# On the Essential Instabilities Caused by Fractional-Order Transfer Functions

Authors: Farshad Merrikh-Bayat; Masoud Karimi-Ghartemani
Journal: Mathematical Problems in Engineering
 (2008)

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## Abstract

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.


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## Body

## 1. 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 root-locus 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 example,(1.1)H(s)=tanhssappears in a boundary controlled
and observed diffusion process in a bounded domain [4]. The transfer
function(1.2)H(s)=cosh(sx0)ssinhs,0<x0<1,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,(1.3)H(s)=1s2+1is the transfer function of the
causal Bessel function of the first kind and of order zero J0(t) [5]. The transfer function(1.4)H(s)=2Φ(s)s+Φ(s)es−Φ(s),ε>0,where(1.5)Φ(s)=s2+εs3/2+1, is involved in
the description of a 1D wave equation in a flared duct of finite length, with
viscothermal losses at the boundary [6]. The transfer function(1.6)H(s)=X(s)F(s)=1ms2+cs1/2+kcorresponds 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]. 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.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(1.7)H(s)=b0smα+b1s(m−1)α+⋯+bmsnα+a1s(n−1)α+⋯+an,where m,n∈ℕ,α∈(0,1),
and ak,bl∈ℝ,(k=1,…,n;l=0,…,m), is stable if and only if the roots of the
equation(1.8)wn+a1wn−1+⋯+an=0lie in the sector defined
by(1.9)|arg(w)|>π2α,where w=sα.
The proof of the above fact is based on studying the asymptotic behavior of the
inverse Laplace transform of the basic element (sα−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 involves 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 Section2. 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.


## 2. 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 theℝ− 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.It is a well-known fact that only the singularities of
a multivalued function in the first Riemann sheet are of physical significance
[19]. We also confine
our study in this paper to the first Riemann sheet. The symbols C+ and C− stand for the closed RHP and the open RHP of
the first Riemann sheet, respectively.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, Figure1 shows a case where the BC is ℝ− and connects the two BPs at zero and at infinity.
In such a case, the integration(2.1)∫C′+∫C′′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=∞.Figure 1
The
complex plane with a BP at origin and a BC atℝ−.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(2.2)F(s)=[s+(s2−1)1/2]1/2is 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, considerF(s) as(2.3)F(s)=1s1/2,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(2.4)∫C1s1/2ds,where C is a counterclockwise closed curve encircling
the origin once. For simplicity, consider the curve as a circle of radius ρ.
Assuming s=ρeiϕ,
it follows that(2.5)∫C1s1/2ds=∫−ππ1ρeiϕ/2(iρeiϕ)dϕ=i4ρ.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 functionF(s) is said to have a
pole at origin if lims→0F(s)=∞.Leth(t) denote the impulse response of an LTI causal
system. Then its Laplace transform H(s) (the system transfer function) is  defined
as(2.6)H(s)≜∫0∞h(t)e−stdt.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, s=σ+iω belongs to ROC if(2.7)∫0∞|h(t)e−st|dt=∫0∞|h(t)|e−σtdt<∞.It is obvious that the ROC of
(2.6) is a half-plane to right of the abscissa of convergence σc.
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 functionF(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(2.8)F(s)=e−s−1s+2must be  chosen as ℜ{s}>1. After determining the ROC of  F(s)
, 				its inverse Laplace transform f(t), can be calculated from(2.9)f(t)=ℒ−1{F(s)}=12πi∫BrF(s)estds,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 C1 in Figure 1).


## 3. Extension of the Concept of Stability

The following
theorem presents the necessary and sufficient condition needed for the
stability of a multivalued transfer function.Theorem 3.1.
A given multivalued transfer function is stable if and
only if it has no pole inC+ and no BP in C−.Proof.
Assume the class of bounded input
signalsu∈L∞,
that is, maxt{|u(t)|}<∞.
The system is stable if for every input u∈L∞,
the output y(t)=u(t)*h(t)=∫0∞h(τ)u(t−τ)dτ is also bounded, that is, y∈L∞.
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 that h∈L1 [20], that is,(3.1)∫0∞|h(t)|dt<∞.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 Theorem3.1 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 lengthl.
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−RN. Kirschoff's laws read(3.2)L∂i∂t=−(R−RN)i−∂v∂x,C∂v∂t=−∂i∂x−Gv,where 0≤x≤l,t≥0,
and the boundary conditions are(3.3)v(0,t)=vs,i(l,t)=0.Straight calculations yield(3.4)Vo(s)Vs(s)=1cosh(l(R−RN+Ls)(G+Cs)).The above transfer function has
BPs at s1=(RN−R)/L and s2=−G/C.
Obviously s1 corresponds to an unstable BP if RN>R.Figure 2
A transmission line with negative
resistor.


## 4. Examples

In this
section, two numerical examples are presented to confirm the validity of
Theorem3.1. 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(4.1)H(s)=(s−1)1/2s2+2s+2, 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: s1=2e−j3π/4,s2=2ej3π/4,s3=2ej5π/4,
and s4=2ej11π/4 where s1 and s2 are on the first Riemann sheet and s3 and s4 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. It is concluded from the
residue theorem that (4.2)∮ΓH(s)estds=∫Br+∫c1+∫c2+∫c3+∫c4(4.3)=2πi(Ress=s1+Ress=s2).It can easily be verified
that(4.4)∫c1H(s)estds=∫c4H(s)estds=0.Some algebra yields(4.5)∫c2+∫c3=∫0∞i2r1/2e(1−r)tr2−4r+5dr,(4.6)Ress=s1+Ress=s2=5cos(t−0.5arctan0.5)e−t(4.7)=5cos(t−13.3∘)e−t.Now, it is concluded from (4.3),
(4.4), (4.5),
and (4.7) that(4.8)h(t)=5cos(t−13.3∘)e−t−1π∫0∞r1/2e(1−r)tr2−4r+5dr.The term 5cos(t−13.3∘)e−t represents a damped oscillation but the
integral(4.9)∫0∞r1/2e(1−r)tr2−4r+5drcorresponds 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).Figure 3
Integration path in Example4.1.Figure 4
The integrand of (4.9)
plotted versus r for various values of t.Figure 5
The
impulse response of Example4.1.Example 4.2.
Consider the LTI system governed by the
integrodifferential equation(4.10)(y′′+4y′+4y)⋆(y′′+4y′+4y)=(u′−u)⋆(u′+u), where ⋆ is the convolution operator. The above
equation corresponds to the transfer function(4.11)H(s)=Y(s)U(s)=(s−1)(s+1)(s+2)2.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(4.12)∮ΓH(s)estds=∫Br+∫C1+⋯+∫C10=0.It can easily be verified
that(4.13)∫C1=∫C10=0.Since there is no cut on ℜ{s}<−1,
we can write(4.14)∫C2+∫C9=0,∫C4+∫C7=0.The residue theorem implies
that(4.15)∫C3+∫C8=2πiRes[(s−1)(s+1)(s+2)2,s=−2]=0.In order to do the integration
along C5 and C6,
let us do the parameterization
s=re±iπ±iδ+1 where positive and negative signs correspond
to C5 and C6,
respectively, r∈(0,2) and δ is a small positive number which tends to
zero. Some algebra yields(4.16)∫C5+∫C6=∫r=20ir(2−r)(3−r)2e(1−r)t(−dr)+∫r=02−ir(2−r)(3−r)2e(1−r)t(−dr),or equivalently(4.17)∫C5+∫C6=∫r=02i2r(2−r)(3−r)2e(1−r)tdr.It is concluded from
(4.12), (4.13),
(4.14),  (4.15),
and (4.17) that(4.18)h(t)=1i2π∫BrH(s)estds(4.19)=−∫r=02r(2−r)π(3−r)2e(1−r)tdr.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.Figure 6
Integration path in Example4.2.Figure 7
The integrand of (4.19)
plotted versus r for various values of t.Figure 8
The
impulse response of Example4.2.


## 5. 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.



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