(1) The document discusses the characteristics and effects of feedback control systems. It describes how feedback can reduce sensitivity to parameter variations, improve stability, and reduce the impact of disturbances. (2) Feedback works by sampling the output signal and comparing it to the desired output to generate an error signal. This error signal is used to adjust the system via negative feedback. (3) While feedback provides advantages like improved robustness, it also introduces complexity and reduces the overall system gain. There is therefore a cost associated with incorporating feedback into a control system.

1. Department of Electrical & Computer Engineering
Introduction to Control Systems
Addis Ababa Science &
Technology University
College of Electrical & Mechanical
Engineering
Biruk T. 1

2. Introduction
• A closed-loop / feedback system uses a measurement of the output
signal and a comparison with the desired output to generate an
error signal that is applied to the actuator.
• A feedback system is one in which the output signal is
sampled and then fed back to the input to form an error
signal that drives the system.
Feedback System Block Diagram Model
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Chapter Three
Characteristics Feedback Control System

3. Chapter Three
Characteristics Feedback Control System
Effect of Feedback(FB)
A feedback is provided to bring about improvement in the performance of a
control system. The advantages of FB in a control system are:
1. Reduces the sensitivity of the system to its parameter variations (i.e. Enhance
robustness), Parameters may vary due to ageing, environmental changes, etc.
2. Improves the sensitivity of a control system but there would be reduction in
system gain
3. Improves the stability if properly designed
4. Negative feedback reduces the overall gain of the system.
5. System response to disturbance signal can be reduced with feedback.
6. Improve dynamic performance or adjust the transient response (such as
reduce time constant)
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4. Sensitivity of system to parameter variations
• System are time-varying in its nature because of inevitable
uncertainties such as changing environment , aging , and other
factors that affect a control process.
• All these uncertainties in open-loop system will result in
inaccurate output or low performance. However, a closed-loop
system can overcome this disadvantage.
• A primary advantage of a closed-loop feedback control
system is its ability to reduce the system’s sensitivity
to parameter variation.
Sensitivity analysis Robust control
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• System sensitivity is the ratio of the change in the system
transfer function to the change of a process transfer function
(or parameter) for a small incremental change.

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It is concluded that due to feedback the variation in o/p caused by
the change in the forward path transfer function is reduced by a
factor of 1+ G(s)H(s) in a closed loop.
𝐸 𝑠 =
R(s)
1 + H s 𝐺 s
• Negative feedback reduces the error between the reference
input, R(s) and system output.
𝐸 𝑠 = 𝑅 𝑠 − 𝐵 𝑠 = 𝑅 𝑠 − 𝑌 𝑠 𝐻 𝑠
𝑌 𝑠 = 𝐸 𝑠 𝐺 𝑠 = R s − Y s H s G(s)
Steady-state error is the error after the transient response has decayed, leaving
only the continuous response.

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Effect of parameter variations on FB control system

7. Effect of Feedback on Overall gain
• The transfer function of negative feedback system is
• T 𝑠 =
Y(s)
R s
=
G(s)
1+H s 𝐺 s
, for non feedback system T(s)= G(s)
• we can say that the overall gain of negative feedback closed loop
control system is the ratio of 'G' and (1+GH). So, the overall gain
may increase or decrease depending on the value of (1+GH).
Effect of Feedback on Sensitivity
• Sensitivity of the overall gain of negative feedback closed loop
control system (T) to the variation in open loop gain (G) is defined
as
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𝑆𝐺
𝑇
=
𝜕𝑇(𝑠)/𝑇(𝑠)
𝜕𝐺(𝑠)/𝐺(𝑠)
Do partial differentiation with respect to G on both sides

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So, the sensitivity of the overall gain of closed loop control system as the
reciprocal of (1+GH). So, Sensitivity may increase or decrease depending on
the value of (1+GH).
• In general, 'G' and 'H' are functions of frequency. So, feedback
will increase the sensitivity of the system gain in one frequency
range and decrease in the other frequency range.
• Therefore, we have to choose the values of 'GH' in such a way that
the system is insensitive or less sensitive to parameter variations.
𝐾 = 𝑝𝑎𝑟𝑎𝑚𝑒ter variation of element Such as Gain or Feedback
A = Variable in Control System Which Changes Its Value O/P
𝑆𝐸𝑁𝑆𝐼𝑇𝐼𝑉𝐼𝑇𝑌 =
%𝐶𝐻𝐴𝑁𝐺𝐸 𝐼𝑁 𝐴
%𝐶𝐻𝐴𝑁𝐺𝐸 𝐼𝑁 𝐾
𝐒𝐾
𝐴
=
𝜕𝐴/𝐴
𝜕𝑘/𝑘
Quiz : Obtained the Sensitivity T(s) with Respect to H(s) (𝐒𝐻
𝑇
)

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Effect of Feedback on Stability
•A system is said to be stable, if its output is under control. Otherwise, it is
said to be unstable.
•In T(s), if the denominator value is zero (i.e., GH = -1), then the output of
the control system will be infinite. So, the control system becomes unstable.
Therefore, we have to properly choose the feedback in order to make the
control system stable.

10. • Effect of Feedback on Noise
• To know the effect of feedback on noise, compare the transfer function
relations with and without feedback due to noise signal alone.
• Consider a closed loop control system with noise signal
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Closed loop transfer function due to noise signal alone
Open loop transfer function due to noise signal alone
In the closed loop control system, the gain due to noise signal is decreased
by a factor of (1+GaGbH) provided that the term (1+GaGbH) is greater
than one.

11. Control over system dynamics by using feedback
• Let us consider the simple feedback system
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The open-loop transfer function is, ( )
K
G s
s 
=
+
1
/
( ) ( ) ( )
t
t
c t Ke u t Ke u t

 −
−
= =
( )
K
T s
s K

=
+ +
The impulse response for the non-feedback system would be,
The closed-loop transfer function of the above system is,
The impulse response of the closed-loop system is,
൯
𝑐(𝑡) = 𝐾𝑒−(𝜇+𝐾)𝑡𝑢(𝑡) = 𝐾𝑒− Τ
𝑡 𝜏2𝑢(𝑡
The location of the pole and the dynamic response of the non-feedback and
feedback system are

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1 1/
 
=
2 1/( )
K
 
= +
It is seen that the time-constant of open-loop system is
and that of closed-loop system is
As the time-constant of closed-loop system is less, its dynamic response is
faster than the same of the open-loop system.

13. Control of the effect of disturbance signal by use of feedback
• Disturbance in the forward path Disturbance in the feedback
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𝐶𝑑(𝑠)
𝑇𝑑(𝑠)
=
−𝐺2(𝑠)
1+𝐺1(𝑠)𝐺2(𝑠)𝐻(𝑠)
≅
−1
𝐺1(𝑠)𝐻(𝑠)
;
𝐶𝑑(𝑠) =
)
−𝑇𝑑(𝑠
)
𝐺1(𝑠)𝐻(𝑠
)
𝐶𝑛(𝑠
)
𝑁(𝑠
=
)
−𝐺1(𝑠)𝐺2(𝑠)𝐻2(𝑠
)
1 + 𝐺1(𝑠)𝐺2(𝑠)𝐻1(𝑠)𝐻2(𝑠
≅
−1
)
𝐻1(𝑠
If 𝐺1(𝑠) is made very large, the
effect of disturbance on the
output will be very small
Therefore, the effect of noise on output is,
𝐶𝑛(𝑠) ≅
−1
𝐻1(𝑠)
⋅ 𝑁(𝑠).
For the optimum performance of the system, the measurement sensor
should be designed such that is maximum. This is equivalent to maximizing
the SNR of the sensor.

14. • Example
• A position control system is shown below. Assume, K=10, 𝛼 = 2, 𝛽 = 1.
Evaluate: 𝑆𝐾
𝑇
, 𝑆𝛼
𝑇
, 𝑆𝛽
𝑇
. For 𝑟(𝑡) = 2cos(0.5𝑡) and a 5% change in 𝐾,
evaluate the steady-state response and the change in steady-state
response.
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𝑆𝐾
𝐺
=
𝐾
𝐺
⋅
𝑑𝐺
𝑑𝐾
= 𝑠(𝑠 + 𝛼) ⋅
1
𝑠(𝑠+𝛼)
= 1;
Here, 𝐺(𝑠) =
𝐾
𝑠(𝑠+𝛼)
, and 𝐻(𝑠) = 𝛽
𝑆𝛽
𝐻
=
𝛽
𝐻
⋅
𝑑𝐻
𝑑𝛽
= 1
𝑆𝛼
𝐺 =
𝛼
𝐺
⋅
𝑑𝐺
𝑑𝛼
=
−𝛼
𝑠+𝛼
=
−2
𝑠+2
;

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16. • The use of feedback has several advantages as outlined in the previous
sections. These advantages have an attended cost due to an increased
number of components and complexity in the system.
• In an open-loop system the transfer function is G(s) and is reduced to
G(s)/[1 + G(s) H(s)] in a feedback (closed-loop) system.
• So, the loss of gain by the same factor of 1/[1 + G(s) H(s)] that reduces
the sensitivity of the system to parameter variations is again an added cost
of using feedback.
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THE COST OF FEEDBACK

17. Thank you!
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