Partial-fraction expansion in system analysis

Abstract

Abstract A procedure is given for finding the partial-fraction expansion of strictly proper rational transfer functions that are commonly found in systems analysis. Eight formulae for the expansions are given to include the cases: (a) scalar transfer functions, (b) matrix transfer functions, (c) left and right matrix fraction descriptions, (d) transfer function matrices, and (e) block expansion of left and right matrix fraction descriptions. These formulae are derived from system theory using the observability canonical form and do not require the use of derivatives, the particular nature of the Jordan form, or interpolating polynomials; rather, it is required that the Vandermonde matrix be formed from the roots, latent roots, eigenvalues or solvents of the denominator. The main features of the procedure include that all the residues are found at the same time by simple matrix manipulation and can be used for the repeated and/or non-repeated case. The structured form of the formulae is very simple and fil...