Abstract
In this paper, algebraic matrix Riccati equations are analyzed that have nonnegative-definite quadratic as well as constant terms. Necessary and sufficient conditions are established for the existence of stabilizing solution, which is not necessarily nonnegative-definite unlike the case of standard equations arising in LQG problems. Nonnegative-definite stabilizing solution is shown to exist if and only if the system is asymptotically stable and the H∞ norm of transfer matrix be less than one. For positive-definite solution, we need additional condition that the system be controllable.
Next, necessary and sufficient conditions are established for the existence of the so-called antistabilizing solution for both general and positive-definite cases. It is shown that stabilizing solution is minimal and anti-stabilizing solution is maximal of all the real symmetric solutions. Then, assuming the existence of nonnegative-definite stabilizing solution, all the other (hence nonnegative-definite) solutions with lattice structure are derived, together with the number of nonnegative-definite solutions.
The results are established by employing consistently the so-called algebraic method based on an eigenvalue problem of a Hamiltonian matrix.
