Abstract:
Spectral decomposition of the quadratic Lyapunov function is already known. The modal contribution and modal interaction are the main components of this decomposition, which form the basis of Lyapunov modal analysis. This paper presents the results of a further expansion of these spectral components into individual state variables and into their pairwise combinations. The obtained result can also be considered as a decomposition of the quadratic Lyapunov function not only by elements of the spectrum of the dynamical system (by modes), but also by elements of the state space. On the basis of the proposed decomposition method, new Lyapunov modal analysis indices are formulated, which allow one to evaluate the contribution of individual eigenvalues or their pairwise interaction, but in connection only with the part of the external perturbation that is associated with a particular state variable or a pair of such variables. This, in particular, makes it possible to comprehensively evaluate the joint influence of both the mode and the associated state variable on the energy of the system output signal. It is assumed that the main area of application of the new decompositions will be related to the problems of the model order reduction of the large dynamical systems.
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