Computational Solution of the Algebraic Riccati Equation.
Angelika Bunse‐Gerstner · 1996
The numerical solution of the algebraic Riccati equation is required in a large number of applications like linear quadratic optimal control problems, differential games and computation of Kalman filters. This paper gives a survey of computational method developed and investigated over the last three decades. In particular we discuss Newton's method, the matrix sign function method, defect correction and methods using eigenvalue computations. x1 Introduction The algebraic Riccati equation (ARE) (1:1) R(X) = XA+ A T X +Q \\Gamma XBR \\Gamma1 B T X = 0 arises frequently in control problems. Here A 2 IR n\\Thetan ; B 2 IR n\\Thetam ; Q = Q T 2 IR n\\Thetan is positive semidefinite, R = R T 2 IR m\\Thetam is positive definite and X 2 IR n\\Thetan is the unknown matrix we have to compute. Usually, the desired solution is stabilizing in the sense that the eigenvalues of A \\Gamma BR \\Gamma1 B T X have negative real parts. Under mild assumptions (see Section 2) such a stabi...