A Robust Stability Test Procedure for a Class of Uncertain Lti Fractional Order Systems

For the first time, in this paper, a robust stability test procedure is proposed for linear time-invariant fractional order systems (LTI FOS) of commensurate orders with interval uncertain parameters. For the LTI FOS with no uncertainty, the existing stability test (or check) methods for dynamic systems with integer-orders such as Routh table technique, cannot be directly applied. This is due to the fact that the characteristic equation of the LTI FOS is, in general, not a polynomial but a pseudo-polynomial function of the fractional powers of the complex variable s. Of course, being the characteristic equation a function of a complex variable, stability test based on the argument principle can be applied. On the other hand, it has been shown, by several authors and by using several methods, that for the case of LTI FOS of commensurate order, a geometrical method based on the argument of the roots of the characteristic equation (a polynomial in this particular case) can be used for the stability check in the BIBO sense (bounded-input boundedoutput). In this paper, we demonstrated this technique for the stability check for LTI FOS with parametric interval uncertainties through a worked-out illustrative example. In this example, time-domain analytical expression are available. Therefore, time-domain and frequency-domain stability test results can be cross-validated.