EES42042 Fundamental of Control Systems Bode Plots DR. Ir. Wahidin Wahab M.Sc. Ir. Aries Subiantoro M.Sc.
2 Bode Plots Plot of db Gain and phase vs frequency It is assumed you know how to construct Bode Plots MATLAB program bode.m available for fast Bode plotting useful for determining Gain and Phase margins
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. Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright 2004 by John Wiley & Sons. All rights reserved.
Figure 10.2 Sinusoidal frequency response: a. system; b. transfer function; c. input and output waveforms Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright 2004 by John Wiley & Sons. All rights reserved.
Figure 10.3 System with sinusoidal input Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright 2004 by John Wiley & Sons. All rights reserved.
Figure 10.4 Frequency response plots for G(s) =1/(s + 2): separate magnitude and phase Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright 2004 by John Wiley & Sons. All rights reserved.
Figure 10.5 Frequency response plots for G(s) = 1/(s + 2) : polar plot Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright 2004 by John Wiley & Sons. All rights reserved.
Figure 10.6 Bode plots of G(s)=(s + a): a. magnitude plot; b. phase plot. Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright 2004 by John Wiley & Sons. All rights reserved.
Table 10.1 Asymptotic and actual normalized and scaled frequency response data for G(s) = (s + a) Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright 2004 by John Wiley & Sons. All rights reserved.
Figure 10.7 Asymptotic and actual normalized and scaled magnitude response of G(s) = (s + a) Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright 2004 by John Wiley & Sons. All rights reserved.
Figure 10.8 Asymptotic and actual normalized and scaled phase response of (s + a) Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright 2004 by John Wiley & Sons. All rights reserved.
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) Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright 2004 by John Wiley & Sons. All rights reserved.
13 Gain Margin Factor by which gain has to be increased to encircle (-1,0) point in polar plot ω Define phase crossover frequency arg { G( jω )} G(s) = Gain margin = In db Gain Margin 1 = 180 open loop t.f. G 1 ( jω ) 1 = 20log 10 1 [ G( jω )] such that 1
14 Phase Margin The amount of lag which when applied to the open loop t.f.will cause the polar plot encircle (-1,0) point Define gain crossover frequency ω G ( jω ) 2 = 1or 0db Phase Margin = 180 + arg [ G( jω )] 2 2 such that
Figure 10.54 Effect of delay upon frequency response Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright 2004 by John Wiley & Sons. All rights reserved.
Figure 10.10 Closed-loop unity feedback system Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright 2004 by John Wiley & Sons. All rights reserved.
Figure 10.11 Bode log-magnitude plot for Example 10.2: a. components; b. composite Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright 2004 by John Wiley & Sons. All rights reserved.
Figure 10.12 Bode phase plot for Example 10.2: a. components; b. composite Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright 2004 by John Wiley & Sons. All rights reserved.
19 Example open loop t.f. G( s) = s K ( s + 1)( s + 5) R(s) + - G(s) C(s)
20 Example Positive Gain margin of 21 degrees there system is stable Now try increasing gain from 10 to 100
21 Example 50 0 Magnitude response of open loop t.f. db Magnitude Response Gain crossover frequency DB Gain -50-100 -150 10-1 10 0 10 1 10 2 Angular Frequency - rad/sec
22 Example Angle - degrees Phase Response of open loop t.f. -50-100 -150-200 -250 Phase Response Phase Crossover frequency -180 o -300 10-1 10 0 10 1 10 2 Angular Frequency - rad/sec
23 Example DB Gain Angle - degrees 50 0-50 -100-150 10-50 -100-150 Magnitude response of open loop t.f. Phase margin -1 db Magnitude Response Gain Margin 10 0 10 1 10 2 Phase Angular Response Frequency - rad/sec -200-250 -300 10-1 10 0 10 1 10 2 Angular Frequency - rad/sec
24 Example In this instance gain margin is +8db and the phase margin is +21 0 Therefore system is stable Now try gain K=100
25 Example 50 db Magnitude Response DB Gain -50 0 Negative gain margin Angle - degrees -100-50 -100-150 10-1 10 0 10 1 10 2 Angular Frequency - rad/sec Phase Response Negative phase margin -200-250 -300 10-1 10 0 10 1 10 2 Angular Frequency - rad/sec
26 Example Negative gain and phase margins mean system is unstable for gain K=100 actual values are gain margin = -12dB phase margin = -30 o
Notes on Gain and Phase 27 Margins Measure of nearness of polar plot to (-1,0) point Neither ON THEIR OWN give sufficient description of system stability both must be used together
Notes on Gain and Phase 28 Margins For minimum phase systems both margins should be positive non-minimum phase occurs when poles of OLTF exist in RHP see Ogata pp. 486-487
Notes on Gain and Phase 29 Margins Satisfactory values of gain and phase margin phase margin should be in the range 30 o -60 o gain margin should be >6dB these values lead to satisfactory damping ratios in the closed loop system Bode plot sketches should be enough to give you an idea of potential problems
Closed-Loop Transient M p = 1 2ζ 1 ζ 2 ω p = ω n 1 2ζ 2 ω 2 4 2 ( 1 2ζ ) + 4ζ 4 2 = ω ζ + BW n