Journal of Process Control 34 (2015) 26–34
Contents lists available at ScienceDirect
Journal of Process Control
journal homepage: www.elsevier.com/locate/jprocont
A revision of root locus method with applications
Tomislav B. Šekara a , Milan R. Rapaić b,∗
a
b
School of Electrical Engineering, University of Belgrade, Serbia
Faculty of Technical Sciences, University of Novi Sad, Serbia
a r t i c l e
i n f o
Article history:
Received 14 January 2015
Received in revised form 15 June 2015
Accepted 17 July 2015
Available online 3 August 2015
Keywords:
Root locus method
PID control
Pole placement
Stability analysis
Non-rational systems
Delayed systems
a b s t r a c t
The paper investigates applications of the root-locus (RL) method to analysis and design of closed loop
systems with arbitrary loop transfer functions. Novel analytic sketching rules have been derived first.
These rules are applicable to a wide range of transfer functions: rational, fractional and non-rational
ones in general, those with time-delays incorporated at various locations, as well as transfer functions
describing distributed-parameter systems. An original, straightforward numerical procedure for plotting
the root locus has been proposed next. By means of the derived techniques, a generalization of the poleplacement method, which is applicable to control design of both rational and non-rational processes
has been proposed also. Finally, it has been shown that RL technique can be very effectively used to
investigate the influence of open loop dead-time variations to closed loop poles. It is of particular interest
to stress that the techniques proposed and analyzed in this work are exact, in the sense that no rational
approximations of infinite-dimensional systems have been utilized. All results have been thoroughly
illustrated by numerical examples.
© 2015 Elsevier Ltd. All rights reserved.
1. Introduction
and characteristic equation
The classical root-locus (RL) method is used to analyze and plot
roots of a real polynomial with single adjustable parameter entering the polynomial coefficients in an affine manner. More precisely,
given polynomials an (s) and bm (s) of respective orders n and m ≤ n,
with known, real coefficients and complex argument s, the root
locus is a set of trajectories traversed by the roots of the n-th order
polynomial
F(s) = 1 + k Wn (s) =
fn (s) = an (s) + k bm (s),
(1)
when the adjustable parameter k changes over the set of real numbers.
The primary application of the RL method is in the analysis and
synthesis of control systems with loop transfer function
W (s) = k Wn (s) = k
bm (s)
,
an (s)
(2)
∗ Corresponding author at: Faculty of Technical Sciences, Trg Dositeja Obradovića
6, 21000 Novi Sad, Serbia. Tel.: +381 62 1711982.
E-mail addresses: tomi@etf.rs (T.B. Šekara), rapaja@uns.ac.rs (M.R. Rapaić).
http://dx.doi.org/10.1016/j.jprocont.2015.07.007
0959-1524/© 2015 Elsevier Ltd. All rights reserved.
fn (s)
= 0.
an (s)
(3)
The poles of the closed-loop system are in fact zeros of (3),
i.e. roots of the characteristic polynomial (1). In what follows,
Wn is denoted as the normalized open-loop transfer function. A
schematic diagram of a control system with loop transfer function
(2) is presented in Fig. 1. The adjustable parameter k is interpreted
as the gain of a proportional, error-feedback controller
The root-locus (RL) method was developed in 1948 independently by Walter R. Evans [1,3] and Kazimir F. Theodorchik [2,4].
It has since become a very popular tool for analysis and design of
servomechanisms and control systems with rational transfer functions in general. The method is still a standard topic in most modern
introductory control theory courses and textbooks, e.g. [5,16].
Among the most prominent features of the RL design is availability of a relatively small set of rules allowing for fast and accurate
sketching of the root loci for low-order systems with rational transfer functions [5,16]. General algebraic equations for RL in polar and
Cartesian coordinates have been considered in [7]. Some recent
results in this regard are presented in [8,9]. Several numerical
procedures for RL plotting have also been proposed. For early contributions, the reader is referred to [10] and [11]; the method of
continuation has been proposed by Pan and Chao in [12] and further
refined in [13,14].
T.B. Šekara, M.R. Rapaić / Journal of Process Control 34 (2015) 26–34
Fig. 1. Control loop with proportional error-feedback controller. W(s) = k Wn (s) is
the open-loop transfer function, k is the controller gain. The desired (or reference)
value is denoted by r, while the actual (obtained) output value is denoted by y.
Several generalizations and modifications of the RL method have
been proposed. An extension to the case of a non-affine dependence on the adjustable parameter k has been considered recently
in [15]. The so-called phase root locus and its extensions have been
proposed by Nagurka and Kurfess in [16], and further developed in
[17,18].
Numerous attempts have been made to extend root-locus and
apply them to a more general class of systems. Systems with delay,
having open-loop transfer functions of the form
k Wn (s) = k
bm (s) −s
e ,
an (s)
(4)
are considered in [19–21], and more recently in [24]. Suh and Bien
adopted the continuation method of Pan and Chao to adress systems with time-delay in both control (input) and state variables
[22]. A modification of the RL plots has been used to analyze sensitivity of the delayed systems in [23]. It is also of interest to note
that the aforementioned work of Williamson [11] addresses RL for
systems with “distributed phase lag”, i.e. systems with open-loop
transfer functions
k Wn (s) = k
bm (s) −√s
e
.
an (s)
(5)
In the past few decades fractional-order systems (FOS) captured
a significant attention among researches. General theory of fractional calculus and non-integer order systems can be found in [25].
Transfer functions of FOS contain non-integer powers of the Laplace
variable s, for example
sˇm + bm−1 sˇm−1 + . . . + b0
k Wn (s) = k ˛
,
s n + an−1 s˛n−1 + . . . + a0
(6)
with n, m ∈ N, a0 , . . ., an−1 , b0 , . . ., bm−1 ∈ R, 0 < ˛1 < . . . < ˛n and
0 < ˇ1 < . . . < ˇm . In a special, but practically very important case,
all powers are integer multiples of a common factor,
k Wn (s) = k
bm (s˛ )
.
an (s˛ )
(7)
Such FOS are known as commensurate order systems. Another
common form of fractional order systems is
k Wn (s) = k
(s − z1 )ˇ1 · · ·(s − zm )ˇm
(s − p1 )˛1 · · ·(p − zn )˛n
,
(8)
with all powers ˛1 to ˛n and ˇ1 to ˇm being positive,
real numbers. Complex numbers pi (i ∈ 1, ..., n ) and zi (i ∈ 1, .., m ) are
fractional transmission poles and zeros. RL sketching rules for FOS
have been investigated in [26–29]. An investigation of FOS stability,
backed by the RL method, has been presented in [30].
Recent work of Wang et al. (2009) [31] suggests the possibility
of PID controller design for complex systems using pole-placement
techniques. This technique relies on auxiliary transfer functions
obtained by suitable transformations of the original process model.
In practice, off course, most processes exhibit dead-time, and in this
cases the auxiliary transfer function is of the form Ĝ(s, e−s ), with
e−s appearing at various places. General RL techniques proposed in
our present work enable effective applications of pole-placement
design proposed in [31] for delayed systems and other types of
27
infinite-dimensional linear processes also. Although MIMO systems
have not been addressed in the present paper, in line of [32] we can
claim that the proposed techniques are readily extensible to MIMO
case as well.
The aim of the present paper is to investigate general properties
of root loci for an LTI system described by an arbitrary normalized open-loop transfer function. The results of the present paper
are readily available to rational and non-rational systems, as well
as to the infinite-dimensional and distributed-parameter systems
in general. In particular, processes with transport delay in both
input, output and state variables can all be readily targeted by the
proposed approach. The derived properties can be seen as generalizations of the well-known classical RL sketching rules to the
case of processes described by transfer functions of a more general
form. Due to the generality of the systems under consideration, the
derived rules are somewhat more complex than the corresponding
rules of the traditional RL method. In order to circumvent this fact,
the paper also proposes a straightforward procedure for numerical
plotting of root loci in the general case.
The sequel of the paper is organized as follows. The generalized sketching rules have been derived in the following Section 2.
Section 3 presents a novel, simple and effective direct numerical
procedure for plotting root locus of an arbitrary LTI system. Two
distinct applications are presented in Section 4. Firstly, a generalization of results originaly published by Wang et al in [31] has
been presented. It has been shown that root loci can be effectively
used for PID controller tuning using the dominant pole-placement
approach, even in the case of processes with transport delays and
processes with non-rational transfer functions in general. Secondly,
it has been demonstrated that root loci can be effectively used for
investigation of the influence of the varying open-loop transport
delay to closed-loop poles, much in the same manner as the classical root loci investigate the influence of varying open-loop gain.
The concluding remarks are given in Section 5.
2. Analytic properties and sketching rules
2.1. Basic definitions and properties
Non-rational transfer functions may poses poles at different Riemann sheets. Consequently, RL can span multiple Riemann sheets:
any branch of the root locus can originate on one sheet, cross a
branch cut, and end at another sheet. However, only the Primary
Riemann Sheet has physical significance (see [33]), and only poles
belonging to the Primary Sheet are relevant. Therefore, in the sequel
we will consider only branches (and parts of branches) belonging
to the Primary Sheet.
Definition 1 (RL definition). The root-locus of LTI system described
by a normalized open-loop transfer function Wn (s) is the set of all
points s in the Primary Riemann Sheet of the complex plain C such
that
1 + k Wn (s) = 0 ,
(9)
for some k ∈ R.
There are multiple equivalent ways to define root loci. It is worth
stressing that conditions (9) implies that for the points on the RL
the loop transfer function is real.
Theorem 1 (Alternative definition). Introduce real valued mappings
U, V : R2 → R such that
U(, ω) = R Wn ( + jω)
(10)
V (, ω) = I Wn ( + jω) .
(11)
28
T.B. Šekara, M.R. Rapaić / Journal of Process Control 34 (2015) 26–34
The root locus of LTI system described by a normalized open-loop
transfer function Wn (s) is the set of all points in (, ω) ∈ R2 such that
Having in mind that k is positive, by utilization of (14), one
readily obtains (15). 䊐
V (, ω) = 0 .
(12)
To conclude this section, note that by direct application of conditions (13) and (12) one can generalize the well-known rules
identifying RL branches lying on the real axis.
(13)
Theorem 5. Any point ∈ R such that Wn () ∈ R belongs to the RL
of Wn . The sign of corresponding gain k is opposite to the sign of Wn ().
The corresponding values of k are obtained from
k U(, ω) = −1 .
The branches will be said to flow in the direction of increasing
magnitude of k. A point from which one, or several, RL branches flow
out will be denoted as source or origin, while a point to which one
or more RL branches flow in will be denoted as sink. A point that is
simultaneously a source and a sink will be denoted as break point.
Theorem 2 (Symmetry). Root locus of a system with real-valued
impulse response (kernel function) is symmetric with respect to the
real axis.
Proof. Denote by ¯· the operation of complex conjugation. The
property follows directly from the fact that W (s̄) = W¯(s) whenever
the kernel is real-valued. Thus, if for some k ∈ R s belongs to RL, so
will s̄. 䊐
Off course, the characteristic equation (9) usually has multiple
solutions for a single value of k. The common terminology is that
RL has multiple branches. It is, however, of particular interest to
identify points and intervals of the root locus at which the mapping
s = (k) is bijective.
Theorem 3. Let s0 ∈ C belongs to the RL of a system described by a
normalized open loop transfer function Wn (s) for k = k0 ∈ R. If Wn (s)
is analytic in a neighborhood of s0 and if W (s0 ) =
/ 0 then the RL condition (9) implies that there is a unique analytic mapping s = (k) in a
neighborhood of k0 such that s0 = (k0 ) and
ds
d(k)
Wn (s) 1
=
=− .
Wn (s) k
dk
dk
2.2. An important special case
Let us now investigate root loci of a system with normalized
open-loop transfer function
ds
=0 ,
dk
a
Wn (s) =
Definition 3.
|
+
a(
)=
−
a(
)=
Proof. Since s = (k) is analytic at vicinity of k0 , by virtue of Taylor
arguments
d
(k0 + k) = (k0 ) +
( )k ,
dk
for any sufficiently small k > 0 and some ∈ (k0 , k0 + k). The
argument of the tangent vector under consideration is
ϕ(s0 )
= lim arg (k0 + k) − (k0 )
k→0
= lim arg
k→0
d
dk
( )k
.
−
:l∈Z
:l∈Z
∩ (−, ]
(17)
∩ (−, ]
(18)
+
a(
)| and
Lemma 1. All RL branches of (16) are straight lines, with angles
−
belonging to +
a ( ) for k > 0 and
a ( ) for k < 0. If > 0, the branches
originate at b for k = 0 and sink at infinity as |k| grows to infinity. If
< 0, the branches originate at infinity for k = 0 and sink at b as |k|
grows to infinity.
Rewrite the RL condition (9) as a pair of conditions
(19)
arg (kWn (s)) = ± + 2l ,
l∈Z .
(20)
ej
. It is straightforward to see that (19)
Substitute s = b +
implies |a|k = , and consequently k = /|a|. Simmilarly, (20)
implies
=
1
(argk + arga + (1 + 2l)) ,
l∈Z .
(21)
According to Definition 1, only branches belonging to the primary Riemann sheet are significant, and the desired results readily
follow. An illustration is given by Fig. 2. 䊐
2.3. Asymptotic behavior
In the sequel we will restrict ourselves to the systems with normalized open-loop transfer functions Wn (s) satisfying the following
assumption
Assumption 1.
r
There exist real constants c =
/ 0 and r such that
lim s Wn (s) = c ,
s→∞
−
The cardinality of these sets will be denoted as |
)|, respectively.
|kWn (s)| = 1 ,
(15)
=
/ 0 and a, b ∈ C, a =
/ 0, introduce
−
a(
Definition 2 (Regular and singular points). Any point s0 satisfying
the conditions of Theorem 3 will be denoted as regular point of the
RL. All other points of the RL are denoted as singular (irregular).
Wn (s0 )
− arg k0 .
Wn (s0 )
For any
2l − arga
Proof.
ϕ(s0 ) = + arg
(16)
(1 + 2l) − arga
from which (14) directly follows. 䊐
Theorem 4 (Tangent angles). Let s0 ∈ C be any regular point of the
RL of a system described by a normalized open loop transfer function
Wn (s), and let k0 be the corresponding gain. Let ϕ(s0 ) be the argument
of the tangent vector to RL at s0 drawn in the direction of increasing
gain. Then
,
(s − b)
with a, b ∈ C, a =
/ 0 and ∈ R\{0}. Note that, in general, the kernel function of (16) is not real. In fact, for < 0 the kernel function
can only be defined as a generalized function (a distribution). Nevertheless, the analysis of (16) is crucial for understanding both
asymptotic and local behavior of other systems of greater practical
interest.
(14)
Proof. The existence of the unique, analytic mapping s = (k) such
that s0 = (k0 ) follows directly from the complex Implicit Function Theorem, see [37]. By differentiating (9) with respect to k one
readily obtains
Wn (s) + kWn (s)
Remark 1. An instant consequence of Theorem 5 is that RL of a
process described by rational open-loop transfer function (2) contains the entire real line – some intervals for positive and others for
negative gain. This property does not hold in the general case.
(22)
and sr Wn (s) is analytic at infinity, except possibly along a finite
number of branch-cuts.
T.B. Šekara, M.R. Rapaić / Journal of Process Control 34 (2015) 26–34
Proof. Since the relative degree of Wn is r > 0, Wn is analytic for
|s|→ ∞, (except, possibly, on a finite number of branch-cuts). Let us
introduce complex-valued mapping g such that
4
3
c
Wn (s) =
(s g(s))
2
ω
29
.
r
(26)
1
Note that g must also be analytic (except, possibly, on a finite
number of branch-cuts), and also that lim g(s) = 1. By series
0
expansion of g at infinity,
-1
g(s) = 1 + a
|s|→∞
Wn (s) = -3
-4
c
r .
(28)
s + a + O( 1s )
It is now obvious that Wn (s) behaves like
-4
-3
-2
-1
0
σ
1
2
3
4
Fig. 2. RL of 1/s2.5 . Branches corresponding to positive gain are shown solid, while
branches corresponding to negative gain are shown dashed.
The constant r will be denoted as the relative degree of Wn .
Transfer functions satisfying the Assumption 1 with r ≥ 0 will be
denoted as proper, whereas those with r > 0 as strictly proper. If
many cases, the relative degree can be computed by invoking the
Argument Principle (i.e. Rouché Theorem) [34],
a = lim
d
lim
→∞
sW (s)
W (s)
dϕ .
(23)
s= ejϕ
In the sequel, only the case of strictly proper transfer functions
will be considered.
Remark 2. The case of negative relative degree (r < 0) can be
addressed by considering
1 + k̂ Ŵn (s) = 0 ,
(24)
with k̂ = k−1 , Ŵn (s) = Wn−1 (s). Note that the relative degree of Ŵn
is positive.
Remark 3. The case of zero relative degree (r = 0) can be dealt by
transforming
1
1 (−1)
f (z) = rg r−1 ( )g ( ) 2
.
z
z z
(30)
Finally, due to (29) and since lim g( 1z ) = 1,
f (z)
1
− lim
r z→0 f (z)
By introducing
k
k̂ = 1+k
c
k
(Wn (s) − c)
1+kc
Remark 4.
that
Theorem 6 (Asymptotes). Consider RL of a system described by normalized transfer function Wn satisfying Assumption 1, with r > 0 and
c ∈ R\{0}. The asymptotes intersect at
f (z)
,
f (z) =
Wn (z −1 )
.
(31)
f (z) =
By application of Theorem 6 to (8), one first obtains
(1 − z p1 )˛1 · · ·(1 − z pn )˛n
(1 − z z1 )ˇ1 · · ·(1 − z zm )ˇm
By introducing A(z) =
m
ˇi
n
i=1
.
(1 − z pi )˛i , Aj (z) = B(z)/(1 − z pj )˛j ,
B(z) = i=1 (1 − z zi ) , and Bj (z) = B(z)/(1 − z zj )ˇj , the derivative of
f becomes
n
B(z)
n
j=1
(−pj ˛j )Bj (z) − A(z)
B(z)2
n
and Ŵn (s) = Wn (s) − c, one readily
zr
= a ,
j=1
(−zj ˇj )Bj (z)
.
By noting that B(0) = A(0) = 1, and also that for any admissible j
Bj (0) = Aj (0) = 1, it is not hard to see that (25) reduces to
.
obtains the characteristic equation (24), with Ŵn having positive
relative degree. Note also the singularity in k̂ for k = − c−1 . This will
cause some branches to reach infinity even for finite gains, see [8].
f (z)
−g ( 1z ) −1
z2
= lim
z→0
g( 1z )
which concludes the proof. 䊐
f (z) =
= 1 + k (Wn (s) + c − c)
= (1 + k c) 1 +
1
a = − lim
r z→0
(29)
Since f (z) = g r ( 1z ), by differentiation with respect to z,
1 + k Wn (s)
1 1
1
g( ) = −lim g ( ) 2 .
z
z z
z→0
z→0
2
0
c
. The number
(s−a )r
of asymptotes and their angles now follow directly from Lemma 1
excluding, off course, all asymptotes on which the (s − a )r Wn (s) is
not analytic at infinity.
Further, by construction,
z→0 d z
1
r=−
2
(27)
Substituting (27) into (26), one readily obtains
-2
-5
-5
1
1
+ O( 2 ) .
s
s
(25)
−
Angles of the asymptotes form a subset of +
c (r) ∪
c (r). For each
+
j
in c (r) there is an asymptote A : s = a + Re , provided that
(s − a )r Wn (s) is analytic on A for large values of R. These asymptotes
correspond to positive gains. Under the same conditions, there is an
asymptote corresponding to k < 0 for each in −
c (r).
˛p −
j=1 j j
n
˛ −
j=1 j
a = m
ˇz
j=1 j j
m
j = 1 ˇj
.
(32)
Note that in the special case when all orders are integers, the
obtain result gracefully reduces to the well-known rule valid for
integer order systems: the sum of poles minus the sum of zeros,
divided by the relative degree.
Remark 5.
f (z) =
By application of Theorem 6 to (6), one obtains
A(z)
,
B(z)
with
A(z) = 1 + an−1 z˛n −˛n−1 + · · · + a0 z˛n
and
ˇ
m
B(z) = 1 + bm−1 z −ˇm−1 + · · · + b0 zˇm . By noting that A(0) = B(0) = 1,
the expression (25) becomes
B (z) − A (z)
,
z→0 ˛n − ˇm
a = lim
(33)
30
T.B. Šekara, M.R. Rapaić / Journal of Process Control 34 (2015) 26–34
which is the general expression for the intersection point of
the asymptotes of the root locus of (6). In the case of integer
order systems, both A and B are polynomials, and consequently
˛n − ˛n−1 = ˇm − ˇm−1 = 1, and B (0) − A (0) = bm−1 − am−1 . By Viete’s
rules, bm−1 is the negative sum of all process zeros and likewise an−1
is the negative sum of all process poles. Again, the classical result
has been reestablished.
2.4. Singular points
Theorem 7. If a point p such that limWn (s) = ∞ belongs to RL of Wn ,
s→p
then p is a source (k = 0). Similarly, if a point z such that limW (s) = 0
belongs to RL of Wn , then z is a sink (k→ ∞).
s→z
Proof. By RL condition (9), if at some point belonging to RL Wn
is zero, then k must be infinite. Similarly, if at such a point Wn is
infinite, then the corresponding k must be unbounded. 䊐
Theorem 8. Let p ∈ C be a singularity of Wn (s). Let there exists ˛ > 0
such that (s − p)˛ W(s) is analytic and non-vanishing on in the vicinity
of p, except possibly on a finite set of rays originating at p. Let also,
lim(s − p)˛ W (s) = a =
/ 0 .
(34)
s→p
Then, there is a RL branch exiting p for each ∈ +
a (˛) (k > 0) and
˛
j for
for each ∈ −
a (˛) (k < 0), if (s − p) W(s) is analytic on p + e
∈ [0, ε) and some sufficiently small ε.
Proof. Consider any ∈ (− , ] such that (s − p)˛ W(s) is analytic
and non-vanishing on R (ε) = {p + ej : ∈ [0, ε)} for some sufficiently small ε. Then, there exists W1 , analytic and non-vanishing
on R (ε), such that
Wn (s) =
a
(s − p)˛
W1 (s) ,
s ∈ R (ε) ,
(35)
lim
W1 (s) = 1. Thus, at vicinity of p, on R (ε),
s→p,s∈R (ε)
behaves like a ˛ and the result follows from Lemma 1. 䊐
(s−p)
with
Wn
Theorem 9. Let z ∈ C be a zero of Wn (s), and let there exists ˛ > 0
such that W(s)/(s − z)˛ is analytic and non-vanishing at vicinity of z,
except possibly on a finite set of rays originating at p, and
lim
W (s)
s→z (s − z)
/ 0
˛ =a=
.
(36)
Then, there is a RL branch sinking to z for each ∈ +
a (˛) (k > 0)
−˛ W(s) is analytic on z + ej for
and each ∈ −
a (˛) (k < 0), if (s − z)
∈ [0, ε) and some sufficiently small ε.
Proof. The proof follows the same line as the proof of previous
Theorem 8, and is omitted here for brevity. 䊐
In the general case, RL branches need not originate at points
where Wn is unbounded, nor must they sink at zeros of Wn . In fact,
branches may originate or terminate at any singular point of Wn .
To conclude this subsection, we present the following well-known
and obvious claim:
Theorem 10.
one has
At any break-in or break-away point of the real axis,
d
1
=0 ,
d Wn ()
(37)
where d/d denotes the first derivative along the real line.
Example 1.
Consider
process described by a normalized transfer
function Wn (s) =
(s−1)2 +2
√
3 . Root loci are shown in Fig. 3, Left.
s( s+2)
3
Having in mind that for large |s|, Wn (s) behaves like s− 2 , it is not
hard to see that the given transfer function satisfies Assumption
1 with relative degree r = 32 and c = 1. Considering that (25) yields
f (z) =
(2z+1)3
3z 2 −2z+1
, the intersection point of the asymptotes can be
readily established. Simple calculations give a = − 83 . By further
application of Theorem 6, root loci are found to have two asymp, and a single asymptote with angle
totes for k > 0, with angles ± 2
3
2
0 for k < 0. Indeed, it is not hard to show that +
c (r) = {± 3 } and
+
(r)
=
{0}.
c
The transfer function
√ under consideration has singularities at 0
and −2, zeros at 1 ± j 2, and
√ branch cuts along the lines s = + 0j
( < −2) and s = 1 ± ωj (ω > 2).
W (s) satisfies conditions of Theorem 8 with p = −2, ˛ =
Near −2,
√ n
3
11
and
a
=
≈ 1.66. Since, +
( 3 ) = {± 23 } and −
( 3 ) = {0},
2
2
1.66 2
1.66 2
we conclude that 2 branches leave −2 for positive and one for negative gain, with angles ± 23 and 0, respectively. At pole p = 0, Theorem
3
8 should be applied with a =
≈ 0.61 and ˛ = 1. Simple compu8
+
(1) = {0}: two branches are
tations give 0.61 (1) = {} and −
0.61
exiting p = 0, one for positive k going to the left, and another for
negative k going to the right.
sinking to the two zeros, we apply
In order to identify branches
√
Theorem 9, with s = 1 ± j 2, a ≈ −0.1084 ∓ 0.1187j and ˛ = 12 . Hav1
ing in mind that arg a ≈ ∓0.264, one readily obtains +
a (2) =
{∓0.5288}, i.e. branches sink to zero almost vertically, as seen in
Fig. 3, Left.
Finally, it is not hard to see that the entire real axis right of −2 is
part of the root locus, since ImG() = 0 for all > −2. The break-away
point at s = 0 ≈ −0.675 is also easily identified as a local extreme
point (in this case maximum) of ReG(). This is illustrated in Fig. 3,
Middle and Right.
3. Numerical construction of root loci
Alternative RL definition given by Theorem 1 can readily be used
for numerical construction of root loci for an arbitrary open loop
transfer function. In particular, the root locus can be seen as set {s ∈
R : ImW (s) = 0}, i.e. as a set of zero-level contour lines of ImW(s).
Contour-detection algorithms of this kind are commonly implemented in most currently available numerical packages, both open
and commercial. (See for example contourc and contour functions in MATLAB® or contour function in Python’s matplotlib
library. Similar functions can be found also in Mathematica® ,
Maple® , Octave® , and others.)
The problem of using available contour-detection algorithms to
draw root loci, is that for non-rational transfer functions possessing
branching points, branch cuts will be often falsely identified as parts
of the root loci. The reason is that ImW(s) is non-differentiable at
the branch cut, and that it often abruptly changes sign. This abrupt
sign change is, unfortunately, identified as a regular zero crossing
by most contour-detection routines.
The solution to the above mentioned problem is to traverse all
primarily identified contours and to eliminate intervals for which
the RL condition (12) is not satisfied. During this procedure, it is
convenient to compute gains using (13) and separate branches corresponding to positive and negative k. Further, one can also detect
poles as branch origin at which |U(, ω)| is very big. Similarly, zeros
can be identified as branch sinks for which |U(, ω)| is very small.
Note that in general case, not all branch sources are poles and not
all sinks are zeros, since branches can originate and terminate at
singular points of other types.
Example 2. √ Consider normalized open-loop transfer function
W (s) = √1s e− s . Various stages of the numerical root locus construction are illustrated by plots in Fig. 4. Plot Fig. 4a shows V(,
T.B. Šekara, M.R. Rapaić / Journal of Process Control 34 (2015) 26–34
2
ω
1
2
0
0
1
-2
-1
0
V(σ)-4
-2
-3
-4
-8
-2
-3
-3
U(σ)
-6
-1
31
-5
-10
-2
-1
0
σ
1
2
-3
-2
-1
0
1
σ
2
3
-3
-2
-1
0
σ
1
2
3
Fig. 3. (Left) Root locus of Wn (s) of Example 1. Branches corresponding to positive gain are shown solid, while those corresponding to the negative gain are shown dashed.
(Middle) Imaginary part of G() for ∈ [−3, 3]. (Right) Real part of G(). The point at which RL branches escape the real line are identified using vertical dashed line at
= 0 ≈ −0.675.
ω) = ImG( + jω). The branch cut at the negative real-line is apparent at this figure. The raw contour lines obtained by detecting sign
changes of V are shown in Fig. 4b. The third plot Fig. 4c shows
the actual root locus, which is obtained by identifying and removing fake contours. Notice that the process under consideration is
infinite-dimensional, and that its root locus possesses an infinite
number of branches. Only those belonging to the square [− 10,
10] × [−10, 10] are highlighted in Fig. 4. It is not difficult to show
that only isolated points of the negative real axis satisfy the RL condition (9). The negative real axis is, however, a branch cut in this
particular case, and one can easily verify that, except at discrete
set of points, V(, ) and V(, − ) have opposite sign for arbitrary
small > 0. Consequently, most contour-detection algorithms will
identify negative real axis as a contour line, although it does not
belong to the root locus.
4. Applications
4.1. Guaranteed dominant pole-placement design
In a recent publication by Wang et al. [31], an original, guaranteed dominant pole-placement design procedure for PID controller
tuning has been proposed. In cases when there is no transport
delay in the process model, Wang et al. utilize the RL procedure,
yet in cases when the transport delay is present they are forced
to apply a technique based on the Nyquist stability criteria. In
the sequel, we demonstrate that it is possible to utilize the root
locus method regardless of the nature of the open loop transfer function under consideration. In fact, not only processes with
delay, but also fractional and infinite dimensional processes in
general can be tackled by the proposed approach in a unified manner.
Let us review the method of Wang et al. Let G(s) be the transfer function of the process under consideration, and let C(s) be the
transfer function of the PID controller, with parameters kp , ki and
kd . The characteristic equation of the closed-loop system is
1 + C(s)G(s) = 0
(38)
Let pd1,2 = d ± jωd be the desired dominant pair of the closed
loop poles. By requiring that pd1,2 satisfies (38), one obtains two
linear equations with respect to the unspecified controller parameters. By means of these equations, it is possible to express two
parameters as linear functions of the third one, i.e.
ki = ki (kp ) = ai kp + bi ,
kd = kd (kp ) = ad kp + bd ,
(39)
where ai , bi , ad and bd are constant factors. A simple rearrangements
now give an auxiliary characteristic equation, equivalent to (38),
with kp , as the only free parameter, appearing in a multiplicative
fashion,
1 + kp Ĝ(s) = 0 .
(40)
The performance of the closed-loop system can now be investigated by RL techniques. In particular, two closed-loop poles are
fixed to pd1,2 , while others vary with kp . The goal is then to find
values of kp such that the real parts of all other poles are less than
m d , with m typically bigger than 3. For further details, see [31].
The position of the dominant poles can be obtained by specifying
performance requirements, such as settling time and overshoot.
If all closed-loop poles and zeros, except the dominant pair, are
positioned left of the vertical line Res = m d , expressions obtained
for second order system with no zeros can be
used
[16]: The ı%
, the peak-time
settling time can be computed as tı% = 1 ln 100
ı
d
is tp = | | , and the overshoot, expressed in percentage, is Mp =
exp
d d
ωd
× 100%.
1
e−5s ,
(s+1)3
k +k /s+k s
and a PID regulator with noise-cancellation filter C(s) = p sTi +1 d .
f
Example 3. Consider a delay-dominated process G(s) =
√
Fig. 4. Illustration of the steps of the numerical procedure for drawing root locus of Wn (s) = √1s e− s . In the rightmost plot 4c, branches corresponding to positive gain are
shown solid, while the branches with negative gain are shown dashed.
32
T.B. Šekara, M.R. Rapaić / Journal of Process Control 34 (2015) 26–34
k =0.2577
1
0.5
k =0.2321
0.8
p
p
response
ω
1
0
0.6
-0.5
0.4
-1
0.2
-1.5
-2
-1.5
-1
-0.5
σ
0
0
0.5
0
20
40
60
80
100
time [s]
Fig. 5. (Left) Root locus of the auxiliary transfer function (41). By specification, all non-dominant poles must lay left of the vertical dashed line at Res = −0.46. Note that the
two branches are captured by the dipoles at pd1,2 = −0.15333 ± j0.12314, and are not shown in the diagram. (Right) Unit step response of the closed loop system, with step
disturbance of amplitude 0.1 starting from t = 50.
By requiring an overshoot less than 2% and 1% settling time 35
seconds, one obtains pd1,2 = −0.15333 ± j0.12314. The pole of the
noise-cancellation filter pf = −1/Tf is placed at 10|Re{pd1,2 }|, giving
Tf = 0.6522. Eq. (39) now gives
kd
= 3.26087kp − 0.60945 ,
ki
= 0.12611kp + 0.05138 ,
and the auxiliary transfer function becomes
Nyquist plot of Ĝ, shown in Fig. 6, one can readily see that the Critical Point must lie between -4.308 and -3.881, resulting in allowable
values of the proportional gain kp ∈ [0.2321, 0.2577]. The results are
in full accordance with the ones obtained by direct utilization of the
root locus method, as proposed in the present paper.
Example 4. Consider an infinite-dimensional process described
1√
by a transfer function G(s) = cosh(
. By requiring that the overs)
shoot should not be bigger than 1%, and also that 1%-settling
(3.260867s2 + s + 0.12611)e−5s
Ĝ(s) = kp
.
(−0.60945s2 + 0.051383)e−5s + 0.652174s5 + 2.95652s4 + 4.95652s3 + 3.65217s2 + s
Root locus of Ḡp is given in Fig. 5, Left. By choosing m = 3, we
see that all non-dominant poles lie left of −m d for kp ∈ [0.2321,
0.2577]. By choosing kp = 0.2577, one obtains ki = 0.0839 and
kd = 0.2309. Provided that the entire PID regulator is implemented
as an error-feedback controller (with both P and D actions in the
direct path) the step and disturbance responses are shown in Fig. 5,
Right.
Comparison with the approach of Wang et al. [31]. To utilize the approach of [31], one must draw the modified Nyquist
curve Ĝ(md + jω), ω ∈ (− ∞ , ∞), first. The number of encirclements
of this curve around the Critical Point −1/kp must be equal to
the number of poles of Ĝ laying to the right of the vertical line
R s = md = −0.46. There are 7 of these poles in total: p1 ≈ −0.1319,
p2,3 ≈ ±0.2229 − 0.9521j, p4,5 ≈ −0.4407 ± 1.892j, and the two
desired dominant poles p6,7 = pd1,2 . Poles p1 and p2,3 can be seen on
the RL diagram shown in Fig. 5, Left. Poles p3,4 are not seen because
they fall outside of the drawn area (they could be seen by extending the range of imaginary axis), while poles p6,7 are captured by
the dominant pole-zero pairs and cannot be seen on numerically
plotted RL diagram (yet their existence is guaranteed by the design
procedure itself). In the approach of Wang et al. the number of these
poles is obtained by a second utilization of the modified Nyquist
criterion, this time to the auxiliary transfer function
Ḡ0 (s) =
−0.60945s2 + 0.051383
e−5s .
4
(0.652174s + 2.95652s3 + 4.95652s2 + 3.65217s + 1)s
By direct evaluation, one can see that the modified Nyquist curve
Ḡ0 (md + jω), ω ∈ (− ∞ , ∞), encircles the fixed Critical Point −1
exactly 6 times. Considering that Ḡ0 (s) has one pole with real part
bigger than m d = −0.46, the total number of poles of Ĝ lying in the
“udesired” area must be 6 + 1 =7.
Since the two dominant poles of Ĝ are fixed, the controller
needs to move 7 − 2 =5 poles left of the line Re = m d , meaning that
the total number of encirclements of the modified Nyquist curve
around point −1/kp must be 5. By investigation of the modified
(41)
time should be 1.4 seconds, the desired locations of the dominant poles are pd1,2 = −3.2857 ± 2.2415j. For simplicity, let us
assume that the noise-cancellation filter is rather soft, with
pf = −1/Tf = −100|Repd1,2 |, giving Tf = 0.00304. Straightforward calculations give
kd
= 0.15217kp − 0.27298 ,
ki
= 2.40741kp − 1.5686 .
3
2.5
2
1.5
1
0.5
0
-0.5
-1
-1/k p, min
-1/k p,max
-1.5
-2
-6
-5
-4
-3
-2
-1
0
1
2
3
Fig. 6. Modified Nyquist curve Ĝ(md + jω), for Ĝ given by (41). Only the part of the
Nyquist curve corresponding to positive values of ω are shown. Consequently, the
total number of encirclements is twice the number obtained from the diagram. One
positive encirclement is far to the left and is not shown. Allowable positions of the
Critical Point are highlighted by arrows.
T.B. Šekara, M.R. Rapaić / Journal of Process Control 34 (2015) 26–34
33
1
40
k=3.5
0.8
ω
response
20
0
0.6
-20
0.4
-40
0.2
-60
-60
-40
-20
0
σ
20
40
0
0
1
2
3
4
5
6
7
8
time [s]
Fig. 7. (Left) Root locus of the auxiliary transfer function (41). By specification, all non-dominant poles must lay left of the vertical dashed line at = −11.04. (Right) Unit step
response of the closed loop system, with unit step disturbance starting from t = 4.
The auxiliary transfer function in the considered case is
Ĝ(s) =
0.15217s2 + s + 2.40741
,
√
s cosh( 2s) (0.00304s + 1) − 0.27298s2 + 1.56857
the corresponding RL diagram is shown in Fig. 7, Left. By choosing
kp = 3.5, which gives kd = 0.2596 and ki = 6.8574, one obtains a satisfactory values of m = 3.4. The zeros of the obtained PID regulator
are z1 = −2.38 and z2 = −11.1. The first zero is too close to the dominant poles, so in this particular case both P and D actions should
act on the filtered measurement only, not on the error signal. By
implementing the obtained regulator as,
1
Y (s) ,
sTf + 1
Yf (s)
=
U(s)
= −kp Yf (s) +
ki R(s) − Yf (s) − kd sYf (s) ,
s
one obtains the unit step and disturbance responses shown in Fig. 7,
Right.
position of all closed-loop poles, i.e. what is the root locus of the
characteristic equation 1 + W(s)e−s when varies across [0, ∞)?
For a given , the characteristic equation of the closed-loop system is
W (s)e−s = −1 .
(42)
By taking the natural logarithm of both sides, one obtains
ln W (s) − s = j + j 2 l ,
(l ∈ Z) .
(43)
Note that, in general, complex logarithm is multivalued function. When analyzing the influence of delay it is important to
consider branches corresponding to all integer values of l. A simple
rearrangements now gives
1+
−s
=0 ,
ln(−W (s)) + j2 l
(l ∈ Z) .
(44)
The root loci of W(s)e−s with respect to variable transport delay
can now be obtained by introducing the auxiliary normalized
open-loop transfer function
4.2. Dependence of closed loop poles on variations of transport
delay
ˇW l (s) =
One of the basic robustness issues in control systems design is
how much additional transport delay can the process suffer before
it becomes unstable. Let W(s) be the open-loop transfer function.
The question is, what is the maximal value of for which the system
with delayed open-loop transfer function W(s)e−s is stable? For a
broad class of systems, this question is answered by investigations
of phase (and delay) margins. A more general question we are posing here is: How does the additional transport delay influences the
and constructing the classical root loci ofˇW l for all l ∈ Z. Note that
/ 0 RL of ˇW l is not symmetric with respect to the real
for fixed l =
axis. However, for any fixed l =
/ 0, ˇW l is conjugate-symmetric to
ˇW −l . Thus, the combined RL, obtained by plotting the ofˇW l for all
l ∈ Z, is symmetric.
Many properties of the closed loop system can now be investigated. Delay margin m , for example, is the smallest positive value
of delay for which some RL branch cross from the left to the
−s
,
ln (−W (s)) + j2 l
(l ∈ Z)
(45)
Fig. 8. An Illustration of Example 5. Fig. 8a shows jointly the RL ofˇW l for different values of l: l = 0 (solid, upper and lower), l = 1, − 1 (dashed, upper and lower) and l = −2, 2
(dash-dotted, upper and lower). Details of the upper plots, important for stability analysis, are shown in the right. Fig. 8b corresponds to l = 0, while Fig. 8c corresponds to
l = 1.
34
T.B. Šekara, M.R. Rapaić / Journal of Process Control 34 (2015) 26–34
right side of the complex plane. The corresponding phase margin
is obtained as
m = ωm m
,
(46)
where ωm is the crossover frequency, i.e. the imaginary part of the
point at which RL intersects the imaginary axis. Note however, that
the complete spectrum of the closed loop system can is effectively
identified for any value of the transport delay.
Example 5. Consider the process with open-loop transfer function
W (s) =
−1.5se−s
.
s1.5 − 1.5s + 4s0.5 + 8
(47)
Let us investigate the values of for which the process is stable
in the closed loop. By introducingˇW l according to (45), one readily
obtains
−s
ˇW l (s) =
(48)
.
ln(− 1.5 −1.5s 0.5 ) + j2 l
s
−1.5s+4s
+8
Root loci are shown in Fig. 8. Fig. 8a illustrates the part of RL
obtained for l ∈ { −2, . . ., 2}. Fig. 8b and c shows interesting details
for l = 0 and l = 1, respectively.
By detail numerical investigation of plot presented in Fig. 8b, one
may notice that the closed loop process is unstable for < ≈0.0499.
A similar investigation of Fig. 8c reveals that the closed-loop process
is also unstable for ∈ (≈0.7855, ≈ 0.9984). Other instability regions
would be revealed by investigation ofˇW l for higher values of l.
This process has previously been investigate in [35,36], where
it has been established that the closed-loop system is stable for
∈ (0.0498686 + 0.9484655k, /4 + k/4) for non-negative, integer
values of k. Having in mind that /4 ≈0.785398, one readily concludes that these results have been verified by our analysis.
5. Conclusions
General properties of root loci constructed for arbitrary, linear time-invariant systems have been investigated in the present
paper. In addition, simple yet powerful and effective numerical
algorithm for construction of the root loci has been presented.
Root loci are important tool in control engineering, both classical and modern. By means of the results presented in the current
paper the influence of varying gain can successfully be analyzed
in the case of an arbitrary normalized open-loop transfer function,
including rational and fractional transfer functions, transfer functions with delays and logarithmic factors, as well as many others.
Two particular application examples in which root locus of
“non-classical” transfer functions appear are considered. First, it
has been established that dominant-pole design technique for PID
controller tuning can be applied by regardless of the form of the
loop transfer function under considerations. Second, it has been
demonstrated that root locus, accompanied by an appropriate
transformation of the original transfer function, allows for effective
analysis of the influence of transport delay to closed loop poles. The
applications presented clearly demonstrate the effectiveness of the
root locus technique in solving some of the most pressing issues in
modern process control.
Acknowledgments
The authors gratefully acknowledge supported of the Ministry
of Education, Science and Technological Development of Republic
of Serbia, Grants TR32018 and TR33013 (MRR), and TR33020 (TBŠ).
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