Chapter 10: Frequency Response Techniques 1 ©2000, John Wiley & Sons, Inc. Nise/Control Systems Engineering, 3/e Chapter 10 Frequency Response Techniques

Chapter 10: Frequency Response Techniques 2 ©2000, John Wiley & Sons, Inc. Nise/Control Systems Engineering, 3/e Figure 10.1 The HP 35670A Dynamic Signal Analyzer obtains frequency response data from a physical system. The displayed data can be used to analyze, design, or determine a mathematical model for the system. Courtesy of Hewlett-Packard.

Chapter 10: Frequency Response Techniques 3 ©2000, John Wiley & Sons, Inc. Nise/Control Systems Engineering, 3/e Figure 10.2 Sinusoidal frequency response: a. system; b. transfer function; c. input and output waveforms

Chapter 10: Frequency Response Techniques 4 ©2000, John Wiley & Sons, Inc. Nise/Control Systems Engineering, 3/e Figure 10.3 System with sinusoidal input

Chapter 10: Frequency Response Techniques 5 ©2000, John Wiley & Sons, Inc. Nise/Control Systems Engineering, 3/e Figure 10.4 Frequency response plots for G(s) = 1/(s + 2): separate magnitude and phase

Chapter 10: Frequency Response Techniques 6 ©2000, John Wiley & Sons, Inc. Nise/Control Systems Engineering, 3/e Figure 10.5 Frequency response plots for G(s) = 1/(s + 2): polar plot

Chapter 10: Frequency Response Techniques 7 ©2000, John Wiley & Sons, Inc. Nise/Control Systems Engineering, 3/e Figure 10.6 Bode plots of (s + a): a. magnitude plot; b. phase plot.

Chapter 10: Frequency Response Techniques 8 ©2000, John Wiley & Sons, Inc. Nise/Control Systems Engineering, 3/e Table 10.1 Asymptotic and actual normalized and scaled frequency response data for (s + a)

Chapter 10: Frequency Response Techniques 9 ©2000, John Wiley & Sons, Inc. Nise/Control Systems Engineering, 3/e Figure 10.7 Asymptotic and actual normalized and scaled magnitude response of (s + a)

Chapter 10: Frequency Response Techniques 10 ©2000, John Wiley & Sons, Inc. Nise/Control Systems Engineering, 3/e Figure 10.8 Asymptotic and actual normalized and scaled phase response of (s + a)

Chapter 10: Frequency Response Techniques 11 ©2000, John Wiley & Sons, Inc. Nise/Control Systems Engineering, 3/e Figure 10.9 Normalized and scaled Bode plots for a. G(s) = s; b. G(s) = 1/s; c. G(s) = (s + a); d. G(s) = 1/(s + a)

Chapter 10: Frequency Response Techniques 12 ©2000, John Wiley & Sons, Inc. Nise/Control Systems Engineering, 3/e Figure Closed-loop unity feedback system

Chapter 10: Frequency Response Techniques 13 ©2000, John Wiley & Sons, Inc. Nise/Control Systems Engineering, 3/e Figure Bode log-magnitude plot for Example 10.2: a. components; b. composite

Chapter 10: Frequency Response Techniques 14 ©2000, John Wiley & Sons, Inc. Nise/Control Systems Engineering, 3/e Table 10.2 Bode magnitude plot: slope contribution from each pole and zero in Example 10.2

Chapter 10: Frequency Response Techniques 15 ©2000, John Wiley & Sons, Inc. Nise/Control Systems Engineering, 3/e Table 10.3 Bode phase plot: slope contribution from each pole and zero in Example 10.2

Chapter 10: Frequency Response Techniques 16 ©2000, John Wiley & Sons, Inc. Nise/Control Systems Engineering, 3/e Figure Bode phase plot for Example 10.2: a. components; b. composite

Chapter 10: Frequency Response Techniques 17 ©2000, John Wiley & Sons, Inc. Nise/Control Systems Engineering, 3/e Figure Bode asymptotes for normalized and scaled G(s) = a. magnitude; b. phase

Chapter 10: Frequency Response Techniques 18 ©2000, John Wiley & Sons, Inc. Nise/Control Systems Engineering, 3/e Table 10.4 Data for normalized and scaled log- magnitude and phase plots for (s  n s +  n 2 ). Mag = 20 log(M/  n 2 ) (table continues) Freq.  n Mag (dB)  = 0.1 Phase (deg)  = 0.1 Mag (dB)  = 0.2 Phase (deg)  = 0.2 Mag (dB)  = 0.3 Phase (deg)  = 0.3

Chapter 10: Frequency Response Techniques 19 ©2000, John Wiley & Sons, Inc. Nise/Control Systems Engineering, 3/e Table 10.4 (continued) Freq.  n Mag (dB)  = 0.5 Phase (deg)  = 0.5 Mag (dB)  = 0.7 Phase (deg)  = 0.7 Mag (dB)  = 1 Phase (deg)  = 1

Chapter 10: Frequency Response Techniques 20 ©2000, John Wiley & Sons, Inc. Nise/Control Systems Engineering, 3/e Figure Normalized and scaled log-magnitude response for

Chapter 10: Frequency Response Techniques 21 ©2000, John Wiley & Sons, Inc. Nise/Control Systems Engineering, 3/e Figure Scaled phase response for

Chapter 10: Frequency Response Techniques 22 ©2000, John Wiley & Sons, Inc. Nise/Control Systems Engineering, 3/e Table 10.5 Data for normalized and scaled log- magnitude and phase plots for 1/(s  n s +  n 2 ). Mag = 20 log(M/  n 2 ) (table continues) Freq.  n Mag (dB)  = 0.1 Phase (deg)  = 0.1 Mag (dB)  = 0.2 Phase (deg)  = 0.2 Mag (dB)  = 0.3 Phase (deg)  = 0.3

Chapter 10: Frequency Response Techniques 23 ©2000, John Wiley & Sons, Inc. Nise/Control Systems Engineering, 3/e Table 10.5 (continued) Freq.  n Mag (dB)  = 0.5 Phase (deg)  = 0.5 Mag (dB)  = 0.7 Phase (deg)  = 0.7 Mag (dB)  = 1 Phase (deg)  = 1

Chapter 10: Frequency Response Techniques 24 ©2000, John Wiley & Sons, Inc. Nise/Control Systems Engineering, 3/e Figure Normalized and scaled log magnitude response for

Chapter 10: Frequency Response Techniques 25 ©2000, John Wiley & Sons, Inc. Nise/Control Systems Engineering, 3/e Figure Scaled phase response for

Chapter 10: Frequency Response Techniques 26 ©2000, John Wiley & Sons, Inc. Nise/Control Systems Engineering, 3/e Figure Bode magnitude plot for G(s) = (s + 3)/[(s + 2) (s 2 + 2s + 25)]: a. components; b. composite

Chapter 10: Frequency Response Techniques 27 ©2000, John Wiley & Sons, Inc. Nise/Control Systems Engineering, 3/e Table 10.6 Magnitude diagram slopes for Example 10.3

Chapter 10: Frequency Response Techniques 28 ©2000, John Wiley & Sons, Inc. Nise/Control Systems Engineering, 3/e Table 10.7 Phase diagram slopes for Example 10.3

Chapter 10: Frequency Response Techniques 29 ©2000, John Wiley & Sons, Inc. Nise/Control Systems Engineering, 3/e Figure Bode phase plot for G(s) = (s + 3)/[(s +2) (s 2 + 2s + 25)]: a. components; b. composite

Chapter 10: Frequency Response Techniques 30 ©2000, John Wiley & Sons, Inc. Nise/Control Systems Engineering, 3/e Figure Closed-loop control system

Chapter 10: Frequency Response Techniques 31 ©2000, John Wiley & Sons, Inc. Nise/Control Systems Engineering, 3/e Figure Mapping contour A through function F(s) to contour B

Chapter 10: Frequency Response Techniques 32 ©2000, John Wiley & Sons, Inc. Nise/Control Systems Engineering, 3/e Figure Examples of contour mapping

Chapter 10: Frequency Response Techniques 33 ©2000, John Wiley & Sons, Inc. Nise/Control Systems Engineering, 3/e Figure Vector representation of mapping

Chapter 10: Frequency Response Techniques 34 ©2000, John Wiley & Sons, Inc. Nise/Control Systems Engineering, 3/e Figure Contour enclosing right half-plane to determine stability

Chapter 10: Frequency Response Techniques 35 ©2000, John Wiley & Sons, Inc. Nise/Control Systems Engineering, 3/e Figure Mapping examples: a. contour does not enclose closed-loop poles; b. contour does enclose closed-loop poles

Chapter 10: Frequency Response Techniques 36 ©2000, John Wiley & Sons, Inc. Nise/Control Systems Engineering, 3/e Figure a. Turbine and generator; b. block diagram of speed control system for Example 10.4

Chapter 10: Frequency Response Techniques 37 ©2000, John Wiley & Sons, Inc. Nise/Control Systems Engineering, 3/e Figure Vector evaluation of the Nyquist diagram for Example 10.4: a. vectors on contour at low frequency; b. vectors on contour around infinity; c. Nyquist diagram

Chapter 10: Frequency Response Techniques 38 ©2000, John Wiley & Sons, Inc. Nise/Control Systems Engineering, 3/e Figure Detouring around open-loop poles: a. poles on contour; b. detour right; c. detour left

Chapter 10: Frequency Response Techniques 39 ©2000, John Wiley & Sons, Inc. Nise/Control Systems Engineering, 3/e Figure a. Contour for Example 10.5; b. Nyquist diagram for Example 10.5

Chapter 10: Frequency Response Techniques 40 ©2000, John Wiley & Sons, Inc. Nise/Control Systems Engineering, 3/e Figure Demonstrating Nyquist stability: a. system; b. contour; c. Nyquist diagram

Chapter 10: Frequency Response Techniques 41 ©2000, John Wiley & Sons, Inc. Nise/Control Systems Engineering, 3/e Figure a. Contour for Example 10.6; b. Nyquist diagram

Chapter 10: Frequency Response Techniques 42 ©2000, John Wiley & Sons, Inc. Nise/Control Systems Engineering, 3/e Figure a. Contour and root locus of system that is stable for small gain and unstable for large gain; b. Nyquist diagram

Chapter 10: Frequency Response Techniques 43 ©2000, John Wiley & Sons, Inc. Nise/Control Systems Engineering, 3/e Figure a. Contour and root locus of system that is unstable for small gain and stable for large gain; b. Nyquist diagram

Chapter 10: Frequency Response Techniques 44 ©2000, John Wiley & Sons, Inc. Nise/Control Systems Engineering, 3/e Figure a. Portion of contour to be mapped for Example 10.7; b. Nyquist diagram of mapping of positive imaginary axis

Chapter 10: Frequency Response Techniques 45 ©2000, John Wiley & Sons, Inc. Nise/Control Systems Engineering, 3/e Figure Nyquist diagram showing gain and phase margins

Chapter 10: Frequency Response Techniques 46 ©2000, John Wiley & Sons, Inc. Nise/Control Systems Engineering, 3/e Figure Bode log-magnitude and phase diagrams for the system of Example 10.9

Chapter 10: Frequency Response Techniques 47 ©2000, John Wiley & Sons, Inc. Nise/Control Systems Engineering, 3/e Figure Gain and phase margins on the Bode diagrams

Chapter 10: Frequency Response Techniques 48 ©2000, John Wiley & Sons, Inc. Nise/Control Systems Engineering, 3/e Figure Second-order closed-loop system

Chapter 10: Frequency Response Techniques 49 ©2000, John Wiley & Sons, Inc. Nise/Control Systems Engineering, 3/e Figure Representative log-magnitude plot of Eq. (10.51)

Chapter 10: Frequency Response Techniques 50 ©2000, John Wiley & Sons, Inc. Nise/Control Systems Engineering, 3/e Figure Closed-loop frequency percent overshoot for a two-pole system

Chapter 10: Frequency Response Techniques 51 ©2000, John Wiley & Sons, Inc. Nise/Control Systems Engineering, 3/e Figure Normalized bandwidth vs. damping ratio for: a. settling time; b. peak time; c. rise time

Chapter 10: Frequency Response Techniques 52 ©2000, John Wiley & Sons, Inc. Nise/Control Systems Engineering, 3/e Figure Constant M circles

Chapter 10: Frequency Response Techniques 53 ©2000, John Wiley & Sons, Inc. Nise/Control Systems Engineering, 3/e Figure Constant N circles

Chapter 10: Frequency Response Techniques 54 ©2000, John Wiley & Sons, Inc. Nise/Control Systems Engineering, 3/e Figure Nyquist diagram for Example and constant M and N circles

Chapter 10: Frequency Response Techniques 55 ©2000, John Wiley & Sons, Inc. Nise/Control Systems Engineering, 3/e Figure Closed-loop frequency response for Example 10.11

Chapter 10: Frequency Response Techniques 56 ©2000, John Wiley & Sons, Inc. Nise/Control Systems Engineering, 3/e Figure Nichols chart

Chapter 10: Frequency Response Techniques 57 ©2000, John Wiley & Sons, Inc. Nise/Control Systems Engineering, 3/e Figure Nichols chart with frequency response for G(s) = K/[s(s + 1)(s + 2)] superimposed. Values for K = 1 and K = 3.16 are shown.

Chapter 10: Frequency Response Techniques 58 ©2000, John Wiley & Sons, Inc. Nise/Control Systems Engineering, 3/e Figure Phase margin vs. damping ratio

Chapter 10: Frequency Response Techniques 59 ©2000, John Wiley & Sons, Inc. Nise/Control Systems Engineering, 3/e Figure Open-loop gain vs. open-loop phase angle for –3 dB closed-loop gain

Chapter 10: Frequency Response Techniques 60 ©2000, John Wiley & Sons, Inc. Nise/Control Systems Engineering, 3/e Figure a. Block diagram (figure continues)