On the Essential Instabilities Caused by Fractional-Order Transfer Functions

The exact stability condition for certain class of fractional-order multivalued transfer functions is presented. Unlike the conventional case that the stability is directly studied by investigating the poles of the transfer function, in the systems under consideration, the branch points must also come into account as another kind of singularities. It is shown that a multivalued transfer function can behave unstably because of the numerator term while it has no unstable poles. So, in this case, not only the characteristic equation but the numerator term is of significant importance. In this manner, a family of unstable fractional-order transfer functions is introduced which exhibit essential instabilities, that is, those which cannot be removed by feedback. Two illustrative examples are presented; the transfer function of which has no unstable poles but the instability occurred because of the unstable branch points of the numerator term. The effect of unstable branch points is studied and simulations are presented.


Introduction
The stability problem of linear systems has been the subject of many studies.In the field of classical linear time-invariant LTI systems, the well-known Routh-Hurwitz criterion is widely used for testing the stability of a given rational transfer function.In the literature, the system stability assessment of a given delayed system is usually performed with a graphical method, for example, Nyquist criterion 1 , Mikhailov criterion 2 , and the rootlocus technique 3 .A common observation is that the open-loop stability can be examined by investigating the denominator of the transfer function for closed right half-plane RHP roots.
Although most LTI systems can be represented by rational transfer functions possibly with delay but there are also some important exceptions.For  Systems such as those described by 1.1 -1.6 are of multivalued nature and this makes the related studies a challenging task.For example, the stability testing of such systems is not yet fully addressed.In the field of fractional-order systems, the most well-known analytic stability test is the one available for the particular case of fractional-order systems commonly known as fractional differential systems of commensurate order 5, 8, 9 .The test resembles the stability condition for classical systems and it concludes that a system described by the multivalued transfer function lie in the sector defined by where w s α .The proof of the above fact is based on studying the asymptotic behavior of the inverse Laplace transform of the basic element s α − λ −j where j ∈ N, α ∈ R , and λ ∈ C. See also 10, 11 for recent developments on this subject.In the literature, most studies on the infinite-dimensional LTI systems are focused on the stability of a class of distributed systems whose transfer functions involve √ s and/or e − √ s .The former group of studies relies on using Pontryagin's theory of quasipolynomials 12 , coprime factorization together with Nyquist-like criterion 13 , pseudodelay transformation 14 , and Routh-like algorithm 15 .A numerical algorithm for stability testing of fractional delay systems can be found in 16 .
The purpose of this paper is to present the necessary and sufficient conditions for the stability of certain class of fractional-order transfer functions.This class is identified by those multivalued transfer functions which are defined on a Riemann surface with limited number of Riemann sheets.It is shown in the paper that not only the poles but also the branch points are crucial in determining the stability.It concludes that not only a pole and/or a singularity originated from the characteristic equation, but the branch points in the numerator of a given fractional-order transfer function are also important for the stability analysis.
The rest of this paper is divided to three sections as follows.An overview of some important properties of the multivalued complex functions is presented in Section 2. Section 3 contains a theorem that provides the necessary and sufficient condition for the stability of multivalued transfer functions.Two illustrative examples are presented in Section 4. Finally, Section 5 concludes the paper.

Preliminaries and background
In this section, a review of the most important features of multivalued functions is presented, which will be instrumental in what follows.Most of the following results can be found in textbooks that study the concept of Riemann surface and multivalued functions see i.e., 17,18 .
The multivalued functions under consideration are defined on a Riemann surface which consists of finite number of Riemann sheets.These sheets are separated by branch cuts BC and each sheet has only one edge at a BC.Branch cuts are the boundaries of discontinuity for the multivalued function.It is also assumed that the transfer functions under consideration have two branch points BPs which belong to all Riemann sheets.The two BPs connect the two endings of a BC.It is a fact that BPs are unique but BCs are not.A BC can be any simple arc connecting the two BPs.It is often selected as the simplest curve which is a straight line connecting the two BPs.A BC on the R − is usually selected when there is a BP at origin and another BP at infinity.Such a selection signifies the causality of the signal in the time domain.One important fact is that the integrations in opposite directions along opposite sides of a BC are not canceled due to the very discontinuity of the function on the BC.For example, Figure 1 shows a case where the BC is R − and connects the two BPs at zero and at infinity.In such a case, the integration is not necessarily zero.The BP of the multivalued function F s at infinity if any can be investigated by examining F 1/s at origin.For instance, F s s −1/2 has BPs of order one at s 0 and s ∞.
The best way to locate the BPs of a multivalued function is to use the property that a multivalued function has fewer values at a BP than at other points.For instance, the function is four-valued, in general, but it reduces to a double-valued at s ±1.Thus, the points s ±1 are BPs depending on which sheet they are in.
In multivalued functions, a point at which the function becomes infinite is not necessarily a pole.For example, consider F s as which has a BP at s 0. Although this function becomes infinite at the BP, this is not a pole.To find the reason, consider the integral where C is a counterclockwise closed curve encircling the origin once.For simplicity, consider the curve as a circle of radius ρ.Assuming s ρe iφ , it follows that It is observed that the integral around BP approaches zero as the radius of integration approaches zero.This would not be true for integration around a pole.We will use the following definition to deal with the singularities at origin.
Definition 2.1.The multivalued transfer function F s is said to have a pole at origin if lim s → 0 F s ∞.
Let h t denote the impulse response of an LTI causal system.Then its Laplace transform H s the system transfer function is defined as The set of all points on the fist Riemann sheet for which the Laplace integral 2.6 is absolutely convergent is called the region of convergence ROC , that is, It is obvious that the ROC of 2.6 is a half-plane to right of the abscissa of convergence σ c .The left-hand boundary of ROC is a line parallel to the imaginary axis.
Cares must be taken when calculating the inverse Laplace transform of a given multivalued function F s .The ROC by definition in the first Riemann sheet must be chosen equal to the right half-plane of the rightmost singularity either pole or BP .For example, the ROC of must be chosen as R{s} > 1.After determining the ROC of F s , its inverse Laplace transform f t , can be calculated from F s e st ds, 2.9 where Br is the Bromwich contour considered in the ROC.Note that choosing the Br contour in the ROC as described before guarantees the zero result for t < 0. Figure 1 shows the Br contour of a multivalued function the rightmost singularity of which is a BP at origin.We are not authorized to use a contour that intersects the BC because such a contour enters to the next Riemann sheet and has a discontinuity at the BC see C 1 in Figure 1 .

Extension of the concept of stability
The following theorem presents the necessary and sufficient condition needed for the stability of a multivalued transfer function.
It is then easy to prove that for a causal LTI system with impulse response h t to be BIBO stable as defined above , the necessary and sufficient condition is 3.1 Comparing to 2.7 , h t corresponds to a stable system if and only if the ROC of H s includes the imaginary axis.According to the previous discussions, it will be the case if and only if H s has no pole in C and no BP in C − because, else the Laplace integral will not be convergent .This completes the proof.
Note that this theorem cannot deal with the stability of the general case of the fractional-order systems.Interesting outcome of Theorem 3.1 is that an LTI system without any unstable pole may be unstable because of the unstable BP.It also implies that in dealing with fractional-order transfer functions, not only the denominator but also the numerator is important from the stability point of view.In fact, an LTI system can be unstable just because of an unstable BP appeared in the numerator term.It is concluded from Theorem 3.1 that the instabilities caused by unstable BPs cannot be removed by feedback because the transfer function of the resulted system will unavoidably have the same unstable BP.For that reason, such instabilities can be called essential instabilities.
Most of the practical fractional-order transfer functions have BPs at the origin and infinity but unstable BPs are not trivial.In the following, we study the propagation of an electric signal through a special electric line of length l.We show that such a system has an essential instability, that is, an unstable BP. Figure 2 shows the model of the unbalanced transmission line under consideration which applies the negative resistor.Per unit of length, the inductance is L, the capacity is C, and the conductance is G.It can also be easily verified that per unit of length the resistance is R − R N .Kirschoff's laws read where 0 ≤ x ≤ l, t ≥ 0, and the boundary conditions are v 0, t v s , i l, t 0.

3.3
Straight calculations yield The above transfer function has BPs at

Examples
In this section, two numerical examples are presented to confirm the validity of Theorem 3.1.The effect of an unstable BP in the numerator term on system instability is studied in the following examples.
Example 4.1.Consider a system with transfer function the domain of definition for which is a Riemann surface with two Riemann sheets where s 1 is a BP of order one.H s has four poles: , and where s 1 and s 2 are on the first Riemann sheet and s 3 and s 4 belong to the second Riemann sheet.Although there is no pole in C , it is concluded from Theorem 3.1 that H s is unstable because of a BP in C − .In the following, the instability caused by the BP at s 1 is verified by direct calculation of the system impulse response.
In order to calculate the inverse Laplace transform of 4.1 , consider the contour Γ shown in Figure 3 corresponds to an unstable function of time.To find out the reason of instability, the integrand is plotted in Figure 4 as a function of r for several values of t.As it is observed, the area under the plot is increased unboundedly by increasing time.Figure 5 shows the system impulse response which is given by 4.8 .
The above transfer function has a stable BP at s −1 and an unstable BP at s 1.The BC is assumed to be the straight line connecting two BPs as depicted in Figure 6.Such a system is unstable according to Theorem 3.1.In order to calculate the system impulse response, consider the contour Γ shown in Figure 6.Since there is no pole inside the contour, it is concluded from the residue theorem that In order to do the integration along C 5 and C 6 , let us do the parameterization s re ±iπ ± iδ 1 where positive and negative signs correspond to C 5 and C 6 , respectively, r ∈ 0, 2 and δ is a small positive number which tends to zero.Some algebra yields   The integrand of 4.19 is plotted in Figure 7 as a function of r for various values of t which indicates that the integrand is an unbounded function of time.The system impulse response is shown in Figure 8 which exhibits instability as expected.

Conclusion
The stability of a certain class of fractional-order multivalued transfer functions is studied.A stability theorem is proved which is applicable for open-loop stability testing of many fractional-order transfer functions.It is shown that in dealing with the stability of multivalued transfer functions, the numerator term is as important as the characteristic function.The effect of unstable BPs is studied and two illustrative examples are presented which are unstable because of the unstable BP appeared in numerator.A family of challenging instabilities is introduced which are called essential instabilities in this paper.

Figure 1 :
Figure 1:The complex plane with a BP at origin and a BC at R − .

Figure 2 :
Figure 2: A transmission line with negative resistor.

8 Figure 4 :Figure 5 :
Figure 4: The integrand of 4.9 plotted versus r for various values of t.

Figure 8 :
Figure 8: The impulse response of Example 4.2.
corresponds to the heat equation with Neumann boundary control 4 .As a general observation, systems governed by the heat equation commonly lead to multivalued transfer functions; the domain of definition for which is a Riemann surface with two Riemann sheets where the origin is a branch point of order one.As another example, Many other real-world examples of the similar type can be found in 4, 8 .All transfer functions that contain noninteger powers of the Laplace variable, s, are commonly entitled fractional-order systems.
corresponds to the fractionally damped model where m, c, and k represent the mass, damping, and stiffness, respectively, f t L −1 {F s } is the externally applied force, and x t L −1 {X s } is the displacement 7 .
Theorem 3.1.A given multivalued transfer function is stable if and only if it has no pole in C and no BP in C − .Proof.Assume the class of bounded input signals u ∈ L ∞ , that is, max t {|u t |} < ∞.The system is stable if for every input u ∈ L ∞ , the output y t u t * h t ∞ 0 . It is concluded from the residue theorem that The integrand of 4.19 plotted versus r for various values of t.