Hindawi Publishing Corporation
Mathematical Problems in Engineering
Volume 2009, Article ID 727908, 14 pages
doi:10.1155/2009/727908

Research Article
On Complementary Root Locus of
Biproper Transfer Functions

Marcelo C. M. Teixeira,1 Edvaldo Assunção,1 Rodrigo Cardim,1
Neusa A. P. Silva,2 and Erica R. M. D. Machado2

1 Department of Electrical Engineering (DEE), Faculdade de Engenharia de Ilha Solteira,
UNESP—São Paulo State University, 15385-000 Ilha Solteira, SP, Brazil

2 Department of Mathematics (DM), Faculdade de Engenharia de Ilha Solteira,
UNESP—São Paulo State University, 15385-000 Ilha Solteira, SP, Brazil

Correspondence should be addressed to Marcelo C. M. Teixeira, marcelo@dee.feis.unesp.br

Received 12 February 2009; Revised 22 May 2009; Accepted 11 August 2009

Recommended by J. Rodellar

This paper addresses the root locus (locus of positive gain) and the complementary root locus
(locus of negative gain) of biproper transfer functions (transfer functions with the same number
of poles and zeros). It is shown that the root locus and complementary root locus of a biproper
transfer function can be directly obtained from the plot of a suitable strictly proper transfer
function (transfer function with more poles than zeros). There exists a lack of sources on the
complementary root locus plots. The proposed procedure avoids the problems pointed out by
Eydgahi and Ghavamzadeh, is a new method to plot complementary root locus of biproper
transfer functions, and offers a better comprehension on this subject. It also extends to biproper
open-loop transfer functions, previous results about the exact plot of the complementary root locus
using only the well-known root locus rules.

Copyright q 2009 Marcelo C. M. Teixeira et al. This is an open access article distributed under
the Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.

1. Introduction

The analysis of the location of the closed-loop poles in the s-plane and in function of a positive
gain in the open-loop transfer function, proposed by Evans [1, 2], has found wide application
in the analysis and design of linear time-invariant systems. Recently, the root locus method
was extended to fractional-order systems [3, 4]. More details about the design of controllers
based on root-locus method or depending on one parameter can be found in [5–12]. The
term complementary root locus, that means locus of the closed-loop poles in function of a
negative open-loop gain, was proposed by Narendra, considering that the root locus and the
complementary root locus together form a complete algebraic curve [13].



2 Mathematical Problems in Engineering

In the design of control systems, for changing time constants, and when large loop gain
is required in inner loops, the complementary root locus is useful tool [14]. Further results
about the complementary root locus can be found in [14].

There exists a lack of sources on the complementary root locus and many textbooks
about control systems present the root-locus method and its application in control design,
but do not present information about complementary root locus [15, 16]. The textbooks [17–
21] present both methods.

In [14], the authors have shown that some of the steps in the method of constructing
complementary root locus (construction rules) are not valid for open-loop transfer functions
with the same number of poles and zeros, referred to as biproper transfer functions.
Considering this fact, this problem was studied and new definitions and analyses were
presented in [14] to plot the complementary root locus. This fact has motivated researches
to find new methods to avoid these problems and to offer a better comprehension for the
designers about this subject.

For instance, in [22] a method to plot complementary root locus using only the well-
known root-locus construction rules to plot root locus was proposed. This new technique
allows an exact plot when the sum of the number of poles and zeros is odd and an
approximated plot otherwise. An example in that paper illustrated the application of this
method to approximately plot the complementary root locus of biproper transfer functions.

In this paper an alternative procedure to plot the complementary root locus of transfer
functions that are biproper is presented. The main idea was to show that the root locus and
complementary root locus of a biproper transfer function can be directly and exactly obtained
from the plot of the root locus and complementary root locus of a suitable strictly proper
transfer function (transfer function with more poles than zeros). This procedure removes the
problem cited above, because the plots are done only with strictly proper open-loop transfer
functions. The proposed method extends the results presented in [22], because it can also
allow an exact plot of the complementary root locus of biproper transfer functions, using
the well-known root-locus construction rules. An example illustrates this fact. Literature did
not present many papers about the complementary root locus, and this paper offers a better
comprehension on this subject.

2. Statement of the Problem

Given the feedback system described in Figure 1, consider that the open-loop transfer
function is biproper:

G(s)H(s) =
bns

n + bn−1s
n−1 + · · · + b1s + b0

sn + an−1sn−1 + · · · + a1s + a0
, (2.1)

where K0, bn, bi, ai ∈ R, for i = 0, . . . , n − 1 and bn /= 0.
Now, define

K = K0bn. (2.2)

Then, draw the root locus and complementary root locus of the closed-loop system given in
(2.1) and Figure 1 for −∞ < K <∞, using only the plots of the root locus and complementary
root locus of an equivalent system with a strictly proper open-loop transfer function.



Mathematical Problems in Engineering 3

R(s) Y (s)
K0 G(s)

H(s)

−

Figure 1: A closed-loop controlled system.

3. Root Locus and Complementary Root Locus of
Biproper Transfer Functions

In this section a solution of the problem stated in Section 2 is presented.
From (2.1), (2.2), and Figure 1, the closed-loop transfer function is

Y (s)
R(s)

=
K0G(s)

1 +K0G(s)H(s)

=
K
(
sn + cn−1s

n−1 + · · · + c1s + c0
)

K
(
sn + cn−1sn−1 + · · · + c1s + c0

)
+
(
sn + an−1sn−1 + · · · + a1s + a0

) ,

(3.1)

where ci = bi/bn, i = 0, 1, . . . , n − 1.
Therefore, the characteristic equation of this feedback system is

Δ(s) = K
(
sn + cn−1s

n−1 + · · · + c1s + c0

)
+
(
sn + an−1s

n−1 + · · · + a1s + a0

)

= KN(s) +D(s) = 0,
(3.2)

where N(s) = (sn + cn−1s
n−1 + · · · + c1s + c0) and D(s) = (sn + an−1s

n−1 + · · · + a1s + a0).
Note that if K defined in (2.2) is positive, then one can directly apply the classical

root-locus construction rules to plot the root locus of (3.2). The reader can find these rules,
for instance, in [15, 16, 18–20, 23].

In the case where K < 0, Eydgahi and Ghavamzadeh [14] showed that one cannot
directly apply the classical rules to plot the complementary root locus. It is necessary to
consider additional properties, related to the roots at infinity [14]. In this case, a method
presented in [22] allows an approximated plot, using only the root-locus construction rules.

The next theorem offers an alternative method to solve the problem stated in Section 2,
without the approximated plot of [22] or the additional definitions and analyses presented
in [14]. It also complements the available results and offers a better understanding about this
subject.

Theorem 3.1. The root-locus and complementary root-locus plots (−∞ < K < ∞) of the system
given in Figure 1 with the characteristic equation (3.2), whose open-loop transfer function is biproper,
can be directly obtained from the root-locus and complementary root-locus plots of the strictly proper
open-loop system K̃Ñ(s)/D(s) given by its characteristic equation:

K̃Ñ(s) +D(s) = 0, (3.3)



4 Mathematical Problems in Engineering

where

D(s) = sn + an−1s
n−1 + · · · + a1s + a0, N(s) = sn + cn−1s

n−1 + · · · + c1s + c0, (3.4)

Ñ(s) =N(s) −D(s) = (cn−1 − an−1)sn−1 + (cn−2 − an−2)sn−2 + · · · + (c0 − a0), (3.5)

K̃ =
K

K + 1
. (3.6)

The gain K can be obtained from the gain K̃, using the following expression:

K =
K̃

1 − K̃
. (3.7)

Proof. From (3.2) and (3.5) it follows that

KN(s) +D(s) = (K + 1)N(s) +D(s) −N(s) = (K + 1)N(s) − Ñ(s) = 0. (3.8)

Now, considering K/= − 1, then, from (3.8),

N(s) − 1
K + 1

Ñ(s) = 0. (3.9)

Again from (3.8), note that KN(s) = −D(s). So, multiplying both sides of (3.9) by K one
obtains

KN(s) − K

K + 1
Ñ(s) = −D(s) − K̃Ñ(s) = 0, (3.10)

where K̃ was defined in (3.6). Then, (3.3) follows from (3.10). Now, note that (3.6) and (3.7)
are equivalent. So, (3.7) can be used to calculate the gainK, given a gain K̃. Finally, from (3.4)
and (3.5), observe that the transfer Ñ(s)/D(s) has more poles than zeros and so it is strictly
proper.

Remark 3.2. The proof of Theorem 3.1 can also be done using block diagram manipulations,
as described in Figure 2. Observe that in Figure 2, from (3.5), the block diagrams (b) and
(c) are equivalent. The block diagrams are very useful in the analysis and design of control
systems [15–21]. A systematic procedure for reducing block diagrams and its implementation
in MATLAB program can be found in [24].

Remark 3.3. Figure 3 describes the relation between K and K̃, from (3.7). Note that: (i) for
K̃ in the interval −∞ < K̃ < ∞, then K presents also values in all interval −∞ < K < ∞;
(ii) K → ∞ when K̃ → 1−, and (iii) K → −∞ when K̃ → 1+. Therefore, the root-locus
plot (0 ≤ K̃ < ∞) and the complementary root locus (−∞ < K̃ ≤ 0) of (3.3) correspond to
the complete root loci plots (−∞ < K < ∞) of the original equation (3.2). From Figure 3 one
can see that the complementary root locus (−∞ < K ≤ 0) of (3.2) can be obtained from the
root-loci plots of (3.3) for −∞ < K̃ ≤ 0 and 1 < K̃ < ∞ can be obtained and the root locus
(0 ≤ K <∞) of (3.2) from the root locus of (3.3) for 0 ≤ K̃ < 1.



Mathematical Problems in Engineering 5

R(s) E(s)
K0 G(s)

Y (s)

H(s)

−

Z(s)

(a)

R(s) E(s)
K = K0bn

G(s)H(s)
bn

=
N(s)
D(s)

Z(s)

−

(b)

R(s) E(s)
K 1 +

Ñ(s)
D(s)

Z(s)

−

(c)

R(s) E(s)

K

KÑ(s)
D(s)

Z2(s)

Z1(s)

Z(s)

−

(d)

R(s) E(s)

K

KÑ(s)
D(s)

Z2(s)

Z1(s)

−−

(e)

R(s)
K̃ =

K

1 +K

Z2(s)Ñ(s)
D(s)

Z1(s)

−

(f)

Figure 2: An alternative proof of Theorem 3.1: the block diagrams (a), (b), (c), (d), (e), and (f) are
equivalent, for K/= − 1 .

Remark 3.4. Figure 3 shows also that for K → −1, |K̃| → ∞. It means that in this situation,
the root-loci plots of (3.3), for the open-loop transfer function K̃Ñ(s)/D(s), will present at
least one asymptote, because it is a strictly proper transfer function. Therefore, at least one
branch of these plots will approach infinity in the s-plane. It was firstly presented in [14] and
the analysis above offers an alternative proof of this fact. The aforementioned gain K = −1
is equivalent to the case where one has the critical gain defined in [14]. Furthermore, from
Figure 3 note that for K̃ → 1, |K| → ∞.

Remark 3.5. The characteristic equations (3.2) and (3.3) present the same open-loop poles,
because for K = 0 and K̃ = 0 it follows from these equations that D(s) = 0. Then, the
procedure proposed in Theorem 3.1 preserves the open-loop poles of the original system,
given in Figure 1.



6 Mathematical Problems in Engineering

43210−1−2−3−4

K̃

−10

−8

−6

−4

−2
−1

0

2

4

6

8

10

K

Complementary root locus
Root
locus

Complementary
root locus

Figure 3: Relation between the gains K and K̃, from (3.7): K = K̃/(1 − K̃).

In order to exemplify the results of the above theorem, the next examples are
presented.

Example 3.6. Consider the open-loop transfer function, that is, biproper, in Figure 1:

K0G(s)H(s) = K0
s2 + 3s − 18

s2 − 4
. (3.11)

In this case, K = K0, N(s) = s2 +3s−18, and D(s) = s2 −4. Then, from Theorem 3.1, and (3.5),
Ñ(s) =N(s) −D(s) = 3s − 14. Therefore, from (3.3) it follows that

K̃Ñ(s) +D(s) = K̃(3s − 14) +
(
s2 − 4

)
= 0. (3.12)

Note that (3.12) is equivalent to the characteristic equation 1 + K̃Ñ(s)/D(s) = 0, where
K̃Ñ(s)/D(s) is a strictly proper transfer function.

It is easy to plot the root locus (0 ≤ K̃ < ∞) of (3.12). The open-loop poles and the
zero are p1 = −2 e p2 = 2 and z1 = 4.6667, respectively. There exists one asymptote with
angle equal to 180◦ and the sections of the real axis that are occupied by this root locus are
2 ≤ Re(s) ≤ 4.6667 and Re(s) ≤ −2. Figure 4 shows this root-locus plot.

Now, the complementary root locus (−∞ < K̃ ≤ 0) of (3.12) can be directly plotted
using the construction rules presented in [17–21], because K̃Ñ(s)/D(s) is a strictly proper
transfer function, and the problems with these rules, pointed out in [14], will not appear.
Following these construction rules one can conclude that the open-loop poles and the zero
are p1 = −2 e p2 = 2 and z1 = 4.6667, respectively. There exists one asymptote with angle
equal to 0◦, and the sections of the real axis that are occupied by this complementary root
locus are −2 ≤ Re(s) ≤ 2 and Re(s) ≥ 4.6667. Furthermore, the breakway points are equal to
0.4503 and 8.8830, and there is only one point (s = 0) where the complementary root locus
intersects the imaginary axis. Figure 5 shows this complementary root-locus plot.



Mathematical Problems in Engineering 7

1086420−2−4−6−8−10

Re(s)

−4

−3

−2

−1

0

1

2

3

4

Im
(s
) K̃ →∞ K̃ = 0 K̃ = 0 K̃ →∞

Figure 4: The root locus plot of Example 3.6 from (3.12).

121086420−2−4−6

Re(s)

−6

−4

−2

0

2

4

6

Im
(s
) K̃ = 0 K̃ = 0 K̃ → −∞ K̃ → −∞

Figure 5: The complementary root locus plot from (3.12).

Another method to plot this complementary root locus, using only the root-locus
construction rules, was proposed in [22]. Following this method, define s = −v. Therefore,
one can get from (3.12)

K̃Ñ(−v) +D(−v) = K̃(−3v − 14) +
(
v2 − 4

)
= 0. (3.13)

Hence, defining Kp = −K̃ it follows that

Kp(v + 14) +
(
v2 − 4

)
= 0. (3.14)



8 Mathematical Problems in Engineering

6420−2−4−6−8−10−12

Re(v)

−6

−4

−2

0

2

4

6

Im
(v
) Kp →∞ Kp →∞ Kp = 0Kp = 0

Figure 6: The root-locus plot in the plane v from (3.14).

1086420−2−4−6−8−10

Re(s)

−6

−4

−2

0

2

4

6

Im
(s
) K̃ →∞ K̃ = 0 K̃ = 0 K̃ → −∞

K̃ → −∞,∞

Figure 7: The complete root-locus plot of (3.12): root locus (· − ·) and complementary root locus (−).

Now, we can easily plot the root locus of the above equation, in the complex plane
v and with the gain 0 < Kp < ∞. Figure 6 displays this plot. Finally, from the expressions
s = −v and K = −Kp, one can obtain the complementary root locus in the complex plane s
and with the gain K, by rotating 180◦ the v plane around the imaginary axis [22]. This plot is
the same obtained with the complementary root-locus construction rules and it is also given
in Figure 5.

The complete root-locus plot (−∞ < K̃ < ∞) of (3.12) can be obtained from the plots
of Figures 4 and 5 and is presented in Figure 7.

Now, from Figure 3 and (3.7) one can directly plot the root locus and complementary
root locus with respect to the gain K. From Figure 3, the root locus (0 ≤ K < ∞) corresponds
to the region 0 ≤ K̃ < 1. The closed-loop poles for K̃ = 1, from (3.12) (or Figure 4 or Figure 7),
are equal to 3 and −6. This root-locus plot is shown in Figure 8 and it was directly obtained



Mathematical Problems in Engineering 9

1086420−2−4−6−8−10

Re(s)

−4

−3

−2

−1

0

1

2

3

4

Im
(s
) K →∞ K = 0 K = 0 K →∞

Figure 8: The root-locus plot of (3.11) for K (0 ≤ K̃ < 1).

from Figure 4 or Figure 7. Now, again from Figure 3, the complementary root locus (−∞ <

K ≤ 0) corresponds to the regions −∞ < K̃ ≤ 0 and 1 < K̃ < ∞. Therefore, from Figure 7 one
can get this complementary root locus, as described in Figure 9. For each gain K̃ in Figure 7,
the correspondent gainK in Figure 8 and Figure 9 can be obtained using the expression given
in (3.7). Finally, observe that the method proposed in [22] cannot be used to exactly plot the
complementary root locus of (3.11). Thus, this example shows that the proposed method
extends the results given in [22].

Remark 3.7. The root locus and complementary root locus plots presented in this paper, that
illustrate the proposed method, were obtained by using the software MATLAB. The main
contribution of this paper is an alternative analysis of the complementary root locus of
biproper transfer functions, without using the theorems presented in [14].

Example 3.8. Consider the open-loop transfer function K0G(s)H(s) in Figure 1:

K0G(s)H(s) = K0
2s4 + 5s3 + 6s2 + 8s + 12

3s4 + 8s3 + 9s2 + 12s − 16
. (3.15)

The open-loop poles are p1 = −2.4643, p2 = 0.6932, and p3,4 = −0.4478 ± 1.7093j, and the zeros
are z1,2 = −1.5750 ± 0.8068j, and z3,4 = 0.3250 ± 1.3455j.

Eydgahi and Ghavamzadeh [14] have shown that the complementary root locus of this
system presents a root at infinity. Then, considering that this system has the same number of
poles and zeros, the authors concluded that some rules to plot complementary root locus are
not valid in this case. In [14] this problem was studied, and new definition and construction
rules for complementary root locus were proposed. In [22] it was proposed a method and its
application to approximately plot this complementary root locus, using only the well-known
root-locus construction rules.



10 Mathematical Problems in Engineering

1086420−2−4−6−8−10

Re(s)

−6

−4

−2

0

2

4

6

Im
(s
) K → −1 K → −∞ K = 0 K = 0 K → −∞ K → −1

Figure 9: The complementary root-locus plot of (3.11) for K (−∞ < K̃ ≤ 0 and 1 < K̃ <∞).

1086420−2−4−6−8−10

Re(s)

−4

−3

−2

−1

0

1

2

3

4

Im
(s
)

Figure 10: The root-locus plot from (3.17) with the gain K̃.

Now, one can use the results of Theorem 3.1 to exactly plot this complementary root
locus. Following this procedure, from (2.1)–(3.2) then N(s) = s4 + 2.5s3 + 3s2 + 4s + 6, D(s) =
s4 + 2.6667s3 + 3s2 + 4s − 5.3333, K = 2K0/3, and from (3.5) it follows that

Ñ(s) =N(s) −D(s) = −0.1667s3 + 11.3333. (3.16)

Therefore, from (3.3),

K̃Ñ(s) +D(s) = K̃
(
−0.1667s3 + 11.3333

)
+
(
s4 + 2.6667s3 + 3s2 + 4s − 5.3333

)
= 0. (3.17)



Mathematical Problems in Engineering 11

1086420−2−4−6−8−10

Re(s)

−4

−3

−2

−1

0

1

2

3

4

Im
(s
)

Figure 11: The complementary root-locus plot from (3.17) with the gain K̃.

1086420−2−4−6−8−10

Re(s)

−4

−3

−2

−1

0

1

2

3

4

Im
(s
)

Figure 12: The complete root-locus plot from (3.17) with the gain K̃.

Note that (3.17) is equivalent to the characteristic equation 1 + K̃Ñ(s)/D(s) = 0, where
K̃Ñ(s)/D(s) is a strictly proper transfer function. Therefore, one can plot the root locus
(0 ≤ K̃ < ∞) and the complementary root locus (−∞ < K̃ ≤ 0) of (3.17), where the gain
is K̃, using the well-known construction rules [17–21]. These two plots are shown in Figures
10 and 11, respectively. The complete root locus (−∞ < K̃ <∞) is given in Figure 12.

Hence, from Figure 10 and (3.7), it is easy to obtain the root locus of (3.2), that
corresponds to the regions where 0 ≤ K̃ < 1 (see Figure 3 for more details). This plot is
presented in Figure 13.

The complementary root locus of (3.2) can also be directly plotted from Figure 11,
Figure 12, and (3.7) for −∞ < K̃ ≤ 0 and 1 < K̃ <∞. Figure 14 shows this plot.



12 Mathematical Problems in Engineering

1086420−2−4−6−8−10

Re(s)

−4

−3

−2

−1

0

1

2

3

4

Im
(s
)

Figure 13: The root-locus plot from (3.2) with the gain K.

1086420−2−4−6−8−10

Re(s)

−4

−3

−2

−1

0

1

2

3

4

Im
(s
)

Figure 14: The complementary root-locus plot from (3.2), with the gain K.

4. Conclusion

A new method for plotting complementary root locus of biproper open-loop transfer
functions has been presented in this article. The main idea was to show that this plot can
be directly obtained from the root locus and complementary root locus of a suitable strictly
proper transfer function. With this method it is not necessary to perform the additional
analysis proposed in [14], that must complete the construction rules to plot complementary
root locus.

The proposed method allows an exact plot of complementary root locus, of biproper
open-loop transfer functions, using only well-known root locus rules. From the method
presented in [22], for this case, one can only obtain an approximated plot. There exists a lack



Mathematical Problems in Engineering 13

of sources about complementary root locus. This paper offers an alternative procedure and
provides a better comprehension about this subject. Finally, the educators that teach block
diagrams can also use the analysis depicted in Figure 2, as an interesting application of this
theory.

Acknowledgments

The authors gratefully acknowledge the partial financial support by FAPESP, CAPES, and
CNPq of Brazil.

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14 Mathematical Problems in Engineering

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