Lecture 29 Margins: Bode, Nyquist Process Control Prof. Kannan M. - - PowerPoint PPT Presentation

Lecture 29 Margins: Bode, Nyquist

Process Control

• Prof. Kannan M. Moudgalya

IIT Bombay Monday, 14 October 2013

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Outline

• 1. One more Nyquist plot example
• 2. Recalling margins from Bode plot
• 3. Margins using Nyquist plots

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• 1. Another Nyquist Plot Example

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Example 2

Find the range of proportional controller K that will make the closed loop system stable for the plant G(s) = 10(s + 1) s(s − 10)

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• 2. Recalling margins from Bode plot

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Instability Problem Statement

◮ G(s) is open loop transfer function ◮ Does not have poles and zeros on RHP ◮ Put in a closed loop with a proportional

controller Kc

◮ As Kc increases, closed loop system

becomes unstable

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Example

Determine the gain and phase margins of an

• pen loop transfer function:

G(s) = 15 (s + 1)(s + 2)(s + 3)

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Recall: Gain Margin

G1 (dB)

• 1

1 10 10 20 15 10 5

• 5
• 10
• 15
• 20
• 25

10 10

• 2

Bode plot

• 250
• 200
• 150
• 100
• 50

10 10 10 10

• 2
• 1

1 Phase

• mag. plot goes through 0dB

When Kc is increased to 4 ωpc Gain Margin = 4 = 12dB

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Method used to get gain margin

◮ Locate ωc, where ∠G(jωc) = −180◦ ◮ Find |G(jωc)| at that point ◮ Can increase gain of the system by Kc

until Kc|G(jωc)| = 1

◮ Can verify that we can increase Kc until 4 ◮ This method was followed in the previous

slide

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Phase margin

G1 (dB)

• 1

1

10 10 20 15 10 5

• 5
• 10
• 15
• 20
• 25

10 10

• 2

Bode plot Phase

• 2
• 1

1

10

• 50
• 100

10

• 150
• 200
• 250
• 300
• 350
• 400

10 10

Phase Margin = 56.8 deg ωgc

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Method used to get phase margin

◮ When φ(ωg) = −123.2◦, |G(jωg)| = 1 ◮ ωg = 1.57 radians ◮ The phase can drop by 56.8◦ for the

system to be unstable

◮ The phase margin is 56.8◦ ◮ Call this ω as ωgc or ωg

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• 3. Margins using Nyquist plots

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Nomenclature used in Nyquist Plot

◮ C1 contour covers RHP in the s-plane ◮ C2 contour is a plot of 1 + KcG(s)

evaluated along C1 and plotted on G plane

◮ C3 is C2 − 1 ◮ Stability in terms of C3 contour:

◮ For stability, C3 contour should encircle −1

point −P times

◮ P is the number of open loop unstable poles

(or poles of G within C1)

◮ If P = 0, stability condition is, N = 0

◮ We will consider only systems with P = 0

for margins discussion

◮ So we demand no encirclement of (−1, 0)

point by C3

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C3 with Kc = 1 or Nyquist Plot

G(s) = 30 (s + 1)(s + 2)(s + 3)

R2

Im(G(s)) Re(G(s)) C41 C43 (−0.5, 0) at ω = √ 11 (5, 0) At ω = 1, (0, −3)

R1 R3 R4

Gain margin is 2. What about phase margin?

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Nyquist Plot

Re(G) Im(G)

This is only an approximate plot. The real plot will be a smooth one.

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GM: Nyquist Plot and Unit Circle

α Re(G) Im(G)

The gain margin is

• 1. α
• 2. 1/alpha
• 3. -1/alpha
• 4. Inadequate information is given

Answer: 2

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PM: Nyquist Plot and Unit Circle

φ Re(G) Im(G)

The Phase margin is

• 1. φ
• 2. −φ
• 3. 180 − φ
• 4. Insufficient

information Answer: 1

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Nyquist Plot: Gain and Phase Margin

φ Re(G) Im(G) α

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Efficacy of GM, PM Metric

◮ Have to use GM and PM together ◮ One alone is insufficient ◮ Examples ◮ Modulus/vector margin

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Can we draw Nyquist plot from Bode plot?

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What we learnt today

◮ Margins ◮ Relation between Nyquist and Bode plots

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Thank you

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