadratic Regulator problem
Linear-quadratic regulator
The theory of optimal control is concerned with operating a dynamic system at minimum cost. The case where the system dynamics are described by a set of linear differential equations and the cost is described by a quadratic functional is called the LQ problem...
(LQR) as well as that of the infinite horizon time-invariant Linear-Quadratic-Gaussian control problem
Linear-quadratic-Gaussian control
In control theory, the linear-quadratic-Gaussian control problem is one of the most fundamental optimal control problems. It concerns uncertain linear systems disturbed by additive white Gaussian noise, having incomplete state information and undergoing control subject to quadratic costs...
(LQG). These are two of the most fundamental problems in control theory
Control theory
Control theory is an interdisciplinary branch of engineering and mathematics that deals with the behavior of dynamical systems. The desired output of a system is called the reference...
.
A solution to the algebraic Riccati equation can be obtained by matrix factorizations or by iterating on the Riccati equation.