# Extending the Root-Locus Method to Fractional-Order Systems

Authors: Farshad Merrikh-Bayat; Mahdi Afshar
Journal: Journal of Applied Mathematics
 (2008)

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## Abstract

The well-known root-locus method is developed for special subset of linear time-invariant systems known as fractional-order systems. Transfer functions of these systems are rational functions with polynomials of rational powers of the Laplace variables. Such systems are defined on a Riemann surface because of their multivalued nature. A set of rules for plotting the root loci on the first Riemann sheet is presented. The important features of the classical root-locus method such as asymptotes, roots condition on the real axis, and breakaway points are extended to fractional case. It is also shown that the proposed method can assess the closed-loop stability of fractional-order systems in the presence of a varying gain in the loop. Three illustrative examples are presented to confirm the effectiveness of the proposed algorithm.


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## Body

## 1. Introduction

The root-locus
method of Evans is one of the most popular and powerful tools for both analysis
and design of single-input single-output (SISO) linear time-invariant (LTI)
systems.  There are two main application areas for this method [1] as follows. (1) Stability: to obtain sufficient
conditions on a real parameter k under which the closed-loop system in Figure 1
remains stable.  (2) Design: the root-locus method offers an efficient tool for
design of lead-lag compensators. There have been further advances to the
root-locus method since its origin in 1948. Krall [2, 3] developed the method for
delayed systems, Bahar and Fitzwater [4] studied the problem from the numerical point of view
and finally, Byrnes et al. [5] presented the root-locus method for distributed parameter
systems.Figure 1
Standard closed-loop system.For typical systems, there are several easy-to-use
rules for plotting the root loci that do not generally suffice to determine it
uniquely [1]. These
rules serve only as hints and often the intuitive insight of a control engineer
is needed for completing the root-locus plot. The root-locus method will
apparently become more difficult to apply as the system's order becomes higher
[5].In recent years, there has been an increasing
attention to fractional-order systems. These systems are of interest for both
modelling  and
controller design purposes. In the fields of
continuous-time modelling, fractional derivatives have proved useful in linear
viscoelasticity, acoustics, rheology, polymeric chemistry, biophysics,…
[6, 7]. In general,
fractional-order systems are useful to model various stable physical phenomena
(commonly diffusive systems) with anomalous decay.An interesting study of fractional differential
systems appeared in [8] using a stochastic framework. The idea of fractional
powers is also used for identification purposes. Tsao et al. [9] and Poinot and Trigeassou
[10] clarify the
identification method when the members of model set are of fractional order.  Two
applications of such identifications can be found in [11, 12]. Fractional-order systems
are also used in control field. Podlubny [13] and Valério and Sá da Costa [14] discussed methods of
designing PIλDμ controllers, Raynaud and Zergaïnoh [15] studied fractional-order
lead-lag compensators, and Oustaloup et al. [16, 17] introduced the so-called CRONE controllers.In this paper, the systems under consideration are
described by rational transfer functions and the powers of the Laplace
variable,s,
are limited to rational numbers. Such systems lend themselves well to some
algebraic tools [18, 19]. Practical examples of such systems can be found in
[11, 12, 19]. The problem of plotting the
root loci for these systems is treated in this paper. Unlike [5] that deals with
infinite-dimensional systems, in the problem we are going to solve, the systems
are assumed to be of finite dimension and this makes the problem simple enough
to deal with analytically.The proposed method can be used to examine whether a
given closed-loop system, as shown in Figure1, remains stable for large k's or not, where P(s) is a fractional-order transfer function. In
[20], a generalization
of the Routh-Hurwitz criterion for fractional-order systems is presented.
However, this method can deal with the stability problem for such systems but
it is a very complicated algorithm.The rest of this paper is arranged as follows. Section2 provides some basic definitions and notations together with the problem
statement. In Section 3, the rules for plotting the root loci in fractional
case are presented. Three illustrative examples are presented in Section 4.
Finally, some conclusions end the paper.Notation
Blackboard capitals denote sets and spaces: especiallyℕ the natural numbers
(without zero), ℝ the real numbers, ℚ the rational numbers, ℤ the integer
numbers, and ℂ the complex numbers. The symbol ⌊x⌋, where x∈ℝ, denotes
the biggest integer that is less than or equal to x.


## 2. Problem Statement and Preliminaries

Before
introducing the main problem, some basic definitions and notations are provided.
It is assumed that the reader is familiar with the concepts of “Riemann
surface,” “Riemann sheet,” “branch point,” and “branch
cut” (see, e.g., [21], or [22] for deeper analysis).Definition 2.1.
The
functionQ(s)=a1sq1+a2sq2+⋯+ansqn is a fractional-order polynomial, if and only
if qi∈ℚ+∪{0},ai∈ℝ, for i=1,…,n.Definition 2.2.
Consider the fractional-order
polynomial(2.1)Q(s)=a1sα1/β1+a2sα2/β2+⋯+ansαn/βn,ai∈ℝ,αi∈ℕ∪{0},βi∈ℕ,where αi,βi are relatively prime for i=1,…,n.
(If for some i, αi=0 then by definition βi=1.)
Let λ be the least common multiple (lcm) of β1,β2,…,βn denoted as λ=lcm{β1,β2,…,βn}.
Then Q(s) can be written as(2.2)Q(s)=a1(s1/λ)λ1+a2(s1/λ)λ2+⋯+an(s1/λ)λn.Now the fractional degree (fdeg)
of Q(s) is defined as(2.3)fdeg{Q(s)}=max{λ1,λ2,…,λn}.The functionQ as defined in (2.2) is a multivalued relation of s the domain of definition for which is a
Riemann surface with λ Riemann sheets where the origin is a branch
point [21]. In this
paper, the branch cut is assumed at ℝ− and the first Riemann sheet is denoted by 𝒫 and defined as(2.4)𝒫:={reiθ∣r>0,−π<θ≤π}.Note that each Riemann sheet has
only one edge at branch cut. The following proposition gives the roots number
of Q(s)=0.Proposition 2.3.
LetQ(s) be a fractional-order polynomial with fdeg{Q(s)}=n.
Then the equation Q(s)=0 has exactly n roots on the Riemann surface.Proof.
Consider(2.5)Q(s)=a1(s1/v)n+a2(s1/v)n−1+⋯+an(s1/v)1+an+1,for an appropriate v∈ℕ.
Assuming w:=s1/v,
we have(2.6)Q˜(w)=a1wn+a1wn−1+⋯+anw+an+1.The fundamental theorem of
algebra gives n roots for Q˜(w)=0,
say w1,w2,…,wn.
Consequently, Q(s)=0 has n roots at s1=w1v,s2=w2v,…,sn=wnv.Definition 2.4.
The fractional-order polynomialQ(s)=a0sn/v+a1s(n−1)/v+⋯+an−1s1/v+an is minimal if fdeg{Q(s)}=n.Now consider the standard closed-loop system in Figure1 where the transfer function of plant is given by(2.7)P(s)=N(s)D(s)=sm/v+b1s(m−1)/v+⋯+bm−1s1/v+bmsn/v+a1s(n−1)/v+⋯+an−1s1/v+an,(v>1),and k is assumed to be a positive real constant.
Note that the domain of definition of P(s) is a Riemann surface with v Riemann sheets [21].Definition 2.5.
With
the above notations,P(s) is called strictly proper for n>m,
proper for n≥m,
nonproper for n<m,
and biproper for n=m.Definition 2.6.
The roots of the equationsN(s)=0 and D(s)=0 on 𝒫 are called open-loop zeros and open-loop poles, respectively.It is a fact that when a minimal fractional-order
polynomial is represented in a nonminimal form, the number of its zeros is
increased but the location and the order of zeros on𝒫 remain unchanged. For example, consider the fractional-order
polynomials f(s)=s1/2−1 (minimal) and g(s)=s2/4−1 (nonminimal). The equation f(s)=0 has only one root at s=ei0 (on the first Riemann sheet) while g(s)=0 has a root at s=ei0 (on the first Riemann sheet), and another root
at s=ei4π (on the third Riemann sheet),
although  f(s)  and g(s) have different number of zeros but the
location and the order of their zero on 𝒫 are identical. It concludes that Definition 2.6 is not ambiguous;
representing N(s) and D(s) in a nonminimal form will not affect the
open-loop poles and zeros.Note thats=0 is not a pole of P(s) even if D(0)=0. The following definition deals with the singularities at the origin.Definition 2.7.
The points=0 is defined to be a pole of order r of P(s) (as defined in (2.7)) if the point w=0 is a pole of order r of P˜(w):=P(s)∣w=s1/v.The characteristic equation of the closed-loop
system shown in Figure1 is(2.8)Δ(s)=D(s)+kN(s)=sn/v+a1s(n−1)/v+⋯+an−1s1/v+an+k[sm/v+b1s(m−1)/v+⋯+bm−1s1/v+bm]=0.It is desired to address the
generalized root-locus problem that is to plot the root loci of (2.8) on 𝒫 when k varies. The reason for concerning about the
first Riemann sheet is that the time-domain behavior and stability properties
of the closed-loop system are determined only by those roots of the
characteristic equation that lie on the first
Riemann sheet [19, 23]. Note that a system with characteristic equation (2.8)
is stable (in the sense of bounded-input bounded-output) if and only if it has
no roots in the closed right half plane (CRHP) of 𝒫 [24, 25]. In this paper, we restrict
ourselves to the following conditions.
(i)
The transfer
function of the plant is strictly proper. Final results can easily be extended
for nonproper systems but such an extension to biproper transfer functions is
complicated. This is similar to the difficulty that occurs when the root-locus
plot for an integer-order system with biproper transfer function is involved
[26].(ii)
Parameterk is a positive real number. The results can
easily be extended for negative real k's.(iii)
BothN(s) and D(s) are monic fractional-order polynomials. This
does not lose the generality and simplifies the notations.(iv)
N(s) and D(s) have no common roots. Otherwise, the
characteristic equation will have root(s) that does (do) not vary by changing k.
In the rest of
this paper, the single-valued function obtained by replacing every s1/v in (2.8) with w is denoted by Δ˜(w),
that is,(2.9)Δ˜(w)=wn+a1wn−1+⋯+an−1w+an+k[wm+b1wm−1+⋯+bm−1w+bm]=0.We do the same for other multivalued relations.


## 3. Root Loci in Fractional Case

### 3.1. Properties of the Root Loci in Fractional Case

The root-locus plot ofΔ˜(w)=0 provides a very good insight to the root-locus
plot of Δ(s). Figure 2 shows the relationship between w-plane and sheets of the s Riemann surface. In this figure, the sector −π/v<argw≤π/v corresponds to 𝒫. In the following, some important features of the root loci of Δ(s) are presented.Figure 2
The correspondence betweenw-plane and s Riemann sheets.


#### 3.1.1. Symmetry with Respect to Real Axis

Considering the fact that the root loci ofΔ˜(w)=0 is symmetric with respect to real axis, it is
concluded that the roots loci of Δ(s) on 𝒫 are also symmetric with respect to real axis. Note that, in general,
this explanation is not correct for the root loci on other Riemann sheets.


#### 3.1.2. Number of Branches

A branch by definition is the loci of a single root of the characteristic
equation whenk varies from zero to infinity. In classical
case, the root-locus branches start from open-loop poles and terminate at zeros
(finite zeros or zeros at infinity) [1]. In fractional case, it is concluded from Proposition
2.3 that the characteristic equation (2.8) has n roots distributed on v Riemann sheets. Considering the root-locus
plot of Δ˜(w) in w-plane, it is obvious that not all root-locus
branches on necessarily start from open-loop poles and terminate at open-loop
zeros. In fact, a branch may cross the branch cut and enter to another Riemann
sheet.There is another point that should be noted here.
Clearly,r branches start from the open-loop pole s0∈𝒫 which is of order r.
When s0∉ℝ−,
all these branches are on 𝒫 for k→0+.
Otherwise, they belong to different Riemann sheets. One important case is due
to the poles at the origin. If s1=0 is a pole of order r of (2.7), then r branches start it, which are not
necessarily on for k→0+.
In order to find the number of branches that start from s1 and are on 𝒫 for k→0+,
let p and q stand for the number of positive real
open-loop poles and zeros, respectively. Then according to the angle condition,
the angle of departure from the pole at the origin is obtained
as(3.1)ϕh=2h+1−p+qrvπ,h=⌊p−q−(n/v)−12⌋+1,…,⌊p−q+(n/v)−12⌋.As a result, if (2.7) has a pole
of order r at the origin, then ⌊(p−q+(n/v)−1)/2⌋−⌊(p−q−(n/v)−1)/2⌋ branches (on 𝒫) start from s=0 the angle of departure of which is calculated
from (3.1).


#### 3.1.3. Roots Conditions on the Real Axis

In the classical root-locus algorithm, any point on
the real axis, the total number of real poles and zeros to the right of which
is odd, lies on a root locus. Clearly, the line segments lying on the positive
real axis ofw-plane are mapped to the line segments lying
on the positive real axis of 𝒫. So, according to the root-locus plot in w-plane, any point on the positive real axis of 𝒫, the total number of real poles and zeros to the right of which is odd,
lies on a root locus. But for P(s) given in (2.7), no line segment on ℝ− can belong to the root locus. The reason is as
follows. If such a line exists then it should necessarily lie on the ray reiπ/v(r>0) in w-plane. It is a well-known classical result
that the semiinfinite line reiπ/v(r>xv),
which is a root loci branch of a system with transfer function P˜(w)=(1/(wv+x))(x∈ℝ+),
is the only object in w-plane that can lie on this ray. But P˜(w) corresponds to P(s)=1/((s1/v)v+x)=1/(s+x) which is not multivalued. Consequently, the
root-locus plot of the multivalued transfer function P(s) can never have a branch at ℝ−.


#### 3.1.4. Asymptotes and Their Directions

Asymptotes are very important in drawing a root-locus
plot as they exhibit directions of the branches for largek's. The asymptotes of the root-locus plot in
integer case are first studied in [21]. Another explanation with more details can be found
in [27]. The approach
used in [21, 27] cannot directly be
applied for fractional case. Here we develop an alternative approach to find
the asymptotes to the root-locus curves of a fractional-order transfer
function. The following theorem is the main result of this paper because it can
be used to examine the closed-loop stability for large gains.Theorem 3.1.
Asymptotes of the root-locus plot are straight lines
all passing through the origin and their directions are given
by(3.2)φh=(2h+1)vn−m180°,h=⌊m−n−v2v⌋+1,…,⌊m−n−v2v⌋+n−m.Proof.
See Appendix.Note that in Theorem3.1, h may belong to any sequence of n−m successive integer numbers but the sequence we
have used makes a relevant correspondence between asymptotes and Riemann
sheets. This sequence guarantees that −180°<φh<360°v−180°.
Note also that for h=⌊(m−n−v)/2v⌋+1,…,⌊(n−m−v)/2v⌋ the resulting asymptotes lie on 𝒫. In fractional case, however, all asymptotes pass through the origin.
Since the open-loop system is assumed to be
strictly proper, the
root-locus plot will always have at least one asymptote.Remark 3.2.
In integer case, if the negative
real axis is an asymptote for the root-locus plot, then it is the asymptote for
one and only one of the branches. Moreover, the corresponding branch of the
root-locus plot coincides with the asymptote. In the fractional case, however,
the negative real axis can be an asymptote line for more than one root and it
is not superposed on any branch. As a matter of fact, according to (3.2) the
negative real axis is the asymptote for all infinite roots when n−m=v.


#### 3.1.5. Breakaway and Break-in Points

The breakaway and break-in points on a root-locus
plot are points where two or more branches intersect and then go apart. Ifs0 is a breakaway (break-in) point then it
necessarily satisfies both Δ(s0)=0 and dΔ(s0)/ds=0 for some s0 on 𝒫. These two equations imply that D(s0)+kN(s0)=0 and dD(s0)/ds+kdN(s0)/ds=0 or equivalently, N(s0)dD(s0)/ds−D(s0)dN(s0)/ds=0.
The latter equation can be interpreted in terms of the open-loop transfer
function, P(s),
as dP(s0)/ds=0.
So, every breakaway (break-in) point must satisfy the equation(3.3)dP(s)ds=0,on 𝒫 (see, e.g., [21] for differentiation of multivalued relations). The
roots of (3.3) are breakaway (break-in) points if the corresponding k 's are positive real numbers. Equation (3.3)
can also be interpreted in terms of P˜(w) as dP˜(w)/dw=0.


### 3.2. Comprehensive Algorithm

The following
is a summarization of the general rules for constructing the root loci in
fractional case.(1)
Locate the open-loop poles and zeros.(2)
Determine the order of the pole at the origin
(if any) and calculate the angle of departure using (3.1).(3)
Determine the root loci on the positive real
axis. Note that no line segment on the negative real axis can belong to the
root loci.(4)
Determine the directions of asymptotes from
(3.2).(5)
Find the breakaway and break-in points (in
any) from (3.3).(6)
Complete the root-locus plot.


## 3.1. Properties of the Root Loci in Fractional Case

The root-locus plot ofΔ˜(w)=0 provides a very good insight to the root-locus
plot of Δ(s). Figure 2 shows the relationship between w-plane and sheets of the s Riemann surface. In this figure, the sector −π/v<argw≤π/v corresponds to 𝒫. In the following, some important features of the root loci of Δ(s) are presented.Figure 2
The correspondence betweenw-plane and s Riemann sheets.


### 3.1.1. Symmetry with Respect to Real Axis

Considering the fact that the root loci ofΔ˜(w)=0 is symmetric with respect to real axis, it is
concluded that the roots loci of Δ(s) on 𝒫 are also symmetric with respect to real axis. Note that, in general,
this explanation is not correct for the root loci on other Riemann sheets.


### 3.1.2. Number of Branches

A branch by definition is the loci of a single root of the characteristic
equation whenk varies from zero to infinity. In classical
case, the root-locus branches start from open-loop poles and terminate at zeros
(finite zeros or zeros at infinity) [1]. In fractional case, it is concluded from Proposition
2.3 that the characteristic equation (2.8) has n roots distributed on v Riemann sheets. Considering the root-locus
plot of Δ˜(w) in w-plane, it is obvious that not all root-locus
branches on necessarily start from open-loop poles and terminate at open-loop
zeros. In fact, a branch may cross the branch cut and enter to another Riemann
sheet.There is another point that should be noted here.
Clearly,r branches start from the open-loop pole s0∈𝒫 which is of order r.
When s0∉ℝ−,
all these branches are on 𝒫 for k→0+.
Otherwise, they belong to different Riemann sheets. One important case is due
to the poles at the origin. If s1=0 is a pole of order r of (2.7), then r branches start it, which are not
necessarily on for k→0+.
In order to find the number of branches that start from s1 and are on 𝒫 for k→0+,
let p and q stand for the number of positive real
open-loop poles and zeros, respectively. Then according to the angle condition,
the angle of departure from the pole at the origin is obtained
as(3.1)ϕh=2h+1−p+qrvπ,h=⌊p−q−(n/v)−12⌋+1,…,⌊p−q+(n/v)−12⌋.As a result, if (2.7) has a pole
of order r at the origin, then ⌊(p−q+(n/v)−1)/2⌋−⌊(p−q−(n/v)−1)/2⌋ branches (on 𝒫) start from s=0 the angle of departure of which is calculated
from (3.1).


### 3.1.3. Roots Conditions on the Real Axis

In the classical root-locus algorithm, any point on
the real axis, the total number of real poles and zeros to the right of which
is odd, lies on a root locus. Clearly, the line segments lying on the positive
real axis ofw-plane are mapped to the line segments lying
on the positive real axis of 𝒫. So, according to the root-locus plot in w-plane, any point on the positive real axis of 𝒫, the total number of real poles and zeros to the right of which is odd,
lies on a root locus. But for P(s) given in (2.7), no line segment on ℝ− can belong to the root locus. The reason is as
follows. If such a line exists then it should necessarily lie on the ray reiπ/v(r>0) in w-plane. It is a well-known classical result
that the semiinfinite line reiπ/v(r>xv),
which is a root loci branch of a system with transfer function P˜(w)=(1/(wv+x))(x∈ℝ+),
is the only object in w-plane that can lie on this ray. But P˜(w) corresponds to P(s)=1/((s1/v)v+x)=1/(s+x) which is not multivalued. Consequently, the
root-locus plot of the multivalued transfer function P(s) can never have a branch at ℝ−.


### 3.1.4. Asymptotes and Their Directions

Asymptotes are very important in drawing a root-locus
plot as they exhibit directions of the branches for largek's. The asymptotes of the root-locus plot in
integer case are first studied in [21]. Another explanation with more details can be found
in [27]. The approach
used in [21, 27] cannot directly be
applied for fractional case. Here we develop an alternative approach to find
the asymptotes to the root-locus curves of a fractional-order transfer
function. The following theorem is the main result of this paper because it can
be used to examine the closed-loop stability for large gains.Theorem 3.1.
Asymptotes of the root-locus plot are straight lines
all passing through the origin and their directions are given
by(3.2)φh=(2h+1)vn−m180°,h=⌊m−n−v2v⌋+1,…,⌊m−n−v2v⌋+n−m.Proof.
See Appendix.Note that in Theorem3.1, h may belong to any sequence of n−m successive integer numbers but the sequence we
have used makes a relevant correspondence between asymptotes and Riemann
sheets. This sequence guarantees that −180°<φh<360°v−180°.
Note also that for h=⌊(m−n−v)/2v⌋+1,…,⌊(n−m−v)/2v⌋ the resulting asymptotes lie on 𝒫. In fractional case, however, all asymptotes pass through the origin.
Since the open-loop system is assumed to be
strictly proper, the
root-locus plot will always have at least one asymptote.Remark 3.2.
In integer case, if the negative
real axis is an asymptote for the root-locus plot, then it is the asymptote for
one and only one of the branches. Moreover, the corresponding branch of the
root-locus plot coincides with the asymptote. In the fractional case, however,
the negative real axis can be an asymptote line for more than one root and it
is not superposed on any branch. As a matter of fact, according to (3.2) the
negative real axis is the asymptote for all infinite roots when n−m=v.


### 3.1.5. Breakaway and Break-in Points

The breakaway and break-in points on a root-locus
plot are points where two or more branches intersect and then go apart. Ifs0 is a breakaway (break-in) point then it
necessarily satisfies both Δ(s0)=0 and dΔ(s0)/ds=0 for some s0 on 𝒫. These two equations imply that D(s0)+kN(s0)=0 and dD(s0)/ds+kdN(s0)/ds=0 or equivalently, N(s0)dD(s0)/ds−D(s0)dN(s0)/ds=0.
The latter equation can be interpreted in terms of the open-loop transfer
function, P(s),
as dP(s0)/ds=0.
So, every breakaway (break-in) point must satisfy the equation(3.3)dP(s)ds=0,on 𝒫 (see, e.g., [21] for differentiation of multivalued relations). The
roots of (3.3) are breakaway (break-in) points if the corresponding k 's are positive real numbers. Equation (3.3)
can also be interpreted in terms of P˜(w) as dP˜(w)/dw=0.


## 3.1.1. Symmetry with Respect to Real Axis

Considering the fact that the root loci ofΔ˜(w)=0 is symmetric with respect to real axis, it is
concluded that the roots loci of Δ(s) on 𝒫 are also symmetric with respect to real axis. Note that, in general,
this explanation is not correct for the root loci on other Riemann sheets.


## 3.1.2. Number of Branches

A branch by definition is the loci of a single root of the characteristic
equation whenk varies from zero to infinity. In classical
case, the root-locus branches start from open-loop poles and terminate at zeros
(finite zeros or zeros at infinity) [1]. In fractional case, it is concluded from Proposition
2.3 that the characteristic equation (2.8) has n roots distributed on v Riemann sheets. Considering the root-locus
plot of Δ˜(w) in w-plane, it is obvious that not all root-locus
branches on necessarily start from open-loop poles and terminate at open-loop
zeros. In fact, a branch may cross the branch cut and enter to another Riemann
sheet.There is another point that should be noted here.
Clearly,r branches start from the open-loop pole s0∈𝒫 which is of order r.
When s0∉ℝ−,
all these branches are on 𝒫 for k→0+.
Otherwise, they belong to different Riemann sheets. One important case is due
to the poles at the origin. If s1=0 is a pole of order r of (2.7), then r branches start it, which are not
necessarily on for k→0+.
In order to find the number of branches that start from s1 and are on 𝒫 for k→0+,
let p and q stand for the number of positive real
open-loop poles and zeros, respectively. Then according to the angle condition,
the angle of departure from the pole at the origin is obtained
as(3.1)ϕh=2h+1−p+qrvπ,h=⌊p−q−(n/v)−12⌋+1,…,⌊p−q+(n/v)−12⌋.As a result, if (2.7) has a pole
of order r at the origin, then ⌊(p−q+(n/v)−1)/2⌋−⌊(p−q−(n/v)−1)/2⌋ branches (on 𝒫) start from s=0 the angle of departure of which is calculated
from (3.1).


## 3.1.3. Roots Conditions on the Real Axis

In the classical root-locus algorithm, any point on
the real axis, the total number of real poles and zeros to the right of which
is odd, lies on a root locus. Clearly, the line segments lying on the positive
real axis ofw-plane are mapped to the line segments lying
on the positive real axis of 𝒫. So, according to the root-locus plot in w-plane, any point on the positive real axis of 𝒫, the total number of real poles and zeros to the right of which is odd,
lies on a root locus. But for P(s) given in (2.7), no line segment on ℝ− can belong to the root locus. The reason is as
follows. If such a line exists then it should necessarily lie on the ray reiπ/v(r>0) in w-plane. It is a well-known classical result
that the semiinfinite line reiπ/v(r>xv),
which is a root loci branch of a system with transfer function P˜(w)=(1/(wv+x))(x∈ℝ+),
is the only object in w-plane that can lie on this ray. But P˜(w) corresponds to P(s)=1/((s1/v)v+x)=1/(s+x) which is not multivalued. Consequently, the
root-locus plot of the multivalued transfer function P(s) can never have a branch at ℝ−.


## 3.1.4. Asymptotes and Their Directions

Asymptotes are very important in drawing a root-locus
plot as they exhibit directions of the branches for largek's. The asymptotes of the root-locus plot in
integer case are first studied in [21]. Another explanation with more details can be found
in [27]. The approach
used in [21, 27] cannot directly be
applied for fractional case. Here we develop an alternative approach to find
the asymptotes to the root-locus curves of a fractional-order transfer
function. The following theorem is the main result of this paper because it can
be used to examine the closed-loop stability for large gains.Theorem 3.1.
Asymptotes of the root-locus plot are straight lines
all passing through the origin and their directions are given
by(3.2)φh=(2h+1)vn−m180°,h=⌊m−n−v2v⌋+1,…,⌊m−n−v2v⌋+n−m.Proof.
See Appendix.Note that in Theorem3.1, h may belong to any sequence of n−m successive integer numbers but the sequence we
have used makes a relevant correspondence between asymptotes and Riemann
sheets. This sequence guarantees that −180°<φh<360°v−180°.
Note also that for h=⌊(m−n−v)/2v⌋+1,…,⌊(n−m−v)/2v⌋ the resulting asymptotes lie on 𝒫. In fractional case, however, all asymptotes pass through the origin.
Since the open-loop system is assumed to be
strictly proper, the
root-locus plot will always have at least one asymptote.Remark 3.2.
In integer case, if the negative
real axis is an asymptote for the root-locus plot, then it is the asymptote for
one and only one of the branches. Moreover, the corresponding branch of the
root-locus plot coincides with the asymptote. In the fractional case, however,
the negative real axis can be an asymptote line for more than one root and it
is not superposed on any branch. As a matter of fact, according to (3.2) the
negative real axis is the asymptote for all infinite roots when n−m=v.


## 3.1.5. Breakaway and Break-in Points

The breakaway and break-in points on a root-locus
plot are points where two or more branches intersect and then go apart. Ifs0 is a breakaway (break-in) point then it
necessarily satisfies both Δ(s0)=0 and dΔ(s0)/ds=0 for some s0 on 𝒫. These two equations imply that D(s0)+kN(s0)=0 and dD(s0)/ds+kdN(s0)/ds=0 or equivalently, N(s0)dD(s0)/ds−D(s0)dN(s0)/ds=0.
The latter equation can be interpreted in terms of the open-loop transfer
function, P(s),
as dP(s0)/ds=0.
So, every breakaway (break-in) point must satisfy the equation(3.3)dP(s)ds=0,on 𝒫 (see, e.g., [21] for differentiation of multivalued relations). The
roots of (3.3) are breakaway (break-in) points if the corresponding k 's are positive real numbers. Equation (3.3)
can also be interpreted in terms of P˜(w) as dP˜(w)/dw=0.


## 3.2. Comprehensive Algorithm

The following
is a summarization of the general rules for constructing the root loci in
fractional case.(1)
Locate the open-loop poles and zeros.(2)
Determine the order of the pole at the origin
(if any) and calculate the angle of departure using (3.1).(3)
Determine the root loci on the positive real
axis. Note that no line segment on the negative real axis can belong to the
root loci.(4)
Determine the directions of asymptotes from
(3.2).(5)
Find the breakaway and break-in points (in
any) from (3.3).(6)
Complete the root-locus plot.


## 4. Examples

Example 4.1.
Consider the closed-loop system in
Figure1 with(4.1)P(s)=s1/2−1s2−3s3/2−2s+2s1/2+12.The open-loop poles of this
system are located at s=4ei0 and s=9ei0,
and there is an open-loop zero located at s=1ei0.
The line segments 0≤ℜ{s}≤1 and 4≤ℜ{s}≤9 belong to the root loci because the total
number of poles and zeros to the right of any point on them is an odd number.
The directions of the asymptotes (on the first Riemann sheet) are φ−1=−120° and φ0=120°.
The roots of the equation dP˜(w)/dw are 2.4820, −0.9599,
and 0.9056±i1.0671.
The only feasible solution is w=2.4820 which corresponds to s=6.1603ei0.
Figure 3 shows the root-locus plot of this system.Figure 3
Root-locus plot of Example4.1.Example 4.2.
Consider the closed-loop system in
Figure1 with(4.2)P(s)=s1/2−2s2(s1/2−1)3.This system has an open-loop
zero at s=2ei0 and an open-loop pole of order three at s=1ei0.
There is also a pole of order four at the origin. The line segment 1≤ℜ{s}≤2 on the positive real axis belongs to the root
loci. According to (3.1), the angles of departure from the pole at the origin are ϕ0=−π/2 and ϕ1=π/2.
Letting n=7, m=1,
and v=2, it is concluded from (3.2) that the directions
of the asymptotes on 𝒫 are φ−1=−60°, φ0=60°,
and φ1=180°.
Figure 4 shows the root-locus plot of this system.Figure 4
Root-locus plot of Example4.2.Example 4.3.
According to [28], the fractional-order model
of a heating furnace is given by(4.3)P(s)=114994s1.131+6009.5s0.97+1.69,which compared to (2.7) results in n=131, m=0,
and v=100.
Assuming P(s) in the connection of Figure 1, the root-locus
plot has two asymptotes on the directions of which are φ−1≈−137.4° and φ0≈137.4°,
thanks to (3.2). Since none of these asymptotes lie on the CRHP of 𝒫 and P(s) has no zeros, one can choose k arbitrarily large to arrive at a closed-loop
system with the ability of tracking the command input. Figure 5 shows the
closed-loop system response for k=5,10,20 together with the open-loop system response
when the system is subjected to a unit step. As it is expected, the closed-loop
system is stable and its bandwidth is increased by increasing k.
Podlubny [29] proposed
the fractional-order model(4.4)P(s)=10.7943s2.5708+5.2385s0.8372+1.5560,for another heating furnace.
This transfer function can be represented in the equivalent form(4.5)P(s)=10.7943s6427/2500+5.2385s2093/2500+1.5560,which corresponds to n=6427, m=0,
and v=2500.
The root-locus plot of this system has two unstable infinite branches the
directions of which are φ−1≈−70.02° and φ0≈70.02°.
Hence, it is not possible to control the system by means of a simple
proportional action. One possible approach is to use more complex structures
such as lead-lag compensators.Figure 5
Step responses corresponding to Example4.3.


## 5. Conclusion

In this paper,
an approach for constructing the root-locus plot for fractional-order systems
is developed. Important features of the root loci are studied and a
comprehensive algorithm is presented. Although the rules for plotting the root
loci in fractional case are somehow similar to those available in integer case,
there are also some major differences. For example, it is shown that in
fractional case no line segment onℝ− can belong to the root loci. It is also shown
that the asymptotes of all infinite branches of the root-locus plot pass
through the origin. As another difference, the well-known Routh-Hurwitz
stability test cannot be used to find the points at which the root-locus plot
intersects with the imaginary axis. Three illustrative examples are presented
to confirm the effectiveness of the proposed algorithm.



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*Source: 528934-2008-06-15.xml*