TRANSFER FUNCTION (4).pptx

Control system :- Transfer Function

1. Sub:- Control System Topic:- U se 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 σ ј ω ј ω 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 G 1 (s) and G 2 (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. G 1 (s) G 2 (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. G 1 (s) G 3 (s) G 2 (s) R(s) C 1 (s) C(s) C 2 (s) G 1 (s)*G 2 (s)*G 3 (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. G 1 (s) G 3 (s) G 2 (s) R(s) C(s) G 1 (s)+G 2 (s)+G 3 (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 R 1 (s) input Fig. (c):- Considering only R 2 (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 G 2 to a position before block G 2.

35. Eliminate the summing point after block G 2 shift the take off point before block (G 2 + G 3 ).

36. Eliminate the summing point before block (G 2 + G 3 )

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 G 3 to a position after the block G 3

40. Eliminate the summing point placed before block G 3

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

42. Interchange consecutive summing point & then eliminate the summing point before block G 1