Spectral Decompositions of Gramians of Continuous Stationary Systems Given by Equations of State in Canonical Forms

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.