1. Chapter 6 Frequency Response

2. http://kunst.gymszbad.de/kunstgeschichte/ motivgeschichte/altaere/frame-menue.htm Hieronymus Bosch Garden of Delights

3. http://kunst.gymszbad.de/kunstgeschichte/ motivgeschichte/altaere/frame-menue.htm Master of Flemalle ( Robert Campin) Mérode-Altar The central panel shows the Annunciation. The child is already on his way on golden rays…

4. http://kunst.gymszbad.de/kunstgeschichte/ motivgeschichte/altaere/frame-menue.htm Master of Flemalle ( Robert Campin) Mérode-Altar The Cloisters New York, NY Detail: The child is already on his way on golden rays, carrying the cross of the passion with him.

5. Master of Flemalle ( Robert Campin) Mérode-Altar The Metropolitan Museum of Art, The Cloisters New York, NY Another Detail: St. Joseph the carpenter (right panel) has just completed a mousetrap (on the table), possibly to trap the devil.

6. Time domain signals: Square Wave and triangular wave. An Example: Analysis of Sound Waves

7. Time domain signal analysis: Spectrum of Square Wave Fundam. freq HARMONICS The second harmonic is twice the fundam- ental frequency, the third harmonic is three times the fundam. frequency, and so forth. Analysis of Sound Waves

12. Fourier Transform: Let period T  infinity The interval between Discrete frequencies  0 The Fourier series becomes the Fourier Transform

13. The A(  ) and B(  ) terms of the Fourier Transform can be combined into the complex term C(j  ) becomes

14. where Compare with the definition of the Laplace Transform

18. A Sound Wave and its Spectrum

19. Question: How do we recognize voices or musical instruments?

20. Answer: Our brains perform a real time spectral analysis of the incoming sound signal. The spectrum, not the signal itself, informs us about the source.

21. Question: How do we recognize color?

30. Bode Plots: Same content as polar plot, just a different mode of presentation. Bode Plots: Logarithmic  -axis. Logarithmic | F | (magnitude axis) Why? Phase values are entered directly Why?

31. Basic Bode Plot (First Order) f = -45deg. at  b Break Frequency  b G(s)= K/(  s+1) 2.  b at -45 deg. And |F| = 0.707

32. 1. Note K and  b 2. Draw |F| from low freq to  b 3. Draw |F| from  b, slope -1/decade Bode Magnitude Plot K = 2  b =5

33. Bode Phase Plot 1. Phase = -45 0 at  b 2. Draw  from 0 to  b /10, slope =0 3. Draw  from  b /10 freq to 10*  b 4. Min Phase is -90 0 from 10*  b

34. Decibels An alternate unit of Magnitude or Gain Definition: x dB = 20* lg(x) dB Notation is widely used in Filter theory and Acoustics xlg(x)X(db) 10120 100240 0.1-20

35. Decibels An alternate unit of Magnitude or Gain Definition: x dB = 20* lg(x)

36. Bode Plot of Integrator G(s) = 1 / (s) |F|= 1/   = -tan-1(  /0) = -90 0 Memorize!

37. Underdamped second order systems and Resonance

38. Asymptote Slope = -2 Phase is -90 deg. at  n

39. Bode Plot Construction G(s) = 2/ (s)(s+1) 1. Construct each Element plot Integrator Slope = -1 Integrator Phase = -90 deg. 2. Graphical Summation Gain = 2. Slope = -2

40. Bode Plot of 1/(s(s+1)): Matlab Plot

41. Bode Plot Construction G(s) = 5* (s+1) / (10s+1)(100s+1) 1. Construct each Element plot 2. Graphical Summation: Complete plot. Note beginning and final values K = 5 Slope = -1 Slope = -2 Slope = -1

42. Phase Plot Construction G(s) = 5* (s+1) / (10s+1)(100s+1) 2. Graphical Summation of phase angles. Note beginning and final phase values. Here:  = 0 at  = 0, and  = -90 final angle K = 5 Initial Phase is zero to 0.001, follows the first Phase up to 0.01 - 90 deg./decade 0 deg./decade +45 deg./decade Final phase: Constant - 90 deg

43. Bode Plot Construction: Matlab Plot

44. Nyquist Criterion: Closed Loop Stability: Evaluate Frequency response at Phase of -180 degrees

45. Nyquist Stability Criterion

46. Nyquist Criterion: Stability in the Frequency Domain

47. Nyquist Criterion in the Bode Plot: Gain Margin and Phase Margin Phase Margin Gain Margin

48. Nyquist Criterion in the Bode Plot: Gain Margin and Phase Margin

49. Bode Lead Design 1. Select Lead zero such that the phase margin increases while keeping the gain crossover frequency as low as reasonable. 2. Adjust Gain to the desired phase margin.

50. Lead compensator |p| = 10*z G(s) = 1. Construct each Element plot Slope = 0 2. Graphical Summation Gain = 1 Slope = +1 Slope = 0 Phase = 0 Slope = 0 Note Break Frequencies

51. Bode Lead Design Objectives: Increase Loop Gain and damping by raising the phase margin at the 0dB crossover frequency. Phase Margin = 45 deg. Phase Margin Try: Lead Zero at 0.9, pole at 9Draw new Phase and Mag. Plots Phase with Lead. The new crossover freq. is 3 rad/s. Magn. with Lead. Final Step: Adjust Gain. Here K is raised approx. 3-fold

52. Bode Lead Design Bode plot with Lead Zero at 0.9, pole at 9 (in Red). Phase Margin Note phase crossing at  =3 with -135 deg. phase margin Adjust gain at  =3 phase crossing. Here: raise gain by about 10 dB or by a factor of 3

53. Bode Lead Design Final Design: Raise Gain K = 3 From Matlab: Phase Margin = 38.3356 degrees

54. Bode Lag Design 1. All other design should be complete. Gain K and phase margin are fixed 2. Select Lag zero such that the phase margin does not drop further. (Slow) 3. Steady State Gain should now be about 10 times larger than without Lag.

55. Lag compensator |p| = 0.1*z G(s) = Construct each Element plot Slope = 0Gain = 0.1 Slope = 0 Phase = 0 Slope = 0 Slope = -1 Slope = 0 Note Break Frequencies

56. Bode Lag Design

58.  -margin = 39 deg. K = 10

59. Lead Design Example (a) P-control for phase margin of 45 degrees. Controller gain K = 0.95

60. (b) Lead-control for phase margin of 45 degrees. Lead zero and pole in RED. Initial design: Lead is too slow Lead is too slow. Lead Zero should be near the phase margin. Here: Place Lead zero around 3 rad/s.

61. (b) Lead-control for phase margin of 45 degrees. Lead zero and pole in RED. Improved design: Lead zero at 3, pole at 30 rad/s Lead zero at 3. Lead pole at 30. New gain crossover at 5 rad/s Final step: adjust gain K such that |F| = 0 dB at  cr. Result: The controller gain is now K = 3.4 (4 times better than P- control)

62. Bode Lead and Lag Design: General placement rules