Systems Analysis and Control


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1 Systems Analysis and Control Matthew M. Peet Illinois Institute of Technology Lecture 2: Drawing Bode Plots, Part 2
2 Overview In this Lecture, you will learn: Simple Plots Real Zeros Real Poles Complex Zeros Complex Poles M. Peet Lecture 2: Control Systems 2 / 33
3 Review Recall: Frequency Response Input: u(t) = M sin(ωt + φ) Output: Magnitude and Phase Shift Linear Simulation Results y(t) = G(ıω) M sin(ωt + φ + G(ıω)) Amplitude Time (sec) Frequency Response to sin ωt is given by G(ıω) M. Peet Lecture 2: Control Systems 3 / 33
4 Review Recall: Bode Plot Definition 1. The Bode Plot is a pair of loglog and semilog plots: 1. Magnitude Plot: 2 log 1 G(ıω) vs. log 1 ω 2. Phase Plot: G(ıω) vs. log 1 ω BiteSize Chucks: G(ıω) = i (ıωτ zi + 1) i (ıωτ pi + 1) 2 log G(ıω) = i 2 log ıωτ zi + 1 i 2 log ıωτ pi + 1 M. Peet Lecture 2: Control Systems 4 / 33
5 Plotting Simple Terms Plotting Normal Zeros Behaves as G 1 (s) = (τs + 1) { ıωτ ω << 1 τ = ıωτ ω >> 1 τ A constant at low ω. A zero at high ω. Break Point at ω = 1 τ. Magnitude: db ω << 1 τ 2 log ıωτ + 1 = 3.1 db ω = 1 τ 2 log ω ω >> 1 τ ω = τ M. Peet Lecture 2: Control Systems 5 / 33
6 Plotting Real Zeros Phase Geometry: (ıωτ + 1) = tan 1 ( ωτ 1 ω << 1 τ (ıωτ + 1) = 45 ω = 1 τ 9 ω >> 1 τ ) 1+ω 2 τ 2 1 Im(s) } ωτ } Re(s) ω = τ M. Peet Lecture 2: Control Systems 6 / 33
7 Plotting Real Zeros Phase We can improve our sketch using a line (ıωτ + 1) ω <.1 τ = 45.1 (1 + log ωτ) τ < ω < 1 τ 9 ω > 1 τ 9 45 ω =.1 τ 1 ω = τ 1 ω = 1 τ M. Peet Lecture 2: Control Systems 7 / 33
8 Plotting Real Poles Magnitude We can plot real zeros Now lets do the poles. Poles are the opposite of zeros G zero = G 1 pole 1 G pole (ıω) = ıωτ + 1 G zero (ıω) = ıωτ + 1 Magnitude: G pole (ıω) = 1 ıωτ log G pole (ıω) = 2 log ıωτ M. Peet Lecture 2: Control Systems 8 / 33
9 Plotting Real Poles Magnitude Our approximation is also a reflection of the zero 2 log 1 ıωτ + 1 = db ω << 1 τ 3.1 db ω = 1 τ 2 log ω ω >> 1 τ ω = τ ω = τ M. Peet Lecture 2: Control Systems 9 / 33
10 Plotting Real Poles Phase We can plot real zeros Now lets do the poles. Poles are the opposite of zeros G pole (ıω) = 1 ıωτ + 1 = G 1 zero(ıω) Phase: G pole (ıω) = tan 1 ωτ = G zero (ıω) M. Peet Lecture 2: Control Systems 1 / 33
11 Plotting Real Poles Phase Our approximation is also a reflection of the zero (ıωτ + 1) 1 ω <.1 τ = 45.1 (1 + log ωτ) τ < ω < 1 τ 9 ω > 1 τ ω =.1 τ 1 ω = τ 1 ω = 1 τ ω =.1 τ ω = τ 19 ω = 1 τ M. Peet Lecture 2: Control Systems 11 / 33
12 Plotting Real Poles and Zeros LeadLag We now have the pieces for a LeadLag Compensator. Example: Lead Compensator First Step: Standard Form G(s) = s + 1 s + 1 G(s) = s + 1 s + 1 = 1 s s + 1 Second Step: Plot the Constant c =.1 = 2dB M. Peet Lecture 2: Control Systems 12 / 33
13 Plotting Real Poles and Zeros LeadLag Third Step: Plot the Zero at ω = 1. Magnitude: db ω << 1 2 log ıω + 1 = 3.1 db ω = 1 2 log ω ω >> ω = 1 ω = Phase: (ıω + 1) ω <.1 = 45 (1 + log ω).1 < ω < 1 9 ω > ω = 1 zero M. Peet Lecture 2: Control Systems 13 / 33
14 Plotting Real Poles and Zeros LeadLag Fourth Step: Plot the Pole at ω = 1. Magnitude: 2 log ıω db ω << 1 = 3.1 db ω = 1 2 log ω ω >> ω = 1 ω = Phase: ( ıω ) ω < 1 = 45 ( ) 1 + log ω 1 1 < ω < 1 9 ω > ω = 1 ω = 145 pole M. Peet Lecture 2: Control Systems 14 / 33
15 Plotting Real Poles and Zeros LeadLag Add them all together 2 1 ω = ω = zero ω = 1 ω = 1 total pole M. Peet Lecture 2: Control Systems 15 / 33
16 Plotting Real Poles and Zeros LeadLag Compare with the True Plot 2 1 ω = True ω = True 45 ω = 1 ω = M. Peet Lecture 2: Control Systems 16 / 33
17 Plotting Real Poles and Zeros LeadLag Example: Lag Compensator G(s) = 1 s s + 1 = s s Pole at ω = 1 Zero at ω = 1 Output Phase Lags behind the input. M. Peet Lecture 2: Control Systems 17 / 33
18 Plotting Real Poles and Zeros LeadLag Put them together to get a LeadLag Compensator. Zero at ω = 1 Pole at ω = 1 Pole at ω = 1 Zero at ω = 1 Magnitude: Changes in Slope. ω = 1: +2 ω = 1: 2 ω = 1: 2 ω = 1: +2 G(s) = s + 1 s + 1 s + 1 s + 1 = s + 1 s s s M. Peet Lecture 2: Control Systems 18 / 33
19 Plotting Real Poles and Zeros LeadLag G(s) = s + 1 s + 1 s + 1 s + 1 Phase: Changes in Slope. ω =.1: +45 ω = 1: 45 ω = 1: 9 ω = 1: +9 ω = 1: +45 ω = 1: M. Peet Lecture 2: Control Systems 19 / 33
20 Complex Poles and Zeros Now we move on to the final topic: Complex Poles and Zeros Complex Zeros: G(s) = s 2 + 2ζω n s + ω 2 n First Step: Put in standard form ( ( s G(s) = ωn 2 ω n ) ) 2 + 2ζ s + 1 ω n So y ss = 1 Ignoring the constant G 1 (ıω) = ( ω ω n ) 2 + 2ζ ω ω n ı + 1 = 2ζı ( 1 ω ) 2 ω ω n ω n << 1 ω ω n = 1 ω ω n >> 1 M. Peet Lecture 2: Control Systems 2 / 33
21 Complex Poles and Zeros Magnitude: ( ) 2 ω 2 log G 1 (ıω) = 2 log + 2ζ sω ı + 1 ω n ω n ω db ω n << 1 ω = 2 log(2ζ) ω n = 1 ω 4(log ω log ω n ) ω n >> M. Peet Lecture 2: Control Systems 21 / 33
22 Complex Poles and Zeros The Behavior near ω = ω n depends on ζ. When ζ =, 2 log G 1 (ıω) = 2 log(2ζ) = When ζ =.5, When ζ = 1, 2 log G 1 (ıω) = 2 log 1 = 2 log G 1 (ıω) = 2 log 2 = 6.2dB M. Peet Lecture 2: Control Systems 22 / 33
23 Complex Poles and Zeros Phase: ( ( ) ) 2 ω G 1 (ıω) = 1 + 2ζ ω ı ω n ω n ( ) = tan 1 2ζωωn ωn 2 ω 2 ω ω n << 1 = 9 ω ω n = 1 18 ω ω n >> 1 Re(s) Im(s) } 2 ζ ω/ω n < G(iω) } 1(ω/ω n ) M. Peet Lecture 2: Control Systems 23 / 33
24 Complex Poles and Zeros The Behavior near ω = ω n depends on ζ. For ω = ω n = 1, When ζ =, d G 1(ıω) = dω When ζ =.1, d G 1(ıω) dω When ζ =.5, d G 1(ıω) dω When ζ = 1, d G 1(ıω) dω = 1 = 2 = 1 M. Peet Lecture 2: Control Systems 24 / 33
25 Complex Poles and Zeros Comparison vs. True for ω n = 1, ζ = Good for magnitude, bad for phase M. Peet Lecture 2: Control Systems 25 / 33
26 Complex Poles and Zeros Comparison vs. True for ω n = 1, ζ = M. Peet Lecture 2: Control Systems 26 / 33
27 Complex Poles Treatment of Complex Poles is similar Complex Poles: 1 G(s) = ( ( ) ) 2 s ω n + 2ζ s ω n Magnitude: ( ) 2 ω 2 log G 1 (ıω) = 2 log 1 + 2ζ sω ı ω n ω n ω db ω n << 1 ω = 2 log(2ζ) ω n = 1 ω 4(log ω log ω n ) ω n >> 1 M. Peet Lecture 2: Control Systems 27 / 33
28 Complex Poles The Behavior near ω = ω n depends on ζ. When ζ =, 2 log G 1 (ıω) = 2 log(2ζ) = + When ζ =.5, When ζ = 1, 2 log G 1 (ıω) = 2 log 1 = 2 log G 1 (ıω) = 2 log 2 = 6.2dB M. Peet Lecture 2: Control Systems 28 / 33
29 Complex Poles Phase: ( ( ω G 1 (ıω) = 1 ω n ω ω n << 1 = 9 ω ω n = 1 18 ω ω n >> 1 ) ) 2 + 2ζ ω ı ω n M. Peet Lecture 2: Control Systems 29 / 33
30 Complex Poles The Behavior near ω = ω n depends on ζ. For ω = ω n = 1, When ζ =, d G 1(ıω) = dω When ζ =.1, d G 1(ıω) dω When ζ =.5, d G 1(ıω) dω When ζ = 1, d G 1(ıω) dω = 1 = 2 = 1 M. Peet Lecture 2: Control Systems 3 / 33
31 Complex Poles Comparison vs. True for ω n = 1, ζ = Good for magnitude, bad for phase M. Peet Lecture 2: Control Systems 31 / 33
32 Complex Poles Comparison vs. True for ω n = 1, ζ = Phase improves, Magnitude gets worse. M. Peet Lecture 2: Control Systems 32 / 33
33 Summary What have we learned today? Simple Plots Real Zeros Real Poles Complex Zeros Complex Poles Next Lecture: Compensation in the Frequency Domain M. Peet Lecture 2: Control Systems 33 / 33
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