**2. Preliminaries and properties in FCO-LTI systems**

#### **2.1 Preliminaries to fractional-order calculus**

Based on [13, 15], fractional-order calculus can be viewed as a generalization of the regular (integer-order) calculus, including integration and differentiation. The basic idea of fractional-order calculus is as old as the regular one and can be traced back to 1695 when Leibniz and L'Hôpital discussed what they termed the half-order derivative. The exact definition formula for the so-called *r*-order calculus was well established then by Riemann and Liouville in the form of

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

$$\_aD\_t^r f(t) = \frac{1}{\Gamma(n-r)} \frac{d^n}{dt^n} \int\_a^t \frac{f(\tau)}{(t-\tau)^{r-n+1}} d\tau \tag{1}$$

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 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> <sup>0</sup> *<sup>e</sup>*�*<sup>τ</sup> τn*�*r*�1*dτ* and it is convergent for each *n* � *r*>0.

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


$${}\_{a}D\_{t}^{r}[\mathfrak{q}f(t) + b\mathfrak{g}(t)] = a\left[{}\_{a}D\_{t}^{r}f(t)f(t)\right] + b\left[{}\_{a}D\_{t}^{r}\mathfrak{g}(t)\right].$$

• Under some additional assumptions about *f t*ð Þ, the following additive index relation (or the semigroup property) holds true:

$${}\_aD\_t^{r\_1} \left[ {}\_aD\_t^{r\_2}f(t) \right] = {}\_aD\_t^{r\_2} \left[ {}\_aD\_t^{r\_1}f(t) \right] = {}\_aD\_t^{r\_1+r\_2}f(t)$$

• If *f* ð Þ*<sup>k</sup>* ð Þ*<sup>t</sup>* � � � *<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 fractional-order derivative commutes with integer-order derivative:

$$\frac{d^m}{dt^m} \left[ {}\_a D\_t^r f(t) \right] = \,\_a D\_t^r \left[ \frac{d^m}{dt^m} f(t) \right] = \,\_a D\_t^{r+m} f(t)$$

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 frequently used, which is given by

$$\mathcal{L}\{\,\_0D\_t^r f(t)\} = \int\_0^\infty e^{-st} \,\_0D\_t^r f(t)dt = s^r F(s) - \sum\_{k=0}^{n-1} s^k \,\_0D\_t^r f(t)\bigg|\_{t=0} \tag{2}$$

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 L <sup>0</sup>*D<sup>r</sup> <sup>t</sup> f t*ð Þ � � <sup>¼</sup> *<sup>s</sup> r F s*ð Þ. To simplify our notations, we denote <sup>0</sup>*D<sup>r</sup> <sup>t</sup> f t*ð Þ*<sup>S</sup>* by *<sup>D</sup><sup>r</sup> <sup>t</sup> f t*ð Þ in the following if nothing otherwise is meant.

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

A scalar fractional-order linear time-invariant system can be described with a fractional-order state-space equation in the form of

$$\begin{cases} D\_t^r \mathfrak{x}(t) = A\mathfrak{x}(t) + Bu(t) \\ \mathfrak{y}(t) = \mathbf{C}\mathfrak{x}(t) + Du(t) \end{cases} \tag{3}$$

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

(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

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

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

*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

Based on [13, 15], fractional-order calculus can be viewed as a generalization of the regular (integer-order) calculus, including integration and differentiation. The basic idea of fractional-order calculus is as old as the regular one and can be traced back to 1695 when Leibniz and L'Hôpital discussed what they termed the half-order derivative. The exact definition formula for the so-called *r*-order calculus was well

freedoms (due to its distributed parameter features [4, 11–13]).

complex/frequency-domain facts.

*Control Theory in Engineering*

conclusions are given in Section 5.

counterclockwise encirclements.

**32**

**2. Preliminaries and properties in FCO-LTI systems**

established then by Riemann and Liouville in the form of

**2.1 Preliminaries to fractional-order calculus**

and *y t*ð Þ, *<sup>A</sup>* <sup>∈</sup> <sup>R</sup>*<sup>n</sup>*�*n*, *<sup>B</sup>*<sup>∈</sup> <sup>R</sup>*<sup>n</sup>*�<sup>1</sup> , *C*∈ R<sup>1</sup>�*n*, and *D* ∈ R<sup>1</sup>�<sup>1</sup> are constant matrices. We denote:

$$D\_t^r \mathfrak{x}(t) = \begin{bmatrix} D\_t^{r\_n} \mathfrak{x}\_1(t) \\ \vdots \\ D\_t^{r\_1} \mathfrak{x}\_n(t) \end{bmatrix} \in \mathcal{R}^n$$

where we have used *<sup>d</sup>*

Then, if we set *z* ¼ *s*

form of

*<sup>s</sup>* ! exp log *zl*

*r*

may not be rational.

**35**

*ds* log *s* ¼ *s*

*G z*ð Þ¼ *;r*

(

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

*r*

8 ><

>:

C\ 0f g is holomorphic. Thus, Δð Þ *s;rn;* ⋯*;r*<sup>1</sup> is holomorphic on C\ 0f g.

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

isolated zeros and poles, let us assume that there exists 0 <*r*≤1 such that

above actually say by the definition ([16], p. 8) that each term in Δð Þ *s;rn;* ⋯*;r*<sup>1</sup> on

To see under what conditions Δð Þ *s;rn;* ⋯*;r*<sup>1</sup> (respectively, *G s*ð Þ) possesses only

*α<sup>n</sup>* ¼ *knr*>*αn*�<sup>1</sup> ¼ *kn*�1*r*> ⋯ >*α*<sup>1</sup> ¼ *k*1*r <sup>β</sup><sup>m</sup>* <sup>¼</sup> *lmr*>*βm*�<sup>1</sup> <sup>¼</sup> *lm*�1*r*<sup>&</sup>gt; <sup>⋯</sup> <sup>&</sup>gt;*β*<sup>1</sup> <sup>¼</sup> *<sup>l</sup>*1*<sup>r</sup>*

where *kn* >*kn*�<sup>1</sup> > ⋯ >*k*<sup>1</sup> ≥ 0, *lm* >*lm*�<sup>1</sup> > ⋯ >*l*<sup>1</sup> ≥0, and *kn* ≥*lm* are integers.

<sup>Δ</sup>ð Þ¼ *<sup>z</sup>;<sup>r</sup> anzkn* <sup>þ</sup> *an*�<sup>1</sup>*zkn*�<sup>1</sup> <sup>þ</sup> <sup>⋯</sup> <sup>þ</sup> *<sup>a</sup>*1*z<sup>k</sup>*<sup>1</sup>

Obviously, *G z*ð Þ *;r* is a so-called rational function on the *z*-complex plane and possesses only finitely many isolated zeros and poles. More precisely, *G z*ð Þ *;r* is a special meromorphic function that is determined up to a multiplicative constant by prescribing the locations and multiplicities of its zeros and poles ([16], p. 87);

has finitely many isolated zeros. Thus, *G z*ð Þ *;r* and Δð Þ *z;r* can be written in the

*bm* Q *l* ð Þ *z* þ *zl μl*

Q

Under the assumption that (7) and suppose that *G z*ð Þ *;r* and Δð Þ *z;r* have no zero,

*<sup>μ</sup><sup>l</sup>* <sup>¼</sup> *lm* and <sup>P</sup>

*<sup>l</sup> <sup>s</sup>*ð Þ *<sup>r</sup>* <sup>þ</sup> *zl <sup>μ</sup><sup>l</sup>*

Q *k s <sup>r</sup>* <sup>þ</sup> *pk* � �*<sup>ν</sup><sup>k</sup>*

*<sup>k</sup> sr* þ *pk* � �*<sup>ν</sup><sup>k</sup>*

latter says specifically by Corollary 3.2 of [16] that *<sup>s</sup>* <sup>¼</sup> exp log *pk=<sup>r</sup>* � � is a pole of

In the sequel, when the assumption (7) is true and Δð Þ *z;r* and *G z*ð Þ *;r* have no zeros and poles at the origin, then Δð Þ *s;rn;* ⋯*;r*<sup>1</sup> and *G s*ð Þ are well-defined on C\ 0f g with respect to fractional commensurate order *r*. Only fractional commensurate

*<sup>k</sup> z* þ *pk* � �*<sup>ν</sup><sup>k</sup>*

*<sup>k</sup> z* þ *pk* � �*<sup>ν</sup><sup>k</sup>*

*<sup>k</sup>ν<sup>k</sup>* ¼ *kn*.

*<sup>r</sup>* ! *pk* (or equivalently, *<sup>s</sup>* ! exp log *pk*

*<sup>r</sup>* ! *zl* (or equivalently by (6),

*<sup>k</sup>* and has only finitely

*an* Q

or equivalently it is in the form of a complex fraction with regular-order polynomials as its nominator and denominator. Also, Δð Þ *z;r* is a regular-order

polynomial that is holomorphic on the whole *z*-complex plane and

*G z*ð Þ¼ *;r*

8 ><

>:

*μ<sup>l</sup>* ≥ 1*, ν<sup>k</sup>* ≥1 are integers such that P

Δð Þ¼ *z;r an*

zeros, and zero singularities, it holds that *zl* 6¼ 0*, pk* 6¼ 0∈C. In addition,

*G s*ðÞ¼ *bm*

8 ><

>:

By (10), it is not hard to see that *G s*ðÞ¼ 0 as *s*

*G s*ð Þ. Then, *G s*ð Þ is holomorphic on <sup>C</sup>\ exp log *pk=<sup>r</sup>* � � � �

n o) and <sup>∣</sup>*G s*ð Þ<sup>∣</sup> ! <sup>∞</sup> as *<sup>s</sup>*

systems are considered in this study.

*an* Q

Δð Þ¼ *s;rn;* ⋯*;r*<sup>1</sup> *an*

many isolated zeros and poles. It follows by ([16], pp. 86–87) that *G s*ð Þ is meromorphic on C\ 0f g. Confined to the discussion of this paper, *G s*ð Þ may or

*l*

Based on (6) and (9), the *s*-domain relationships can be rewritten by

Q

, then *G s*ð Þ and Δð Þ *s;rn;* ⋯*;r*<sup>1</sup> can be re-expressed as

*bmzlm* <sup>þ</sup> *bm*�1*zlm*�<sup>1</sup> <sup>þ</sup> <sup>⋯</sup> <sup>þ</sup> *<sup>b</sup>*1*zl*<sup>1</sup> *anzkn* þ *an*�<sup>1</sup>*zkn*�<sup>1</sup> þ ⋯ þ *a*1*zk*<sup>1</sup>

�<sup>1</sup> for any *<sup>s</sup>* 6¼ 0 (see [27], p. 36). The deductions

(7)

(8)

(9)

(10)

*r* n o). The

For simplicity, we employ *r* to stand for the fractional-order indices set f g *rn;* ⋯*;r*<sup>1</sup> with 0 ≤*ri* <1 with a little abuse of notations. The corresponding transfer function follows as

$$\mathcal{G}(s) = \mathcal{C}(\text{diag}[s^{r\_n}, \dots, s^{r\_1}] - A)^{-1}B + D = \frac{b\_m s^{\beta\_m} + b\_{m-1} s^{\beta\_{m-1}} + \dots + b\_1 s^{\beta\_1}}{a\_n s^{a\_n} + a\_{n-1} s^{a\_{n-1}} + \dots + a\_1 s^{a\_1}} \tag{4}$$

which is the fractional-order transfer function defined from *U s*ð Þ to *Y s*ð Þ. In (4), diag *s rn ;* ⋯*; s <sup>r</sup>*<sup>1</sup> ½ �∈C*n*�*<sup>n</sup>* stands for a diagonal matrix. Also, *ak* ð Þ *<sup>k</sup>* <sup>¼</sup> <sup>1</sup>*;* <sup>⋯</sup>*; <sup>n</sup>* and *bk* ð Þ *k* ¼ 1*;* ⋯*; m* are constants, while *α<sup>k</sup>* ð Þ *k* ¼ 1*;* ⋯*; n* and *β<sup>k</sup>* ð Þ *k* ¼ 1*;* ⋯*; m* are nonnegative real numbers satisfying

$$\begin{cases} a\_n > a\_{n-1} > \cdots > a\_1 \ge 0 \\ \beta\_m > \beta\_{m-1} > \cdots > \beta\_1 \ge 0 \end{cases}$$

In the following, the fractional-order polynomial

$$\Delta(s, r\_n, \dots, r\_1) = \det(\text{diag}[s^{r\_n}, \dots, s^{r\_1}] - A) = a\_n s^{a\_n} + a\_{n-1} s^{a\_{n-1}} + \dots + a\_1 s^{a\_1} \tag{5}$$

is called the characteristic polynomial of the state-space equation (3). We note by complex analysis ([16], p. 100) that *s <sup>α</sup>* is well-defined and satisfies

$$
\sigma^a = e^{a \log s}, \quad \forall s \in \mathcal{C} \backslash \{0\}, \forall a \in \mathcal{C} \tag{6}
$$

where log *s* is the principal branch of the complex logarithm of *s* or the principal sheet of the Riemann surface in the sense of �*π* <arg*s*≤*π*. In view of (6), we see that Δð Þ *s;rn;* ⋯*;r*<sup>1</sup> as fractional-order polynomial and *G s*ð Þ as fractional-order fraction are well-defined only on C\ 0f g for all *ri, αi, β<sup>i</sup>* ∈C, whenever at least one of f g *ri* , f g *α<sup>i</sup>* , and f g *β<sup>i</sup>* is a fraction number. Both are well-defined on the whole complex plane C, if all f g *ri* , f g *α<sup>i</sup>* , and f g *β<sup>i</sup>* are integers.

Bearing (6) in mind, our questions are (i) under what conditions Δð Þ *s;rn;* ⋯*;r*<sup>0</sup> is holomorphic and has only isolated zeros and (ii) under what conditions *G s*ð Þ is meromorphic and has only isolated zeros and poles?

To address (i), let us return to (6) and observe for any *s* ∈C\0 and *α*∈C that

$$\frac{d}{ds}s^a = \frac{d}{ds}e^{a\log s} = \frac{d}{ds}\left[\sum\_{k=0}^{\infty} \frac{1}{k!} (a\log s)^k\right]$$

$$= \sum\_{k=0}^{\infty} \frac{1}{k!} \frac{d}{ds} (a\log s)^k$$

$$= \sum\_{k=0}^{\infty} \frac{1}{(k-1)!} (a\log s)^{k-1} a \frac{d}{ds} \log s$$

$$= a \sum\_{k=1}^{\infty} \frac{1}{(k-1)!} (a\log s)^{k-1} s^{-1}$$

$$= a e^{a \log s} s^{-1} = a e^{a \log s} e^{-\log s}$$

$$= a e^{(a-1)\log s} = a (e^{\log s})^{a-1} = a s^{a-1}$$

**34**

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

where we have used *<sup>d</sup> ds* log *s* ¼ *s* �<sup>1</sup> for any *<sup>s</sup>* 6¼ 0 (see [27], p. 36). The deductions above actually say by the definition ([16], p. 8) that each term in Δð Þ *s;rn;* ⋯*;r*<sup>1</sup> on C\ 0f g is holomorphic. Thus, Δð Þ *s;rn;* ⋯*;r*<sup>1</sup> is holomorphic on C\ 0f g.

To see under what conditions Δð Þ *s;rn;* ⋯*;r*<sup>1</sup> (respectively, *G s*ð Þ) possesses only isolated zeros and poles, let us assume that there exists 0 <*r*≤1 such that

$$\begin{cases} a\_n = k\_n r > a\_{n-1} = k\_{n-1} r > \cdots > a\_1 = k\_1 r \\ \beta\_m = l\_m r > \beta\_{m-1} = l\_{m-1} r > \cdots > \beta\_1 = l\_1 r \end{cases} \tag{7}$$

where *kn* >*kn*�<sup>1</sup> > ⋯ >*k*<sup>1</sup> ≥ 0, *lm* >*lm*�<sup>1</sup> > ⋯ >*l*<sup>1</sup> ≥0, and *kn* ≥*lm* are integers. Then, if we set *z* ¼ *s r* , then *G s*ð Þ and Δð Þ *s;rn;* ⋯*;r*<sup>1</sup> can be re-expressed as

$$\begin{cases} G(z,r) = \frac{b\_m z^{l\_m} + b\_{m-1} z^{l\_{m-1}} + \dots + b\_1 z^{l\_1}}{a\_n z^{k\_n} + a\_{n-1} z^{k\_{n-1}} + \dots + a\_1 z^{k\_1}} \\ \Delta(z,r) = a\_n z^{k\_n} + a\_{n-1} z^{k\_{n-1}} + \dots + a\_1 z^{k\_1} \end{cases} \tag{8}$$

Obviously, *G z*ð Þ *;r* is a so-called rational function on the *z*-complex plane and possesses only finitely many isolated zeros and poles. More precisely, *G z*ð Þ *;r* is a special meromorphic function that is determined up to a multiplicative constant by prescribing the locations and multiplicities of its zeros and poles ([16], p. 87); or equivalently it is in the form of a complex fraction with regular-order polynomials as its nominator and denominator. Also, Δð Þ *z;r* is a regular-order polynomial that is holomorphic on the whole *z*-complex plane and has finitely many isolated zeros. Thus, *G z*ð Þ *;r* and Δð Þ *z;r* can be written in the form of

$$\begin{cases} G(z,r) = \frac{b\_m \prod\_l (z+z\_l)^{\mu\_l}}{a\_n \prod\_k (z+p\_k)^{\nu\_k}}\\ \Delta(z,r) = a\_n \prod\_k (z+p\_k)^{\nu\_k} \end{cases} \tag{9}$$

Under the assumption that (7) and suppose that *G z*ð Þ *;r* and Δð Þ *z;r* have no zero, zeros, and zero singularities, it holds that *zl* 6¼ 0*, pk* 6¼ 0∈C. In addition, *μ<sup>l</sup>* ≥ 1*, ν<sup>k</sup>* ≥1 are integers such that P *l <sup>μ</sup><sup>l</sup>* <sup>¼</sup> *lm* and <sup>P</sup> *<sup>k</sup>ν<sup>k</sup>* ¼ *kn*.

Based on (6) and (9), the *s*-domain relationships can be rewritten by

$$\begin{cases} G(s) = \frac{b\_m \prod\_l (s^r + z\_l)^{\mu\_l}}{a\_n \prod\_k (s^r + p\_k)^{\nu\_k}} \\ \Delta(s, r\_n, \dots, r\_1) = a\_n \prod\_k (s^r + p\_k)^{\nu\_k} \end{cases} \tag{10}$$

By (10), it is not hard to see that *G s*ðÞ¼ 0 as *s <sup>r</sup>* ! *zl* (or equivalently by (6), *<sup>s</sup>* ! exp log *zl r* n o) and <sup>∣</sup>*G s*ð Þ<sup>∣</sup> ! <sup>∞</sup> as *<sup>s</sup> <sup>r</sup>* ! *pk* (or equivalently, *<sup>s</sup>* ! exp log *pk r* n o). The latter says specifically by Corollary 3.2 of [16] that *<sup>s</sup>* <sup>¼</sup> exp log *pk=<sup>r</sup>* � � is a pole of *G s*ð Þ. Then, *G s*ð Þ is holomorphic on <sup>C</sup>\ exp log *pk=<sup>r</sup>* � � � � *<sup>k</sup>* and has only finitely many isolated zeros and poles. It follows by ([16], pp. 86–87) that *G s*ð Þ is meromorphic on C\ 0f g. Confined to the discussion of this paper, *G s*ð Þ may or may not be rational.

In the sequel, when the assumption (7) is true and Δð Þ *z;r* and *G z*ð Þ *;r* have no zeros and poles at the origin, then Δð Þ *s;rn;* ⋯*;r*<sup>1</sup> and *G s*ð Þ are well-defined on C\ 0f g with respect to fractional commensurate order *r*. Only fractional commensurate systems are considered in this study.

and *y t*ð Þ, *<sup>A</sup>* <sup>∈</sup> <sup>R</sup>*<sup>n</sup>*�*n*, *<sup>B</sup>*<sup>∈</sup> <sup>R</sup>*<sup>n</sup>*�<sup>1</sup>

*Control Theory in Engineering*

*Dr tx t*ð Þ ≕

�

*rn ;* ⋯*; s <sup>r</sup>*<sup>1</sup> <sup>ð</sup> ½ �� *<sup>A</sup>*Þ ¼ *ans*

is called the characteristic polynomial of the state-space equation (3).

In the following, the fractional-order polynomial

We note by complex analysis ([16], p. 100) that *s*

*s <sup>α</sup>* <sup>¼</sup> *<sup>e</sup> α* log *s*

plane C, if all f g *ri* , f g *α<sup>i</sup>* , and f g *β<sup>i</sup>* are integers.

meromorphic and has only isolated zeros and poles?

<sup>¼</sup> <sup>X</sup><sup>∞</sup> *k*¼0

<sup>¼</sup> <sup>X</sup><sup>∞</sup> *k*¼0

¼ *α* X∞ *k*¼1

¼ *αe α* log *s s* �<sup>1</sup> <sup>¼</sup> *<sup>α</sup><sup>e</sup>*

¼ *αe*

1 *k*! *d ds*ð Þ *<sup>α</sup>* log *<sup>s</sup>*

*d ds s <sup>α</sup>* <sup>¼</sup> *<sup>d</sup> dse*

*rn ;* ⋯*; ; s <sup>r</sup>*<sup>1</sup> ð Þ ½ �� *<sup>A</sup>* �<sup>1</sup>

nonnegative real numbers satisfying

Δð Þ¼ *s;rn;* ⋯*;r*<sup>1</sup> det diag *s*

*Drn <sup>t</sup> x*1ð Þ*t* ⋮ *Dr*<sup>1</sup> *<sup>t</sup> xn*ð Þ*t*

For simplicity, we employ *r* to stand for the fractional-order indices set f g *rn;* ⋯*;r*<sup>1</sup> with 0 ≤*ri* <1 with a little abuse of notations. The corresponding transfer function

*<sup>B</sup>* <sup>þ</sup> *<sup>D</sup>* <sup>¼</sup> *bms*

which is the fractional-order transfer function defined from *U s*ð Þ to *Y s*ð Þ. In (4),

*α<sup>n</sup>* >*α<sup>n</sup>*�<sup>1</sup> > ⋯ >*α*<sup>1</sup> ≥0 *<sup>β</sup><sup>m</sup>* <sup>&</sup>gt;*β<sup>m</sup>*�<sup>1</sup> <sup>&</sup>gt; <sup>⋯</sup> <sup>&</sup>gt;*β*<sup>1</sup> <sup>≥</sup><sup>0</sup>

where log *s* is the principal branch of the complex logarithm of *s* or the principal sheet of the Riemann surface in the sense of �*π* <arg*s*≤*π*. In view of (6), we see that Δð Þ *s;rn;* ⋯*;r*<sup>1</sup> as fractional-order polynomial and *G s*ð Þ as fractional-order fraction are well-defined only on C\ 0f g for all *ri, αi, β<sup>i</sup>* ∈C, whenever at least one of f g *ri* , f g *α<sup>i</sup>* , and f g *β<sup>i</sup>* is a fraction number. Both are well-defined on the whole complex

Bearing (6) in mind, our questions are (i) under what conditions Δð Þ *s;rn;* ⋯*;r*<sup>0</sup> is

X∞ *k*¼0

*k*

ð Þ *α* log *s*

ð Þ *α* log *s*

*α* log *s e* � log *s*

ð Þ *<sup>α</sup>*�<sup>1</sup> log *<sup>s</sup>* <sup>¼</sup> *<sup>α</sup> <sup>e</sup>*log *<sup>s</sup>* � �*<sup>α</sup>*�<sup>1</sup>

1 *k*!

" #

*k*�1 *α d ds* log *<sup>s</sup>*

> *k*�1 *s* �1

> > ¼ *αs α*�1

ð Þ *α* log *s k*

holomorphic and has only isolated zeros and (ii) under what conditions *G s*ð Þ is

*<sup>α</sup>* log *<sup>s</sup>* <sup>¼</sup> *<sup>d</sup> ds*

1 ð Þ *k* � 1 !

> 1 ð Þ *k* � 1 !

To address (i), let us return to (6) and observe for any *s* ∈C\0 and *α*∈C that

*<sup>r</sup>*<sup>1</sup> ½ �∈C*n*�*<sup>n</sup>* stands for a diagonal matrix. Also, *ak* ð Þ *<sup>k</sup>* <sup>¼</sup> <sup>1</sup>*;* <sup>⋯</sup>*; <sup>n</sup>* and *bk* ð Þ *k* ¼ 1*;* ⋯*; m* are constants, while *α<sup>k</sup>* ð Þ *k* ¼ 1*;* ⋯*; n* and *β<sup>k</sup>* ð Þ *k* ¼ 1*;* ⋯*; m* are

2 6 4

denote:

follows as

diag *s*

**34**

*G s*ðÞ¼ *C* diag *s*

*rn ;* ⋯*; s*

, *C*∈ R<sup>1</sup>�*n*, and *D* ∈ R<sup>1</sup>�<sup>1</sup> are constant matrices. We

*<sup>β</sup><sup>m</sup>* <sup>þ</sup> *bm*�1*<sup>s</sup>*

*<sup>α</sup><sup>n</sup>* <sup>þ</sup> *an*�<sup>1</sup>*<sup>s</sup>*

*,* ∀*s* ∈C\ 0f g*,* ∀*α*∈C (6)

*ans<sup>α</sup><sup>n</sup>* þ *an*�1*s<sup>α</sup>n*�<sup>1</sup> þ ⋯ þ *a*1*s<sup>α</sup>*<sup>1</sup>

*<sup>β</sup>m*�<sup>1</sup> <sup>þ</sup> <sup>⋯</sup> <sup>þ</sup> *<sup>b</sup>*1*<sup>s</sup>*

*<sup>α</sup>n*�<sup>1</sup> <sup>þ</sup> <sup>⋯</sup> <sup>þ</sup> *<sup>a</sup>*1*<sup>s</sup>*

*<sup>α</sup>* is well-defined and satisfies

*β*1

(4)

*<sup>α</sup>*<sup>1</sup> (5)

3 7 <sup>5</sup><sup>∈</sup> <sup>R</sup>*<sup>n</sup>*

### **2.3 Closed-loop configuration with FCO-LTI systems**

Consider the feedback system illustrated in **Figure 1**, in which we denote by Σ*<sup>G</sup>* and Σ*H*, respectively, an FCO-LTI plant and an FCO-LTI feedback subsystem that possess the following fractional-order state-space equations.

$$\Sigma\_G: \begin{cases} D\_t^r \mathfrak{x} = A\mathfrak{x} + B\mathfrak{e} \\ \mathfrak{y} = \mathbb{C}\mathfrak{x} + D\mathfrak{e} \end{cases}, \quad \Sigma\_H: \begin{cases} D\_t^\mathfrak{q} \zeta = \Lambda \zeta + \Gamma \mu \\ \eta = \Theta \zeta + \Pi \mu \end{cases} \tag{11}$$

In the closed-loop system, we can write the closed-loop state-space equation as

*ζ*

. To explicate the Nyquist approach, we begin with the

*A* � *B*ΞΠ*C* �*B*ΞΘ

Γ*D*ΞΠ*C* Γ*D*ΞΘ

" #�<sup>1</sup> *B*ΞΠ*C B*ΞΘ

0 " #

þ

*B*Ξ Γ*D*Ξ

> 1 A

Γ*D*ΞΠ*C* Γ*D*ΞΘ

<sup>Γ</sup>*C I*ð Þ *<sup>n</sup>*ð Þ� *<sup>s</sup>;<sup>r</sup> <sup>A</sup>* �<sup>1</sup> *Ip*ð Þ� *<sup>s</sup>; <sup>q</sup>* <sup>Λ</sup> � ��<sup>1</sup>

*u*

(14)

1 A

(15)

(16)

" #

" #

<sup>Γ</sup>*<sup>C</sup>* � <sup>Γ</sup>*D*ΞΠ*<sup>C</sup>* <sup>Λ</sup> � <sup>Γ</sup>*D*ΞΘ " # *<sup>x</sup>*

þ Π*D*Ξ*u*

conventional return difference equation in the feedback configuration Σ*C*.

By definition, the characteristic polynomial for the closed-loop system Σ*<sup>C</sup>* is

�

" #�<sup>1</sup> *B*ΞΠ*C B*ΞΘ

*In*ð Þ� *s;r A* 0

�Γ*C Ip*ð Þ� *s; q* Λ

*In*ð Þ� *s;r A* 0

<sup>¼</sup> detð Þ *In*ð Þ� *<sup>s</sup>;<sup>r</sup> <sup>A</sup>* det *Ip*ð Þ� *<sup>s</sup>; <sup>q</sup>* <sup>Λ</sup> � � <sup>¼</sup> <sup>Δ</sup>*G*ð Þ *<sup>s</sup>;<sup>r</sup>* <sup>Δ</sup>*H*ð Þ *<sup>s</sup>; <sup>q</sup>*

Δ*C*ð Þ *s;r; q*

*D*ΞΠ *D*Ξ

¼ Δ*O*ð Þ *s;r; q* detð*Il*þ*<sup>m</sup>*

� � ð Þ *In*ð Þ� *<sup>s</sup>;<sup>r</sup> <sup>A</sup>* �<sup>1</sup> <sup>0</sup>

� � *C* 0

" #

<sup>¼</sup> <sup>Δ</sup>*O*ð Þ *<sup>s</sup>;r; <sup>q</sup>* det *In*þ*<sup>p</sup>* <sup>þ</sup> ð Þ *In*ð Þ� *<sup>s</sup>;<sup>r</sup> <sup>A</sup>* �<sup>1</sup> <sup>0</sup> *Ip*ð Þ� *<sup>s</sup>; <sup>q</sup>* <sup>Λ</sup> � ��<sup>1</sup>

� � ΞΠ Ξ

with <sup>Δ</sup>*G*ð Þ¼ *<sup>s</sup>;<sup>r</sup>* detð Þ *In*ð Þ� *<sup>s</sup>;<sup>r</sup> <sup>A</sup>* and <sup>Δ</sup>*H*ð Þ¼ *<sup>s</sup>; <sup>q</sup>* det *Ip*ð Þ� *<sup>s</sup>; <sup>q</sup>* <sup>Λ</sup> � �. Clearly, Δ*O*ð Þ *s;r; q* is the characteristic polynomial for Σ*O*, while Δ*G*ð Þ *s;r* and Δ*H*ð Þ *s; q* are the characteristic polynomials for the subsystems Σ*<sup>G</sup>* and Σ*H*, respectively, in the feed-

�Γ*C Ip*ð Þ� *s; q* Λ

" #

0 Θ � �

<sup>Γ</sup>*C I*ð Þ *<sup>n</sup>*ð Þ� *<sup>s</sup>;<sup>r</sup> <sup>A</sup>* �<sup>1</sup> *Ip*ð Þ� *<sup>s</sup>; <sup>q</sup>* <sup>Λ</sup> � ��<sup>1</sup>

Þ

! " #

0 " #

<sup>Γ</sup>*<sup>C</sup>* � <sup>Γ</sup>*D*ΞΠ*<sup>C</sup>* <sup>Λ</sup> � <sup>Γ</sup>*D*ΞΘ ! " #

<sup>¼</sup> *<sup>A</sup>* � *<sup>B</sup>*ΞΠ*<sup>C</sup>* �*B*ΞΘ

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

*x ζ*

*In*ð Þ *s;r* 0

�Γ*C Ip*ð Þ� *s; q* Λ

where detð Þ� means the determinant of ð Þ� and

Let us return to (15) and continue to observe that

� *<sup>B</sup>* <sup>0</sup> 0 Γ

*Ip*ð Þ� *<sup>s</sup>; <sup>q</sup>* <sup>Λ</sup> � ��<sup>1</sup>

Δ*O*ð Þ *s;r; q* ≔ det

*In*ð Þ� *s;r A* 0

0 *Ip*ð Þ *s; q*

�Γ*C Ip*ð Þ� *s; q* Λ

" #

" #

Σ*<sup>C</sup>* :

*Dr tx Dq t ζ*

where <sup>Ξ</sup> <sup>¼</sup> ð Þ *Im* <sup>þ</sup> <sup>Π</sup>*<sup>D</sup>* �<sup>1</sup>

8 >>>>><

>>>>>:

Δ*C*ð Þ *s;r; q* ≔ det

¼ det

�det *In*þ*<sup>p</sup>* þ

¼ Δ*O*ð Þ *s;r; q* det *In*þ*<sup>p</sup>* þ

back configuration of **Figure 1**.

þ

**37**

*C* 0 0 Θ @

@

" #

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

*η* ¼ ½ � ΞΠ*C* ΞΘ

*In*ð Þ� *s;r A* 0

! " #

where *A* ∈ R*<sup>n</sup>*�*n*, *B* ∈ R*<sup>n</sup>*�*m*, *C*∈ R*<sup>l</sup>*�*n*, and *D* ∈ R*<sup>l</sup>*�*m*, respectively, are constant matrices, while Λ∈ R*<sup>p</sup>*�*p*, Γ∈ R*<sup>p</sup>*�*<sup>l</sup>* , Θ ∈ R*<sup>m</sup>*�*p*, and Π ∈ R*<sup>m</sup>*�*<sup>l</sup>* are constant.

Fractional-order transfer functions for Σ*<sup>G</sup>* and Σ*<sup>H</sup>* are given as follows:

$$\begin{cases} G(s) = \mathcal{C}(\text{diag}\left[s^{r\_n}, \dots, s^{r\_1}\right] - A)^{-1}B + D\\ \qquad \coloneqq \mathcal{C}(I\_n(s, r) - A)^{-1}B + D =: \hat{G}(s) + D\\ \qquad \coloneqq \frac{\mathcal{C}\,\text{adj}\left(I\_n(s, r) - A\right)B + \det(I\_n(s, r) - A)D}{\det(I\_n(s, r) - A)}\\ \qquad \coloneqq \hat{G}(s)/\det(I\_n(s, r) - A)\\ \qquad\qquad\qquad \coloneqq \Theta(\text{diag}\left[s^{q\_r}, \dots, s^{q\_1}\right] - \Lambda)^{-1}\Gamma + \Pi\\ \qquad\qquad\qquad\qquad\qquad \coloneqq \Theta(I\_p(s, q) - \Lambda)^{-1}\Gamma + \Pi =: \hat{H}(s) + \Pi\\ \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad \left(\begin{aligned} &\text{if } (I\_p(s, q) - \Lambda)\Gamma & \dashv\text{if } (I\_p(s, q) - \Lambda)\Pi\\ &\text{det}(I\_p(s, q) - \Lambda) & \end{aligned}\right) \end{cases} \tag{12}$$

$$\begin{aligned} \text{where } \text{adj}\left(I\_p(s, q) - \Lambda\right)\Gamma &\coloneqq \det\left(I\_p(s, q) - \Lambda\right)\Pi\\ \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad \left(\begin{aligned} &\text{if } (I\_p(s, q) - \Lambda) \end{aligned}\right) \end{cases}$$

where *G s* ^ ð Þ, *H s* ^ ð Þ, *G s* � ð Þ, and *H s* � ð Þ are obvious and *adj*ð Þ� is the adjoint. Also

$$I\_n(s, r) = \text{diag}[\mathfrak{s}^{r\_n}, \dots, \mathfrak{s}^{r\_1}], \quad I\_p(s, q) = \text{diag}[\mathfrak{s}^{q\_p}, \dots, \mathfrak{s}^{q\_1}]$$

Now we construct the state-space equations for the open- and closed-loop systems of **Figure 1**. The open-loop system can be expressed by the fractional-order state-space equation:

$$\begin{aligned} \boldsymbol{\Sigma\_{O}} : \begin{cases} \begin{bmatrix} D\_t^{\boldsymbol{\varsigma}} \boldsymbol{\varkappa} \\ D\_t^{\boldsymbol{\varsigma}} \boldsymbol{\zeta} \end{bmatrix} = \begin{bmatrix} \boldsymbol{A} & \mathbf{0} \\ \boldsymbol{\Gamma} \mathbf{C} & \boldsymbol{\Lambda} \end{bmatrix} \begin{bmatrix} \boldsymbol{\varkappa} \\ \boldsymbol{\zeta} \end{bmatrix} + \begin{bmatrix} \boldsymbol{B} \\ \boldsymbol{\Gamma} \boldsymbol{D} \end{bmatrix} \boldsymbol{u} \\ \boldsymbol{\eta} = \begin{bmatrix} \boldsymbol{\Pi} \mathbf{C} & \boldsymbol{\Theta} \end{bmatrix} \begin{bmatrix} \boldsymbol{\varkappa} \\ \boldsymbol{\zeta} \end{bmatrix} + \boldsymbol{\Pi} \boldsymbol{D} \boldsymbol{u} \end{aligned} \tag{13}$$

**Figure 1.** *FCO-LTI feedback configuration.*

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

In the closed-loop system, we can write the closed-loop state-space equation as

$$\Sigma\_{C}: \begin{cases} \begin{bmatrix} D\_{t}^{\prime}\boldsymbol{x} \\ D\_{t}^{\prime}\boldsymbol{\zeta} \end{bmatrix} = \begin{bmatrix} A - B\boldsymbol{\Xi}\boldsymbol{\Pi}\boldsymbol{C} & -B\boldsymbol{\Xi}\boldsymbol{\Theta} \\ \boldsymbol{\Gamma}\boldsymbol{C} - \boldsymbol{\Gamma}D\boldsymbol{\Xi}\boldsymbol{\Pi}\boldsymbol{C} & \Lambda - \boldsymbol{\Gamma}D\boldsymbol{\Xi}\boldsymbol{\Theta} \end{bmatrix} \begin{bmatrix} \boldsymbol{x} \\ \boldsymbol{\zeta} \end{bmatrix} + \begin{bmatrix} B\boldsymbol{\Xi} \\ \boldsymbol{\Gamma}D\boldsymbol{\Xi} \end{bmatrix} u \\ \boldsymbol{\eta} = \begin{bmatrix} \boldsymbol{\Xi}\boldsymbol{\Pi}\boldsymbol{C} & \boldsymbol{\Xi}\boldsymbol{\Theta} \end{bmatrix} \begin{bmatrix} \boldsymbol{x} \\ \boldsymbol{\zeta} \end{bmatrix} + \boldsymbol{\Pi}\boldsymbol{D}\boldsymbol{\Xi}u \end{cases} \tag{14}$$

where <sup>Ξ</sup> <sup>¼</sup> ð Þ *Im* <sup>þ</sup> <sup>Π</sup>*<sup>D</sup>* �<sup>1</sup> . To explicate the Nyquist approach, we begin with the conventional return difference equation in the feedback configuration Σ*C*.

By definition, the characteristic polynomial for the closed-loop system Σ*<sup>C</sup>* is

$$
\Delta\_C(s, r, q) := \det\left( \begin{bmatrix} I\_n(s, r) & 0 \\ 0 & I\_p(s, q) \end{bmatrix} - \begin{bmatrix} A - B\Xi \Pi C & -B\Xi \Theta \\ \Gamma C - \Gamma D\Xi \Pi C & \Lambda - \Gamma D\Xi \Theta \end{bmatrix} \right)
$$

$$
\begin{aligned}
&= \det\left( \begin{bmatrix} I\_n(s, r) - A & 0 \\ -\Gamma C & I\_p(s, q) - \Lambda \end{bmatrix} \right) \\
&= \det\begin{pmatrix} I\_n(s, r) - A & 0 \\ -\Gamma C & I\_p(s, q) - \Lambda \end{pmatrix}^{-1} \begin{bmatrix} B\Xi \Pi C & B\Xi \Theta \\ \Gamma D\Xi \Pi C & \Gamma D\Xi \Theta \end{bmatrix} \right) \\
&= \Delta\_O(s, r, q) \det\begin{pmatrix} I\_n(s, r) - A & 0 \\ -\Gamma C & I\_p(s, q) - \Lambda \end{pmatrix}^{-1} \begin{bmatrix} B\Xi \Pi C & B\Xi \Theta \\ \Gamma D\Xi \Pi C & \Gamma D\Xi \Theta \end{pmatrix} \end{aligned} \tag{15}
$$

where detð Þ� means the determinant of ð Þ� and

$$\begin{aligned} \Delta\_{\mathcal{O}}(s, r, q) &:= \det \begin{pmatrix} \begin{bmatrix} I\_n(s, r) - A & \mathbf{0} \\ -\Gamma \mathbf{C} & I\_p(s, q) - \Lambda \end{bmatrix} \\ \mathbf{0} &= \det(I\_n(s, r) - A)\det(I\_p(s, q) - \Lambda) = \Delta\_{\mathcal{G}}(s, r)\Delta\_H(s, q) \end{aligned} \tag{16}$$

with <sup>Δ</sup>*G*ð Þ¼ *<sup>s</sup>;<sup>r</sup>* detð Þ *In*ð Þ� *<sup>s</sup>;<sup>r</sup> <sup>A</sup>* and <sup>Δ</sup>*H*ð Þ¼ *<sup>s</sup>; <sup>q</sup>* det *Ip*ð Þ� *<sup>s</sup>; <sup>q</sup>* <sup>Λ</sup> � �. Clearly, Δ*O*ð Þ *s;r; q* is the characteristic polynomial for Σ*O*, while Δ*G*ð Þ *s;r* and Δ*H*ð Þ *s; q* are the characteristic polynomials for the subsystems Σ*<sup>G</sup>* and Σ*H*, respectively, in the feedback configuration of **Figure 1**.

Let us return to (15) and continue to observe that

$$\Delta\_C(s, r, q)$$

$$= \Delta\_O(s, r, q) \det\begin{pmatrix} I\_{n+p} + \begin{bmatrix} (I\_n(s, r) - A)^{-1} & 0\\ (I\_p(s, q) - \Lambda)^{-1} \Gamma C(I\_n(s, r) - A)^{-1} & (I\_p(s, q) - \Lambda)^{-1} \end{bmatrix} \end{pmatrix}$$

$$\cdot \begin{bmatrix} B & \mathbf{0} \\ \mathbf{0} & \Gamma \end{bmatrix} \begin{bmatrix} \Xi \Pi & \Xi \\ D \Xi \Pi & D \Xi \end{bmatrix} \begin{bmatrix} C & \mathbf{0} \\ \mathbf{0} & \Theta \end{bmatrix}$$

$$= \Delta\_O(s, r, q) \det(I\_{l+m}$$

$$+ \begin{bmatrix} C & \mathbf{0} \\ \mathbf{0} & \Theta \end{bmatrix} \begin{bmatrix} (I\_n(s, r) - A)^{-1} & \mathbf{0} \\ (I\_p(s, q) - \Lambda)^{-1} \Gamma C(I\_n(s, r) - A)^{-1} & (I\_p(s, q) - \Lambda)^{-1} \end{bmatrix}$$

**2.3 Closed-loop configuration with FCO-LTI systems**

possess the following fractional-order state-space equations.

*G s*ðÞ¼ *C* diag *s*

≕ *G s*

≕ *H s*

� ð Þ, and *H s*

*Dr tx Dq t ζ*

8 >>>>><

>>>>>:

" #

*η* ¼ ½ � Π*C* Θ

*H s*ðÞ¼ Θ diag *s*

<sup>≕</sup> *C I*ð Þ *<sup>n</sup>*ð Þ� *<sup>s</sup>;<sup>r</sup> <sup>A</sup>* �<sup>1</sup>

<sup>≕</sup> <sup>Θ</sup> *Ip*ð Þ� *<sup>s</sup>; <sup>q</sup>* <sup>Λ</sup> � ��<sup>1</sup>

� ð Þ*=*detð Þ *In*ð Þ� *<sup>s</sup>;<sup>r</sup> <sup>A</sup>*

� ð Þ*=*det *Ip*ð Þ� *<sup>s</sup>; <sup>q</sup>* <sup>Λ</sup> � �

*rn ;* ⋯*; s*

*tx* ¼ *Ax* þ *Be*

*<sup>y</sup>* <sup>¼</sup> *Cx* <sup>þ</sup> *De ,* <sup>Σ</sup>*<sup>H</sup>* : *<sup>D</sup><sup>q</sup>*

Fractional-order transfer functions for Σ*<sup>G</sup>* and Σ*<sup>H</sup>* are given as follows:

*rn ;* ⋯*; ; s <sup>r</sup>*<sup>1</sup> ð Þ ½ �� *<sup>A</sup>* �<sup>1</sup>

*qp ;* ⋯*; ; s <sup>q</sup>*<sup>1</sup> ð Þ ½ �� <sup>Λ</sup> �<sup>1</sup>

( (

<sup>Σ</sup>*<sup>G</sup>* : *<sup>D</sup><sup>r</sup>*

matrices, while Λ∈ R*<sup>p</sup>*�*p*, Γ∈ R*<sup>p</sup>*�*<sup>l</sup>*

*Control Theory in Engineering*

8

>>>>>>>>>>>>>>>>>>>>>>>>><

>>>>>>>>>>>>>>>>>>>>>>>>>:

^ ð Þ, *H s*

^ ð Þ, *G s*

*In*ð Þ¼ *s;r* diag *s*

Σ*<sup>O</sup>* :

where *G s*

state-space equation:

**Figure 1.**

**36**

*FCO-LTI feedback configuration.*

Consider the feedback system illustrated in **Figure 1**, in which we denote by Σ*<sup>G</sup>* and Σ*H*, respectively, an FCO-LTI plant and an FCO-LTI feedback subsystem that

where *A* ∈ R*<sup>n</sup>*�*n*, *B* ∈ R*<sup>n</sup>*�*m*, *C*∈ R*<sup>l</sup>*�*n*, and *D* ∈ R*<sup>l</sup>*�*m*, respectively, are constant

*B* þ *D* ≕ *G s*

Γ þ Π ≕ *H s*

<sup>≕</sup> <sup>Θ</sup> adj *Ip*ð Þ� *<sup>s</sup>; <sup>q</sup>* <sup>Λ</sup> � �<sup>Γ</sup> <sup>þ</sup> det *Ip*ð Þ� *<sup>s</sup>; <sup>q</sup>* <sup>Λ</sup> � �<sup>Π</sup> det *Ip*ð Þ� *<sup>s</sup>; <sup>q</sup>* <sup>Λ</sup> � �

*<sup>r</sup>*<sup>1</sup> ½ �*, Ip*ð Þ¼ *<sup>s</sup>; <sup>q</sup>* diag *<sup>s</sup>*

" # *x*

*ζ*

þ Π*Du*

þ

*B* Γ*D*

*u*

" #

" #

Now we construct the state-space equations for the open- and closed-loop systems of **Figure 1**. The open-loop system can be expressed by the fractional-order

> <sup>¼</sup> *<sup>A</sup>* <sup>0</sup> Γ*C* Λ

> > *x ζ*

" #

<sup>≕</sup> *<sup>C</sup>* adjð Þ *In*ð Þ� *<sup>s</sup>;<sup>r</sup> <sup>A</sup> <sup>B</sup>* <sup>þ</sup> detð Þ *In*ð Þ� *<sup>s</sup>;<sup>r</sup> <sup>A</sup> <sup>D</sup>* detð Þ *In*ð Þ� *s;r A*

*<sup>t</sup> ζ* ¼ Λ*ζ* þ Γ*μ*

(11)

(12)

(13)

*η* ¼ Θ*ζ* þ Π*μ*

, Θ ∈ R*<sup>m</sup>*�*p*, and Π ∈ R*<sup>m</sup>*�*<sup>l</sup>* are constant.

*B* þ *D*

^ ðÞþ *<sup>D</sup>*

Γ þ Π

^ ðÞþ <sup>Π</sup>

� ð Þ are obvious and *adj*ð Þ� is the adjoint. Also

*qp ;* ⋯*; s <sup>q</sup>*<sup>1</sup> ½ �

� *<sup>B</sup>* <sup>0</sup> 0 Γ � � ΞΠ Ξ *D*ΞΠ *D*Ξ � �� ¼ Δ*O*ð Þ *s;r; q* det *Il*þ*<sup>m</sup>* þ *G s* ^ ð Þ <sup>0</sup> *H s* ^ ð Þ*G s* ^ ð Þ*, H s* ^ ð Þ " # ΞΠ Ξ *D*ΞΠ *D*Ξ ! � � <sup>¼</sup> <sup>Δ</sup>*O*ð Þ *<sup>s</sup>;r; <sup>q</sup>* det *Im* <sup>þ</sup> <sup>Π</sup>*G s* ^ ð ÞΞ Π*G s* ^ ð Þ<sup>Ξ</sup> *H s* ^ ð Þ*G s*ð ÞΞ*, Im* <sup>þ</sup> *H s* ^ ð Þ*G s*ð Þ<sup>Ξ</sup> ! " # <sup>¼</sup> <sup>Δ</sup>*O*ð Þ *<sup>s</sup>;r; <sup>q</sup>* det *Im* <sup>Π</sup>*G s* ^ ð Þ<sup>Ξ</sup> �*Im Im* <sup>þ</sup> *H s* ^ ð Þ*G s*ð Þ<sup>Ξ</sup> ! " # <sup>¼</sup> <sup>Δ</sup>*O*ð Þ *<sup>s</sup>;r; <sup>q</sup>* det *Im* <sup>Π</sup>*G s* ^ ð Þ<sup>Ξ</sup> <sup>0</sup> *Im* <sup>þ</sup> *H s* ^ ð Þ*G s*ð Þ<sup>Ξ</sup> <sup>þ</sup> <sup>Π</sup>*G s* ^ ð Þ<sup>Ξ</sup> ! " # <sup>¼</sup> <sup>Δ</sup>*O*ð Þ *<sup>s</sup>;r; <sup>q</sup>* det *Im* <sup>þ</sup> *H s* ^ ð Þ*G s*ð Þ<sup>Ξ</sup> <sup>þ</sup> <sup>Π</sup>*G s* ^ ð Þ<sup>Ξ</sup> � � <sup>¼</sup> <sup>Δ</sup>*O*ð Þ *<sup>s</sup>;r; <sup>q</sup>* detð Þ <sup>Ξ</sup> det *Im* <sup>þ</sup> <sup>Π</sup>*<sup>D</sup>* <sup>þ</sup> *H s* ^ ð Þ*G s*ð Þþ <sup>Π</sup>*G s* ^ ð Þ � � ¼ Δ*O*ð Þ *s;r; q* detð Þ Ξ detð Þ *Im* þ *H s*ð Þ*G s*ð Þ (17)

Firstly, the simply closed curve defined on the *z*-domain as illustrated by the dashed-line in **Figure 2** is the standard contour for a Nyquist-like stability criterion in terms of Δð Þ *z;r* . The contour portions of N *<sup>z</sup>* along the two slopes actually overlap the slope lines. Clearly, the contour N *<sup>z</sup>* is actually the boundary of the sector region encircled by the dashed-line. The radiuses of the two arcs in the sector are sufficiently small and large, respectively, or simply *γ* ! 0 and *R* ! ∞. The sector is

Secondly, the simply closed curve defined on the *s*-domain as illustrated by **Figure 3** presents the standard contour used for a Nyquist-like stability criterion in terms of Δð Þ *s;rn;* ⋯*;r*<sup>1</sup> . Again the contour is plotted with dashed-lines; in particular, the contour portion along the imaginary axis is actually overlapping the imaginary axis. More precisely, N *<sup>s</sup>* is the boundary of the open right-half complex plane

N *<sup>s</sup>* ∪ C<sup>þ</sup> ¼ f g *s*∈C : R*e s*ð Þ≥0 *,* Intð Þ¼ N *<sup>s</sup>* C<sup>þ</sup>

where I*nt*ð Þ� denotes the interior of a closed set. Similar to the contour N *<sup>z</sup>*, the radiuses of the two half-circles in N *<sup>s</sup>* are sufficiently small and large, respectively,

• In both cases, the origin of the complex plane is excluded from the contours themselves and their interiors. The reason for these specific contours is that *G s*ð Þ and Δð Þ *s;rn;* ⋯*;r*<sup>1</sup> (respectively, *G z*ð Þ *;r* and Δð Þ *z;r* ) are well-defined merely on

*r* . 2 *r*.

symmetric with respect to the real axis, whose half angle is *<sup>π</sup>*

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

C<sup>þ</sup> ¼ f g *s* ∈C : R*e s*ð Þ>0 in the sense that

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

Remarks about the contours N *<sup>z</sup>* and N *<sup>s</sup>*:

C\ 0f g due to the relation (6) and *z* ¼ *s*

namely, *γ* ! 0 and *R* ! ∞.

**Figure 2.**

**Figure 3.**

**39**

*The standard z-domain contour* N *<sup>z</sup>.*

*The standard s-domain contour* N *<sup>s</sup>.*

In deriving (17), the determinant equivalence detð Þ¼ *I*<sup>1</sup> þ *XY* detð Þ *I*<sup>2</sup> þ *YX* is repeatedly used, where *X* and *Y* are matrices of compatible dimensions and *I*<sup>1</sup> and *I*<sup>2</sup> are identities of appropriate dimensions. By (17), we have

$$\frac{\Delta\_C(s, r, q)}{\Delta\_O(s, r, q)} = \frac{\Delta\_C(s, r, q)}{\Delta\_G(s, r)\Delta\_H(s, q)} = \frac{\det(I\_m + H(s)G(s))}{\det(I\_m + \Pi D)}\tag{18}$$

which is nothing but the return difference relationship for the fractional-order feedback system Σ*C*. By the definitions, it is clear that Δ*O*ð Þ *s;r; q* , Δ*O*ð Þ *s;r; q* , Δ*G*ð Þ *s;r* , and Δ*H*ð Þ *s; q* are all fractional-order. It is based on (18) that Nyquist-like criteria will be worked out. However, in order to get rid of any open-loop structure and spectrum, let us instead work with

$$
\Delta\_C(s, r, q) = \Delta\_G(s, r)\Delta\_H(s, q)\frac{\det(I\_m + H(s)G(s))}{\det(I\_m + \Pi D)}\tag{19}
$$

**Remark 1.** Recalling our discussion in Section 2.2 and assuming that there exists a number 0< *ρ*≤1 such that Σ*<sup>G</sup>* and Σ*<sup>H</sup>* are fractionally commensurate with respect to the same commensurate order *ρ*, it follows that *G s*ð Þ and *H s*ð Þ are meromorphic on C\ 0f g, while Δ*C*ð Þ *s;r; q* , Δ*G*ð Þ *s;r* , and Δ*H*ð Þ *s; q* are holomorphic on the whole complex plane. These complex functional facts will play a key role for us to apply the argument principle to (18) as well as (19).

### **3. Main results**

#### **3.1 Nyquist contours in the** *z***-/***s***-domains**

As another preparation for stability analysis in fractional-order systems by means of the argument principle for meromorphic functions, we need to choose appropriate Nyquist contours.

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

Firstly, the simply closed curve defined on the *z*-domain as illustrated by the dashed-line in **Figure 2** is the standard contour for a Nyquist-like stability criterion in terms of Δð Þ *z;r* . The contour portions of N *<sup>z</sup>* along the two slopes actually overlap the slope lines. Clearly, the contour N *<sup>z</sup>* is actually the boundary of the sector region encircled by the dashed-line. The radiuses of the two arcs in the sector are sufficiently small and large, respectively, or simply *γ* ! 0 and *R* ! ∞. The sector is symmetric with respect to the real axis, whose half angle is *<sup>π</sup>* 2 *r*.

Secondly, the simply closed curve defined on the *s*-domain as illustrated by **Figure 3** presents the standard contour used for a Nyquist-like stability criterion in terms of Δð Þ *s;rn;* ⋯*;r*<sup>1</sup> . Again the contour is plotted with dashed-lines; in particular, the contour portion along the imaginary axis is actually overlapping the imaginary axis. More precisely, N *<sup>s</sup>* is the boundary of the open right-half complex plane C<sup>þ</sup> ¼ f g *s* ∈C : R*e s*ð Þ>0 in the sense that

$$\mathcal{N}\_s \cup \mathcal{C}^+ = \{ s \in \mathcal{C} : \text{Re}(s) \ge 0 \}, \quad \text{Int}(\mathcal{N}\_s) = \mathcal{C}^+$$

where I*nt*ð Þ� denotes the interior of a closed set. Similar to the contour N *<sup>z</sup>*, the radiuses of the two half-circles in N *<sup>s</sup>* are sufficiently small and large, respectively, namely, *γ* ! 0 and *R* ! ∞.

Remarks about the contours N *<sup>z</sup>* and N *<sup>s</sup>*:

• In both cases, the origin of the complex plane is excluded from the contours themselves and their interiors. The reason for these specific contours is that *G s*ð Þ and Δð Þ *s;rn;* ⋯*;r*<sup>1</sup> (respectively, *G z*ð Þ *;r* and Δð Þ *z;r* ) are well-defined merely on C\ 0f g due to the relation (6) and *z* ¼ *s r* .

**Figure 3.** *The standard s-domain contour* N *<sup>s</sup>.*

� *<sup>B</sup>* <sup>0</sup> 0 Γ

<sup>¼</sup> <sup>Δ</sup>*O*ð Þ *<sup>s</sup>;r; <sup>q</sup>* det *Im* <sup>þ</sup> <sup>Π</sup>*G s*

¼ Δ*O*ð Þ *s;r; q* det *Il*þ*<sup>m</sup>* þ

*Control Theory in Engineering*

� � ΞΠ Ξ

*H s* ^ ð Þ*G s*

*H s*

<sup>¼</sup> <sup>Δ</sup>*O*ð Þ *<sup>s</sup>;r; <sup>q</sup>* det *Im* <sup>Π</sup>*G s*

0 *Im* þ *H s*

In deriving (17), the determinant equivalence detð Þ¼ *I*<sup>1</sup> þ *XY* detð Þ *I*<sup>2</sup> þ *YX* is repeatedly used, where *X* and *Y* are matrices of compatible dimensions and *I*<sup>1</sup> and *I*<sup>2</sup>

which is nothing but the return difference relationship for the fractional-order feedback system Σ*C*. By the definitions, it is clear that Δ*O*ð Þ *s;r; q* , Δ*O*ð Þ *s;r; q* , Δ*G*ð Þ *s;r* , and Δ*H*ð Þ *s; q* are all fractional-order. It is based on (18) that Nyquist-like criteria will be worked out. However, in order to get rid of any open-loop structure and spec-

**Remark 1.** Recalling our discussion in Section 2.2 and assuming that there exists a number 0< *ρ*≤1 such that Σ*<sup>G</sup>* and Σ*<sup>H</sup>* are fractionally commensurate with respect to the same commensurate order *ρ*, it follows that *G s*ð Þ and *H s*ð Þ are meromorphic on C\ 0f g, while Δ*C*ð Þ *s;r; q* , Δ*G*ð Þ *s;r* , and Δ*H*ð Þ *s; q* are holomorphic on the whole complex plane. These complex functional facts will play a key role for us to apply

As another preparation for stability analysis in fractional-order systems by means of the argument principle for meromorphic functions, we need to choose

<sup>¼</sup> <sup>Δ</sup>*O*ð Þ *<sup>s</sup>;r; <sup>q</sup>* det *Im* <sup>Π</sup>*G s*

¼ Δ*O*ð Þ *s;r; q* det *Im* þ *H s*

are identities of appropriate dimensions. By (17), we have

<sup>Δ</sup>*O*ð Þ *<sup>s</sup>;r; <sup>q</sup>* <sup>¼</sup> <sup>Δ</sup>*C*ð Þ *<sup>s</sup>;r; <sup>q</sup>*

Δ*C*ð Þ¼ *s;r; q* Δ*G*ð Þ *s;r* Δ*H*ð Þ *s; q*

Δ*C*ð Þ *s;r; q*

the argument principle to (18) as well as (19).

**3.1 Nyquist contours in the** *z***-/***s***-domains**

appropriate Nyquist contours.

trum, let us instead work with

**3. Main results**

**38**

¼ Δ*O*ð Þ *s;r; q* detð Þ Ξ det *Im* þ Π*D* þ *H s*

*G s* ^ ð Þ <sup>0</sup>

*D*ΞΠ *D*Ξ � ��

^ ð Þ*, H s* ^ ð Þ " # ΞΠ Ξ

^ ð Þ*G s*ð ÞΞ*, Im* <sup>þ</sup> *H s*

�*Im Im* þ *H s*

! � �

^ ð ÞΞ Π*G s*

^ ð Þ<sup>Ξ</sup>

^ ð Þ<sup>Ξ</sup>

^ ð Þ*G s*ð Þ<sup>Ξ</sup> <sup>þ</sup> <sup>Π</sup>*G s*

� �

¼ Δ*O*ð Þ *s;r; q* detð Þ Ξ detð Þ *Im* þ *H s*ð Þ*G s*ð Þ (17)

^ ð Þ*G s*ð Þ<sup>Ξ</sup> <sup>þ</sup> <sup>Π</sup>*G s*

^ ð Þ*G s*ð Þ<sup>Ξ</sup>

! " #

! " #

! " #

� �

<sup>Δ</sup>*G*ð Þ *<sup>s</sup>;<sup>r</sup>* <sup>Δ</sup>*H*ð Þ *<sup>s</sup>; <sup>q</sup>* <sup>¼</sup> detð Þ *Im* <sup>þ</sup> *H s*ð Þ*G s*ð Þ

*D*ΞΠ *D*Ξ

^ ð Þ<sup>Ξ</sup>

^ ð Þ*G s*ð Þ<sup>Ξ</sup>

^ ð Þ<sup>Ξ</sup>

^ ð Þ

detð Þ *Im* <sup>þ</sup> <sup>Π</sup>*<sup>D</sup>* (18)

detð Þ *Im* <sup>þ</sup> <sup>Π</sup>*<sup>D</sup>* (19)

^ ð Þ<sup>Ξ</sup>

^ ð Þ*G s*ð Þþ <sup>Π</sup>*G s*

detð Þ *Im* þ *H s*ð Þ*G s*ð Þ

• One might suggest that in order to detour the origin, the small arc in N *<sup>z</sup>* and the small half-circle in N *<sup>s</sup>* can also be taken from the left-hand side around the origin so that possible sufficiency deficiency in the subsequent stability conditions can be dropped. In fact, if the origin is in the interior of N *<sup>s</sup>*, *G s*ð Þ and Δð Þ *s;rn;* ⋯*;r*<sup>1</sup> have no definition at the origin. Therefore, the argument principle does not apply rigorously.

#### **3.2 Stability conditions related to FCO-LTI systems**

Stability conditions in terms of the zeros distribution of Δð Þ¼ *s;rn;* ⋯*;r*<sup>1</sup> 0 and that of Δð Þ¼ *z;r* 0 are given by the following proposition [13, 15].

**Proposition 1.** Consider the fractionally commensurate system with commensurate order 0 <*r*≤1 defined by the fractional-order differential equation (1) or the fractional-order state-space equation (3). The system is stable if and only if all the zeros of Δð Þ¼ *s;rn;* ⋯*;r*<sup>1</sup> 0, denoted by f g *sk <sup>k</sup>*, have negative real parts or all the zeros of Δð Þ¼ *z;r* 0, denoted by f g *zl <sup>l</sup>* , satisfy

$$|\operatorname{Arg}(z\_l)| > \frac{\pi}{2}r, \quad \forall z\_l \tag{20}$$

*Proof of Theorem 1*. By introducing any fractional-order commensurately

Bearing these facts in mind, let us apply the argument principle to (22) counterclockwisely with N *<sup>s</sup>* being the Cauchy integral contour, and then the

*Nz*ðΔð Þ *s;rn;* ⋯*;r*<sup>1</sup> Þ � *Nz*ð Þ¼ *β*ð Þ *s;r N fs* ð ð Þ *; β*ð Þ *s;r* j

Under the given assumption about the concerned characteristic polynomial and the fact that *β*ð Þ *s;r* is holomorphic, we can assert that *f s*ð Þ *; β*ð Þ *s;r* is well-defined in the sense that it is meromorphic and without singularities at the origin. This says in particular that the argument principle applies to (22) as long as *f s*ð Þ *; β*ð Þ *s;r* 6¼ 0 over *s* ∈ N *<sup>s</sup>*. Apparently, Eq. (22) holds even if there are any factor cancelations between Δð Þ *s;rn;* ⋯*;r*<sup>1</sup> and *β*ð Þ *s;r* . This says that all unstable poles in I*nt*ð Þ N *<sup>s</sup>* ∪ N *<sup>s</sup>*, if

More precisely, since *f s*ð Þj *; β*ð Þ *s;r <sup>s</sup>*<sup>∈</sup> <sup>N</sup> *<sup>s</sup>* vanishes nowhere over *s* ∈ N *<sup>s</sup>*, we conclude

where *Nz*ð Þ� denotes the zero number of ð Þ� in Intð Þ N *<sup>s</sup>* ∪ N *<sup>s</sup>* and *N*ð Þ� denotes the

Note that all the roots of *β*ð Þ *s;r* are beyond Intð Þ N *<sup>s</sup>* . It follows that *N*ð Þ¼ *β*ð Þ *s;r* 0.

 

¼ 0. The desired results are verified if we mention that

*<sup>β</sup>*ð Þ *<sup>s</sup>;<sup>r</sup>* <sup>¼</sup> *f s*ð Þ *; <sup>β</sup>*ð Þ *<sup>s</sup>;<sup>r</sup>* (22)

*s*∈ N *<sup>s</sup>* 

*s*∈*C<sup>γ</sup>*

ð Þ *s;r* with *L*>0 be con-

� *Nc*ð *f s*ð Þ *; β*ð Þ *s;r* j

(23)

Δð Þ *s;rn;* ⋯*;r*<sup>1</sup>

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

Hurwitz polynomial *β*ð Þ *s;r* into Δð Þ *s;rn;* ⋯*;r*<sup>1</sup> , we obtain

any, remain in the left-hand side of (22).

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

desired assertion in terms of (25) follows.

net number of the locus encirclements around the origin.

Together with *N*ðÞ¼ � *Nc*ðÞ�� *Nc*ð Þ� , it follows by (17) that

*Nz*ðΔð Þ *s;rn;* ⋯*;r*<sup>1</sup> Þ ¼ *Nc f s*ð Þ *; β*ð Þ *s;r <sup>s</sup>*∈*C<sup>γ</sup>*

I*nt*ð Þ N *<sup>s</sup>* ∪ N *<sup>s</sup>*, and thus the concerned system is stable.

from 0ð Þ *; j*0 . When c-degð Þ¼ *β*ð Þ *s;r kn* and let *β*ð Þ¼ *s;r Lβ*<sup>0</sup>

ð Þ *s;r* monic, it follows from (25) that

The above equation says that *Nz*ðΔð Þ *s;rn;* ⋯*;r*<sup>1</sup> Þ ¼ 0 if and only if

*Nz*ðΔð Þ *s;rn;* ⋯*;r*<sup>1</sup> Þ ¼ 0 is equivalent to the assertion that Δð Þ¼ *s;rn;* ⋯*;r*<sup>1</sup> 0 has no roots in I*nt*ð Þ N *<sup>s</sup>* ∪ N *<sup>s</sup>*. This says exactly that Δð Þ¼ *s;rn;* ⋯*;r*<sup>1</sup> 0 has no roots in

• Theorem 1 is independent of the contour and locus orientations; or the locus orientations can be alternatively defined after the locus is already drawn. The fractionally commensurate Hurwitz polynomial *β*ð Þ *s;r* can be arbitrarily prescribed so that no existence issues exist. In addition, the stability locus is not unique. The polynomial *β*ð Þ *s;r* actually provides us additional freedom in frequency-domain

• When the stability locus with respect to the infinite portion of N *<sup>s</sup>* is concerned,

it is most appropriate to let the commensurate degree of *β*ð Þ *s;r* , namely, cdegð Þ¼ *β*ð Þ *s;r lp*, satisfy *lp* ¼ *kn*, although *β*ð Þ *s;r* can be arbitrary as long as it is fractionally commensurate Hurwitz. For example, in the sense of (25),

if c-degð Þ *β*ð Þ *s;r* >*kn*, the stability locus approaches the origin 0ð Þ *; j*0 as *s* ! ∞. It may be graphically hard to discern possible encirclements around 0ð Þ *; j*0 ; if

c-degð Þ *β*ð Þ *s;r* < *kn*, the stability locus contains portions that are plotted infinitely far

readily that

*N fs* ð ð Þ *; β*ð Þ *s;r* j

analysis and synthesis.

stant and *β*<sup>0</sup>

**41**

*s*∈*C<sup>γ</sup>* 

Several remarks about Theorem 1.

where *Arg* ð Þ� is the principal branch argument of ð Þ� on the Riemann surface.

#### **3.3 Stability criterion in FCO-LTI systems**

In what follows, a fractional-order polynomial *β*ð Þ *s;r* is said to be commensurately Hurwitz if *β*ð Þ *s;r* is fractionally commensurate order *r* with some 0< *r*≤ 1 in the sense that all the roots of *β*ð Þ¼ *s;r* 0 possess negative real parts. More specifically, we write

$$\beta(s,r) = c\_p s^{\tau l\_p} + c\_{p-1} s^{\tau l\_{p-1}} + \dots + c\_1 e^{\tau l\_1}$$

where *lp* >*lp*�<sup>1</sup> > ⋯ >*l*<sup>1</sup> ≥0 are integers and Reð Þ*s* < 0 for any *β*ð Þ¼ *s;r* 0. It is straightforward to see by definition that a fractionally commensurate order Hurwitz polynomial must be holomorphic on the whole complex plane.

**Theorem 1.** Consider the fractional-order system with commensurate order 0<*r*≤1 defined by the differential equation (1) or the state-space equation (3). The concerned system is stable if and only if for any *γ* >0 sufficiently small and *R*>0 sufficiently large, any prescribed commensurate order Hurwitz polynomial *β*ð Þ *s;r* , the stability locus

$$f(s, \beta(s, r))\Big|\_{s \in \mathcal{N}\_r} =: \frac{\Delta(s, r\_n, \cdots, r\_1)}{\beta(s, r)}\Big|\_{s \in \mathcal{N}\_r} \tag{21}$$

vanishes nowhere over N *<sup>s</sup>*, namely, *f s*ð Þ *; β*ð Þ *s;r* 6¼ 0 for all *s*∈ N *<sup>s</sup>*; and the number of its clockwise encirclement around the origin is equal to that of its counterclockwise ones, namely,

$$N\left(f(s,\boldsymbol{\beta}(\boldsymbol{s},\boldsymbol{r})|\_{\boldsymbol{s}\in\boldsymbol{N}\_{\boldsymbol{s}}})\right) = N\_{\boldsymbol{\varepsilon}}\left(f(s,\boldsymbol{\beta}(\boldsymbol{s},\boldsymbol{r}))\Big|\_{\boldsymbol{s}\in\boldsymbol{N}\_{\boldsymbol{s}}}\right) - N\_{\boldsymbol{\varepsilon}}\left(f(s,\boldsymbol{\beta}(\boldsymbol{s},\boldsymbol{r}))\Big|\_{\boldsymbol{s}\in\boldsymbol{N}\_{\boldsymbol{s}}}\right) = \mathbf{0}$$

In the above, the clockwise/counterclockwise orientation of *f s*ð Þj *; β*ð Þ *s;r <sup>s</sup>*<sup>∈</sup> <sup>N</sup> *<sup>s</sup>* can be self-defined.

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

*Proof of Theorem 1*. By introducing any fractional-order commensurately Hurwitz polynomial *β*ð Þ *s;r* into Δð Þ *s;rn;* ⋯*;r*<sup>1</sup> , we obtain

$$\frac{\Delta(s, r\_n, \dots, r\_1)}{\beta(s, r)} = f(s, \beta(s, r))\tag{22}$$

Under the given assumption about the concerned characteristic polynomial and the fact that *β*ð Þ *s;r* is holomorphic, we can assert that *f s*ð Þ *; β*ð Þ *s;r* is well-defined in the sense that it is meromorphic and without singularities at the origin. This says in particular that the argument principle applies to (22) as long as *f s*ð Þ *; β*ð Þ *s;r* 6¼ 0 over *s* ∈ N *<sup>s</sup>*. Apparently, Eq. (22) holds even if there are any factor cancelations between Δð Þ *s;rn;* ⋯*;r*<sup>1</sup> and *β*ð Þ *s;r* . This says that all unstable poles in I*nt*ð Þ N *<sup>s</sup>* ∪ N *<sup>s</sup>*, if any, remain in the left-hand side of (22).

Bearing these facts in mind, let us apply the argument principle to (22) counterclockwisely with N *<sup>s</sup>* being the Cauchy integral contour, and then the desired assertion in terms of (25) follows.

More precisely, since *f s*ð Þj *; β*ð Þ *s;r <sup>s</sup>*<sup>∈</sup> <sup>N</sup> *<sup>s</sup>* vanishes nowhere over *s* ∈ N *<sup>s</sup>*, we conclude readily that

$$N\_x(\Delta(\mathfrak{s}, r\_n, \dots, r\_1)) - N\_x(\beta(\mathfrak{s}, r)) = N(f(\mathfrak{s}, \beta(\mathfrak{s}, r))|\_{\mathfrak{s} \in \mathcal{N}\_\mathfrak{s}}) \tag{23}$$

where *Nz*ð Þ� denotes the zero number of ð Þ� in Intð Þ N *<sup>s</sup>* ∪ N *<sup>s</sup>* and *N*ð Þ� denotes the net number of the locus encirclements around the origin.

Note that all the roots of *β*ð Þ *s;r* are beyond Intð Þ N *<sup>s</sup>* . It follows that *N*ð Þ¼ *β*ð Þ *s;r* 0. Together with *N*ðÞ¼ � *Nc*ðÞ�� *Nc*ð Þ� , it follows by (17) that

$$N\_x(\Delta(\mathfrak{s}, r\_n, \dots, r\_1)) = N\_{\mathfrak{c}}\left(f(\mathfrak{s}, \beta(\mathfrak{s}, r))|\_{\mathfrak{s} \in C\_{\mathfrak{r}}}\right) - N\_{\mathfrak{c}}(f(\mathfrak{s}, \beta(\mathfrak{s}, r))|\_{\mathfrak{s} \in C\_{\mathfrak{r}}}),$$

The above equation says that *Nz*ðΔð Þ *s;rn;* ⋯*;r*<sup>1</sup> Þ ¼ 0 if and only if *N fs* ð ð Þ *; β*ð Þ *s;r* j *s*∈*C<sup>γ</sup>* ¼ 0. The desired results are verified if we mention that *Nz*ðΔð Þ *s;rn;* ⋯*;r*<sup>1</sup> Þ ¼ 0 is equivalent to the assertion that Δð Þ¼ *s;rn;* ⋯*;r*<sup>1</sup> 0 has no roots in I*nt*ð Þ N *<sup>s</sup>* ∪ N *<sup>s</sup>*. This says exactly that Δð Þ¼ *s;rn;* ⋯*;r*<sup>1</sup> 0 has no roots in I*nt*ð Þ N *<sup>s</sup>* ∪ N *<sup>s</sup>*, and thus the concerned system is stable.

Several remarks about Theorem 1.

• Theorem 1 is independent of the contour and locus orientations; or the locus orientations can be alternatively defined after the locus is already drawn. The fractionally commensurate Hurwitz polynomial *β*ð Þ *s;r* can be arbitrarily prescribed so that no existence issues exist. In addition, the stability locus is not unique. The polynomial *β*ð Þ *s;r* actually provides us additional freedom in frequency-domain analysis and synthesis.

• When the stability locus with respect to the infinite portion of N *<sup>s</sup>* is concerned, it is most appropriate to let the commensurate degree of *β*ð Þ *s;r* , namely, cdegð Þ¼ *β*ð Þ *s;r lp*, satisfy *lp* ¼ *kn*, although *β*ð Þ *s;r* can be arbitrary as long as it is fractionally commensurate Hurwitz. For example, in the sense of (25), if c-degð Þ *β*ð Þ *s;r* >*kn*, the stability locus approaches the origin 0ð Þ *; j*0 as *s* ! ∞. It may be graphically hard to discern possible encirclements around 0ð Þ *; j*0 ; if c-degð Þ *β*ð Þ *s;r* < *kn*, the stability locus contains portions that are plotted infinitely far from 0ð Þ *; j*0 . When c-degð Þ¼ *β*ð Þ *s;r kn* and let *β*ð Þ¼ *s;r Lβ*<sup>0</sup> ð Þ *s;r* with *L*>0 be constant and *β*<sup>0</sup> ð Þ *s;r* monic, it follows from (25) that

• One might suggest that in order to detour the origin, the small arc in N *<sup>z</sup>* and the small half-circle in N *<sup>s</sup>* can also be taken from the left-hand side around the origin so that possible sufficiency deficiency in the subsequent stability conditions can be dropped. In fact, if the origin is in the interior of N *<sup>s</sup>*, *G s*ð Þ and Δð Þ *s;rn;* ⋯*;r*<sup>1</sup> have no definition at the origin. Therefore, the argument principle does not apply

Stability conditions in terms of the zeros distribution of Δð Þ¼ *s;rn;* ⋯*;r*<sup>1</sup> 0 and

**Proposition 1.** Consider the fractionally commensurate system with commensurate order 0 <*r*≤1 defined by the fractional-order differential equation (1) or the fractional-order state-space equation (3). The system is stable if and only if all the zeros of Δð Þ¼ *s;rn;* ⋯*;r*<sup>1</sup> 0, denoted by f g *sk <sup>k</sup>*, have negative real parts or all the

, satisfy

2

where *Arg* ð Þ� is the principal branch argument of ð Þ� on the Riemann surface.

In what follows, a fractional-order polynomial *β*ð Þ *s;r* is said to be commensurately Hurwitz if *β*ð Þ *s;r* is fractionally commensurate order *r* with some 0< *r*≤ 1 in the sense that all the roots of *β*ð Þ¼ *s;r* 0 possess negative real parts. More specifi-

*rlp* <sup>þ</sup> *cp*�<sup>1</sup>*<sup>s</sup>*

straightforward to see by definition that a fractionally commensurate order Hurwitz polynomial must be holomorphic on the whole complex plane.

where *lp* >*lp*�<sup>1</sup> > ⋯ >*l*<sup>1</sup> ≥0 are integers and Reð Þ*s* < 0 for any *β*ð Þ¼ *s;r* 0. It is

**Theorem 1.** Consider the fractional-order system with commensurate order 0<*r*≤1 defined by the differential equation (1) or the state-space equation (3). The concerned system is stable if and only if for any *γ* >0 sufficiently small and *R*>0 sufficiently large, any prescribed commensurate order Hurwitz polynomial *β*ð Þ *s;r* ,

*f s*ð Þ *; <sup>β</sup>*ð Þ *<sup>s</sup>;<sup>r</sup> <sup>s</sup>*<sup>∈</sup> <sup>N</sup> *<sup>s</sup>* <sup>≕</sup> <sup>Δ</sup>ð Þ *<sup>s</sup>;rn;* <sup>⋯</sup>*;r*<sup>1</sup>

vanishes nowhere over N *<sup>s</sup>*, namely, *f s*ð Þ *; β*ð Þ *s;r* 6¼ 0 for all *s*∈ N *<sup>s</sup>*; and the num-

 

In the above, the clockwise/counterclockwise orientation of *f s*ð Þj *; β*ð Þ *s;r <sup>s</sup>*<sup>∈</sup> <sup>N</sup> *<sup>s</sup>* can

 

ber of its clockwise encirclement around the origin is equal to that of its

¼ *Nc f s*ð Þ *; β*ð Þ *s;r <sup>s</sup>*<sup>∈</sup> <sup>N</sup> *<sup>s</sup>*

*rlp*�<sup>1</sup> <sup>þ</sup> <sup>⋯</sup> <sup>þ</sup> *<sup>c</sup>*1*<sup>e</sup>*

*β*ð Þ *s;r*

 � *Nc* 

 *s*∈ N *<sup>s</sup>*

*f s*ð Þ *; β*ð Þ *s;r*

 *s*∈ N *<sup>s</sup>* (21)

¼ 0

*rl*1

*r,* ∀*zl* (20)

<sup>∣</sup>Argð Þ *zl* <sup>∣</sup><sup>&</sup>gt; *<sup>π</sup>*

**3.2 Stability conditions related to FCO-LTI systems**

zeros of Δð Þ¼ *z;r* 0, denoted by f g *zl <sup>l</sup>*

**3.3 Stability criterion in FCO-LTI systems**

*β*ð Þ¼ *s;r cps*

that of Δð Þ¼ *z;r* 0 are given by the following proposition [13, 15].

rigorously.

*Control Theory in Engineering*

cally, we write

the stability locus

counterclockwise ones, namely,

*s*∈ N *<sup>s</sup>*

*N fs*ð *; β*ð Þ *s;r* j

be self-defined.

**40**

$$\lim\_{s \to \infty} f(s, \beta(s, r)) = \lim\_{s \to \infty} \frac{\Delta(s, r\_n, \dots, r\_1)}{L\beta'(s, r)} = \mathbf{1}/L < \infty \tag{24}$$

Then, the closed-loop system is stable if and only if for any *s*-domain contour N *<sup>s</sup>* � ϵ with *R*>0 sufficiently large and *γ* >0, ϵ≥0 sufficiently small, any prescribed com-

> detð Þ *Im* þ *H s*ð Þ*G s*ð Þ detð Þ *Im* þ Ξ*D*

> > �

� � � � *s*∈ N *<sup>s</sup>*�ϵ

*f <sup>C</sup>*ð Þ *s; β*ð Þ *s; ρ*

� � � *s*∈ N *<sup>s</sup>*�ϵ (26)

� ¼ 0.

*β*ð Þ *s; ρ*

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

Here, N *<sup>s</sup>* � ϵ stands for the contour by shifting N *<sup>s</sup>* to its left with distance ϵ.

(19) is well-defined on N *<sup>s</sup>* � ϵ and Intð Þ N *<sup>s</sup>* � ϵ . Then, applying the argument principle to (19) and repeating some arguments similar to those in the proof for

To complete the proof, it remains to only show why we must work with the contour N *<sup>s</sup>* � ϵ in general, rather than the standard contour N *<sup>s</sup>* itself directly. To see this, we notice that the shifting factor ϵ≥0 is introduced for detouring possible openloop zeros and poles on the imaginary axis but excluding the origin. Since all zeros and poles, if any, are isolated, ϵ≥0 is always available. Furthermore, we have by (12) that

> detð Þ *Im* þ *H s*ð Þ*G s*ð Þ detð Þ *Im* þ Ξ*D*

> > *<sup>G</sup>*ð Þ *<sup>s</sup>; <sup>ρ</sup>* <sup>Δ</sup>*<sup>m</sup>*

On the contrary, if there exist no imaginary open-loop poles, it is not hard to see that working with N *<sup>s</sup>* � ϵ in numerically computing the locus yields no sufficiency

**Theorem 4.** Under the same assumptions of Theorem 3; the closed-loop system is stable if and only if for any *s*-domain contour N *<sup>z</sup>* � ϵ with *R* >0 large sufficiently and *γ* >0, ϵ≥ 0 sufficiently small, any prescribed *πr*-sector Hurwitz polynomial

*α*ð Þ *z; ρ*

¼ 0. Here, we have

which says clearly that if there are imaginary zeros of Δ*G*ð Þ *s; ρ* Δ*H*ð Þ *s; ρ* , then factor cancelation will happen between Δ*G*ð Þ *s; ρ* Δ*H*ð Þ *s; ρ* and detð Þ *Im* þ *H s*ð Þ*G s*ð Þ . Such factor cancelation, if any, can be revealed analytically as in the above algebras, whereas it does bring us trouble in numerically computing the locus (obviously, numerical computation cannot reflect any existence of factor cancelation rigorously). Consequently, when the standard contour N *<sup>s</sup>* is adopted, if there do exist imaginary open-loop poles, the corresponding stability locus cannot be well-defined with respect to N *<sup>s</sup>*. When this happens, one need to know the exact positions of imaginary zeros and/or poles and modify the contour accordingly to detour them before computing the locus; in other words, working with N *<sup>s</sup>* brings sufficiency

Δ*<sup>m</sup>*

det <sup>Δ</sup>*G*ð Þ *<sup>s</sup>; <sup>ρ</sup>* <sup>Δ</sup>*H*ð Þ *<sup>s</sup>; <sup>ρ</sup> Im* <sup>þ</sup> *H s* � ð Þ*G s* � ð Þ � �

*<sup>H</sup>*ð Þ *s; ρ* detð Þ *Im* þ Ξ*D*

det *Im* <sup>þ</sup> *H z* <sup>~</sup> ð Þ*G z* <sup>~</sup> ð ÞÞ detð Þ *Im* þ Ξ*D* �

� � � *z* ∈ N *<sup>z</sup>*�ϵ

(27)

*Proof of Theorem 3*. Under the given assumptions, the return difference equation

mensurate order Hurwitz polynomial *β*ð Þ *s;r* ; the locus

satisfies: (i) *f <sup>C</sup>*ð Þ *s; β*ð Þ *s; ρ* 6¼ 0 for all *s*∈ N *<sup>s</sup>* � ϵ; (ii) *N*

*<sup>f</sup> <sup>C</sup>*ð Þ *<sup>s</sup>; <sup>β</sup>*ð Þ *<sup>s</sup>; <sup>ρ</sup> <sup>s</sup>*<sup>∈</sup> <sup>N</sup> *<sup>s</sup>*�<sup>ϵ</sup> <sup>≕</sup> <sup>Δ</sup>*G*ð Þ *<sup>s</sup>; <sup>ρ</sup>* <sup>Δ</sup>*H*ð Þ *<sup>s</sup>; <sup>ρ</sup>*

Theorem 1, the desired results follow readily.

Δ*G*ð Þ *s; ρ* Δ*H*ð Þ *s; ρ*

¼ Δ*G*ð Þ *s; ρ* Δ*H*ð Þ *s; ρ*

deficiency when imaginary open-loop poles exist.

redundance as *ε* ! 0, noting that ϵ≥0 always exists. As a *z*-domain counterpart to Theorem 3, we have.

*gC*ð Þ *<sup>z</sup>; <sup>α</sup>*ð Þ *<sup>z</sup>; <sup>ρ</sup> <sup>z</sup>* <sup>∈</sup> <sup>N</sup> *<sup>z</sup>*�<sup>ϵ</sup> <sup>≕</sup> <sup>Δ</sup><sup>~</sup> *<sup>G</sup>*ð Þ *<sup>z</sup>; <sup>ρ</sup>* <sup>Δ</sup><sup>~</sup> *<sup>H</sup>*ð Þ *<sup>z</sup>; <sup>ρ</sup>*

�

satisfies: (i) *gC*ð Þ *z; α*ð Þ *z; ρ* 6¼ 0 for all *z*∈ N *<sup>z</sup>* � ϵ; (ii)

*α*ð Þ *s;r* ; the stability locus

*gC*ð Þ *z; α*ð Þ *z; ρ*

*N* �

**43**

� � � � � �

� � � *z* ∈ N *<sup>z</sup>*�ϵ

� � � �

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

Since *f s*ð Þ *; β*ð Þ *s;r* is continuous in *s*, (27) says that ∣*f s*ð Þ *; β*ð Þ *s;r* ∣ ≤ *M* over *s*∈ N *<sup>s</sup>* for some 0 < *M* < ∞ and *f s*ð Þj *; β*ð Þ *s;r <sup>s</sup>*<sup>∈</sup> <sup>N</sup> *<sup>s</sup>* can be plotted in a bounded region around the origin. Thus, no prior frequency sweep is needed when dealing with the stability locus (25).

• Each and all the conditions in Theorem 1 can be implemented only by numerically integrating ∠*f s*ð Þ *; β*ð Þ *s;r* for computing the argument incremental ∇∠*f s*ð Þ *; β*ð Þ *s;r* along the Cauchy integral contour N *<sup>s</sup>*, and then checking if ∇∠*f s*ð Þ *; β*ð Þ *s;r =*2*π* ¼ 0 holds. In this way, numerically implementing Theorem 1 entails no graphical locus plotting. This is also the case for the following results.

• The clockwise/counterclockwise orientation of *f s*ð Þj *; β*ð Þ *s;r <sup>s</sup>*<sup>∈</sup> <sup>N</sup> *<sup>s</sup>* can be selfdefined. This is also the case in all the subsequent results.

Next, a regular-order polynomial *α*ð Þ *z;r* is said to be *πr*-sector Hurwitz if all the roots of *α*ð Þ¼ *z;r* 0 satisfy (20). More specifically, we write

$$a(z,r) = c\_p z^{l\_p} + c\_{p-1} z^{l\_{p-1}} + \cdots + c\_1 z^{l\_1}$$

where *lp* >*lp*�<sup>1</sup> > ⋯ >*l*<sup>1</sup> ≥0 are integers and Argð Þ*z* <0 for any *α*ð Þ¼ *z;r* 0. It is easy to see that a *πr*-sector Hurwitz polynomial is holomorphic.

**Theorem 2.** Consider the fractional-order system with commensurate order 0<*r*≤1 of the differential equation (1) or the state-space equation (3). The system is stable if and only if for any *γ* >0 small and *R*>0 large sufficiently, any prescribed *πr*-sector Hurwitz polynomial *α*ð Þ *z;r* , the stability locus

$$\left. \mathcal{g}(z, a(z, r)) \right|\_{z \in \mathcal{N}\_x} =: \frac{\Delta(z, r)}{a(z, r)} \Big|\_{z \in \mathcal{N}\_x} \tag{25}$$

vanishes nowhere over N *<sup>z</sup>*, namely, *g z*ð Þ *; α*ð Þ *z;r* 6¼ 0 for all *z*∈ N *<sup>z</sup>*; and the number of its clockwise encirclement around the origin is equal to that of its counterclockwise ones, namely, *N gz*ð *; α*ð Þ *z;r* j *z*∈ N *<sup>z</sup>* <sup>¼</sup> 0.

*Proof of Theorem 2.* Repeating those for Theorem 1 but in terms of *g z*ð Þ *; α*ð Þ *z;r* rather with *f s*ð Þ *; β*ð Þ *s;r* , while the contour N *<sup>s</sup>* is replaced with N *<sup>z</sup>*.

**Remark 2.** The proof arguments can also be understood by using the transformation *z* ¼ *s <sup>r</sup>* to (22) and then applying the argument principle to the resulting *z*domain relationship. Note that *z* ¼ *s <sup>r</sup>* is holomorphic on <sup>C</sup>\ 0f g. The angle preserving property ([16], pp. 255–256) leads immediately that the stability conditions of Theorem 2 are equivalent to those of Theorem 1.

#### **3.4 Stability criteria for closed-loop FCO-LTI systems**

Based on the return difference equation (31) claimed in the feedback configuration of **Figure 1**, together with the argument principle, the following *s*-domain criterion follows readily.

**Theorem 3.** Consider the feedback system as in **Figure 1** with the fractionalorder subsystems Σ*<sup>G</sup>* and Σ*<sup>H</sup>* defined in (22). Assume that both subsystems are fractionally commensurate with respect to a same commensurate order 0 <*ρ*≤1. *Nyquist-Like Stability Criteria for Fractional-Order Linear Dynamical Systems DOI: http://dx.doi.org/10.5772/intechopen.88119*

Then, the closed-loop system is stable if and only if for any *s*-domain contour N *<sup>s</sup>* � ϵ with *R*>0 sufficiently large and *γ* >0, ϵ≥0 sufficiently small, any prescribed commensurate order Hurwitz polynomial *β*ð Þ *s;r* ; the locus

$$\left.f\_{C}(\boldsymbol{s},\boldsymbol{\beta}(\boldsymbol{s},\boldsymbol{\rho}))\right|\_{\boldsymbol{s}\in\mathcal{N}\_{\boldsymbol{s}}-\boldsymbol{e}} = \frac{\Delta\_{G}(\boldsymbol{s},\boldsymbol{\rho})\Delta\_{H}(\boldsymbol{s},\boldsymbol{\rho})}{\boldsymbol{\rho}(\boldsymbol{s},\boldsymbol{\rho})} \frac{\det(I\_{m}+H(\boldsymbol{s})G(\boldsymbol{s}))}{\det(I\_{m}+\Xi D)}\bigg|\_{\boldsymbol{s}\in\mathcal{N}\_{\boldsymbol{s}}-\boldsymbol{e}}\tag{26}$$

satisfies: (i) *f <sup>C</sup>*ð Þ *s; β*ð Þ *s; ρ* 6¼ 0 for all *s*∈ N *<sup>s</sup>* � ϵ; (ii) *N* � *f <sup>C</sup>*ð Þ *s; β*ð Þ *s; ρ* � � � *s*∈ N *<sup>s</sup>*�ϵ � ¼ 0.

Here, N *<sup>s</sup>* � ϵ stands for the contour by shifting N *<sup>s</sup>* to its left with distance ϵ.

*Proof of Theorem 3*. Under the given assumptions, the return difference equation (19) is well-defined on N *<sup>s</sup>* � ϵ and Intð Þ N *<sup>s</sup>* � ϵ . Then, applying the argument principle to (19) and repeating some arguments similar to those in the proof for Theorem 1, the desired results follow readily.

To complete the proof, it remains to only show why we must work with the contour N *<sup>s</sup>* � ϵ in general, rather than the standard contour N *<sup>s</sup>* itself directly. To see this, we notice that the shifting factor ϵ≥0 is introduced for detouring possible openloop zeros and poles on the imaginary axis but excluding the origin. Since all zeros and poles, if any, are isolated, ϵ≥0 is always available. Furthermore, we have by (12) that

$$\begin{aligned} &\Delta\_G(s,\rho)\Delta\_H(s,\rho)\frac{\det(I\_m+H(s)\check{G}(s))}{\det(I\_m+\Xi D)}\\ &=\Delta\_G(s,\rho)\Delta\_H(s,\rho)\frac{\det\Big(\Delta\_G(s,\rho)\Delta\_H(s,\rho)I\_m+\check{H}(s)\check{G}(s)\Big)}{\Delta\_G^m(s,\rho)\Delta\_H^m(s,\rho)\det(I\_m+\Xi D)} \end{aligned}$$

which says clearly that if there are imaginary zeros of Δ*G*ð Þ *s; ρ* Δ*H*ð Þ *s; ρ* , then factor cancelation will happen between Δ*G*ð Þ *s; ρ* Δ*H*ð Þ *s; ρ* and detð Þ *Im* þ *H s*ð Þ*G s*ð Þ . Such factor cancelation, if any, can be revealed analytically as in the above algebras, whereas it does bring us trouble in numerically computing the locus (obviously, numerical computation cannot reflect any existence of factor cancelation rigorously). Consequently, when the standard contour N *<sup>s</sup>* is adopted, if there do exist imaginary open-loop poles, the corresponding stability locus cannot be well-defined with respect to N *<sup>s</sup>*. When this happens, one need to know the exact positions of imaginary zeros and/or poles and modify the contour accordingly to detour them before computing the locus; in other words, working with N *<sup>s</sup>* brings sufficiency deficiency when imaginary open-loop poles exist.

On the contrary, if there exist no imaginary open-loop poles, it is not hard to see that working with N *<sup>s</sup>* � ϵ in numerically computing the locus yields no sufficiency redundance as *ε* ! 0, noting that ϵ≥0 always exists.

As a *z*-domain counterpart to Theorem 3, we have.

**Theorem 4.** Under the same assumptions of Theorem 3; the closed-loop system is stable if and only if for any *s*-domain contour N *<sup>z</sup>* � ϵ with *R* >0 large sufficiently and *γ* >0, ϵ≥ 0 sufficiently small, any prescribed *πr*-sector Hurwitz polynomial *α*ð Þ *s;r* ; the stability locus

$$\mathcal{g}\_{\mathcal{C}}(\boldsymbol{z},a(\boldsymbol{z},\rho))\Big|\_{\boldsymbol{z}\in\mathcal{N}\_{\boldsymbol{z}}-\boldsymbol{e}} =: \frac{\tilde{\Delta}\_{\boldsymbol{G}}(\boldsymbol{z},\rho)\tilde{\Delta}\_{H}(\boldsymbol{z},\rho)}{a(\boldsymbol{z},\rho)}\det\left(\frac{\boldsymbol{I}\_{m}+\tilde{H}(\boldsymbol{z})\tilde{\mathcal{G}}(\boldsymbol{z})}{\det(\boldsymbol{I}\_{m}+\Xi\boldsymbol{D})}\right)\Big|\_{\boldsymbol{z}\in\mathcal{N}\_{\boldsymbol{z}}-\boldsymbol{e}}\tag{27}$$

satisfies: (i) *gC*ð Þ *z; α*ð Þ *z; ρ* 6¼ 0 for all *z*∈ N *<sup>z</sup>* � ϵ; (ii) *N* � *gC*ð Þ *z; α*ð Þ *z; ρ* � � � *z* ∈ N *<sup>z</sup>*�ϵ � ¼ 0. Here, we have

lim*s*!<sup>∞</sup> *f s*ð Þ¼ *; <sup>β</sup>*ð Þ *<sup>s</sup>;<sup>r</sup>* lim*s*!<sup>∞</sup>

defined. This is also the case in all the subsequent results.

roots of *α*ð Þ¼ *z;r* 0 satisfy (20). More specifically, we write

easy to see that a *πr*-sector Hurwitz polynomial is holomorphic.

rather with *f s*ð Þ *; β*ð Þ *s;r* , while the contour N *<sup>s</sup>* is replaced with N *<sup>z</sup>*.

*πr*-sector Hurwitz polynomial *α*ð Þ *z;r* , the stability locus

counterclockwise ones, namely, *N gz*ð *; α*ð Þ *z;r* j

Theorem 2 are equivalent to those of Theorem 1.

**3.4 Stability criteria for closed-loop FCO-LTI systems**

domain relationship. Note that *z* ¼ *s*

criterion follows readily.

mation *z* ¼ *s*

**42**

locus (25).

*Control Theory in Engineering*

Δð Þ *s;rn;* ⋯*;r*<sup>1</sup> *Lβ* 0

Since *f s*ð Þ *; β*ð Þ *s;r* is continuous in *s*, (27) says that ∣*f s*ð Þ *; β*ð Þ *s;r* ∣ ≤ *M* over *s*∈ N *<sup>s</sup>* for some 0 < *M* < ∞ and *f s*ð Þj *; β*ð Þ *s;r <sup>s</sup>*<sup>∈</sup> <sup>N</sup> *<sup>s</sup>* can be plotted in a bounded region around the origin. Thus, no prior frequency sweep is needed when dealing with the stability

• Each and all the conditions in Theorem 1 can be implemented only by numer-

• The clockwise/counterclockwise orientation of *f s*ð Þj *; β*ð Þ *s;r <sup>s</sup>*<sup>∈</sup> <sup>N</sup> *<sup>s</sup>* can be self-

Next, a regular-order polynomial *α*ð Þ *z;r* is said to be *πr*-sector Hurwitz if all the

*<sup>α</sup>*ð Þ¼ *<sup>z</sup>;<sup>r</sup> cpzlp* <sup>þ</sup> *cp*�<sup>1</sup>*zlp*�<sup>1</sup> <sup>þ</sup> <sup>⋯</sup> <sup>þ</sup> *<sup>c</sup>*1*z<sup>l</sup>*<sup>1</sup>

where *lp* >*lp*�<sup>1</sup> > ⋯ >*l*<sup>1</sup> ≥0 are integers and Argð Þ*z* <0 for any *α*ð Þ¼ *z;r* 0. It is

**Theorem 2.** Consider the fractional-order system with commensurate order 0<*r*≤1 of the differential equation (1) or the state-space equation (3). The system is stable if and only if for any *γ* >0 small and *R*>0 large sufficiently, any prescribed

*g z*ð Þ *; <sup>α</sup>*ð Þ *<sup>z</sup>;<sup>r</sup> <sup>z</sup>*<sup>∈</sup> <sup>N</sup> *<sup>z</sup>* <sup>≕</sup> <sup>Δ</sup>ð Þ *<sup>z</sup>;<sup>r</sup>*

vanishes nowhere over N *<sup>z</sup>*, namely, *g z*ð Þ *; α*ð Þ *z;r* 6¼ 0 for all *z*∈ N *<sup>z</sup>*; and the number of its clockwise encirclement around the origin is equal to that of its

*Proof of Theorem 2.* Repeating those for Theorem 1 but in terms of *g z*ð Þ *; α*ð Þ *z;r*

**Remark 2.** The proof arguments can also be understood by using the transfor-

Based on the return difference equation (31) claimed in the feedback configura-

**Theorem 3.** Consider the feedback system as in **Figure 1** with the fractionalorder subsystems Σ*<sup>G</sup>* and Σ*<sup>H</sup>* defined in (22). Assume that both subsystems are fractionally commensurate with respect to a same commensurate order 0 <*ρ*≤1.

property ([16], pp. 255–256) leads immediately that the stability conditions of

tion of **Figure 1**, together with the argument principle, the following *s*-domain

 

*α*ð Þ *z;r*

*z*∈ N *<sup>z</sup>*

*<sup>r</sup>* to (22) and then applying the argument principle to the resulting *z*-

 *z* ∈ N *<sup>z</sup>*

¼ 0.

*<sup>r</sup>* is holomorphic on <sup>C</sup>\ 0f g. The angle preserving

(25)

ically integrating ∠*f s*ð Þ *; β*ð Þ *s;r* for computing the argument incremental ∇∠*f s*ð Þ *; β*ð Þ *s;r* along the Cauchy integral contour N *<sup>s</sup>*, and then checking if ∇∠*f s*ð Þ *; β*ð Þ *s;r =*2*π* ¼ 0 holds. In this way, numerically implementing Theorem 1 entails no graphical locus plotting. This is also the case for the following results.

ð Þ *<sup>s</sup>;<sup>r</sup>* <sup>¼</sup> <sup>1</sup>*=L*<sup>&</sup>lt; <sup>∞</sup> (24)

$$\begin{cases} \tilde{\Delta}\_G(\boldsymbol{z}, \boldsymbol{\rho}) = \Delta\_G(\boldsymbol{s}, \boldsymbol{\rho})|\_{\boldsymbol{s}^\circ = \boldsymbol{x}}, \quad \tilde{\Delta}\_H(\boldsymbol{z}, \boldsymbol{\rho}) = \Delta\_H(\boldsymbol{s}, \boldsymbol{\rho})\big|\_{\boldsymbol{s}^\circ = \boldsymbol{x}}, \\\ \tilde{H}(\boldsymbol{z}) = H(\boldsymbol{s})|\_{\boldsymbol{s}^\circ = \boldsymbol{x}}, \quad \tilde{\mathcal{G}}(\boldsymbol{z}) = G(\boldsymbol{s})\big|\_{\boldsymbol{s}^\circ = \boldsymbol{x}} \end{cases}$$

• *a*>0, *b*>0, and *a*<sup>2</sup> ≥*b*. By examining the *s*-loci of **Figure 4** graphically, no

*f s*ð Þ *; β*ð Þ *s;r*

The same conclusions can be drawn by examining the *z*-loci of **Figure 4**. More

• *a*>0, *b*>0, and *a*<sup>2</sup> <*b*. By the *s*-loci of **Figure 5**, no encirclements around the

The same conclusions can be drawn by examining the *z*-loci of **Figure 5**. More

• *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

 *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 revealed by

 *s*∈ N *<sup>s</sup>* 

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

∣ ¼ 1 for each

¼ 0 for each

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

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

 *f* 

*z; α*ð Þ *z;r*

 *z* ∈ N *<sup>z</sup>* ¼ 1

*r*∈f g 0*:*2*;* 0*:*4*;* 0*:*6*;* 0*:*8*;* 1*:*0 can be verified numerically without locus plotting.

origin are counted in each case of *r*∈f g 0*:*2*;* 0*:*4*;* 0*:*6*;* 0*:*8*;* 1*:*0 graphically; or

encirclements around the origin are counted in each case of

*f z*ð Þ *; α*ð Þ *z;r*

*z; α*ð Þ *z;r*

 *f* 

means of the *z*-loci of **Figure 6**. More precisely, we have *N*

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

 *z* ∈ N *<sup>z</sup>* 

*s; β*ð Þ *s;r* 

 *z*∈ N *<sup>z</sup>* 

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

*r*∈f g 0*:*2*;* 0*:*4*;* 0*:*6*;* 0*:*8*;* 1*:*0 ; indeed, *N*

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

precisely, we have *N*

*s; β*ð Þ *s;r*

precisely, we have *N*

numerically.

**Figure 5.**

**Figure 6.**

**45**

 *s*∈ N *<sup>s</sup>* 

graphically; alternatively, *N*

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

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

fore, the system is stable in each case.

 *f* 

numerically.

*N f* 

Therefore, the system is stable in each case.

In the above, N *<sup>z</sup>* � ϵ is the contour by shifting N *<sup>z</sup>* to its left with distance ϵ. Several remarks about Theorems 3 and 4:

• The shifted contour N *<sup>s</sup>* � ϵ reduces to the standard contour N *<sup>s</sup>* when ϵ ¼ 0. This is also the case for the shifted contour N *<sup>z</sup>* � ϵ and N *<sup>z</sup>*.

• 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 involved.
