**4. Numerical illustrations**

#### **4.1 Example description for Theorems 1 and 2**

Consider a single fractional-order commensurate system [15] with the characteristic polynomial

$$s^{2r} + 2as^r + b = 0, \quad a, b \in \mathcal{R}$$

where the commensurate order 0<sup>&</sup>lt; *<sup>r</sup>* <sup>¼</sup> *<sup>k</sup> <sup>m</sup>* ≤ 1 with *k*, *m*, and *k*≤ *m* being positive integers as appropriately. In all the numerical simulations based on Theorem 1, the fractionally commensurate Hurwitz polynomial *<sup>β</sup>*ð Þ¼ *<sup>s</sup>;<sup>r</sup>* ð Þ *<sup>s</sup>* <sup>þ</sup> <sup>1</sup> <sup>2</sup>*<sup>r</sup>* is employed. In all the numerical simulations based on Theorem 2, the *πr*-sector Hurwitz polynomial *<sup>α</sup>*ð Þ¼ *<sup>z</sup>;<sup>r</sup>* ð Þ *<sup>z</sup>* <sup>þ</sup> <sup>0</sup>*:*<sup>1</sup> <sup>2</sup> is adopted.

In what follows, the *s*-domain contour N *<sup>s</sup>* is defined with *γ* ¼ 0*:*01 and *R* ¼ 100000, while the *z*-domain contour N *<sup>z</sup>* is defined with *γ* ¼ 0*:*01 and *R* ¼ 10000.

#### **4.2 Numerical results for Theorems 1 and 2**

The following cases are considered in terms of *a* and *b*. In each figure, the lefthand sub-figure plots the stability locus in terms of *f s*ð Þj *; β*ð Þ *s;r <sup>s</sup>*<sup>∈</sup> <sup>N</sup> *<sup>s</sup>* , or simply the *s*locus, for some fixed *a* and , while the right-hand sub-figure presents the stability locus in terms of *g z*ð Þj *; α*ð Þ *z;r <sup>z</sup>*<sup>∈</sup> <sup>N</sup> *<sup>z</sup>* , or simply the *z*-locus.

**Figure 4.** *Stability loci with a* ¼ *2 and b* ¼ *1.*

*Nyquist-Like Stability Criteria for Fractional-Order Linear Dynamical Systems DOI: http://dx.doi.org/10.5772/intechopen.88119*

• *a*>0, *b*>0, and *a*<sup>2</sup> ≥*b*. By examining the *s*-loci of **Figure 4** graphically, no encirclements around the origin are counted in each case of 

*r*∈f g 0*:*2*;* 0*:*4*;* 0*:*6*;* 0*:*8*;* 1*:*0 ; indeed, *N f s*ð Þ *; β*ð Þ *s;r s*∈ N *<sup>s</sup>* ¼ 0 for each

*r*∈f g 0*:*2*;* 0*:*4*;* 0*:*6*;* 0*:*8*;* 1*:*0 can be verified numerically without locus plotting. Therefore, the system is stable in each case.

The same conclusions can be drawn by examining the *z*-loci of **Figure 4**. More precisely, we have *N f z*ð Þ *; α*ð Þ *z;r z*∈ N *<sup>z</sup>* ¼ 0 for each *r*∈f g 0*:*2*;* 0*:*4*;* 0*:*6*;* 0*:*8*;* 1*:*0 numerically.

• *a*>0, *b*>0, and *a*<sup>2</sup> <*b*. By the *s*-loci of **Figure 5**, no encirclements around the origin are counted in each case of *r*∈f g 0*:*2*;* 0*:*4*;* 0*:*6*;* 0*:*8*;* 1*:*0 graphically; or *N f s; β*ð Þ *s;r s*∈ N *<sup>s</sup>* ¼ 0 for each *r*∈f g 0*:*2*;* 0*:*4*;* 0*:*6*;* 0*:*8*;* 1*:*0 numerically. Therefore, the system is stable in each case.

The same conclusions can be drawn by examining the *z*-loci of **Figure 5**. More precisely, we have *N f z; α*ð Þ *z;r z* ∈ N *<sup>z</sup>* ¼ 0 for each *r*∈f g 0*:*2*;* 0*:*4*;* 0*:*6*;* 0*:*8*;* 1*:*0 numerically.

• *a*<0, *b*<0, and thus *a*<sup>2</sup> ≥*b* holds always. By the *s*-loci of **Figure 6**, one net encirclement around the origin is counted in each case of *r*∈f g 0*:*2*;* 0*:*4*;* 0*:*6*;* 0*:*8*;* 1*:*0 graphically; alternatively, *N f s; β*ð Þ *s;r s*∈ N *<sup>s</sup>* ∣ ¼ 1 for each

*r*∈f g 0*:*2*;* 0*:*4*;* 0*:*6*;* 0*:*8*;* 1*:*0 numerically. Therefore, the system is unstable in each case. The instability conclusions in each *r*∈f g 0*:*2*;* 0*:*4*;* 0*:*6*;* 0*:*8*;* 1*:*0 can be revealed by means of the *z*-loci of **Figure 6**. More precisely, we have *N f z; α*ð Þ *z;r z* ∈ N *<sup>z</sup>* ¼ 1 for each *r*∈f g 0*:*2*;* 0*:*4*;* 0*:*6*;* 0*:*8*;* 1*:*0 numerically.

**Figure 5.** *Stability loci with a* ¼ *1=2 and b* ¼ *1.*

**Figure 6.** *Stability loci with a =* �*1 and b =* �*1.*

<sup>Δ</sup><sup>~</sup> *<sup>G</sup>*ð Þ¼ *<sup>z</sup>; <sup>ρ</sup>* <sup>Δ</sup>*G*ð Þ *<sup>s</sup>; <sup>ρ</sup> <sup>s</sup>ρ*¼*z;* <sup>Δ</sup><sup>~</sup> *<sup>H</sup>*ð*z; <sup>ρ</sup>*Þ ¼ <sup>Δ</sup>*H*ð*s; <sup>ρ</sup>*<sup>Þ</sup> �

<sup>~</sup> ð Þ¼ *G s*ð Þ � � �

In the above, N *<sup>z</sup>* � ϵ is the contour by shifting N *<sup>z</sup>* to its left with distance ϵ.

• The shifted contour N *<sup>s</sup>* � ϵ reduces to the standard contour N *<sup>s</sup>* when ϵ ¼ 0.

• Clearly, the detouring treatments in Theorems 3 and 4 do not exist in Theorems 1 and 2, since the stability conditions in the latter ones are claimed directly on the fractional-order characteristic polynomials, in which transfer functions are not

Consider a single fractional-order commensurate system [15] with the charac-

<sup>2</sup>*<sup>r</sup>* <sup>þ</sup> <sup>2</sup>*asr* <sup>þ</sup> *<sup>b</sup>* <sup>¼</sup> <sup>0</sup>*, a, b*<sup>∈</sup> <sup>R</sup>

integers as appropriately. In all the numerical simulations based on Theorem 1, the fractionally commensurate Hurwitz polynomial *<sup>β</sup>*ð Þ¼ *<sup>s</sup>;<sup>r</sup>* ð Þ *<sup>s</sup>* <sup>þ</sup> <sup>1</sup> <sup>2</sup>*<sup>r</sup>* is employed. In all the numerical simulations based on Theorem 2, the *πr*-sector Hurwitz polynomial

The following cases are considered in terms of *a* and *b*. In each figure, the left-

, or simply the *z*-locus.

locus, for some fixed *a* and , while the right-hand sub-figure presents the stability

In what follows, the *s*-domain contour N *<sup>s</sup>* is defined with *γ* ¼ 0*:*01 and *R* ¼ 100000, while the *z*-domain contour N *<sup>z</sup>* is defined with *γ* ¼ 0*:*01 and

hand sub-figure plots the stability locus in terms of *f s*ð Þj *; β*ð Þ *s;r <sup>s</sup>*<sup>∈</sup> <sup>N</sup> *<sup>s</sup>*

*H z*

Several remarks about Theorems 3 and 4:

**4.1 Example description for Theorems 1 and 2**

where the commensurate order 0<sup>&</sup>lt; *<sup>r</sup>* <sup>¼</sup> *<sup>k</sup>*

**4.2 Numerical results for Theorems 1 and 2**

*s*

(

*Control Theory in Engineering*

**4. Numerical illustrations**

*<sup>α</sup>*ð Þ¼ *<sup>z</sup>;<sup>r</sup>* ð Þ *<sup>z</sup>* <sup>þ</sup> <sup>0</sup>*:*<sup>1</sup> <sup>2</sup> is adopted.

locus in terms of *g z*ð Þj *; α*ð Þ *z;r <sup>z</sup>*<sup>∈</sup> <sup>N</sup> *<sup>z</sup>*

teristic polynomial

*R* ¼ 10000.

**Figure 4.**

**44**

*Stability loci with a* ¼ *2 and b* ¼ *1.*

involved.

<sup>~</sup> ð Þ¼ *H s*ð Þ *<sup>s</sup>ρ*¼*z; G z*

This is also the case for the shifted contour N *<sup>z</sup>* � ϵ and N *<sup>z</sup>*.

� �

� *sρ*¼*z* � *sρ*¼*z*

*<sup>m</sup>* ≤ 1 with *k*, *m*, and *k*≤ *m* being positive

, or simply the *s*-

• *a*>0, *b*<0, and thus *a*<sup>2</sup> ≥*b* holds always. By the *s*-loci of **Figure 7**, one net encirclement around the origin is counted in each case of *r*∈ f g 0*:*2*;* 0*:*4*;* 0*:*6*;* 0*:*8*;* 1*:*0 graphically; alternatively, *N f s; β*ð Þ *s;r s*∈ N *<sup>s</sup>* ∣ ¼ 1 for each

*r*∈f g 0*:*2*;* 0*:*4*;* 0*:*6*;* 0*:*8*;* 1*:*0 numerically. Therefore, the system is unstable in each case. The instability conclusions in each *r*∈f g 0*:*2*;* 0*:*4*;* 0*:*6*;* 0*:*8*;* 1*:*0 can be drawn again by means of the *z*-loci of **Figure 7**. More precisely, we have

*N f z; α*ð Þ *z;r z*∈ N *<sup>z</sup>* ¼ 1 for each *r*∈f g 0*:*2*;* 0*:*4*;* 0*:*6*;* 0*:*8*;* 1*:*0 numerically.

• *a*<0, *b*>0, and *a*<sup>2</sup> ≥*b*. By the *s*-loci of **Figure 8**, one net encirclement around the origin is counted in the case of *r* ¼ 0*:*2 graphically, or *N f s; β*ð Þ *s;r s*∈ N *<sup>s</sup>* ∣ ¼ 1 for *r* ¼ 0*:*2 numerically; two net encirclements around the origin are counted in each case of *r*∈f g 0*:*4*;* 0*:*6*;* 0*:*8*;* 1*:*0 , or *N f s; β*ð Þ *s;r s*∈ N *<sup>s</sup>* ∣ ¼ 2 for each *r*∈f g 0*:*4*;* 0*:*6*;* 0*:*8*;* 1*:*0 numerically. Therefore, the system is unstable in each case of *r*∈f g 0*:*2*;* 0*:*4*;* 0*:*6*;* 0*:*8*;* 1*:*0 .

The instability conclusions in each *r*∈f g 0*:*2*;* 0*:*4*;* 0*:*6*;* 0*:*8*;* 1*:*0 can be drawn again by means of the *z*-loci of **Figure 8**. More specifically, we have

*N f z; α*ð Þ *z;r z*∈ N *<sup>z</sup>* ¼ 1 for *r* ¼ 0*:*2 and *N f z; α*ð Þ *z;r z*∈ N *<sup>z</sup>* ¼ 2 for each *r*∈f g 0*:*4*;* 0*:*6*;* 0*:*8*;* 1*:*0 numerically.

• *a*<0, *b*>0, and *a*<sup>2</sup> <*b*. By the *s*-loci of **Figure 9**, no encirclements around the origin are counted in each case of *r*∈f g 0*:*2*;* 0*:*4*;* 0*:*6 graphically, or *N f s; β*ð Þ *s;r s*∈ N *<sup>s</sup>* ∣ ¼ 0 for each *r*∈f g 0*:*2*;* 0*:*4*;* 0*:*6 numerically. Therefore, the system is stable in each case of *r*∈f g 0*:*2*;* 0*:*4*;* 0*:*6 . However, two net encirclements around the origin are counted in each case of *r*∈f g 0*:*8*;* 1*:*0 graphically or

*N f* 

**Table 1.**

**Figure 9.**

have *N f* 

*N f* 

*s; β*ð Þ *s;r* 

*PID controller parameters.*

*z; α*ð Þ *z;r*

more generally.

*G s*ðÞ¼ *KP* þ

*KP* þ *KI= s*

1

**47**

 *s*∈ N *<sup>s</sup>* 

*Stability loci with a* ¼ �*1=2 and b* ¼ *1.*

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

*z; α*ð Þ *z;r*

 *z*∈ N *<sup>z</sup>* 

 *z* ∈ N *<sup>z</sup>* 

**4.3 Example description for Theorem 3**

*KI <sup>s</sup><sup>λ</sup>* <sup>þ</sup> <sup>0</sup>*:*<sup>0001</sup> <sup>þ</sup>

*<sup>λ</sup>* <sup>þ</sup> <sup>0</sup>*:*<sup>0001</sup> <sup>þ</sup> <sup>100</sup>*KDs*

consisting of a fractional-order PID in the form of

definition problem at the origin when it is in the form of *KI=s*

system is unstable in either case of *r*∈f g 0*:*8*;* 1*:*0 .

∣ ¼ 2 for each *r*∈f g 0*:*8*;* 1*:*0 numerically. Therefore, the

*KP KI KD μ λ*

Stability in each case of *r*∈f g 0*:*2*;* 0*:*4*;* 0*:*6 and instability for either case of *r*∈f g 0*:*8*;* 1*:*0 can be verified by the *z*-loci of **Figure 9** as appropriately. Indeed, we

Case 1 1.2623 0.5531 0.2382 1.2555 1.1827 Case 2 1.2623 0.5526 0.2381 1.2559 1.1832

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

¼ 0 for *r*∈f g 0*:*2*;* 0*:*4*;* 0*:*6 and

¼ 2 for each *r*∈ f g 0*:*8*;* 1*:*0 numerically.

Consider the feedback configuration of **Figure 1** used for automatic voltage

where *H s*ð Þ is a regular-order sensor model and *G s*ð Þ is a cascading model

*<sup>μ</sup><sup>=</sup> <sup>s</sup>*ð Þ *<sup>μ</sup>* <sup>þ</sup> <sup>100</sup> <sup>1</sup>

ð Þ 1 þ 0*:*1*s* ð Þ 1 þ 0*:*4*s* ð Þ 1 þ *s*

*λ*.

, an amplifier modeled as

*,H s*ðÞ¼ <sup>1</sup>

*<sup>λ</sup>* <sup>þ</sup> <sup>0</sup>*:*<sup>0001</sup> in order to avoid

1 þ 0*:*01*s*

regulator (AVR) in generators, which is formed by the subsystems [6]:

100*KDs μ*

*<sup>s</sup><sup>μ</sup>* <sup>þ</sup> <sup>100</sup> <sup>10</sup>

The fractional-order integral portion in the PID is approximated by *KI= s*

Based on the numerical results, the stability/instability conclusions based on the *s*-loci completely coincide with those drawn based on the *z*-loci. This reflects the fact of Remark 2. These numerical results are also in accordance with those by [15] about the same example, which are summarized by working with solving polynomial roots. It is worth mentioning that polynomial roots are not always solvable in general. Fortunately, the suggested Nyquist-like criteria can be implemented graphically and numerically, independent of any polynomial root solution and inter-complex-plane transformation. Hence, the suggested technique is applicable

**Figure 7.** *Stability loci with a = 1 and b =* �*1.*

**Figure 8.** *Stability loci with a* ¼ �*2 and b* ¼ *1.*

*Nyquist-Like Stability Criteria for Fractional-Order Linear Dynamical Systems DOI: http://dx.doi.org/10.5772/intechopen.88119*

**Figure 9.** *Stability loci with a* ¼ �*1=2 and b* ¼ *1.*


**Table 1.** *PID controller parameters.*

• *a*>0, *b*<0, and thus *a*<sup>2</sup> ≥*b* holds always. By the *s*-loci of **Figure 7**, one net encirclement around the origin is counted in each case of *r*∈ f g 0*:*2*;* 0*:*4*;* 0*:*6*;* 0*:*8*;* 1*:*0

 *s*∈ N *<sup>s</sup>* 

*r*∈f g 0*:*2*;* 0*:*4*;* 0*:*6*;* 0*:*8*;* 1*:*0 numerically. Therefore, the system is unstable in each case. The instability conclusions in each *r*∈f g 0*:*2*;* 0*:*4*;* 0*:*6*;* 0*:*8*;* 1*:*0 can be drawn again

• *a*<0, *b*>0, and *a*<sup>2</sup> ≥*b*. By the *s*-loci of **Figure 8**, one net encirclement around

for *r* ¼ 0*:*2 numerically; two net encirclements around the origin are counted in

 *f* 

*r*∈f g 0*:*4*;* 0*:*6*;* 0*:*8*;* 1*:*0 numerically. Therefore, the system is unstable in each case of

The instability conclusions in each *r*∈f g 0*:*2*;* 0*:*4*;* 0*:*6*;* 0*:*8*;* 1*:*0 can be drawn again

• *a*<0, *b*>0, and *a*<sup>2</sup> <*b*. By the *s*-loci of **Figure 9**, no encirclements around the

system is stable in each case of *r*∈f g 0*:*2*;* 0*:*4*;* 0*:*6 . However, two net encirclements

around the origin are counted in each case of *r*∈f g 0*:*8*;* 1*:*0 graphically or

 *f* 

∣ ¼ 1 for each

 *f* 

 *z*∈ N *<sup>z</sup>* 

 *s*∈ N *<sup>s</sup>* 

*z; α*ð Þ *z;r*

∣ ¼ 0 for each *r*∈f g 0*:*2*;* 0*:*4*;* 0*:*6 numerically. Therefore, the

*s; β*ð Þ *s;r* 

∣ ¼ 2 for each

 *s*∈ N *<sup>s</sup>* ∣ ¼ 1

¼ 2 for each

¼ 1 for each *r*∈f g 0*:*2*;* 0*:*4*;* 0*:*6*;* 0*:*8*;* 1*:*0 numerically.

*s; β*ð Þ *s;r* 

 *f* 

by means of the *z*-loci of **Figure 7**. More precisely, we have

the origin is counted in the case of *r* ¼ 0*:*2 graphically, or *N*

by means of the *z*-loci of **Figure 8**. More specifically, we have

¼ 1 for *r* ¼ 0*:*2 and *N*

origin are counted in each case of *r*∈f g 0*:*2*;* 0*:*4*;* 0*:*6 graphically, or

*s; β*ð Þ *s;r* 

graphically; alternatively, *N*

*Control Theory in Engineering*

 *z*∈ N *<sup>z</sup>* 

*r*∈f g 0*:*2*;* 0*:*4*;* 0*:*6*;* 0*:*8*;* 1*:*0 .

 *z*∈ N *<sup>z</sup>* 

*r*∈f g 0*:*4*;* 0*:*6*;* 0*:*8*;* 1*:*0 numerically.

 *s*∈ N *<sup>s</sup>* 

*z; α*ð Þ *z;r*

*s; β*ð Þ *s;r* 

each case of *r*∈f g 0*:*4*;* 0*:*6*;* 0*:*8*;* 1*:*0 , or *N*

*z; α*ð Þ *z;r*

*N f* 

*N f* 

> *N f*

**Figure 7.**

**Figure 8.**

**46**

*Stability loci with a = 1 and b =* �*1.*

*Stability loci with a* ¼ �*2 and b* ¼ *1.*

*N f s; β*ð Þ *s;r s*∈ N *<sup>s</sup>* ∣ ¼ 2 for each *r*∈f g 0*:*8*;* 1*:*0 numerically. Therefore, the system is unstable in either case of *r*∈f g 0*:*8*;* 1*:*0 .

Stability in each case of *r*∈f g 0*:*2*;* 0*:*4*;* 0*:*6 and instability for either case of *r*∈f g 0*:*8*;* 1*:*0 can be verified by the *z*-loci of **Figure 9** as appropriately. Indeed, we have *N f z; α*ð Þ *z;r z* ∈ N *<sup>z</sup>* ¼ 0 for *r*∈f g 0*:*2*;* 0*:*4*;* 0*:*6 and *N f z; α*ð Þ *z;r z*∈ N *<sup>z</sup>* ¼ 2 for each *r*∈ f g 0*:*8*;* 1*:*0 numerically.

Based on the numerical results, the stability/instability conclusions based on the *s*-loci completely coincide with those drawn based on the *z*-loci. This reflects the fact of Remark 2. These numerical results are also in accordance with those by [15] about the same example, which are summarized by working with solving polynomial roots. It is worth mentioning that polynomial roots are not always solvable in general. Fortunately, the suggested Nyquist-like criteria can be implemented graphically and numerically, independent of any polynomial root solution and inter-complex-plane transformation. Hence, the suggested technique is applicable more generally.

#### **4.3 Example description for Theorem 3**

Consider the feedback configuration of **Figure 1** used for automatic voltage regulator (AVR) in generators, which is formed by the subsystems [6]:

$$G(s) = \left(K\_P + \frac{K\_I}{s^t + 0.0001} + \frac{100K\_D s^\mu}{s^\mu + 100}\right) \frac{10}{(1 + 0.1s)(1 + 0.4s)(1 + s)},\\H(s) = \frac{1}{1 + 0.01s}$$

where *H s*ð Þ is a regular-order sensor model and *G s*ð Þ is a cascading model consisting of a fractional-order PID in the form of *KP* þ *KI= s <sup>λ</sup>* <sup>þ</sup> <sup>0</sup>*:*<sup>0001</sup> <sup>þ</sup> <sup>100</sup>*KDs <sup>μ</sup><sup>=</sup> <sup>s</sup>*ð Þ *<sup>μ</sup>* <sup>þ</sup> <sup>100</sup> <sup>1</sup> , an amplifier modeled as

<sup>1</sup> The fractional-order integral portion in the PID is approximated by *KI= s <sup>λ</sup>* <sup>þ</sup> <sup>0</sup>*:*<sup>0001</sup> in order to avoid definition problem at the origin when it is in the form of *KI=s λ*.

applicable in fractional-order control design and parametrization. This point is significant for practical control applications involving fractional-order plants,

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

The study is completed under the support of the National Natural Science

Department of Control Engineering, School of Energy and Electrical Engineering,

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

which are our perspective topics in the future.

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

Foundation of China under Grant No. 61573001.

**Acknowledgements**

**Author details**

Hohai University, Nanjing, China

provided the original work is properly cited.

\*Address all correspondence to: katsura@hhu.edu.cn

Jun Zhou

**49**

**Figure 10.** *Stability loci for cases 1 and 2.*

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 particle swarm optimization.

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 utilize Theorem 3, the fractionally commensurate Hurwitz polynomial *<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.

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

#### **4.4 Numerical results for Theorem 3**

Based on **Table 1**, the stability loci in the two cases are plotted in **Figure 10**. 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 either case yields that *N f C s; β*ð Þ *s; ρ s*∈ N *<sup>s</sup>*�*ε* ¼ 0. From these facts, Theorem 3 ensures that the closed-loop fractional-order system is stable. This coincides with the results in [6].

### **5. Conclusions**

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 direct in testing methodology.

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

applicable in fractional-order control design and parametrization. This point is significant for practical control applications involving fractional-order plants, which are our perspective topics in the future.
