1. سیستمهای کنترل خطی پاییز 1389 بسم ا... الرحمن الرحيم دکتر حسين بلندي- دکتر سید مجید اسما عیل زاده

2. Recap. Transient Response: First order system transient response –Step response specs and relationship to pole location Second order system transient response 2

3. Error

4. The error signal: e(t) = 1-y(t)=e -at us(t) Normalized time t/ 

5. General First-order system Step response starts at y(0+)=k, final value kz/p 1/p =  is still time constant; in every , y(t) moves 63.2% closer to final value

6. Unit ramp response:

8. Note: In step response, the steady-state tracking error = zero.

9. Unit impulse response:

10. Performance of a second-order system 10

11. Prototype 2 nd order system:

12. Characteristic equation 12

15. Unit step response: 1) Under damped, 0 < ζ < 1

16. cos  =  =-Re/|root|  = cos -1 (Re/|root|)  = tan -1 (-Re/Im) d  =Im =-Re

18. To find y(t) max:

21. Response to unit step input 21

22. Natural frequency  n - the frequency of natural oscillation that would occur for two complex poles if the damping were equal to zero Damping ratio  - a measure of damping for second-order characteristic equation 22

23. Characteristic equation 23

24. Finding  n and  for a second-order system 24

25. Second-order responses for  underdamped 25

26. Unit impulse response R(s)=1 T(s)=Y(s) 26

27. 27

28. Standard performance measures Peak time Settling time Percent overshoot Peak response 28

29. fig_04_14 29

30. Settling time The settling time is defined as the time required for a system to settle within a certain percentage of the input amplitude. 30

31. Settling time 31

32. Rise time The time it takes for a signal to go from 10% of its value to 90% of its final value 32

33. Rise time 33

34. Peak time Peak time is the time required by a signal to reach its maximum value. 34

35. Peak time 35

36. Percent overshoot Percent Overshoot is defined as: P.O. = [(M pt – fv) / fv] * 100% M pt = The peak value of the time response fv = Final value of the response 36

37. Percent overshoot 37

38. Finding transient response 38

39. 39

40. Gain design for transient response 40

41. 41

42.  Controllable Canonical Form

43.  Observable Canonical Form  Diagonal Form

44.  Jordan Form

45. حل معادلات فضاي حالت در محدوده فركانس

46. حل معادلات فضاي حالت در محدوده زمان عكس تبديل لاپلاس مي گيريم: از معادله

47. سؤال: چگونه ماتريس را محاسبه كنيم؟ (2). (3). (1).

48. Example 1)

49. Example2) If

51. روش سوم: تابع يك ماتريس مربعي ) Function of a square matrix) يك تابع باشد كه در Spectrum (دامنه يا طيف) و اگر نيز يك چند جمله اي باشد كه داراي مقادير مساوي مانند در دامنه باشد سپس تابع مي باشد، بشكل زير تعريف مي گردد: اگر تعريف شده باشد كه يك تابع مقدار-ماتريسي ( Matrix Valued )

52. يعني كه اگر يك ماتريس باشد، اگر كه مقدار از روي شود، ما مي توانيم يك چندجمله اي پيدا كنيم كه داراي درجه داده مي باشد بطوريكه: بطوريكه برابر با در دامنه كه تمامي توابع عبارتند از: مي باشد. از اين تعريف مي دانيم

53. طريقه عمل : 1. مقادير ويژه را محاسبه نمائيد. 2. 3. مثال : اگر را محاسبه نمائيد.

54. 1) 2)

55. مثال:.1.2

57. مثال: (1

59. From

61. مثال:

62. حل معادلات حالت

63. حل : (1 (2 (3 (4

64. مثال: حل:

67. حل:

69. مثال : حل :