Eigenvalue spectra and bounds for certain classes of dynamic systems having tree bond graphs

For a class of linear, time-invariant dynamic systems, information about the eigenvalues of the system can be obtained directly from a graphical model of the system, namely the bond graph model. In contrast to the standard approach, which starts from the state matrix of the system, we use a canonical form of the bond graph, namely the gyrobondgraph, to estimate bounds on the eigenvalues of the associated system. For some special cases, the entire spectrum of the system can be obtained by comparing the graph structure to existing graphs with known spectra. For more general cases, bounds are obtained on the largest real and imaginary parts of the eigenvalues as functions of the system's gyrobondgraph topology and its parameters. For some classes of systems, we present a simple graph-based recurrence formula for deriving the characteristic polynomial of a system from a simplified form of its gyrobondgraph. These results, when suitably automated, can reduce the time required to estimate the system performance.