Abstract: The application of transformations of the state equations of continuous linear and bilinear
systems to the canonical form of controllability allows one to simplify the computation of Gramians
of these systems. In this paper, we develop the method and obtain algorithms for computation of
the controllability and observability Gramians of continuous linear and bilinear stationary systems
with many inputs and one output, based on the method of spectral expansion of the Gramians and
the iterative method for computing the bilinear systems Gramians. An important feature of the
concept is the idea of separability of the Gramians expansion: separate computation of the scalar
and matrix parts of the spectral Gramian expansion reduces the sub-Gramian matrices computation
to calculation of numerical sequences of their elements. For the continuous linear systems with one
output the method and the algorithm of the spectral decomposition of the controllability Gramian
are developed in the form of Xiao matrices. Analytical expressions for the diagonal elements of the
Gramian matrices are obtained, and by making use of which the rest of the elements can be
calculated. For continuous linear systems with many outputs the spectral decompositions of the
Gramians in the form of generalized Xiao matrices are obtained, which allows us to significantly
reduce the number of calculations. The obtained results are generalized for continuous bilinear
systems with one output. Iterative spectral algorithms for computation of elements of Gramians of
these systems are proposed. Examples are given that illustrate theoretical results.