Spectral decompositions of gramians and energy metrics of continuous unstable control systems

Deterministic continuous finite-dimensional stationary linear dynamic control systems with many inputs and many outputs are considered. Authors assume that the dynamics matrix can be both stable and unstable, but its eigenvalues are different, do not belong to the imaginary axis, and their pairwise sum is not equal to 0. The problems of constructing spectral solutions of the equations of state and matrices of gramian controllability of these systems, as well as the associated energy functionals of the degree of stability and reachability with the aim of optimal placement of sensors and actuators of multi-connected control systems and complex networks are considered. To solve the listed problems, the article uses various models of the system in state space: a general representation, as well as a representation in various canonical forms. To calculate the spectral decompositions of controllability gramians, pseudo-Hankel matrices (Xiao matrices) are used. New methods have been proposed and algorithms have been developed for calculating controllability gramians and energy metrics of linear systems. The research results can be used for the optimal placement of sensors and actuators of multi-connected control systems or for control with minimal energy in complex networks of various natures.