School of Electrical Engineering Design via Root Locus Problems Homework solve 3 questions linear controls CHAPTER 9 (Nise, 6thEdition) attached Example

School of Electrical Engineering Design via Root Locus Problems Homework solve 3 questions linear controls

CHAPTER 9 (Nise, 6thEdition) attached

Example 9.5

Required Compensated New Peak Time =(Uncompensated Peak Time @ New % OS) * (factor in column below)

factor = 0.5.

New % Overshoot = 15%

use value provided for factor and over shoot and recalculate the ex 9.5

after that go to example 9.9 since its continue for 9.5 calculations you got

(Using New Designed PID Values) from 9.5

Skil lAssessment Exercise # 9.3 solve the original

I will provide of sample to help you .

handwritten must be clean and clear.

check grading rubric .

please no plagiarism !!! 0% similarity

please provide me the % of similarity in trunit in . Design via Root Locus
9
Chapter Learning Outcomes
After completing this chapter the student will be able to:

Use the root locus to design cascade compensators to improve the steady-state error
(Sections 9.1–9.2)
Use the root locus to design cascade compensators to improve the transient
response (Section 9.3)
Use the root locus to design cascade compensators to improve both the steady-state
error and the transient response (Section 9.4)
Use the root locus to design feedback compensators to improve the transient
response (Section 9.5)
Realize the designed compensators physically (Section 9.6)
Case Study Learning Outcomes
You will be able to demonstrate your knowledge of the chapter objectives with case
studies as follows:

Given the antenna azimuth position control system shown on the front endpapers,
you will be able to design a cascade compensator to meet transient response and
steady-state error specifications.

Given the pitch or heading control system for the UFSS vehicle shown on the back
endpapers, you will be able to design a cascade or feedback compensator to meet
transient response specifications.
455
456
Chapter 9
Design via Root Locus
9.1 Introduction
In Chapter 8, we saw that the root locus graphically displayed both transient response
and stability information. The locus can be sketched quickly to get a general idea of the
changes in transient response generated by changes in gain. Specific points on the
locus also can be found accurately to give quantitative design information.
The root locus typically allows us to choose the proper loop gain to meet a
transient response specification. As the gain is varied, we move through different
regions of response. Setting the gain at a particular value yields the transient
response dictated by the poles at that point on the root locus. Thus, we are limited
to those responses that exist along the root locus.
Improving Transient Response
Flexibility in the design of a desired transient response can be increased if we can
design for transient responses that are not on the root locus. Figure 9.1(a) illustrates
the concept. Assume that the desired transient response, defined by percent overshoot and settling time, is represented by point B. Unfortunately, on the current root
locus at the specified percent overshoot, we only can obtain the settling time
represented by point A after a simple gain adjustment. Thus, our goal is to speed
up the response at A to that of B, without affecting the percent overshoot. This
increase in speed cannot be accomplished by a simple gain adjustment, since point B
does not lie on the root locus. Figure 9.1(b) illustrates the improvement in the
transient response we seek: The faster response has the same percent overshoot as
the slower response.
j?
B
A
s-plane
?
= Closed-loop pole
= Open-loop pole
(a)
c(t)
FIGURE 9.1 a. Sample root
locus, showing possible design
point via gain adjustment (A)
and desired design point that
cannot be met via simple gain
adjustment (B); b. responses
from poles at A and B
Poles at B
Poles at A
t
(b)
9.1 Introduction
One way to solve our problem is to replace the existing system with a system
whose root locus intersects the desired design point, B. Unfortunately, this replacement is expensive and counterproductive. Most systems are chosen for characteristics other than transient response. For example, an elevator cage and motor are
chosen for speed and power. Components chosen for their transient response may
not necessarily meet, for example, power requirements.
Rather than change the existing system, we augment, or compensate, the
system with additional poles and zeros, so that the compensated system has a root
locus that goes through the desired pole location for some value of gain. One of the
advantages of compensating a system in this way is that additional poles and zeros
can be added at the low-power end of the system before the plant. Addition of
compensating poles and zeros need not interfere with the power output requirements of the system or present additional load or design problems. The compensating poles and zeros can be generated with a passive or an active network.
A possible disadvantage of compensating a system with additional open-loop
poles and zeros is that the system order can increase, with a subsequent effect on the
desired response. In Chapters 4 and 8, we discussed the effect of additional closedloop poles and zeros on the transient response. At the beginning of the design
process discussed in this chapter, we determine the proper location of additional
open-loop poles and zeros to yield the desired second-order closed-loop poles.
However, we do not know the location of the higher-order closed-loop poles until the
end of the design. Thus, we should evaluate the transient response through simulation after the design is complete to be sure the requirements have been met.
In Chapter 12, when we discuss state-space design, the disadvantage of finding
the location of higher-order closed-loop poles after the design will be eliminated by
techniques that allow the designer to specify and design the location of all the closedloop poles at the beginning of the design process.
One method of compensating for transient response that will be discussed later
is to insert a differentiator in the forward path in parallel with the gain. We can
visualize the operation of the differentiator with the following example. Assuming a
position control with a step input, we note that the error undergoes an initial large
change. Differentiating this rapid change yields a large signal that drives the plant.
The output from the differentiator is much larger than the output from the pure gain.
This large, initial input to the plant produces a faster response. As the error
approaches its final value, its derivative approaches zero, and the output from
the differentiator becomes negligible compared to the output from the gain.
Improving Steady-State Error
Compensators are not only used to improve the transient response of a system; they
are also used independently to improve the steady-state error characteristics.
Previously, when the system gain was adjusted to meet the transient response
specification, steady-state error performance deteriorated, since both the transient
response and the static error constant were related to the gain. The higher the gain,
the smaller the steady-state error, but the larger the percent overshoot. On the other
hand, reducing gain to reduce overshoot increased the steady-state error. If we use
dynamic compensators, compensating networks can be designed that will allow us to
meet transient and steady-state error specifications simultaneously.1 We no longer
1
The word dynamic describes compensators with noninstantaneous transient response. The transfer
functions of such compensators are functions of the Laplace variable, s, rather than pure gain.
457
458
Chapter 9
Design via Root Locus
Cascade
compensator
Original
controller
Plant
G1(s)
G2(s)
G3(s)
R(s) +
C(s)
–
(a)
R(s) +
Original
controller
Plant
G1(s)
G2(s)
C(s)
–
Feedback
compensator
FIGURE 9.2 Compensation
techniques: a. cascade;
b. feedback
H1(s)
(b)
need to compromise between transient response and steady-state error, as long as
the system operates in its linear range.
In Chapter 7, we learned that steady-state error can be improved by adding an
open-loop pole at the origin in the forward path, thus increasing the system type and
driving the associated steady-state error to zero. This additional pole at the origin
requires an integrator for its realization.
In summary, then, transient response is improved with the addition of differentiation, and steady-state error is improved with the addition of integration in the
forward path.
Configurations
Two configurations of compensation are covered in this chapter: cascade compensation and feedback compensation. These methods are modeled in Figure 9.2. With
cascade compensation, the compensating network, G1(s), is placed at the low-power
end of the forward path in cascade with the plant. If feedback compensation is used,
the compensator, H1(s), is placed in the feedback path. Both methods change the
open-loop poles and zeros, thereby creating a new root locus that goes through the
desired closed-loop pole location.
Compensators
Compensators that use pure integration for improving steady-state error or pure
differentiation for improving transient response are defined as ideal compensators.
Ideal compensators must be implemented with active networks, which, in the case of
electric networks, require the use of active amplifiers and possible additional power
sources. An advantage of ideal integral compensators is that steady-state error is
reduced to zero. Electromechanical ideal compensators, such as tachometers, are
often used to improve transient response, since they can be conveniently interfaced
with the plant.
Other design techniques that preclude the use of active devices for compensation can be adopted. These compensators, which can be implemented with passive
elements such as resistors and capacitors, do not use pure integration and differentiation and are not ideal compensators. Advantages of passive networks are that they
9.2 Improving Steady-State Error via Cascade Compensation
are less expensive and do not require additional power sources for their operation.
Their disadvantage is that the steady-state error is not driven to zero in cases where
ideal compensators yield zero error.
Thus, the choice between an active or a passive compensator revolves around
cost, weight, desired performance, transfer function, and the interface between the
compensator and other hardware. In Sections 9.2, 9.3, and 9.4, we first discuss
cascade compensator design using ideal compensation and follow with cascade
compensation using compensators that are not implemented with pure integration
and differentiation.
9.2 Improving Steady-State Error
via Cascade Compensation
In this section, we discuss two ways to improve the steady-state error of a feedback
control system using cascade compensation. One objective of this design is to
improve the steady-state error without appreciably affecting the transient response.
The first technique is ideal integral compensation, which uses a pure integrator
to place an open-loop, forward-path pole at the origin, thus increasing the system
type and reducing the error to zero. The second technique does not use pure
integration. This compensation technique places the pole near the origin, and
although it does not drive the steady-state error to zero, it does yield a measurable
reduction in steady-state error.
While the first technique reduces the steady-state error to zero, the compensator must be implemented with active networks, such as amplifiers. The second
technique, although it does not reduce the error to zero, does have the advantage
that it can be implemented with a less expensive passive network that does not
require additional power sources.
The names associated with the compensators come either from the method of
implementing the compensator or from the compensator’s characteristics. Systems
that feed the error forward to the plant are called proportional control systems.
Systems that feed the integral of the error to the plant are called integral control
systems. Finally, systems that feed the derivative of the error to the plant are called
derivative control systems. Thus, in this section we call the ideal integral compensator
a proportional-plus-integral (PI) controller, since the implementation, as we will see,
consists of feeding the error (proportional) plus the integral of the error forward to
the plant. The second technique uses what we call a lag compensator. The name of
this compensator comes from its frequency response characteristics, which will be
discussed in Chapter 11. Thus, we use the name PI controller interchangeably with
ideal integral compensator, and we use the name lag compensator when the cascade
compensator does not employ pure integration.
Ideal Integral Compensation (PI)
Steady-state error can be improved by placing an open-loop pole at the origin,
because this increases the system type by one. For example, a Type 0 system
responding to a step input with a finite error responds with zero error if the system
type is increased by one. Active circuits can be used to place poles at the origin. Later
in this chapter, we show how to build an integrator with active electronic circuits.
To see how to improve the steady-state error without affecting the transient
response, look at Figure 9.3(a). Here we have a system operating with a desirable
459
460
Chapter 9
Design via Root Locus
Gain
Plant
K
G(s)
R(s) +
C(s)
Compensator
Plant
K
s
G(s)
R(s) +
–
–
j?
A
C(s)
j?
s-plane
s-plane
A
?3
?1
?2
?
?3
– ? 1 – ? 2 – ? 3 – ?c
– ?1 – ? 2 – ? 3 = (2k + 1)180°
(a)
?2
?1
?c
?
(2k + 1)180°
(b)
R(s) +
–
Compensator
Plant
K(s + a)
s
G(s)
C(s)
j?
A
s-plane
? zc
?3
?2
?1
– ? 1 – ? 2 – ? 3 – ? pc + ? zc
? pc
?
(2k + 1)180°
(c)
FIGURE 9.3 Pole at A is a. on the root locus without compensator; b. not on the root locus with
compensator pole added; c. approximately on the root locus with compensator pole and zero
added
transient response generated by the closed-loop poles at A. If we add a pole at the
origin to increase the system type, the angular contribution of the open-loop poles at
point A is no longer 180, and the root locus no longer goes through point A, as
shown in Figure 9.3(b).
To solve the problem, we also add a zero close to the pole at the origin, as shown
in Figure 9.3(c). Now the angular contribution of the compensator zero and compensator pole cancel out, point A is still on the root locus, and the system type has been
increased. Furthermore, the required gain at the dominant pole is about the same as
461
9.2 Improving Steady-State Error via Cascade Compensation
before compensation, since the ratio of lengths from the compensator pole and the
compensator zero is approximately unity. Thus, we have improved the steady-state
error without appreciably affecting the transient response. A compensator with a pole
at the origin and a zero close to the pole is called an ideal integral compensator.
In the example that follows, we demonstrate the effect of ideal integral compensation. An open-loop pole will be placed at the origin to increase the system type and
drive the steady-state error to zero. An open-loop zero will be placed very close to the
open-loop pole at the origin so that the original closed-loop poles on the original root
locus still remain at approximately the same points on the compensated root locus.
Example 9.1
Effect of an Ideal Integral Compensator
PROBLEM: Given the system of Figure 9.4(a), operating with a damping ratio of
0.174, show that the addition of the ideal integral compensator shown in Figure 9.4(b)
reduces the steady-state error to zero for a step input without appreciably affecting
transient response. The compensating network is chosen with a pole at the origin
to increase the system type and a zero at 0:1, close to the compensator pole, so that
the angular contribution of the compensator evaluated at the original, dominant,
second-order poles is approximately zero. Thus, the original, dominant, second-order
closed-loop poles are still approximately on the new root locus.
SOLUTION: We first analyze the uncompensated system and determine the location of the dominant, second-order poles. Next we evaluate the uncompensated
steady-state error for a unit step input. The root locus for the uncompensated
system is shown in Figure 9.5.
A damping ratio of 0.174 is represented by a radial line drawn on the s-plane at
100:02 . Searching along this line with the root locus program discussed in Appendix
H at www.wiley.com/college/nise, we find that the dominant poles are 0:694 j3:926
for a gain, K, of 164.6. Now look for the third pole on the root locus beyond 10 on
the real axis. Using the root locus program and searching for the same gain as that of
the dominant pair, K ¼ 164:6, we find that the third pole is approximately at 11:61.
This gain yields K p ¼ 8:23. Hence, the steady-state error is
eð1Þ ¼
R(s) +
1
1
¼
¼ 0:108
1 þ K p 1 þ 8:23
Gain
Plant
K
1
(s+1)(s+2)(s+10)
E(s)
–
ð9:1Þ
C(s)
(a)
R(s) +
E(s)
–
Compensator
Plant
K(s+0.1)
s
1
(s+1)(s+2)(s+10)
(b)
C(s)
FIGURE 9.4 Closed-loop
system for Example 9.1:
a. before compensation;
b. after ideal integral
compensation
462
Chapter 9
Design via Root Locus
j?
? = 0.174
j4
–0.694 + j3.926
K = 164.6
s-plane
j2
100.02°
?
–11.61
–10
–6
–8
–4
–2
0
–1
–j2
= Closed-loop pole
= Open-loop pole
–j4
FIGURE 9.5 Root locus for uncompensated system of Figure 9.4(a)
Adding an ideal integral compensator with a zero at 0:1, as shown in Figure
9.4(b), we obtain the root locus shown in Figure 9.6. The dominant second-order
poles, the third pole beyond 10, and the gain are approximately the same as for
the uncompensated system. Another section of the compensated root locus is
between the origin and 0:1. Searching this region for the same gain at the
dominant pair, K ¼ 158:2, the fourth closed-loop pole is found at 0:0902, close
j?
? = 0.174
–0.678 + j3.837
K = 158.2
j4
s-plane
j2
100.02°
?
–11.55
–10
–8
–6
–4
–2
–1
0
–0.1
–j2
= Closed-loop pole
= Open-loop pole
–j4
FIGURE 9.6 Root locus for compensated system of Figure 9.4(b)
Fourth closed-loop
pole at – 0.0902
c(t)
9.2 Improving Steady-State Error via Cascade Compensation
2.0
1.8
1.6
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0
Ideal integral
compensated
Uncompensated
0
5
10
Time (seconds)
15
20
FIGURE 9.7 Ideal integral
compensated system response
and the uncompensated system
response of Example 9.1
enough to the zero to cause pole-zero cancellation. Thus, the compensated
system’s closed-loop poles and gain are approximately the same as the uncompensated system’s closed-loop poles and gain, which indicates that the
transient response of the compensated system is about the same as the uncompensated system. However, the compensated system, with its pole at the
origin, is a Type 1 system; unlike the uncompensated system, it will respond to a
step input with zero error.
Figure 9.7 compares the uncompensated response with the ideal integral
compensated response. The step response of the ideal integral compensated system
approaches unity in the steady state, while the uncompensated system approaches
0.892. Thus, the ideal integral compensated system responds with zero steady-state
error. The transient response of both the uncompensated and the ideal integral
compensated systems is the same up to approximately 3 seconds. After that time the
integrator in the compensator, shown in Figure 9.4(b), slowly compensates for the
error until zero error is finally reached. The simulation shows that it takes 18 seconds
for the compensated system to reach to within 2% of the final value of unity, while
the uncompensated system takes about 6 seconds to settle to within 2% of its final
value of 0.892. The compensation at first may appear to yield deterioration in the
settling time. However, notice that the compensated system reaches the uncompensated system’s final value in about the same time. The remaining time is
used to improve the steady-state error over that of the uncompensated system.
A method of implementing an ideal integral compensator is shown in Figure 9.8.
The compensating network precedes G(s) and is an ideal integral compensator since

K2
K 2 K 1 s þ K1
ð9:2Þ
¼
Gc ðsÞ ¼ K 1 þ
s
s
Integral (I)
K2
s
Proportional (P)
R(s) +
K1
Plant
+
+
G(s)
C(s)
–
FIGURE 9.8 PI controller
463
464
Chapter 9
Design via Root Locus
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