Spectral Decompositions of Inverse Gramian Matrices and Energy Metrics of Continuous Dynamic Systems

The article is aimed at developing new algorithms for element-by-element calculation of matrices of direct and inverse gramians for stable continuous linear MIMO LTI systems based on spectral decompositions of gramians in the form of Hadamard products. It is shown that the multiplier matrices in the Hadamard product are invariant under various canonical transformations of linear continuous systems. Spectral decompositions of inverse matrices of gramians of continuous dynamic systems from the spectra of gramian matrices and the original dynamics matrices are also obtained. The properties of the multiplier matrices in spectral decompositions of gramians are studied. Using these results, spectral decompositions of the following energy metrics were obtained: of the volumes of attraction ellipsoids, of the matrix traces of direct and inverse controllability gramians, of the input and output system energies, of the centrality indices of energy controllability metrics and of the average minimum energy. The practical applicability of the results is considered.