1. Chapter 1: Introduction to Control Systems
In this chapter we describe a general process for designing a control system.
A control system consisting of interconnected components is designed to achieve a desired purpose. To understand the purpose of a control system, it is useful to examine examples of control systems through the course of history. These early systems incorporated many of the same ideas of feedback that are in use today.
Modern control engineering practice includes the use of control design strategies for improving manufacturing processes, the efficiency of energy use, advanced automobile control, including rapid transit, among others.
We also discuss the notion of a design gap. The gap exists between the complex physical system under investigation and the model used in the control system synthesis.
The iterative nature of design allows us to handle the design gap effectively while accomplishing necessary tradeoffs in complexity, performance, and cost in order to meet the design specifications.

2. Introduction
System – An interconnection of elements and devices for a desired purpose.
Control System – An interconnection of components forming a system configuration that will provide a desired response.
Process – The device, plant, or system under control. The input and output relationship represents the cause-and-effect relationship of the process.

3. Introduction
Open-Loop Control Systems utilize a controller or control actuator to obtain the desired response.
Closed-Loop Control Systems utilizes feedback to compare the actual output to the desired output response.
Multivariable Control System

4. History Greece (BC) – Float regulator mechanism
Holland (16th Century)– Temperature regulator
Watt’s Flyball Governor
(18th century)

5. History

6. History
18th Century James Watt’s centrifugal governor for the speed control of a steam engine.
1920s Minorsky worked on automatic controllers for steering ships.
1930s Nyquist developed a method for analyzing the stability of controlled systems
1940s Frequency response methods made it possible to design linear closed-loop control systems
1950s Root-locus method due to Evans was fully developed
1960s State space methods, optimal control, adaptive control and
1980s Learning controls are begun to investigated and developed.
Present and on-going research fields. Recent application of modern control theory includes such non-engineering systems such as biological, biomedical, economic and socio-economic systems

7. Examples of Modern Control Systems
(a) Automobile steering control system.
(b) The driver uses the difference between the actual and the desired direction of travel
to generate a controlled adjustment of the steering wheel.
(c) Typical direction-of-travel response.

8. Examples of Modern Control Systems

9. Examples of Modern Control Systems

10. Examples of Modern Control Systems

11. Examples of Modern Control Systems

12. Examples of Modern Control Systems

13. The Future of Control Systems

14. The Future of Control Systems

15. Control System Design

17. Design Example

18. Design Example
CVN(X) FUTURE AIRCRAFT CARRIER

19. Design Example

20. Design Example

21. Design Example

23. Sequential Design Example

24. Block Diagram fundamentals & reduction techniques

25. Introduction
Block diagram is a shorthand, graphical representation of a physical system, illustrating the functional relationships among its components. OR
A Block Diagram is a shorthand pictorial representation of the cause-and-effect relationship of a system.
A cause-effect relationship is a relationship in which one event (the cause) makes another event happen (the effect). The loud sound of the alarm was the cause. Without the alarm, you probably would have overslept. In this scenario the alarm had the effect of you waking up at a certain time. 

26. Introduction
The simplest form of the block diagram is the single block, with one input and one output.
The interior of the rectangle representing the block usually contains a description of or the name of the element, or the symbol for the mathematical operation to be performed on the input to yield the output.
The arrows represent the direction of information or signal flow.

27. Introduction
The operations of addition and subtraction have a special representation.
The block becomes a small circle, called a summing point, with the appropriate plus or minus sign associated with the arrows entering the circle.
Any number of inputs may enter a summing point.
The output is the algebraic sum of the inputs.
Some books put a cross in the circle.

28. Components of a Block Diagram for a Linear Time Invariant System
System components are alternatively called elements of the system.
Block diagram has four components:
Signals
System/ block
Summing junction
Pick-off/ Take-off point

30. In order to have the same signal or variable be an input to more than one block or summing point, a takeoff point is used.
Distributes the input signal, undiminished, to several output points.
This permits the signal to proceed unaltered along several different paths to several destinations.

31. Example-1
Consider the following equations in which x1, x2, x3, are variables, and a1, a2 are general coefficients or mathematical operators.

32. Block Diagram fundamentals & reduction techniques

33. Introduction
Block diagram is a shorthand, graphical representation of a physical system, illustrating the functional relationships among its components. OR
A Block Diagram is a shorthand pictorial representation of the cause-and-effect relationship of a system.
A cause-effect relationship is a relationship in which one event (the cause) makes another event happen (the effect). The loud sound of the alarm was the cause. Without the alarm, you probably would have overslept. In this scenario the alarm had the effect of you waking up at a certain time. 

34. Introduction
The simplest form of the block diagram is the single block, with one input and one output.
The interior of the rectangle representing the block usually contains a description of or the name of the element, or the symbol for the mathematical operation to be performed on the input to yield the output.
The arrows represent the direction of information or signal flow.

35. Introduction
The operations of addition and subtraction have a special representation.
The block becomes a small circle, called a summing point, with the appropriate plus or minus sign associated with the arrows entering the circle.
Any number of inputs may enter a summing point.
The output is the algebraic sum of the inputs.
Some books put a cross in the circle.

36. Components of a Block Diagram for a Linear Time Invariant System
System components are alternatively called elements of the system.
Block diagram has four components:
Signals
System/ block
Summing junction
Pick-off/ Take-off point

38. In order to have the same signal or variable be an input to more than one block or summing point, a takeoff point is used.
Distributes the input signal, undiminished, to several output points.
This permits the signal to proceed unaltered along several different paths to several destinations.

39. Example-1
Consider the following equations in which x1, x2, x3, are variables, and a1, a2 are general coefficients or mathematical operators.

40. Example-1
Consider the following equations in which x1, x2, x3, are variables, and a1, a2 are general coefficients or mathematical operators.

41. Example-2
Consider the following equations in which x1, x2,. . . , xn, are variables, and a1, a2,. . . , an , are general coefficients or mathematical operators.

42. Example-3
Draw the Block Diagrams of the following equations.

43. Topologies
We will now examine some common topologies for interconnecting subsystems and derive the single transfer function representation for each of them.
These common topologies will form the basis for reducing more complicated systems to a single block.
Topolgy: the way in which constituent parts are interrelated or arranged

44. CASCADE
Any finite number of blocks in series may be algebraically combined by multiplication of transfer functions.
That is, n components or blocks with transfer functions G1 , G2, , Gn, connected in cascade are equivalent to a single element G with a transfer function given by

45. Example Multiplication of transfer functions is commutative; that is,
GiGj = GjGi
for any i or j .

46. Cascade: Figure: a) Cascaded Subsystems.
b) Equivalent Transfer Function.
The equivalent transfer function is

47. Parallel Form:
Parallel subsystems have a common input and an output formed by the algebraic sum of the outputs from all of the subsystems.
Figure: Parallel Subsystems.

48. Parallel Form: Figure: a) Parallel Subsystems.
b) Equivalent Transfer Function.
The equivalent transfer function is

49. Feedback Form:
The third topology is the feedback form. Let us derive the transfer function that represents the system from its input to its output. The typical feedback system, shown in figure:
Figure: Feedback (Closed Loop) Control System.
The system is said to have negative feedback if the sign at the summing junction is negative and positive feedback if the sign is positive.

50. Feedback Form: Figure: Feedback Control System.
Simplified Model or Canonical Form.
c) Equivalent Transfer Function.
The equivalent or closed-loop
transfer function is

51. Characteristic Equation
The control ratio is the closed loop transfer function of the system.
The denominator of closed loop transfer function determines the characteristic equation of the system.
Which is usually determined as:
Refer to Nise-5th edition

52. Canonical Form of a Feedback Control System
The system is said to have negative feedback if the sign at the summing junction is negative and positive feedback if the sign is positive.

53. Open loop transfer function
Feed Forward Transfer function
control ratio
feedback ratio
error ratio
closed loop transfer function
characteristic equation
closed loop poles and zeros if K=10.

54. Characteristic Equation

55. Unity Feedback System

56. Reduction techniques 1. Combining blocks in cascade
2. Combining blocks in parallel

57. Reduction techniques
3. Moving a summing point behind a block

58. Reduction techniques 3. Moving a summing point ahead of a block
4. Moving a pickoff point behind a block
5. Moving a pickoff point ahead of a block

59. Reduction techniques 6. Eliminating a feedback loop
7. Swap with two neighboring summing points

60. Block Diagram Transformation Theorems
The letter P is used to represent any transfer function, and W, X , Y, Z denote any transformed signals.

61. Transformation Theorems Continue:

62. Transformation Theorems Continue:

63. Reduction of Complicated Block Diagrams:

64. Example-4: Reduce the Block Diagram to Canonical Form.

65. Example-4: Continue. However in this example step-4 does not apply.

66. Example-5: Simplify the Block Diagram.

67. Example-5: Continue.

68. Example-6: Reduce the Block Diagram.

69. Example-6: Continue.

70. Example-7: Reduce the Block Diagram. (from Nise: page-242)

71. Example-7: Continue.

72. Example-8: For the system represented by the following block diagram determine:
Open loop transfer function
Feed Forward Transfer function
control ratio
feedback ratio
error ratio
closed loop transfer function
characteristic equation
closed loop poles and zeros if K=10.

73. Example-8: Continue
First we will reduce the given block diagram to canonical form

74. Example-8: Continue

75. Example-8: Continue Open loop transfer function
Feed Forward Transfer function
control ratio
feedback ratio
error ratio
closed loop transfer function
characteristic equation
closed loop poles and zeros if K=10.

76. Example-9: For the system represented by the following block diagram determine:
Open loop transfer function
Feed Forward Transfer function
control ratio
feedback ratio
error ratio
closed loop transfer function
characteristic equation
closed loop poles and zeros if K=100.

77. Example-10: Reduce the system to a single transfer function
Example-10: Reduce the system to a single transfer function. (from Nise:page-243).

78. Example-10: Continue.

79. Example-10: Continue.

80. Example-11: Simplify the block diagram then obtain the close-loop transfer function C(S)/R(S). (from Ogata: Page-47)

81. Example-11: Continue.

82. Example-12: Reduce the Block Diagram._ + _ + +

83. Example-12: _ _ + + +

84. Example-12: _ _ + + +

85. Example-12: _ _ + + + +

86. Example-12: _ _ + +

87. Example-12: _ _ + +

88. Example-12: _ +

89. Example-12:

90. Example 13: Find the transfer function of the following block diagrams.

91. Solution:
1. Eliminate loop I
2. Moving pickoff point A behind block
Not a feedback loop

92. 3. Eliminate loop II

93. Superposition of Multiple Inputs

94. Example-14: Multiple Input System
Example-14: Multiple Input System. Determine the output C due to inputs R and U using the Superposition Method.

95. Example-14: Continue.

96. Example-14: Continue.

97. Example-15: Multiple-Input System
Example-15: Multiple-Input System. Determine the output C due to inputs R, U1 and U2 using the Superposition Method.

98. Example-15: Continue.

99. Example-15: Continue.

100. Example-16: Multi-Input Multi-Output System
Example-16: Multi-Input Multi-Output System. Determine C1 and C2 due to R1 and R2.

101. Example-16: Continue.

102. Example-16: Continue.
When R1 = 0,
When R2 = 0,

103. Skill Assessment Exercise:

104. Answer of Skill Assessment Exercise:
Schaum’s series- solved problems- Pg#163
Nise-5th edition- problems- pg# 242

105. SIGNAL FLOW GRAPH

106. Outline Introduction to Signal Flow Graphs Mason’s Gain Formula
Definitions
Terminologies
Mason’s Gain Formula
Examples
Signal Flow Graph from Block Diagrams
Design Examples

107. Signal Flow Graph (SFG)
Alternative method to block diagram representation, developed by Samuel Jefferson Mason.
Advantage: the availability of a flow graph gain formula, also called Mason’s gain formula.
A signal-flow graph consists of a network in which nodes are connected by directed branches.
It depicts the flow of signals from one point of a system to another and gives the relationships among the signals.

108. Fundamentals of Signal Flow Graphs
Consider a simple equation below and draw its signal flow graph:
The signal flow graph of the equation is shown below;

109. Important terminology :
Branches :-
line joining two nodes is called branch.
Branch
Dummy Nodes:-
A branch having one can be added at i/p as well as o/p.
Dummy Nodes

110. Input & output node Input node:-
It is node that has only outgoing branches.
Output node:-
It is a node that has incoming branches. b x4 x3 x2 x1 x0 h f g e d c a
Input node
Out put node

111. Forward path:-
Any path from i/p node to o/p node.
Forward path

112. Loop :-
A closed path from a node to the same node is called loop.

113. Self loop:-
A feedback loop that contains of only one node is called self loop.
Self loop

114. Loop gain:- The product of all the gains forming a loop
Loop gain = A32 A23

115. Path & path gain
Path:-
A path is a traversal of connected branches in the direction of branch arrow.
Path gain:-
The product of all branch gains while going through the forward path it is called as path gain.

116. Feedback path or loop :-
it is a path to o/p node to i/p node.

117. Touching loops:-
when the loops are having the common node that the loops are called touching loops.

118. Non touching loops:-
when the loops are not having any common node between them that are called as non- touching loops.

119. Non-touching loops for forward paths

120. Chain Node :-
it is a node that has incoming as well as outgoing branches.
Chain node

121. SFG terms representation
input node (source)
Chain node
Chain node
forward path
path
loop
branch
node
transmittance
input node (source)
Output node

122. Mason’s Rule (Mason, 1953)
The block diagram reduction technique requires successive application of fundamental relationships in order to arrive at the system transfer function.
On the other hand, Mason’s rule for reducing a signal-flow graph to a single transfer function requires the application of one formula.
The formula was derived by S. J. Mason when he related the signal-flow graph to the simultaneous equations that can be written from the graph.

123. Mason’s Rule :-
The transfer function, C(s)/R(s), of a system represented by a signal-flow graph is;
Where
n = number of forward paths.
Pi = the i th forward-path gain.
∆ = Determinant of the system
∆i = Determinant of the ith forward path

124. ∆ is called the signal flow graph determinant or characteristic function. Since ∆=0 is the system characteristic equation.
∆ = 1- (sum of all individual loop gains) + (sum of the products of the gains of all possible two loops that do not touch each other) – (sum of the products of the gains of all possible three loops that do not touch each other) + … and so forth with sums of higher number of non-touching loop gains
∆i = value of Δ for the part of the block diagram that does not touch the i-th forward path (Δi = 1 if there are no non-touching loops to the i-th path.)

125. Example1: Apply Mason’s Rule to calculate the transfer function of the system represented by following Signal Flow Graph
Therefore,
There are three feedback loops
Continue……

126. There are no non-touching loops, therefore
∆ = 1- (sum of all individual loop gains)
There are no non-touching loops, therefore
Continue……

127. ∆1 = 1- (sum of all individual loop gains)+... ∆1 = 1
Eliminate forward path-1
∆1 = 1
∆2 = 1- (sum of all individual loop gains)+...
Eliminate forward path-2
∆2 = 1
Continue……

129. From Block Diagram to Signal-Flow Graph Models－
C(s)
R(s) G1 G2 H2 H1 G4 G3 H3
E(s) X1 X2 X3
Example2
R(s)
－H2 1 G4 G3 G2 G1
C(s)
－H1
－H3 X1 X2 X3
E(s)
Continue……

130. R(s)
－H2 1 G4 G3 G2 G1
C(s)
－H1
－H3 X1 X2 X3
E(s)

131. Design example Example 3

132. Example 4
Continue……

133. Continue……