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IN 227 Control Systems Design
Lecture 10
Instructor: G R Jayanth
Department of Instrumentation and Applied Physics
Ph: 22933197
E-mail: [email protected]
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Motivation for plotting the root locus
In the last couple of lectures, we learnt how the frequency
domain representation of the open-loop transfer functions of the
controller and the plant can be used for purposes of design of
closed-loop control systems.
Frequency domain design using Bode plots is both powerful and
intuitive-it allows us to achieve the desired dynamic response for
the overall system (say to a step input) in addition to the desired
input tracking/ disturbance rejection properties.
The dynamic response of a plant is decided primarily by its
dominant poles, whose locations can be inferred approximately by
using Bode plots. However, one of the (few) drawbacks of frequency
domain based design is that it does not reveal the location of all
the closed-loop poles. It would be advantageous to have a tool that
enables direct visualization of the trajectory of all the
closed-loop poles when our controller/ plant characteristics change
so that we can ensure that the closed-loop performance variation is
tolerable even when the open-loop characteristics change.
Root locus plots do precisely that: they are plots of the
trajectory of the closed-loop poles of a system with a given
open-loop transfer function, when one of the parameters of the
open-loop system varies. However, their application extends beyond
just studying the changes in closed-loop pole locations when the
open-loop system changes: they can also be used as design
tools.
In one sense, root locus plots are an improvement over Nyquist
stability theory (and chronologically too, was proposed by W. R.
Evans a couple of decades after H. Nyquist proposed his
theory)-Nyquist stability theory seeks to verify if a particular
region in the s-plane has a closed-loop pole or not. However, Root
locus plots directly specify all the different regions in the
s-plane where the closed-loop poles can possibly lie.
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The root locus Consider an open-loop system possessing a single
variable parameter K and open-loop transfer function G(s).
The closed loop system is given by- Gcl(s)=KG(s)/(1+KG(s)). The
root locus seeks to find all the regions in the s-plane where the
poles of Gcl(s) can lie, i.e., all possible locations of the roots
of 1+KG(s)=0 when the gain K changes from (typically) 0 to
infinity.
To find this, we note that if s0 is a root of 1+KG(s)=0, then
G(s0)=-1/K. G(s0) is a complex number. Since K is variable, nothing
definite can be said about the magnitude of G(s0). However, we can
say something definite about its phase. The right-side of the
equation (-1/K) can be expressed as (1/K)e-i. Thus, we see that the
phase of G(s0) is fixed, and equal to either or its cyclic
equivalents: -2l (l=0,1,2).
Thus for a particular complex number s to be a root of
1+KG(s)=0, the necessary and sufficient condition is that
. As a consequence, an equivalent but mathematically more useful
definition emerges for the root locus: the root locus is the set of
points in the s-plane for which .
Significance of the parameter K: From the way that the
closed-loop transfer function has been written, viz.,
Gcl(s)=KG(s)/(1+KG(s)), we get the impression that K should
represent some gain of the controller that we can tune. However,
root locus techniques can also be used to study the effect of
variation in any other parameter (such as the location of poles and
zeros of G(s)) by suitable rearrangement of the denominator
polynomial.
Example: Let us choose . Suppose there did indeed exist a gain K
which we could tune, then the root locus studies the variation of
the zeros of . However, if the gain K is a constant (say equal to
1), but the pole at -1 varies, the effect of the variation of this
pole can be studied by looking at the zeros of (where p is
nominally =1). If a particular s is a root of the numerator
polynomial, this equation is not in a form where we can make
definite predictions about the phase of G(s). However, when we
rearrange, we see that the closed-loop transfer function can be
written as
Now we see that it is possible to study the variation of the
pole p by plotting the locus of points where the phase of equals .
Notice that G1(s) looks very different from G(s) and has nothing to
do with the dynamics of the actual open-loop system. It is
important to note that in each case, the parameter whose effect has
to be studied has been brought out and the denominator has been
expressed in the form 1+KG(s)=0, where G(s) does not contain the
parameter whose variation we wish to study.
Needless to say, root-loci study the effect of changes in just
one parameter at a time.
( ) 2G s
( ) 2G s
( ) 1 [( 1)( 2)]G s s s
1 [( 1)( 2)] 0K s s
1 1 [( )( 2)] 0s p s
2 2
1 1( ) [ ]
( 2 1) 1 ( 2) ( 2 1)clG s
s s p s s s
2
1( ) ( 2) ( 2 1)G s s s s 2
-
k
-pk -zl
Guidelines for drawing root-loci- A review
Geometric interpretation of G(s): Suppose . We see that
for any general point s in the s-plane, s+pk (k=1..n) and s+zl
(l=1..m) are complex numbers shown on the right. Thus G(s) is the
product (or ratio) of all such complex numbers. Thus, the magnitude
of G(s) is the product of the magnitudes of each of the complex
numbers in the numerator divided by the product of the magnitudes
of complex numbers in the denominator. Likewise, the phase of G(s)
is the sum of the phases of each of the numerator complex numbers
and the difference of the phases of the denominator complex
numbers: . This is useful in drawing the root loci.
It is assumed in these lectures that students are already
familiar with drawing root-loci. Thus, only the important
properties are reviewed here. For more detailed guidelines, any
standard textbook in Control Systems ,such as the reference
indicated at the end, can be used.
Rule 0: In general, if the closed-loop denominator polynomial is
of nth degree, it has n distinct roots. As the parameter K varies,
each of these roots trace out their own trajectory in the s-plane.
Thus. there are as many branches to Root-locus as the degree of the
denominator polynomial. Furthermore, since complex roots appear in
conjugate pairs, the root locus is symmetric about the
real-axis.
Rule 1: Gcl(s)=KG(s)/(1+KG(s))=KN(s)/(D(s)+KN(s)), where
G(s)=N(s)/D(s). Thus, as K0, the denominator polynomial tends to
D(s). Thus, the branches of root locus start at the poles of the
open-loop system. Likewise, as K, the denominator would be
dominated by N(s). Thus, some of the branches of the root locus end
in the zeros of the open-loop system.
Rule 2: One the real axis, the regions of existence of the root
locus can deduced from the geometric interpretation of G(s). They
exist in regions which have an odd number of poles+zeros to their
right.
1
1
( )..( )..( )( )
( )..( )..( )
l m
k n
s z s z s zG s
s p s p s p
s
ks pls z
l
1 1
( )m n
l kG s
-
-pk -zl
Rule 3: Those branches of root-locus that do not end in an
open-loop zero go towards infinity as K. Thus, for an open-loop
transfer-function whose numerator is of degree m and denominator is
of degree n, there are n-m branches that tend to infinity. We now
need to find where they exist: If we sit on one of those branches
very far away from the origin, all the open-loop zeros and poles
appear bunched together, i.e., they all appear to subtend the same
angle at the far away location (schematic on the right). The
algebraic sum of the angles subtended by poles and zeros is given
by (n-m). For the point s to lie on the root locus, we require
(n-m)= -2l , i.e., = (2l) /(n-m). Thus we see that, very far away
from the origin, the branches are straight lines inclined at angles
. It can be shown that all these straight lines intersect at a
point on the real-axis given by
Rule 4: The points on the real-axis at which the root loci break
away into the complex plane of break in from the complex plane are
obtained by solving the algebraic equation: where,
G(s)=N(s)/D(s).
Guidelines for drawing root-loci- A review
s
ks p
ls z
Far away point s
0 ( ) ( )i in m
s p z n m
( ) ( )( ) ( ) 0
dN s dD sD s N s
ds ds
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Root locus as a tool for design
If the root-locus plots were used just to study the effect of
variation of a particular parameter on a closed-loop system, it
would just be an analysis tool. However, it can also be used for
design. The goal of this lecture is to discuss the use of
root-locus in designing a control system.
Example 1: Consider a plant given by P(s)=1/[(s+1)(s+2)(s+3)].
The plant is controlled by a proportional controller K. We require
the dominant closed-loop poles to have either the desired damping
or the desired natural frequency n.
We first draw the root-locus of the plant. This is shown below.
Since the plant is of 3rd order with no zeros, we see that the root
locus has three branches going towards infinity at 60 and -180
respectively. The locus of poles in the s-plane that have constant
damping is a straight line tilted at an angle -cos-1() to the real
axis. We find the point P1 at which this line intersects the
root-locus. This point and its complex conjugate represent the
poles of the closed-loop system. The gain K is calculated by
evaluating P(s) at the point P1.
To find the gain at which the closed-loop system will have the
desired natural frequency, we see that the locus of poles in the
s-plane that have constant n is a circle of radius n. Therefore, we
find the point P2 at which this circle intersects the root-locus.
This point and its complex conjugate represent the poles of the
closed-loop system. The gain K is calculated by evaluating P(s) at
the point P2.
Notice that since our system is of third order, we would have
another closed-loop pole for each case above. They would be
situated on the third branch of the locus, at points P10 and P20
respectively as shown below.
-8 -7 -6 -5 -4 -3 -2 -1 0 1-5
-4
-3
-2
-1
0
1
2
3
4
5Root Locus
Real Axis
Imagin
ary
Axis
=constant
n=constant
Root locus of P(s)
P1
P2
P10 P20
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Example 2: In example 1, we see that we cannot simultaneously
choose the desired damping ratio and the desired natural frequency.
Suppose, however, we want our closed-loop dominant poles to be at a
specified location, then our controller should be designed to
ensure that the root-locus passes through this location. In this
example, we assume that the desired dominant closed-loop poles
should be at the points -1.52i.
Since a root-locus is symmetric about the real-axis, we consider
only one of the desired roots, namely, -1.5+2i. We note that the
root-locus of the plant does not pass through -1.5+2i. This is
because the phase of P(s) at s= -1.5+2i is not -180. Instead it is
-233. However, the root-locus will pass through -1.5+2i if the
controller can somehow force the phase of the open-loop transfer
function C(j)P(j) to be -180 at -1.5+2i. Thus, our controller
should be designed to add a phase of 53.
Any transfer-function whose phase is 53 at -1.5+2i can
potentially serve as our controller. However, we seek simplicity in
the design of the controller, because simplicity promotes
understanding.
The simplest controller that can add a phase of 53 is of the
form C(s)=s+z0, where the zero z0 subtends an angle 53 at
-1.5+2i. This would imply that z0=3. Since this controller is
not causal it cannot unfortunately be realized. However it does
give us a clue: if our controller is of the form K(s+z0)/(s+p0),
where the pole p0 is placed very far back along the imaginary axis,
then the controller can not only be realized but will also adds
approximately 53 at -1.5+2i , since the dominant contribution is
only from z0. Thus, we choose (arbitrarily) that p0=11. For this
choice, p0 adds a phase -12 at -1.5+2i. Thus, the zero z0 should
add a phase of 53+12=65 for the root-locus to pass through -1.5+2i.
Using simple trigonometry, this places the zero at -2.44. Finally,
the gain K is calculated by evaluating |C(s)P(s)| at s= -1.5+2i.
|C(s)P(s)|=0.021. Thus, K=1/ |C(s)P(s)| =46.67. Thus our final
controller is C(s)=46.67(s+2.44)/(s+11). This is a lead
compensator. The root locus, plotted below, is now seen to pass
through the desired points.
Root locus as a tool for design
~ 53z
-z0 -p0
~ 0p
1.5 2i
-10 -8 -6 -4 -2 0
-6
-4
-2
0
2
4
6
Root Locus
Real Axis
Imagin
ary
Axis 1.5 2i
Desired characteristics of the
controller transfer function
-
-6 -4 -2 0
-10
-8
-6
-4
-2
0
2
4
6
8
10
Root Locus
Real Axis
Imagin
ary
Axis
Example 3: Example 2 clearly illustrated the use of root-locus
in designing the location of dominant poles: We compute the phase
of G(s) at the desired location of the closed-loop pole and find a
controller C(s) that supplies the deficit phase to ensure that the
total phase of the open-loop system is -180 (360l). The gain K of
the controller is finally calculated using the magnitude criterion,
namely |KC(s)P(s)| =1. In example 2 just a single dominant zero was
used to supply the requisite phase. However, if the deficit is very
large we may need to use two or more zeros (and correspondingly add
two far away poles in order to realize a causal system). Thus, for
example, if we desired our closed-loop dominant pole to be at -33i,
we note that the total phase of G(s) at -3+3i is -322, which is
-142 away from -180.
If a single zero has to supply this phase, it has to be situated
well into the right-half plane. This is undesirable because this
would guarantee that the root-locus would exist along the real-axis
in the right-half plane and thus ensure that one of the poles may
be unstable. Thus, we choose this to be supplied by two zeros, both
in the left half plane. We choose one to supply +90 and the other
to supply the reminder, viz., +52 .
This places the first zero at -3 and the second zero at -5.34.
To realize a causal controller, we need add two poles. We choose to
add these poles far away, at -30. The total phase added by these
two poles at -3+3i is about 13. To compensate, we correct the
position of the second zero so that it subtends 52+13=65 at -3+3i .
Thus, in the second iteration, we obtain its position to be at
-4.4. Thus, the controller is of the form
C(s)=K(s+3)(s+4.4)/(s+30)2. The gain K is found by setting
|KC(s)P(s)| =1. This gives us K=2542, so that the controller is
given by C(s)=2542(s+3)(s+4.4)/(s+30)2. The root locus is shown
below.
Root locus as a tool for design
3 3i
- Example 4: We see that root-locus based design is excellent for
designing compensators that decide the dominant dynamics. However,
as control engineers, our prime area of interest is not the
dominant dynamics but the low-frequency region (low relative to the
poles of the plant), where we want to track a specified input or
reject some disturbance. The low frequency gain of an open-loop
system is given by . In the schematic on the right we see that .
Generally we would like the low frequency gain to be adequately
high. If the gain is not high enough, we can add a pole p0 close to
the origin (i.e., p0
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When we examine what we focused the most on while designing in
frequency domain and the root locus, we notice that we spent the
maximum time achieving high gain in frequency domain-based design.
Any attention we paid to phase was done at the end of the design
process where we used zero-placement to improve the phase-margin.
In contrast, root-locus based design focused a lot on placement of
the dominant poles and relatively little on achieving high
gain.
However, since control engineers generally ought to focus on
high gain much more than dominant dynamics, frequency domain design
tools are preferable over root-locus. Nevertheless, the two
complement one another: we can first design the system for low
frequency control performance using Bode plots and then shift to
root-locus to achieve the desired dominant dynamics by placing the
closed-loop poles at desired locations.
Unlike the frequency domain magnitude and phase plots of the
controller and the plant that add up, root-loci of the controller
and the plant do not add up. Indeed, the root-locus of the combined
open-loop system could be dramatically different from that of each
individual component. Thus, root-locus demands that we invest much
more effort in building our intuition about the characteristics of
root-loci before we use them as design tools.
When we have such an excellent and intuitive tool as Bode plot
for the design of one degree of freedom (1-DOF) control systems it
is hard to think of why one may want to invest extra energy and
learn to design a control system using root-locus. However, we
shall see in Lecture 11 that this is not the case for 2-DOF control
systems root locus offers key insights for design of robust control
systems that are not as intuitively apparent in the case Bode plots
for 2DOF control systems.
Control design using root locus vis--vis Bode plots: some
comments
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While our opinion on root-locus as a design tool for 1-DOF
control systems was not wholly enthusiastic, the same cannot be
said about its virtues as a tool for analysis. For example, in the
previous lecture, we designed a PI-based control system whose step
response has been shown on the right (the blue trace). We notice a
long tail in the response, i.e., after quickly reaching a value
close to the reference, the closed-loop systems response is
dominated by that of a very slow pole. There is simply no way for
us to directly anticipate that closed-loop response would have such
a slow pole simply by looking at the Bode plot of the open-loop
system.
However, if we turn to root locus, we can immediately see why
such a response can occur: we notice that if we employ PI-control,
a segment of the root-locus exists between the pole and the zero of
the PI-controller. Thus, one of the roots of the closed-loop system
has to lie on this segment. Since z0, p0 ~0 , this is a very slow
pole. Now, if the open-loop gain is sufficiently high, the
closed-loop pole will lie very close to the open-loop zero and the
effect of the two on the closed-loop response will nearly cancel
each other. However, if the gain is insufficient, the closed-loop
pole is somewhere midway between the two ends of the segment and
can contribute substantially to the overall response. It is this
situation that we find in the PI-control that was designed in the
last lecture.
Thus, even if we prefer to design a controller using
frequency-domain techniques, it is always prudent to look at the
root-locus of the open-loop system later and identify the
possibilities for maladies such as the one above.
Root locus as a tool for analysis
0 0.5 1 1.5 2 2.50
0.2
0.4
0.6
0.8
1
1.2
1.4Step Response
Time (sec)
Am
plit
ude
Step response of PI-
controlled closed-loop
system
Long tail
-1 -0.5 0
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
Root Locus
Real Axis
Imagin
ary
Axis
0z 0p
Segment of root-locus
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References
(1) G. F. Franklin, J. David Powell and A. Emami-Naeini,
Feedback Control of Dynamic Systems, Pearson Education (2002)
(2) K Ogata, Modern Control Engineering, Pearson (2003)
