This paper provides a new approach to computing infinite and finite time Gramians and cross-Gramians relying on the usage of the Laplacian transformation of matrix exponential time functions and expansion of the product of these functions. The expansions are bilinear and quadratic forms of sequences of the Faddeev matrices generated by resolvents of original matrices. Matrix identities are obtained for bilinear and quadratic forms of these Faddeev sequences. Asymptotic expansions of
controllability and observability Gramians of dynamic systems at the proximity of stability limit are found. Illustrative examples are given to demonstrate the perspective of using Gramians for the smallsignal stability analysis of the electric power systems.
