Mobius and Delta Transforms in The Unification of Continous-Disrete Space

Abstract

Abstract It is well-known that in control theory the stability region of continuoustime system is laid in the left half plane of complex space, while that of discrete-time system is dwelled inside a unit circle. The former fact might be shown by exploiting the Laplace transform and the later by utilizing the corresponding zeta transform. In this paper we revealed the connectivity of both regions by employing Mobius transform. We also used the same transform to derive continuous/discrete-time counterpart of several existing results, including Bode integral and Poisson-Jensen formula. We then demonstrated their unication property by using delta transform. Some numerical examples were provided to verify our results. Keywords: continuous-discrete unication, Mobius transform, delta transform, stability