Delta-operator based efficient stability tests

Three different kinds of stability test algorithms are proposed when a real polynomial formulated in the delta-operator is given. The method used is a simple and clear linear transformation based on some known discrete-time tests including the Schur-Cohn test. But it results in several new tests in both the discrete-time with the S-operator formulation and the continuous-time. It is shown that as the sampling interval vanishes all new algorithms converge to their continuous-time limits, therefore providing smooth transitions from the shift-operator based discrete-time algorithms to the continuous-time ones. This establishes three separate but related families of stability tests, including the well known Schur-Cohn test and Routh test. Examples are given to demonstrate the δ-operator based algorithms which show clear numerical advantages over the traditional shift-operator based ones such as the Schur-Cohn test in case of fast sampling