MOBIUS AND DELTA TRANSFORMS IN THE UNIFICATION OF CONTINUOUS-DISCRETE SPACES

It is well-known that in control theory the stability region of continuous- time 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 M¨obius 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 unification property by using delta transform. Some numerical examples were provided to verify our results

A Novel Delta Operator Kalman Filter Design and Convergence Analysis

This paper focuses on the development of a delta operator Kalman filter and its convergence analysis. The delta operator Kalman filter is designed to estimate the state vectors of a delta operator system. Note that the designed delta operator Kalman filter can express both continuous-time and discrete-time cases. Then, the convergence analysis of the delta operator Kalman filter is also investigated by using Lyapunov approach in delta domain. Furthermore, this paper gives fundamental results for the analysis and application of the delta operator Kalman filter as a state observer in an inverted pendulum model. Some experimental results of an inverted pendulum on a laboratory-scale setup are presented to illustrate the effectiveness of the designed Kalman filter and its implementation