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

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.

Spectral Decomposition of Gramians of Continuous Linear Systems in the Form of Hadamard Products

“…In [26], the general formulas for computing the spectral expansion of Gramians are derived, which are also applicable to the modal equations of MIMO LTI systems…”

“…This paper shows that the Gramians' Hadamard decomposition and their multiplier matrices play an important role in the problems of analyzing structural properties for a wide class of continuous linear dynamical systems given by their different equations of state. In the Introduction, we noted the important role of the scientific direction of works [22][23][24][25][26][27] based on the use of Rouse tables and Xiao matrices to compute finite and infinite Gramians by transforming the equations of state into canonical forms. Let us discuss the question, what is the difference between the methods and algorithms developed in this paper and those of the first direction?…”

“…Note that most of the mentioned works use the spectrum of the system dynamics matrix, which makes it quite natural to apply spectral analysis methods to solve the above problems. The use of canonical forms of controllability previously gave rise to a new approach to Gramian computation based on the use of Rouse-Gurwitz tables and Xiao matrices [22][23][24][25][26]. Almost 30 years ago, a paper [23] appeared in which a method and algorithm for computing solutions to Lyapunov equations are proposed for the case when the equations of state of a linear system are given in the canonical form of controllability or observability.…”

“…The singularity of solutions to modal equations follows from the stability of these equations. The coincidence of the solution matrices of the modal equations follows from the coincidence of the solution matrices of the original equations P c and P o [26]. In this article, analytical expressions of the solution matrices in the form of spectral expansions for a simple spectrum were derived as…”

“…Let us prove the fulfillment of the first property of Xiao matrices (3). The fulfillment for zero elements of matrices is proved in [26]. The alternation of signs of the side diagonal elements passing through diagonal element p jj follows from the sequence of these elements in the form…”