Section 2 Stability Analysis

**Chapter 2**

*Jun Zhou*

**Abstract**

**1. Introduction**

**31**

Nyquist-Like Stability Criteria

In this chapter, we propose several Nyquist-like stability criteria for linear dynamical systems that are described by fractional commensurate order linear time-invariant (FCO-LTI) state-space equations (thus endowed with fractional-order transfer functions) by means of the argument principle for complex analysis. Based on the standard Cauchy integral contour or its shifting ones, the stability conditions are necessary and sufficient, independent of any intermediate poles computation, domain transformation, and distribution investigation, which can be implemented graphically with locus plotting or numerically without any locus plotting. The proposed criteria apply to both single and multiple fractional cases as well and can be exploited in regular-order

**Keywords:** fractional-order, commensurate, stability, meromorphic/holomorphic,

Fractional-order calculus possesses a long history in pure mathematics. In recent

One of the major difficulties for us to exploit the fractional-order models is the absence of solution formulas for fractional-order differential equations. Lately, lots of numerical methods for approximate solution of fractional-order derivative and integral are suggested such that fractional-order calculus can be solved numerically. As far as fractional-order systems and their control are concerned, there are mainly three schools related to fractional-order calculus in terms of system configuration:

decades, its involvements in systems, control, and engineering have attracted great attention; in the latest years, its significant extensions in various aspects of systems and control are frequently encountered [1–8]. It turns out that phenomena modeled with fractional-order calculus much more widely exist than those based on regular-order ones. It has been shown that fractional-order calculus describes realworld dynamics and behaviors more accurately than the regular-order counterparts and embraces many more analytical features and numerical properties of the observed things; indeed, many practical plants and objects are essentially fractionalorder. Without exhausting the literature, typical examples include the so-called non-integer-order system of the voltage–current relation of semi-infinite lossy transmission line [9] and diffusion of the heat through a semi-infinite solid, where

heat flow is equal to the half-derivative of the temperature [10].

for Fractional-Order Linear

Dynamical Systems

systems without any modification. Case study is included.

argument principle, Cauchy integral contour
