On the Theory of Linear Multi-Loop Feedback Systems

It is well known that the concept of return difference plays a central role in the classical theory of linear feedback systems developed by Black, 1 Nyquist, 2 Blackmail 3 and Bode. 1 This concept, which relates to the influence of a single algebraic system-constraint of the f o r m / 2 = 7/1 where/1 and / 2 respectively may lie regarded as a controlling signal and a controlled signal, retains its prominence in the subsequent signal-flow graph theoretic extensions by Mason. 5 It is particularly pertinent to the study of the stability of the system, its degree of immunity from parameter variations, and the determination of its transmission and drivingpoint properties. This paper reports on a generalization of Blackmail's equation 3 and 011 some generalizations of Bode's return difference theorems. 4 Here attention is focused not on a single constraint of the form f2 = 7/1 , but instead 011 a set of constraining equations. For this reason t he results 355 356 T H E BELL SYSTEM TECHNICAL J O U R N A L , MARCH 1963