The Encirclement Condition, An Approach using Algebraic Topology.

摘要

The usual proof of the Nyquist theorem depends heavily on the argument principle. The argument principle, however, supplies unneeded information in that it counts the number of encirclements of '-1'. System stability requires only a binary decision on the number of encirclements. Using homotopy theory, a branch of algebraic topology, new proofs of the classical Nyquist criteria are constructed along with the more recent linear multivariable results. In both cases the proof is essentially the same. Moreover, this point of view to be more intuitive and capable of generalization to systems characterized by functions of several complex variables.