On the Essential Instabilities Caused by Fractional-Order Transfer Functions

摘要

The exact stability condition for certain class offractional-order (multivalued) transfer functions is presented.Unlike the conventional case that the stability is directly studiedby investigating the poles of the transfer function, in the systemsunder consideration, the branch points must also come intoaccount as another kind of singularities. It is shown that amultivalued transfer function can behave unstably because ofthe numerator term while it has no unstable poles. So, in thiscase, not only the characteristic equation but the numeratorterm is of significant importance. In this manner, a family ofunstable fractional-order transfer functions is introduced whichexhibit essential instabilities, that is, those which cannot beremoved by feedback. Two illustrative examples are presented;the transfer function of which has no unstable poles but theinstability occurred because of the unstable branch points ofthe numerator term. The effect of unstable branch points isstudied and simulations are presented.