Return-difference and return-ratio matrices and their use in analysis and design of multivariable feedback control systems

Bode's concepts of return difference and return ratio are shown to play a fundamental role in the analysis of multivariable feedback control systems. Matrix transfer functions are regarded as operators on linear vector spaces over the field of rational functions in the complex variable s. The eigenvalues of such operators are identified as characteristic transfer functions. The corresponding characteristic frequency responses provide a simple and natural link between classical single-loop design techniques and multivariable-system feedback theory. These concepts then serve as a unifying thread in a coherent and systematic discussion of multivariable-feedback-system design techniques.