Automatic Control Systems

Transcription

1 Automatic Control Sytem Lecture- Block Diagram Reduction Emam Fathy Department of Electrical and Control Engineering

2 Introduction A Block Diagram i a horthand pictorial repreentation of the caue-and-effect relationhip of a ytem. The interior of the rectangle repreent the mathematical operation to be performed on the input to yield the output. The arrow repreent the direction of information or ignal flow. x d dt y

3 Introduction The operation of addition and ubtraction have a pecial repreentation. The block become a mall circle, called a umming point, with the appropriate plu or minu ign aociated with the arrow entering the circle. The output i the algebraic um of the input. Any number of input may enter a umming point. Some book put a cro in the circle.

4 Introduction In order to have the ame ignal or variable be an input to more than one block or umming point, a takeoff point i ued. Thi permit the ignal to proceed unaltered along everal different path to everal detination.

5 Example- Conider the following equation in which x, x, x, are variable, and a, a are general coefficient or mathematical operator. x a x a x 5

6 Canonical Form of A Feedback Control Sytem

7 Characteritic Equation The control ratio i the cloed loop tranfer function of the ytem. C R H The denominator of cloed loop tranfer function determine the characteritic equation of the ytem. Which i uually determined a: H 0

8 Example-4 H. Open loop tranfer function. Feed Forward Tranfer function. cloed loop tranfer function 4. characteritic equation E C H R C 0 H H E B

9 In order to analyze the ytem, we want to repreent multiple ubytem a a ingle tranfer function.

10 Reduction technique. Combining block in cacade. Combining block in parallel feed-forward

11 . Eliminating a feedback loop H H OR A B A B AH H

12 4. Moving a pickoff point behind a block 5. Moving a pickoff point ahead of a block

13 6. Moving a umming point ahead of a block 7. Moving a umming point behind a block

14 Note A B B A

15 Example H R + _ _ C H

16 Example H R + _ _ C H

17 Example H R + _ _ C H

18 Example H R + _ + _ + + C H

19 Example H R + _ + _ H C

20 Example H R + _ + _ H C

21 Example R + _ H H C

22 Example R H H C

23 Example Find the tranfer function of the following block diagram 4 R Y H H

24 I R B 4 A Y H H Solution:. Moving pickoff point A ahead of block. Eliminate loop I & implify B 4

25 R B 4 4 A Y H H. Moving pickoff point B behind block R B 4 4 II C Y H H / 4

26 4. Eliminate loop III R 4 4 H H Y H H R Y T R C 4 H Y 4 H C 4 4 H

27 Example Find the tranfer function of the following block diagram R Y H H H

28 Solution:. Eliminate loop I R H A H H I B Y. Moving pickoff point A behind block R H A H H H H H B H Y II Not a feedback loop H H

29 . Eliminate loop II R Y H H H H H H H H H R Y T

30 Example 4 Find the tranfer function of the following block diagram R H 4 4 Y H H H

31 Solution:. Moving pickoff point A behind block 4 I R H 4 4 A B Y H H H 4 H H 4 4 4

32 . Eliminate loop I and Simplify R 4 H 4 H H II B Y III H II feedback III Not feedback 4 H H 4 4 H 4 4 H

33 R Y 4 4 H H H H. Eliminate loop II & IIII H H H H R Y T

34 Example 5 Find the tranfer function of the following block diagram H R A B Y H 4

35 Solution:. Moving pickoff point A behind block I H R A B Y H H 4

36 . Eliminate loop I & Simplify H B B H II H H R H H Y H 4

37 . Eliminate loop II R H H H Y 4 T Y R 4 H H H

38 End of Lec

39 Example-5: Continue.

40 Example-6: Reduce the Block Diagram.

41 Example-6: Continue.

42 Example-7: Reduce the Block Diagram. from Nie: page-4

43 Example-7: Continue.

44 Example-8: Reduce the ytem to a ingle tranfer function. from Nie:page-4.

45 Example-9: Simplify the block diagram then obtain the cloeloop tranfer function CS/RS. from Ogata: Page-47

46 Example-0: Multiple Input Sytem. Determine the output C due to input R and U uing the Superpoition Method.

47 Example-: Continue.

48 Example-: Continue.

49 Example-9: Multiple-Input Sytem. Determine the output C due to input R, U and U uing the Superpoition Method.

50 Example-9: Continue.

51 Example-9: Continue.

52 Example-0: Multi-Input Multi-Output Sytem. Determine C and C due to R and R.

53 Example-0: Continue.

54 Example-0: Continue. When R = 0, When R = 0,

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56 Example-5: Reduce the Block Diagram to Canonical Form.

57 Example-5: Continue. However in thi example tep-4 doe not apply. However in thi example tep-6 doe not apply.

58 Example-6 For the ytem repreented by the following block diagram determine:. Open loop tranfer function. Feed Forward Tranfer function. control ratio 4. feedback ratio 5. error ratio 6. cloed loop tranfer function 7. characteritic equation 8. cloed loop pole and zero if K=0.

59 Example-6 Firt we will reduce the given block diagram to canonical form K

60 Example-6 K H K K

61 Example-7 For the ytem repreented by the following block diagram determine:. Open loop tranfer function. Feed Forward Tranfer function. control ratio 4. feedback ratio 5. error ratio 6. cloed loop tranfer function 7. characteritic equation 8. cloed loop pole and zero if K=00.

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63 Block Diagram of Armature Controlled D.C Motor R a L a c e b V a i a T J L a R a I a K b V a J c K I ma a

64 Block Diagram of Armature Controlled D.C Motor L R I K E a a a b a

65 Block Diagram of Armature Controlled D.C Motor J c K I ma a

66 Block Diagram of Armature Controlled D.C Motor

67 Block Diagram of liquid level ytem q h R h C dh q q dt q h R C dh q q dt

68 Block Diagram of liquid level ytem Q Q H C R H H Q R H Q Q Q H C q q dt dh C R h h q q q dt dh C R h q L L L L

69 Block Diagram of liquid level ytem Q Q H C R H H Q R H Q Q Q H C

70 Block Diagram of liquid level ytem