Reliable algorithm for modal decomposition

This paper describes a reliable, general algorithm for modal decomposition in real arithmetic and its use in analyzing and synthesizing control logic for linear dynamic systems. We describe the numerical difficulties associated with computing the Jordan canonical form when the system has repeated, or nearly repeated, eigenvalues. A new algorithm is described that satisfactorily solves these numerical difficulties. We discuss its relation and extension to related numerical analysis research to clarify the reliability of the techniques. And finally, its implementation as a practical modal decomposition method for efficiently computing the matrix exponential, transfer functions, and frequency response is also described.