EES42042 Fundamental of Control Systems Bode Plots

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1 EES42042 Fundamental of Control Systems Bode Plots DR. Ir. Wahidin Wahab M.Sc. Ir. Aries Subiantoro M.Sc.

2 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

3 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.

4 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.

6 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.

9 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.

10 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.

11 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.

12 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 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 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 = arg [ G( jω )] 2 2 such that

19 19 Example open loop t.f. G( s) = s K ( s + 1)( s + 5) R(s) + - G(s) C(s)

20 20 Example Positive Gain margin of 21 degrees there system is stable Now try increasing gain from 10 to 100

21 21 Example 50 0 Magnitude response of open loop t.f. db Magnitude Response Gain crossover frequency DB Gain Angular Frequency - rad/sec

22 22 Example Angle - degrees Phase Response of open loop t.f Phase Response Phase Crossover frequency -180 o Angular Frequency - rad/sec

23 23 Example DB Gain Angle - degrees Magnitude response of open loop t.f. Phase margin -1 db Magnitude Response Gain Margin Phase Angular Response Frequency - rad/sec Angular Frequency - rad/sec

24 24 Example In this instance gain margin is +8db and the phase margin is Therefore system is stable Now try gain K=100

25 25 Example 50 db Magnitude Response DB Gain Negative gain margin Angle - degrees Angular Frequency - rad/sec Phase Response Negative phase margin Angular Frequency - rad/sec

26 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

27 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

28 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

29 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

30 Closed-Loop Transient M p = 1 2ζ 1 ζ 2 ω p = ω n 1 2ζ 2 ω ( 1 2ζ ) + 4ζ 4 2 = ω ζ + BW n

Bode Plot for Controller Design

Bode Plot for Controller Design Dr. Bishakh Bhattacharya Professor, Department of Mechanical Engineering IIT Kanpur Joint Initiative of IITs and IISc - Funded by This Lecture Contains Bode Plot for Controller