Numerical Stability Analysis of Linear Incommensurate Fractional Order Systems

We present a method for detecting right half plane (RHP) roots of fractional order polynomials. It is based on a Nyquist-like criterion with a system-dependent contour which includes all RHP roots. We numerically count the number of origin encirclements of the mapped contour to determine the number of RHP roots. The method is implemented in Matlab, and a simple code is given. For validation, we use a Galerkin based strategy, which numerically computes system eigenvalues (Matlab code is given). We discuss how, unlike integer order polynomials, fractional order polynomials can sometimes have exponentially large roots. For computing such roots we suggest using asymptotics, which provide intuition but require human inputs (several examples are given).