Analysis, Modeling and Stability of Fractional Order Differential Systems 2

6Stability of Linear FDEs Using the Nyquist Criterion

6.1. Introduction

In this chapter, we present a frequency technique for the stability analysis of linear differential systems, either with integer order or fractional order derivatives; moreover, these systems can include time delays.

Fundamentally, a linear ordinary differential equation (ODE) (i.e. with integer order derivatives) is stable if the roots of the characteristic polynomial are not situated in the right half complex plane (RHP). Obviously, the stability problem can be solved by calculating these roots. Historically, mathematicians and control engineers preferred techniques avoiding this calculation. Two families of methods, based on the complex variable theory [LEP 80], are available: the Routh–Hurwitz criterion [ROU 77, HUR 95] and the Nyquist criterion [NYQ 32].

Based on polynomial properties, the Routh–Hurwitz criterion indicates the number of RHP roots of the characteristic polynomial (or unstable poles of the ODE) owing to relations between the coefficients of this polynomial. From a practical point of view, this criterion is easy to use, but its proof is difficult to establish [HAN 96]. Moreover, for a stable ODE, it does not quantify its stability degree.

The Nyquist criterion basically deals with closed-loop systems. Based on a curve in the complex plane representing the graph of the open-loop transfer function and on a contour including the RHP, it indicates the number of unstable poles of the closed-loop ...