The document discusses the use of Laplace transforms in control systems to represent dynamic systems as transfer functions. It provides information on: - Representing control systems using differential equations that are transformed into the frequency domain using Laplace transforms. - Defining the transfer function as the ratio of the Laplace transform of the output to the Laplace transform of the input. - Explaining poles and zeros of a transfer function and their graphical representations. - Providing steps to determine the transfer function of a control system from its equations. - Using block diagrams to model control systems, with blocks representing transfer functions and rules for reducing complex diagrams into an overall transfer function.

1. Sub:- Control System
Topic:- Use of Laplace
Transform in Control Systems,
Transfer function

2. Use of Laplace transform in control systems
◻ The control action for a dynamic control system whether
electrical, mechanical, thermal, hydraulic etc. can be
represented by a differential equation and the output response
of such a dynamic system to a specified input can be obtained
by solving the said differential equation.
◻ The system differential equation is derived according to
physical laws governing a system in question.
◻ In order to facilitate the solution of a differential equation
describing a control system the equation is transform in
algebraic form.

3. ◻ The differential equation wherein time being the independent
variable is transformed in to a corresponding algebraic
equation by using Laplace transform technique and the
differential equation thus transformed is known as the equation
in frequency domain.
◻ Hence Laplace transform technique transforms a time domain
differential equation in in to a frequency domain algebraic
equation.
+

4. Transfer Function
◻ Transfer Function is the ratio of Laplace transform of the output
variable to the Laplace transform of the input variable. Consider all
initial conditions to zero.
◻The transfer function is expressed as the ratio of output quantity
to input quantity. Therefore
4
Transfer function, g
Input or Excitation function, r Output or response. c

5. Transfer Function
◻ With reference to control system wherein all mathematical
functions are expressed by their corresponding Laplace
transforms, therefore the transfer function is also expressed as a
ratio of Laplace transform of output to Laplace transform of input.
◻ The transfer function is expressed as:-
5
G(S)
Input R(S) Output C(S)

6. Poles and Zeros of a transfer function
◻ The transfer function of a linear control system can be
expressed in the form of a quotient polynomials in the
following form
◻ The numerator and the denominator can be factored in to n &
m terms respectively,
◻ Where is known as the gain factor of the transfer function.

7. ◻ Poles of the transfer function:- In the transfer function expression
if s is put equals to zero, hence
called poles of the transfer function. The graphical symbol for a
pole is X.
◻ Zeros of the transfer function:- In the transfer function
expression if s is put equals to zero, hence
called zeros of the transfer function. The graphical symbol for a
pole is 0.
◻

8. ◻ Multiple poles or zeros :- It is possible that either poles or
zeros may coincide. Such poles or zeros are called multiple
poles or multiple zeros.
◻ Single poles or zeros:- If poles or zeros may non-coincide
are called simple poles or simple zeros.
S=-2
S=-3 S=-2 S=2
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σ

9. Procedure for determining the transfer function
of a control system
The following steps give a procedure for determining the transfer
function of a control system.
 Step 1 :- Formulate the equation of the system.
 Step 2 :- Take the Laplace transform of the system equations,
assuming all initial conditions as zero.
 Step 3 :- Specify the system output at the input.
 Step 4 :- Take the ratio of Laplace transform of the output and
the Laplace transform of the input.

10. Example
◻ Find the transfer function of the electrical network as shown in
network given below :-

11. ◻ Step 1 :- Formulate the equation of the system.
◻ Step 2 :- Take the Laplace transform of the system equations,
assuming all initial conditions as zero.

12. ◻ Step 3 :- Specify the system output at the input.
◻ Step 4 :-Take the ratio of Laplace transform of the output and
the Laplace transform of the input.

13. BLOCK DIAGRAM
◻ A block diagram is basically modeling of any simple or
complex system. It Consists of multiple Blocks connected
together to represent a system to explain how it is functioning.
◻ We often represent control systems using block diagrams. A
block diagram consists of blocks that represent transfer
functions of the different variables of interest.
◻ If a block diagram has many blocks, not all of which are in
cascade, then it is useful to have rules for rearranging the
diagram such that you end up with only one block.

14. Representation of a control system by block
diagram
◻ The block diagram is to represent a control system in diagram form. In other words,
practical representation of a control system is its block diagram.
◻ It is not always convenient to derive the entire transfer function of a complex control
system in a single function.
◻ It is easier and better to derive the transfer function of the control element connected to the
system, separately.
◻ The transfer function of each element is then represented by a block and they are then
connected together with the path of signal flow.
◻ For simplifying a complex control system, block diagrams are used. Each element of the
control system is represented with a block and the block is the symbolic representation of
the transfer function of that element.
◻ A complete control system can be represented with a required number of interconnected
blocks.

15. Representation of a control system by block
diagram
◻ A block diagram is shown in fig. Wherein
◻ G1(s) and G2(s) represent the transfer function of individual elements of a control system.
◻ As the output signal C(s) is feedback and compared with the input R(s), the difference
E(s)=[R(s)- C(s)] is the actuating signal or error signal.
G1(s) G2(s)
R(s) E(s) C(s)

16. BLOCK DIAGRAM REDUCTION
◻ In order to obtain the overall transfer function a procedure block
diagram reduction technique is followed. Some of the important
rules for block diagram reduction are as given below.
◻ Rule 1:-
◻ Where G(s) is known as the transfer function of the system.

17. BLOCK DIAGRAM
Rule 2 :- Take off point
◻ Application of one input source to two or more systems is
represented by a take off point as shown in fig.

18. Rule 3:- Blocks in cascade
◻ When several blocks are connected in cascade the overall
equivalent transfer function is equal to the multiplication of
transfer function of all individual blocks.
G1(s) G3(s)
G2(s)
R(s) C1(s) C(s)
C2(s)
G1(s)*G2(s)*G3(s)
R(s) C(s)

19. Rule 4:- Summing point:-
◻ Summing point represents summation of two or more input
signals entering in a system.
◻ The output of a summing point being the sum of the entering
inputs.
◻ It is necessary to indicate the sign specifying the input signal
entering a summing point.

20. ◻ Rule 5:- Consecutive summing point can be interchanged, as
this interchange does not alter the output signal.
R(s) R(s) –Y(s) R(s) –Y(s)+X(s)
Y(s) X(s)

21. Rule 6:- Blocks in parallel.
◻ When one or more blocks are connected in parallel as shown
in fig. the overall equivalent transfer function is equal to the
sum of all individual transfer function of all the blocks.
G1(s)
G3(s)
G2(s)
R(s) C(s)
G1(s)+G2(s)+G3(s)
R(s) C(s)

22. ◻ Rule 7:- Shifting of a take off point from a position before a
block to a position after the block

23. ◻ Rule 8:- Shifting of a take off point from a position after a block
to a position before the block.

24. ◻ Rule 9:- Shifting of a summing point from a position before a
block to a position after the block.

25. ◻ Rule 10:- Shifting of a summing point from a position after a
block to a position before the block.

26. ◻ Rule 11:- Shifting of a take off point from a position before a
summing point to a position after the summing point .

27. ◻ Rule 12:- Shifting of a take off point from a position after a
summing point to a position before the summing point .

28. ◻ Rule 13:- (a) Elimination of a summing point in a closed
loop system

29. ◻ Rule 13:- (b) A unity feedback sysem

30. ◻ Rule 13:- (c) The transfer function relating E(s) & R(s) for a
closed loop control system
R(s) E(s)

31. ◻ Rule 14:- When two or more inputs act on a system the total
output is obtained by adding effect of each individual input
separately.
Fig. (a):- Two inputs acting on a system
Fig. (b):- Considering only R1(s) input
Fig. (c):- Considering only R2(s) input
The total output is given
by:-

32. Terms used in Control System

33. Solved examples
◻ Ex. 1:- Determine the transfer function from the block
diagram as shown in fie.

34. ◻ Shift the take off point after block G2 to a position before block
G2.

35. ◻ Eliminate the summing point after block G2 shift the take off
point before block (G2+ G3).

36. Eliminate the summing point before block (G2+ G3)

37. The overall transfer function determined below:-

38. ◻ Ex. 2:- Determine the transfer function from the block
diagram as shown in fie

39. Shift the take off point placed before block G3 to a position after the block G3

40. ◻ Eliminate the summing point placed before block G3

42. ◻ Shift the summing point placed after the block to a position before the block

43. ◻ Interchange consecutive summing point & then eliminate the
summing point before block G1