Analysis of the Lyapunov equation using generalized positive real matrices

In this paper, a representation of the solution matrix P to the Lyapunov matrix equation PF + F´P = -LL´ is derived. We consider the class of m??m matrices Z(??) of real rational functions of a complex variable s, bounded at s = ??, with Z (jw) + Z´ (-j??) equal to a nonnegative definite Hermitian matrix for all real ??, and with ?? + ?? ?? 0 for all poles, not necessarily distinct, of Z(s). This last condition is imposed because (1) has a unique solution if and only if ?? + ?? ?? 0 for all eigenvalues of the matrix F. This means that the class {z(s)} is a proper subset of the generalized positive real matrices defined by Anderson and Moore.