Chapter Three

“Control Systems Design by Root-Locus

Method”

3.1 Introduction:
The primary objective of this chapter is to present procedures for the design and
compensation of single-input-single-output linear time-invariant control systems.
Compensation is the modification of the system dynamics to satisfy the given specifications.
The approach to the control system design and compensation used in this chapter is the root-
locus approach.
Control systems are designed to perform specific tasks. The requirements imposed on the
control system are usually spelled out as performance specifications. The specifications may
be given in terms of transient response requirements (such as the maximum overshoot and
settling time in step response) and of steady-state requirements (such as steady-state error in
following ramp input).
The design by the root-locus method is based on reshaping the root locus of the system by
adding poles and zeros to the system's open loop transfer function and forcing the root loci to
pass through desired closed-loop poles in the s plane. The characteristic of the root-locus
design is its being based on the assumption that the closed-loop system has a pair of dominant
closed-loop poles.
Setting the gain is the first step in adjusting the system for satisfactory performance. In many
practical cases, however, the adjustment of the gain alone may not provide sufficient
alteration of the system behavior to meet the given specifications. As is frequently the case,
increasing the gain value will improve the steady-state behavior but will result in poor
stability or even instability. It is then necessary to redesign the system (by modifying the
structure or by incorporating additional devices or components) to alter the overall behavior
so that the system will behave as desired. Such a redesign or addition of a suitable device is
called compensation. A device inserted into the system for the purpose of satisfying the
specifications is called a compensator. The compensator compensates for deficit performance
of the original system.

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Figures 7.l (a) and (b) show compensation schemes commonly used for feedback control
systems. Figure 7.l (a) shows the configuration where the compensator G,(s) is placed in
series with the plant. This scheme is called series compensation.
An alternative to series compensation is to feed back the signal(s) from some element (s) and
place a compensator in the resulting inner feedback path, as shown in Figure 7.l (b). Such
compensation is called parallel compensation or feedback compensation. In this chapter we
discuss series compensation in detail.

Figure 3.1 (a) Series Compensation.

(b) Parallel Compensation.

3.2 Root-Locus Approach to Control System Design:

The root-locus method is a graphical method for determining the locations of all closed-loop
poles from knowledge of the locations of the open-loop poles and zeros as some parameter
(usually the gain) is varied from zero to infinity. The method yields a clear indication of the
effects of parameter adjustment.
In practice, the root-locus plot of a system may indicate that the desired performance cannot
be achieved just by the adjustment of gain. In fact, in some cases, the system may not be
stable for all values of gain. Then it is necessary to reshape the root loci to meet the
performance specifications.
In designing a control system, if other than a gain adjustment is required, we must modify the
original root loci by inserting a suitable compensator. Once the effects on the root locus of the
addition of poles and/or zeros are fully understood, we can readily determine the locations of
the pole(s) and zero(s) of the compensator that will reshape the root locus as desired. In
essence, in the design by the root-locus method, the root loci of the system are reshaped
through the use of a compensator so that a pair of dominant closed-loop poles can be placed at
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the desired location. (Often, the damping ratio and undamped natural frequency of a pair of
dominant closed-loop poles are specified).

3.2.1 Effects of the Addition of Poles

The addition of a pole to the open-loop transfer function has the effect of pulling the root
locus to the right, tending to lower the system's relative stability and to slow down the settling
of the response. (Remember that the addition of integral control adds a pole at the origin, thus
making the system less stable). Figure 3.2 shows examples of root loci illustrating the effects
of the addition of a pole to a single-pole system and the addition of two poles to a single-pole
system.

Figure 3.2 (a) Root-locus plot of a single-pole system;

(b) root-locus plot of a two-pole system;
(c) root-locus plot of a three-pole system.

3.2.2 Effects of the Addition of Zeros

The addition of a zero to the open-loop transfer function has the effect of pulling the root
locus to the left, tending to make the system more stable and to speed up the settling of the
response. (Physically, the addition of a zero in the feed forward transfer function means the
addition of derivative control to the system. The effect of such control is to introduce a degree
of anticipation into the system and speed up the transient response.) Figure 3.3 (a) shows the
root loci for a system that is stable for small gain but unstable for large gain. Figures 3.3 (b),
(c), and (d) show root-locus plots for the system when a zero is added to the open-loop
transfer function. Notice that when a zero is added to the system of Figure 3.3 (a), it becomes
stable for all values of gain.

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 Figure 3.3 (a) Root-locus plot of a three-pole system; (b), (c), and (d) root-
locus plots showing effects of addition of a zero to the three-pole system.

3.3 Lead Compensation

There are many ways to realize continuous-time (or analogue) lead compensators, such as
electronic networks using operational amplifiers, electrical RC networks, and mechanical
spring-dashpot systems.
Figure 3-4 shows an electronic circuit using operational amplifiers. The transfer function for
this circuit was obtained as follows:

Figure 3.4 Electrical circuit of Lead Compensation

1 1
𝐸0 (𝑠) 𝑅2 𝑅4 𝑅1 𝐶1 𝑠 + 1 𝑅4 𝐶1 𝑠 + 𝑅1 𝐶1 𝑠+𝑇
= = = 𝐾𝑐 (3.1)
𝐸𝑖 (𝑠) 𝑅1 𝑅3 𝑅2 𝐶2 𝑠 + 1 𝑅3 𝐶2 𝑠 + 1 1
𝑠 + 𝛼𝑇
𝑅2 𝐶2
Where:
𝑅4 𝐶1
𝑇 = 𝑅1 𝐶1 , 𝛼𝑇 = 𝑅2 𝐶2 , 𝐾𝑐 =
𝑅3 𝐶2

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From Equation (7-I), we see that this network is a lead network if 𝑅1 𝐶1 > 𝑅2 𝐶2 , or 𝛼 < 1. It is a lag
network if 𝑅1 𝐶1 < 𝑅2 𝐶2 . The pole-zero configurations of this network when 𝑅1 𝐶1 > 𝑅2 𝐶2 and 𝑅1 𝐶1 <
𝑅2 𝐶2 are shown in Figure 3-5(a) and (b), respectively.

Figure 3.5: Zero pole configuration of a) Lead Compensator b) Lag Compensator

3.3.1 Lead Compensation Techniques Based on the Root-Locus

Approach.
The root-locus approach to design is very powerful when the specifications are given in terms
of time-domain quantities, such as the damping ratio and undamped natural frequency of the
desired dominant closed-loop poles, maximum overshoot, rise time, and settling time.
Consider a design problem in which the original system either is unstable for all values of
gain or is stable but has undesirable transient-response characteristics. In such a case, the
reshaping of the root locus is necessary in the broad neighbourhood of the 𝒋𝒘 axis and the
origin in order that the dominant closed-loop poles be at desired locations in the complex
plane. This problem may be solved by inserting an appropriate lead compensator in cascade
with the feed forward transfer function.
The procedures for designing a lead compensator for the system shown in Figure 3.6 by the
root-locus method may be stated as follows:

Figure 3.6 Control System

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1. From the performance specifications, determine the desired location for the dominant
closed-loop poles.
2. By drawing the root-locus plot of the uncompensated system (original system), ascertain
whether or not the gain adjustment alone can yield the desired closed loop poles. If not,
calculate the angle deficiency 4.This angle must be contributed by the lead compensator if the
new root locus is to pass through the desired locations for the dominant closed-loop poles.
3. Assume the lead compensator 𝐺𝑐 (𝑠) to be:
1
𝑠+𝑇
𝐺𝑐 (𝑠) = 𝐾𝑐 (0 < 𝛼 < 1)
1
𝑠 + 𝛼𝑇

where 𝛼 and 𝑇 are determined from the angle deficiency. 𝐾𝑐 is determined from the
requirement of the open-loop gain.
4. If static error constants are not specified, determine the location of the pole and zero of the
lead compensator so that the lead compensator will contribute the necessary angle𝜙. If no
other requirements are imposed on the system, try to make the value of 𝛼 as large as possible.
A larger value of 𝛼 generally results in a larger value of 𝐾𝑣 which is desirable. (If a particular
static error constant is specified, it is generally simpler to use the frequency-response
approach.)
5. Determine the open-loop gain of the compensated system from the magnitude condition.
Once a compensator has been designed, check to see whether all performance specifications
have been met. If the compensated system does not meet the performance specifications, then
repeat the design procedure by adjusting the compensator pole and zero until all such
specifications are met.

Example (3.1): Consider the system shown in Figure 3-7(a).The feed forward transfer
function is:
4
𝐺(𝑠) =
𝑠(𝑠 + 2)
The root-locus plot for this system is shown in Figure 3-7(b). The closed-loop transfer
function becomes:
4 4
𝐺(𝑠) = =
𝑠 2 + 2𝑠 + 4 (𝑠 + 1 + 𝑗√3)(𝑠 + 1 − 𝑗√3)

The closed loop poles are located at: 𝑠 = −1 ∓ 𝑗√3

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The damping ratio of the closed-loop poles is 0.5.The undamped natural frequency of the
closed loop poles is 2 rad/sec. The static velocity error constant is 2 sec-1.
It is desired to modify the closed-loop poles so that an undamped natural frequency 𝜔𝑛 = 4
rad/sec is obtained, without changing the value of the damping ratio, 𝜁 = 0.5.
The damping ratio of 0.5 requires that the complex-conjugate poles lie on the lines drawn
through the origin making angles of ±600 with the negative real axis.
Since the damping ratio determines the angular location of the complex-conjugate closed loop
poles, while the distance of the pole from the origin is determined by the undamped natural
frequency 𝜔𝑛 ,the desired locations of the closed-loop poles of this example problem are:
𝑠 = −2 ∓ 𝑗2√3
A general procedure for determining the lead compensator is as follows: First, find the sum of
the angles at the desired location of one of the dominant closed-loop poles with the open-loop
poles and zeros of the original system, and determine the necessary angle 𝜙 to be added so
that the total sum of the angles is equal to f 180°(2k + 1). The lead compensator must
contribute this angle 𝜙. (If the angle 𝜙 is quite large, then two or more lead networks may be
needed rather than a single one.)

Figure 3.7

If the original system has the open-loop transfer function G(s), then the compensated system
will have the open-loop transfer function:
1
𝑠+𝑇
𝐺𝑐 (𝑠)𝐺(𝑠) = (𝐾𝑐 ) 𝐺(𝑠)
1
𝑠 + 𝛼𝑇

The next step is to determine the locations of the zero and pole of the lead compensator. There
are many possibilities for the choice of such locations. In what follows, we shall introduce a
procedure to obtain the largest possible value for 𝛼. First, draw a horizontal line passing
through point P, the desired location for one of the dominant closed-loop poles. This is shown
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as line PA in Figure 3-8. Draw also a line connecting point P and the origin. Bisect the angle
between the lines PA and PO, as shown in Figure 3-8. Draw two lines PC and PD that make
angles ±𝜙/2 with the bisector PB. The intersections of PC and PD with the negative real axis
give the necessary locations for the pole and zero of the lead network. The compensator thus
designed will make point P a point on the root locus of the compensated system. The open-
loop gain is determined by use of the magnitude condition.
In the present system, the angle of 𝑮(𝒔) at the desired closed-loop pole is:
4
arg ( ) = −2100
s(s + 2) at s=−2+j2√3
Thus, if we need to force the root locus to go through the desired closed-loop pole, the lead
compensator must contribute 𝜙 = 300 at this point. By following the foregoing design
procedure, we determine the zero and pole of the lead compensator, as shown in Figure 3-9, to
be:

Figure (3.8)

𝑍𝑒𝑟𝑜 𝑎𝑡 𝑠 = −2.9, 𝑝𝑜𝑙𝑒 𝑎𝑡 𝑠 = −5.4

Or
1 1
𝑇= = 0.345, 𝛼𝑇 = = 0.185
2.9 5.4
Thus 𝛼 = 0.537. The open-loop transfer function of the compensated system becomes:
𝑠 + 2.9 4 𝐾(𝑠 + 2.9)
𝐺𝑠 (𝑠)𝐺(𝑠) = 𝐾𝑐 =
𝑠 + 5.4 (𝑠 + 2) 𝑠(𝑠 + 2)(𝑠 + 5.4)
Where 𝐾 = 4𝐾𝑐 . The root-locus plot for the compensated system is shown in Figure 3.9.The
gain K is evaluated from the magnitude condition as follows: Referring to the root-locus plot
for the compensated system shown in Figure 3.9, the gain 𝑲 is evaluated from the magnitude
condition as:

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 Figure 3.9 Root-locus plot of the compensated system.

𝐾(𝑠 + 2.9
| | =1
𝑠(𝑠 + 2)(𝑠 + 5.4) 𝑠=−2+𝑗2√3
Or:
K=18.7
It follows that:
18.7(𝑠 + 2.9
𝐺(𝑠) =
𝑠(𝑠 + 2)(𝑠 + 5.4)
The constant 𝐾𝑐 of the lead compensator is:
18.7
𝐾𝑐 = = 4.68
4
Hence 𝐾𝑐 𝛼 = 2.51. The lead compensator, therefore, has the transfer function:
0.345𝑠 + 𝑠 + 2.9
𝐺𝑠 (𝑠) = 2.51 = 4.68
0.185𝑠 + 1 𝑠 + 5.4
If the electronic circuit using operational amplifiers as shown in Figure 3.4 is used as the lead
compensator just designed, then the parameter values of the lead compensator are determined
from:
𝐸0 (𝑠) 𝑅2 𝑅4 𝑅1 𝐶1 𝑠 + 1 0.345𝑠 + 1
= = 2.51
𝐸𝑖 (𝑠) 𝑅1 𝑅3 𝑅2 𝐶2 𝑠 + 1 0.185𝑠 + 1
As shown in Figure 7.10, where we have arbitrarily chosen 𝐶1 = 𝐶2 = 10𝜇𝐹 𝑎𝑛𝑑 𝑅3 = 10𝑘Ω.

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 Figure 3.10 Lead compensator circuit

The static velocity error constant 𝐾𝑣 is obtained from the expression:

18.7(𝑠 + 2.9
𝐾𝑣 = lim 𝑠𝐺𝑐 (𝑠)𝐺(𝑠) = lim 𝑠 = 5.02𝑠𝑒𝑐 −1
𝑠−−0 𝑠−−0 𝑠(𝑠 + 2)(𝑠 + 5.4)
Note that the third closed-loop pole of the designed system is found by dividing the
characteristic equation by the known factors as follows:
𝑠(𝑠 + 2)(𝑠 + 5.4) + 18.7(𝑠 + 2.9) = (𝑠 + 2 + 𝑗2√3)(𝑠 + 2 − 𝑗2√3)(𝑠 + 3.4)
In what follows we shall examine the unit-step responses of the compensated and
uncompensated systems with MATLAB.
The closed-loop transfer function of the compensated system is:
𝐶(𝑠) 18.7(𝑠 + 2.9) 18.7(𝑠 + 2.9)
= = 3
𝑅(𝑠) 𝑠(𝑠 + 2)(𝑠 + 5.4) + 18.7(𝑠 + 2.9) 𝑠 + 7.4𝑠 2 + 29.5𝑠 + 54.23
Hence,
Numc=[0 0 18.7 54.23]
Denc=[1 7.4 29.5 54.23]
For the uncompensated system, the closed-loop transfer function is:
𝐶(𝑠) 4
= 2
𝑅(𝑠) 𝑠 + 2𝑠 + 4

Numc=[0 0 4]
Denc=[1 2 4]
MATLAB Program produces the unit-step response curves for the two systems. The resulting
plot is shown in Figure 3.11. Notice that the compensated system exhibits slightly larger
maximum overshoot. The settling time of the compensated system is one-half that of the
original system, as expected.

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% - - - - - - - - - - Unit-step response ---------
% ***** Unit-step responses of compensated and uncompensated
% systems *****
numc = [0 0 18.7 54.231];
denc = [1 7.4 29.5 54.231];
num = [0 0 41];
den = [ 1 2 41];
t = 0:0.05:5;
[cl ,X1 t]= step(numc,denc,t);
[c2,X2,t] = step(num,den,t);
plot(t,cl ,t,cl ,'o',t,c2,t,c2,'x')
grid
title('Unit-Step Responses of Compensated and Uncompensated Systems')
xlabel('t Sec')
ylabel('0utputs cl and c2')
text(0.7,1.32,'Compensated system')
text(1.3,0.68,'Uncompensated system')

Figure 3.11 Unit-Step Responses of Compensated and Uncompensated Systems

3.4 Lag Compensation

The configuration of the electronic lag compensator using operational amplifiers is the same
as that for the lead compensator shown in Figure 3.4. If we choose 𝑅2 𝐶2 > 𝑅1 𝐶1 in the circuit
shown in Figure 3.4, it becomes a lag compensator. Referring to Figure 3.4, the transfer
function of the lag compensator is given by:

1
𝐸𝑜 (𝑠) 𝑠+𝑇
=𝐾̂𝑐
𝐸𝑖 (𝑠) 1
𝑠+
𝛽𝑇
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