Algebraic Riccati Equations

q**Algebraic Riccati Equation:***
*

q* A*X+XA+XRX+Q=0,
R=R*, Q=Q*
*

An ARE may
have many solutions. We are only interested in symmetric solutions. In particular,
we are interested in the symmetric solution such that *A+RX* is stable. This solution
is called **stabilizing
solution.
**

• Consider the associated
Hamiltonian matrix:

•

Then

so *H* and -*H** are similar. Thus l is an eigenvalue iff - l* is.

qThus we conclude: eig(*H*) ¹ jw Û H has n eigenvalues in Re *s*<0 and n eigenvalues in Re *s*>0.