Analysis of the Lyapunov equation using generalized positive real matrices
Bradley W. Dickinson · 1979
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' (-jω) equal to a nonnegative definite Hermitian matrix for all real ∞, and with λ + µ ≠ 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 λ + µ ≠ 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.