An Algebraic Topological Approach to Stability Theory

Résumé du livre

Stability theory using frequency domain techniques essentially began with the advent of the Nyquist Criterion. The real meaning of this criterion seemed clouded by the complexity of its proof. The usual proof depends heavily on the argument principle. This 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 Criterion along with the more recent multivariable results are constructed. In both cases the proofs are essentially the same. Moreover they offer an intuitively satisfying approach to the stability question. These ideas are then generalized to a Nyquist-like test for the stability of multidimensional digital filters. Basically the test is a series of Nyquist tests, each of which is more or less easily implemented on a computer. In particular a kinship between the distinguished boundary of the polydisk and the classical Nyquist contour is established. A special form of homotopy is also defined. It is this special definition of homotopy that suitably describes the stability of multidimensional digital filters characterized by transfer functions in several complex variables.