**1. Introduction**

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

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:

(i) integer-order plant with fractional-order controller, (ii) fractional-order plant with integer-order controller, and (iii) fractional-order plant with fractional-order controller. The principal reason for us to bother with fractional-order controllers is that fractional-order controllers can outperform the integer-order counterparts in many aspects. For example, it has been confirmed that fractional-order PID can provide better performances and equip designers with more parametrization freedoms (due to its distributed parameter features [4, 11–13]).

*αD<sup>r</sup>*

*DOI: http://dx.doi.org/10.5772/intechopen.88119*

convergent for each *n* � *r*>0.

identity relation *<sup>α</sup>D*<sup>0</sup>

• If *f*

L <sup>0</sup>*D<sup>r</sup>*

**33**

*<sup>t</sup> f t*ð Þ � � <sup>¼</sup> *<sup>s</sup>*

ð Þ*<sup>k</sup>* ð Þ*<sup>t</sup>* � � � *αDr*

relation (or the semigroup property) holds true: *<sup>α</sup>Dr*<sup>1</sup> *<sup>t</sup> <sup>α</sup>Dr*<sup>2</sup>

> *dm dtm <sup>α</sup>D<sup>r</sup>*

frequently used, which is given by

L <sup>0</sup>*D<sup>r</sup>*

*r*

*<sup>t</sup> f t*ð Þ � � <sup>¼</sup>

following if nothing otherwise is meant.

• If *f t*ð Þ is analytical in *<sup>t</sup>*, then *<sup>α</sup>Dr*

*<sup>t</sup> f t*ðÞ¼ <sup>1</sup>

*Nyquist-Like Stability Criteria for Fractional-Order Linear Dynamical Systems*

gamma function at *<sup>n</sup>* � *<sup>r</sup>*; by ([16], p. 160), <sup>Γ</sup>ð Þ¼ *<sup>n</sup>* � *<sup>r</sup>* <sup>Ð</sup> <sup>∞</sup>

• If *<sup>r</sup>*≥0 is an integer and *<sup>n</sup>* <sup>¼</sup> *<sup>r</sup>* <sup>þ</sup> 1, then *<sup>α</sup>Dr*

derivative of *f t*ð Þ with respect to *<sup>t</sup>*; namely, *<sup>α</sup>Dr*

• If *<sup>r</sup>* <sup>¼</sup> 0 and thus *<sup>n</sup>* <sup>¼</sup> 1, the definition formula for *<sup>α</sup>D<sup>r</sup>*

*<sup>t</sup>* <sup>½</sup>*af t*ðÞþ *bg t*ð Þ� ¼ *<sup>a</sup> <sup>α</sup>D<sup>r</sup>*

*<sup>t</sup> f t*ð Þ � � <sup>¼</sup> *<sup>α</sup>D<sup>r</sup>*<sup>2</sup>

fractional-order derivative commutes with integer-order derivative:

*tf t*ð Þ � � <sup>¼</sup> *<sup>α</sup>Dr*

ð<sup>∞</sup> 0 *e* �*st* 0*D<sup>r</sup>*

**2.2 Definition and features of FCO-LTI state-space equations**

*Dr*

input, and output vectors, respectively. In accordance with *x t*ð Þ, *u t*ð Þ

fractional-order state-space equation in the form of

*<sup>t</sup> f t*ðÞ¼ *f t*ð Þ.

Γð Þ *n* � *r*

Basic facts about fractional-order calculus are given as follows [13]:

*dn dtn* ð*t α*

where *α* ≥0 and *r*≥ 0 are real numbers while *n* ≥1 is an integer; more precisely, *n* � 1≤*r*<*n* and *n* is the smallest integer that is strictly larger than *r*. Γð Þ *n* � *r* is the

• Fractional-order differentiation and integration are linear operations. Thus

• Under some additional assumptions about *f t*ð Þ, the following additive index

*<sup>t</sup> <sup>α</sup>Dr*<sup>1</sup>

*t dm*

*<sup>t</sup>*¼*<sup>α</sup>* <sup>¼</sup> 0 for *<sup>k</sup>* <sup>¼</sup> <sup>0</sup>*,* <sup>1</sup>*,* <sup>⋯</sup>*, m* with *<sup>m</sup>* being a positive integer, then

The fractional-order calculus (1) and its properties are essentially claimed in the time domain. Therefore, it is generally difficult to handle these relations directly and explicitly. To surmount such difficulties, the Laplace transform of (1) is

*tf t*ð Þ*dt* ¼ *s*

A scalar fractional-order linear time-invariant system can be described with a

*tx t*ðÞ¼ *Ax t*ð Þþ *Bu t*ð Þ *y t*ðÞ¼ *Cx t*ð Þþ *Du t*ð Þ �

where *x t*ðÞ¼ ½ � *<sup>x</sup>*1ð Þ*<sup>t</sup> ;* <sup>⋯</sup>*; xn*ð Þ*<sup>t</sup> <sup>T</sup>* <sup>∈</sup> <sup>R</sup>*<sup>n</sup>*, *u t*ð Þ<sup>∈</sup> <sup>R</sup>, and *y t*ð Þ<sup>∈</sup> <sup>R</sup> are the state,

where *F s*ðÞ¼ Lf g *f t*ð Þ and *s* is the Laplace transform variable. Under the assumption that the initial conditions involved are zeros, it follows that

*F s*ð Þ. To simplify our notations, we denote <sup>0</sup>*D<sup>r</sup>*

*r*

*f*ð Þ*τ*

*<sup>t</sup> f t*ð Þ is analytical in *t* and *r*.

*<sup>t</sup> f t*ð Þ*f t*ð Þ � � <sup>þ</sup> *<sup>b</sup> <sup>α</sup>Dr*

*<sup>t</sup> f t*ð Þ � � <sup>¼</sup> *<sup>α</sup>Dr*1þ*r*<sup>2</sup>

*dt<sup>m</sup> f t*ð Þ � � <sup>¼</sup> *<sup>α</sup>D<sup>r</sup>*þ*<sup>m</sup> <sup>t</sup> f t*ð Þ

*F s*ðÞ�X*<sup>n</sup>*�<sup>1</sup>

*k*¼0 *s k* 0*Dr <sup>t</sup> f t*ð Þ � � � � � *t*¼0

*<sup>t</sup> f t*ð Þ*<sup>S</sup>* by *<sup>D</sup><sup>r</sup>*

ð Þ *<sup>t</sup>* � *<sup>τ</sup> <sup>r</sup>*�*n*þ<sup>1</sup> *<sup>d</sup><sup>τ</sup>* (1)

*<sup>t</sup> f t*ð Þ reduces to the ð Þ *r* þ 1 th-order

*tf t*ðÞ¼ *<sup>α</sup>D*<sup>0</sup>

*<sup>t</sup> g t*ð Þ � �*:*

*<sup>t</sup> f t*ð Þ

*f t*ð Þ*=dtr*þ<sup>1</sup>

.

*<sup>t</sup> f t*ð Þ yields the

(2)

*<sup>t</sup> f t*ð Þ in the

(3)

*τn*�*r*�1*dτ* and it is

<sup>0</sup> *<sup>e</sup>*�*<sup>τ</sup>*

*<sup>t</sup> f t*ðÞ¼ *dr*þ<sup>1</sup>

An important and unavoidable problem about fractional-order systems is stability [13–15]. As is well known, stability in integer-order LTI systems is determined by the eigenvalues distribution; namely, whether or not there are eigenvalues on the close right-half complex plane. The situation changes greatly in fractional-order LTI systems, due to its specific eigenvalue distribution patterns. More precisely, on the one hand, eigenvalues of fractional-order LTI systems cannot generally be computed in analytical and closed formulas; on the other hand, stability of the fractional-order LTI systems is reflected by the eigenvalue distribution in some case-sensitive complex sectors [13, 15], rather than simply the close right-half complex plane for regular-order LTI systems. In this paper, we revisit stability analysis in fractional commensurate order LTI (FCO-LTI) systems by exploiting the complex scaling methodology, together with the well-known argument principle for complex analysis [16]. This work is inspired by the study for structural and spectral characteristics of LTI systems that is also developed by means of the argument principle [17–19]. The complex scaling technique is a powerful tool in stability analysis and stabilization for classes of linear and/or nonlinear systems; the relevant results by the author and his colleagues can be found in [20–25]. Also around fractional-order systems, the main results of this chapter are several Nyquist-like criteria for stability with necessary and sufficient conditions [26], which can be interpreted and implemented either graphically with loci plotting or numerically without loci plotting, independent of any prior pole distribution and complex/frequency-domain facts.

Outline of the paper. Section 2 reviews basic concepts and propositions about stability in FCO-LTI systems that are depicted by fractional commensurate order differential equations or state-space equations. The main results of the study are explicated in Section 3. Numerical examples are sketched in Section 4, whereas conclusions are given in Section 5.

*Notations and terminologies of the paper*. R and C denote the sets of all real and complex numbers, respectively. *Ik* denotes the *k* � *k* identity matrix, while C<sup>þ</sup> is the open right-half complex plane, namely, <sup>C</sup><sup>þ</sup> <sup>¼</sup> f g *<sup>s</sup>*∈<sup>C</sup> : <sup>R</sup>*e s*½ �><sup>0</sup> . ð Þ� <sup>∗</sup> means the conjugate transpose of a matrix ð Þ� . *N*ð Þ� , *Nc*ð Þ� , and *Nc*ð Þ� stand for the net, clockwise, and counterclockwise encirclements of a closed complex curve ð Þ� around the origin 0ð Þ *; j*0 . By definition, *N*ðÞ¼ � *Nc*ðÞ�� *Nc*ð Þ� . In particular, *N*ðÞ¼ � 0 means that the number of clockwise encirclements of ð Þ� around the origin is equal to that of counterclockwise encirclements.
