On BIBO stability of systems with irrational transfer function

Abstract

We consider the input/output-stability of linear time-invariant single-input/single-output systems in terms of singularities of the transfer function F(s) in Laplace domain. The approach is based on complex analysis. A fairly general class of transfer functions F(s) is considered which can roughly be characterized by two properties: (1) F(s) is meromorphic with a finite number of poles in the open right half plane and (2), on the imaginary axes, F(s) may have at most a finite number of poles and branch points. For this class of systems, a complete and thorough characterization of BIBO stability in terms of the singularities in the closed right half-plane is developed where no necessity arises to differentiate between commensurate and incommensurate orders of branch points as often done in literature. A necessary and sufficient condition for BIBO stability is derived which mainly relies on the asymptotic expansion of the impulse response f(t). The second main result is the generalization of the well-known Nyquist citerion for testing closed-loop stability in terms of the open-loop frequency response locus to the class of systems under consideration which is gained by applying Cauchy's argument principle.