**Acknowledgements**

10*=*ð Þ 1 þ 0*:*1*s* , an exciter modeled as 1*=*ð Þ 1 þ 0*:*4*s* , and a generator with model 1*=*ð Þ 1 þ *s* . Fractional-order PID parametrization is addressed in [6] by means of

utilize Theorem 3, the fractionally commensurate Hurwitz polynomial

The so-called optimal controller parameters are listed in **Table 1**.

In the following, we focus merely on verifying the closed-loop stability based on Theorem 3, based on the parametrization results therein. To this end, the *s*-domain shifting contour N *<sup>s</sup>* � ϵ is defined with *R* ¼ 100000, *γ* ¼ 0*:*1, and *ε* ¼ 0*:*01. To

Based on **Table 1**, the stability loci in the two cases are plotted in **Figure 10**.

¼ 0. From these facts, Theorem 3

The stability loci for the two cases cannot be distinguished from each other graphically. By counting the outmost circle as one clockwise encirclement around the origin, then one can count another counterclockwise encirclement after zooming into the local region around the origin; it follows that the net encirclements number is zero. Indeed, our numerical phase increment computations in

 *s*∈ N *<sup>s</sup>*�*ε*

ensures that the closed-loop fractional-order system is stable. This coincides with

Stability is one of the imperative and thorny issues in analysis and synthesis of various types of fractional-order systems. By the literature [28–30], the frequently adopted approaches are through single/multiple complex transformation such that fractional-order characteristic polynomials are transformed into standard regularorder polynomials, and then stability testing of the concerned fractional-order systems is completed by the root distribution of the corresponding regular-order polynomials. In view of the root computation feature, such existing approaches are

In this paper, we claimed and proved an indirect approach that is meant also in the *s*-complex domain but involves no root computation at all. What is more, the main results can be interpreted and implemented graphically with locus plotting as we do in the conventional Nyquist criteria, as well as numerically without any locus plotting (or simply via complex function argument integration). This implies that the complex scaling approach is numerically tractable so that is much more

*s; β*ð Þ *s; ρ*

particle swarm optimization.

*Stability loci for cases 1 and 2.*

*Control Theory in Engineering*

**Figure 10.**

*<sup>β</sup>*ð Þ¼ *<sup>s</sup>;<sup>r</sup>* ð Þ *<sup>s</sup>* <sup>þ</sup> <sup>1</sup> *<sup>λ</sup>*þ*μ*þ<sup>4</sup> is employed.

either case yields that *N*

direct in testing methodology.

the results in [6].

**5. Conclusions**

**48**

**4.4 Numerical results for Theorem 3**

 *f C* 

The study is completed under the support of the National Natural Science Foundation of China under Grant No. 61573001.
