Module-2:Characteristics of Feedback Systems
Akshya Swain
Department of Electrical, Computer & Software Engineering, The University of Auckland,
Auckland, New Zealand.
Akshya Swain
Module-2:Characteristics of Feedback Systems 1 / 37
Learning Outcome of this Module
After completion of this module the students must have learned the following:
1. Basic advantages of closed-loop control system compared to open loop control system.
2. Sensitivity reduction due to parameter variations.
3. Reduction of effects of noise due to feedback.
4. Control of transient response ( speed) by feedback.
Akshya Swain
Module-2:Characteristics of Feedback Systems 2 / 37
What is a Control System?
Definition-1
A control system is defined as an interconnection of components forming a system that will provide a desired system response.
Definition-2
A control system consists of a set of devices ( subsystems, processes or plants) which are combined (assembled) to obtain a desired output with desired performance, for a specified input.
Definition-3
A control system is a system of devices or set of devices, that manages, commands, directs or regulates the behavior of other devices or systems to achieve desired results.
Akshya Swain
Module-2:Characteristics of Feedback Systems 3 / 37
Open-Loop and Closed-Loop Control Systems
Depending on configuration, control systems can be categorized into mainly two classes:
1. Open-loop Control systems
2. Closed-loop (or feedback) Control systems
The block diagram of typical open-loop control systems is shown in the figure.
A control system, in which the control action is totally independent of the output of the system,is called open loop control system.
Akshya Swain
Module-2:Characteristics of Feedback Systems 4 / 37
Some Examples of Open Loop System
1. Electric Hand Drier – Hot air (output) comes out as long as you keep your hand under the machine, irrespective of how much your hand is dried.
2. Automatic Washing Machine – This machine runs according to the pre-set time, irrespective of washing is completed or not.
3. Bread Toaster – This machine runs as per adjusted time irrespective of toasting is completed or not.
4. Automatic Tea/Coffee Maker – These machines also function for pre adjusted time only.
5. Timer Based Clothes Drier – This machine dries wet clothes for pre-adjusted time, it does not matter how much the clothes are dried.
Akshya Swain
Module-2:Characteristics of Feedback Systems 5 / 37
Some Examples of Open Loop System
1. Electric Dryer: This machine dries wet clothes for pre-adjusted time, it does not matter how much the clothes are dried..
2. Control of Rotating Disc:Uses a battery source to provide a voltage that is proportional to the desired speed.
Akshya Swain
Module-2:Characteristics of Feedback Systems 6 / 37
Advantages and Disadvantages of Open Loop Control
Advantages of Open Loop Disadvantages of Open Loop Control System: Control System.
1. Simple in construction and design.
2. Economical.
3. Easy to maintain.
4. Generally stable.
5 Convenient to use as output is difficult to measure.
1. It can not compensate for any disturbances (which include both input and output disturbances.)
2. They are often unreliable.
3. Any change in output cannot be corrected automatically.
Akshya Swain
Module-2:Characteristics of Feedback Systems 7 / 37
Closed Loop Control System
The schematic of typical closed-loop control systems is shown in the figure
A control system, in which the control action is dependent on the output of the system,is called closed loop control system.
Akshya Swain
Module-2:Characteristics of Feedback Systems 8 / 37
Some Examples of Closed Loop Control Systems
1. Student-Teacher Learning System: The student-teacher learning process is inherently a feedback process intended to reduce the system error to minimum.The feedback model of the learning process is shown in the schematic below.
2. A Driver Controlled Cruise Control System :
Akshya Swain
Module-2:Characteristics of Feedback Systems 9 / 37
Some Examples of Closed Loop Control Systems
3. Tacking a sailboat as the wind shifts:: A sailboat can’t sail directly into the wind and traveling straight downwind is usually slow, the shortest sailing distance is rarely a straight line. Thus sailboats tack upwind—the familiar zigzag course—and jibe downwind. A tactician’s decision of when to tack and where to go can determine the outcome of a race.
4. An automated highway control system merging two lanes of traffic :
Akshya Swain
Module-2:Characteristics of Feedback Systems 10 / 37
Some Examples of Closed Loop Control Systems
5. Unmanned aerial vehicle used for crop monitoring in an autonomous mode: Unmanned aerial vehicles (UAVs) are being developed to operate in the air autonomously for long periods of time (see Section 1.3).The UAV must photograph and transmit the entire land area by flying a pre-specified trajectory as accurately as possible.
6. An automobile interior cabin temperature control system block diagram:
Akshya Swain
Module-2:Characteristics of Feedback Systems 11 / 37
Some Examples of Closed Loop System
7. Automatic Electric Iron-Heating elements are controlled by output temperature of the iron.
8. Automatic Electric Dryer-Heating elements are controlled by monitoring the dryness of the clothes.
Akshya Swain
Module-2:Characteristics of Feedback Systems 12 / 37
Some Examples of Closed Loop Control Systems
9. Nuclear Reactor Control
– The accurate control of a nuclear reactor is important for power system
generators.
– Assuming the number of neutrons present is proportional to the power level, an ionization chamber is used to measure the power level. The current i0 is proportional to the power level.
– The position of the graphite control rods moderates the power level.
Akshya Swain
Module-2:Characteristics of Feedback Systems 13 / 37
Some Examples of Closed Loop Control Systems
10. Control of Home Shower
I This is an example of a two-input control system of a home shower with separate valves for hot and cold water.
I The objective is to obtain
i. a desired temperature of the shower water and
ii. a desired flow of water.
I The block diagram of the closed-loop control system is shown below.
Akshya Swain
Module-2:Characteristics of Feedback Systems 14 / 37
Some Examples of Closed Loop Control Systems
11. Automobile Steering Control System
I The block diagram of an automobile steering control system is shown above. Operating Principle: The desired course is compared with a measurement of the actual course in order to generate a measure of the error, as shown in Figure.
I This measurement is obtained by visual and tactile (body movement) feedback, as provided by the feel of the steering wheel by the hand (sensor). The driver ( controller) rotates the steering which brings the vehicle to the desired course.
Akshya Swain
Module-2:Characteristics of Feedback Systems 15 / 37
Speed Control of Rotating Disk
I Many modern devices employ a rotating disk held at a constant speed. For example, a CD player requires a constant speed of rotation in spite of motor wear and variation and other component changes.
I Our goal is to design a system for rotating disk speed control that will ensure that the actual speed of rotation is within a specified percentage of the desired speed.
I We will consider a system without feedback and a system with feedback.
Figure: Open Loop Control of Rotating
Disk: Uses a battery source to provide a
voltage that is proportional to the desired Figure: Closed Loop Control of Rotating speed Disk
Akshya Swain
Module-2:Characteristics of Feedback Systems 16 / 37
Control of Electric Dryer
I We will consider controlling this system both without feedback and with feedback.
Figure: Open Loop Control of Electric Dryer
Figure: Closed Loop Control of Electric Dryer: Heating Elements are Controlled by Monitoring the Dryness Condition
Akshya Swain
Module-2:Characteristics of Feedback Systems 17 / 37
What is Feedback and What are its Effects?
I The motivation for using feedback, has been illustrated by several examples.
I Essentially, in all these examples, feedback is used to reduce the error between
the reference input and the system output.
I However, the significance of the effects of feedback in control systems is more complex than is demonstrated by these simple examples. The reduction of system error is merely one of the many important effects that feedback may have upon a system.
I We will show that feedback has effects on such system performance characteristics as:
1. Overall Gain
2. Stability
3. Noise Reduction
4. Reducing Effects of Parameter Variations. 5. Sensitivity Reduction
6. Speed of Response
7. Bandwidth
I Before we discuss about the characteristics and benefits of feedback, let us first discuss what are the desired properties of a control system.
Akshya Swain
Module-2:Characteristics of Feedback Systems 18 / 37
Desired Properties of a Control System
1. Accuracy:
2. Sensitivity: Any control system should be insensitive to internal
disturbance but sensitive to input signals only.
3. Noise: An undesired input signal is known as noise. A good control system
should be able to reduce the effects of noise for better performance.
4. Stability: It is an important characteristic of the control system. For the bounded input signal, the output must be bounded(BIBO Stability) and if the input is zero then output must be zero then such a control system is said to be a stable system.
5. Bandwidth: An operating frequency range decides the bandwidth of the control system. Bandwidth should be as large as possible for the frequency response of good control system.
6. Speed: It is the time taken by the control system to achieve its stable output. A good control system possesses high speed. The transient period for such system is very small.
7. Oscillation: A small numbers of oscillation or constant oscillation of output tend to indicate the system to be stable.
Akshya Swain
Module-2:Characteristics of Feedback Systems 19 / 37
Characteristics of Feedback Systems
I Some of the beneficial effects of feedback with high loop gain are:
1. The controlled variable accurately follows the desired value.
2. The effects of external disturbance on the controlled variable are significantly reduced.
4. The I The
speed of response can be improved
cost of achieving these improvements include
effects of variations of process and controller parameters is reduced.
3. The
i. These variations occur due to wear, aging, environmental changes etc.
a. Greater system complexity
b. Need for much larger forward path gain and
c. Possibility of instability. This may mean undesired/persistent oscillations of output variable.
Akshya Swain
Module-2:Characteristics of Feedback Systems 20 / 37
What is Feedback and What are its Effects?
I Before we begin, let us first find the input-output relation in a feedback system. I The block diagram of a feedback system is shown
I From the schematic, the relation between the output and input is computed as: C(s) = G(s)E(s), where E(s) = R(s) − B(s) = R(s) − C(s)H(s)
Simple algebraic manipulation gives
C(s) = G(s)E(s) = G(s) [R(s) − C(s)H(s)] = G(s)R(s) − G(s)C(s)H(s) or,C(s) [1 + G(s)H(s)] = G(s)R(s)
C(s) = G(s) R(s) 1 + G(s)H(s)
Using this basic relationship of the feedback system structure, we can uncover some of the significant effects of feedback.
Akshya Swain
Module-2:Characteristics of Feedback Systems 21 / 37
Effects of Feedback on System Gain
I Consider the block diagram of an open loop system with feed-forward path gain G(s).
The output is related to the input by the following relation:
C(s) = G(s)R(s),
or, C(s) = G(s) R(s)
I The block diagram of a feedback system is shown
I From the schematic, the relation beyween the output and input is computed as:
E(s) = R(s) − B(s) = R(s) − C(s)H(s)
C(s) = G(s)E(s) = G(s) [R(s) − C(s)H(s)] or,C(s) = G(s)R(s) − G(s)C(s)H(s)
or,C(s) [1 + G(s)H(s)] = G(s)R(s)
C(s) = G(s) R(s) 1 + G(s)H(s)
Thus the feedback affects the gain of the open-loop system by a factor 1 + G(s)H(s)
Akshya Swain
Module-2:Characteristics of Feedback Systems 22 / 37
Effect of Feedback on Noise
I To know the effect of feedback on noise, let us compare the transfer function relations with and without feedback due to noise signal alone.
I Consider an open loop control system with noise signal as shown below.
I The output is expressed as:
C(s) = [GaR(s) + N(s)] Gb
= GaGbR(s) + GbN(s)
I The open loop transfer function due to noise alone can be obtained by making R(s) = 0 which gives
C(s) = Gb N (s)
Akshya Swain
Module-2:Characteristics of Feedback Systems 23 / 37
Effects of Feedback on Noise
Consider the closed loop control system with noise signal as shown below.
I The output is expressed as:
C(s) = [(R(s) − HC(s))Ga + N] Gb = [GaR(s) − GaHC(s) + N(s)] Gb
I Further simplification gives
C(s) + GaGbHC(s) = GaGbR(s) + GbN(s)
or, C(s)[1+GaGbH]=GaGbR(S)+GbN(S)
C(s) = GaGb R(s) + Gb N(s) 1 + GaGbH 1 + GaGbH
Thus
Akshya Swain
Module-2:Characteristics of Feedback Systems 24 / 37
Effects of Feedback on Noise
I The closed loop transfer function due to noise alone can be obtained by making R(s) = 0 which gives
C(s) = Gb N(s) 1+GaGbH
N(s)
This shows that in the closed loop control system, the gain due to noise signal is decreased by a factor of (1 + GaGbH). Note that, in most practical control systems, (1 + GaGbH) is greater than one.
Akshya Swain
Module-2:Characteristics of Feedback Systems 25 / 37
Effect of Feedback on Speed of Response
Case-1: Open Loop System: Consider an open-loop system with
G(s)=C(s)= K R(s) s+a
The impulse response of this system is given by c(t)=L−1[C(s)]=L−1 K =Ke−at
s+a
The time constant T associated with this mode of response equals to 1/a. Case-2: Closed Loop System:
When the feedback loop is closed with unity feedback, then
G(s)= C(s) = G(s) = K/(s+a) = K R(s) 1+G(s) 1+K/(s+a) s+(a+K)
The impulse response of this system is given by
c(t) = L−1 [C(s)] = L−1 K = Ke−(a+K)t
s+a+K
The time constant T associated with this mode of response equals to 1/(a + K).
Akshya Swain
Module-2:Characteristics of Feedback Systems 26 / 37
Effect of Feedback on Stability
I Consider the basic feedback system.
I The input-output relation is expressed as:
C(s) = G R(s) 1 + GH
I From this it is obvious that when GH = −1, the output of the system will be infinite for any finite input and the system is said to be unstable.
I Thus, feedback can cause a system that is originally stable to become unstable.
I We will demonstrate that feedback can stabilise an unstable system
I Let us introduce another feedback loop through a negative feedback gain F(s) as shown . Then
C(s) = R(s)
G
1 + GH + GF
I The overall system can be made stable by properly selecting the outer feedback gain F(s)
Akshya Swain
Module-2:Characteristics of Feedback Systems 27 / 37
Reduction of Effects of Parameter Variations
Let us define sensitivity on a quantitative basis. In open loop case
C(s) = G(s)R(s)
I Suppose, due to parameter variations G(s) changes to [G(s) + ∆G(s)].
I The output of open-loop system therefore changes to
C(s) + ∆C(s) = [G(s) + ∆G(s)] R(s)
Thus, ∆C(s) = ∆G(s)R(s
I Similarly, in the closed loop case, the output is given by
C(s) = G(s) R(s) 1 + G(s)H(s)
I Due to variation ∆G(s) in the forward path transfer function, this changes to C(s) + ∆C(s) = G(s) + ∆G(s) R(s) = G(s) + ∆G(s) R(s)
1 + [G(s) + ∆G(s)]H(s) 1 + G(s)H(s) + ∆G(s)H(s) I Since |G(s)| ≫ |∆G(s)|, the variation in the output can be expressed as:
∆C(s) = ∆G(s) R(s) 1 + G(s)H(s)
Akshya Swain
Module-2:Characteristics of Feedback Systems 28 / 37
Reduction of Effects of Parameter Variations
1. Open Loop System
2. Closed Loop System
∆C(s) = ∆G(s)R(s)
∆C(s) = ∆G(s) R(s) 1 + G(s)H(s)
Thus compared to the open loop system, the change in the output of the closed loop system due to variations in G(s) is reduced by a factor of
[1 + G(s)H(s)] which is often much greater than unity.
Akshya Swain
Module-2:Characteristics of Feedback Systems 29 / 37
Sensitivity Reduction Due to Feedback
I The term system sensitivity is used to describe the relative variation in the overall transfer function T(s) = C(s)/R(s) due to variation in G(s) and is defined as:
Sensitivity = Percentage change in T(s) Percentage change in G(s)
– For small incremental variation in G(s), the sensitivity of T with respect to G is expressed quantitatively as:
SGT = ∂T/T = ∂ LnT ∂G/G ∂ LnG
I The sensitivity of the closed-loop system is
SGT = ∂T × G = (1 + GH) − GH × G = 1 ∂G T (1+GH)2 G/(1+GH) 1+GH
Akshya Swain
Module-2:Characteristics of Feedback Systems 30 / 37
Sensitivity Reduction Due to Feedback
I The sensitivity of the closed-loop system is
SGT = ∂T × G = (1 + GH) − GH × G = 1
∂G T (1+GH)2 G/(1+GH) 1+GH I The sensitivity of the open loop system is
SGT=∂T×G=1, (*HereT=G) ∂G T
I Thus due to variations in G, the sensitivity of the closed loop system is reduced by a factor of (1 + GH) compared to open-loop system.
I THe sensitivity of T with respect to the feedback sensor H is given as: T ∂T H −G H −GH
SH =∂H×T =G (1+GH)2 ×G/(1+GH)=1+GH
I This shows that for large values of GH, the sensitivity of feedback system with respect to H is unity.
I This implies that changes in H directly affect the system output. Therefore, it is important to use feedback elementswhich remains substantially constant and do not vary with environmental changes.
Akshya Swain
Module-2:Characteristics of Feedback Systems 31 / 37
Sensitivity Reduction Due to Feedback
I Very often, we are interested to find the sensitivity of a system with respect to variation in a particular parameter or parameters.
I Let the transfer function of the system be expressed as
T (s) = N (s, α) ; α = parameter under consideration
D(s, α)
I The sensitivity of T with respect to parameter α is given by
SαT =∂LnN|α0 −∂LnD|α0 ∂ Lnα ∂ Lnα
= S αN − S αD
where α0 is the nominal value of the parameter around which the variation
occurs.
I To have a highly accurate open-loop system,the components of G(s) must be
selected to rigidly to meet the specifications.
I However, in a closed-loop system G(s) may less rigidly be specified, since the
effects of parameter variations can be mitigated by use of feedback.
I However, a closed-loop system requires careful selection of components of the
feedback sensor H(s).
I Note that H(s is often made up of measuring elements which operates at lower
power levels and less costly.
Akshya Swain
Module-2:Characteristics of Feedback Systems 32 / 37
Effect of High Gain in a Feedback System
It will be shown that by Using high gain in a feedback system can make output track input
I Consider a system with gain K shown in Figure below
I The closed loop output (y) and the error (e) response can be expressed as: y= KG u, e= 1 u
1+KG 1+KG
I Fromthese,itisobviousthatasK→∞,y→uande→0
– Open loop gain: |KG| ≫ 1
– Closed loop gain : |KG/(1 + KG)| ≈ 1
I Thus we can make the output track the input even if we do not know the exact value of the open loop gain.
Akshya Swain
Module-2:Characteristics of Feedback Systems 33 / 37
Advantages and Disadvantages of Closed Loop Control
Advantages of Closed Loop Disadvantages of Closed Loop Control System: Control System:
1. Closed loop control systems are more accurate even in the presence of non-linearity.
2. Highly accurate as any error arising is corrected due to presence of feedback signal.
3. Bandwidth range is large.
4. Facilitates automation.
5. The sensitivity of system may be made small to make system more stable.
6. This system is less affected by noise.
1. They are costlier,complicated to design; Require more maintenance.
2. Feedback leads to oscillatory response.
3. Overall gain is reduced due to presence of feedback.
4. Stability is the major problem and more care is needed to design a stable closed loop system.
Akshya Swain
Module-2:Characteristics of Feedback Systems 34 / 37
Comparison Between Open Loop and Closed Loop Control
Open Loop Control
1. The feedback element is absent.
2. An error detector is not present.
3. Easy to construct.
4. Have small bandwidth
5. Often Stable.
6. Less maintenance
7. Often unreliable
Closed Loop Control
1. The feedback element is present.
2. An error detector is always present. 3. Complicated construction.
4. Have large bandwidth.
5. May become unstable.
6. More maintenance.
7. Reliable.
Akshya Swain
Module-2:Characteristics of Feedback Systems 35 / 37
Summary of Important Characteristics of Feedback System
1. Decreased sensitivity of the system to variations in the process parameters. 2. Improved rejection of disturbances.
3. Improved measurement noise attenuation.
4. Improved reduction in steady state error of the system.
5. Easy control and adjustment of the transient response of the system.
Akshya Swain
Module-2:Characteristics of Feedback Systems 36 / 37
Tour Maps of Control System
Akshya Swain
Module-2:Characteristics of Feedback Systems 37 / 37