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 1/27 Process Control Margins

Outline 1. One more Nyquist plot example 2. Recalling margins from Bode plot 3. Margins using Nyquist plots 2/27 Process Control Margins

1. Another Nyquist Plot Example 3/27 Process Control Margins

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) 4/27 Process Control Margins

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2. Recalling margins from Bode plot 10/27 Process Control Margins

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 K c ◮ As K c increases, closed loop system becomes unstable 11/27 Process Control Margins

Example Determine the gain and phase margins of an open loop transfer function: 15 G(s) = (s + 1)(s + 2)(s + 3) 12/27 Process Control Margins

Recall: Gain Margin Bode plot 20 15 10 5 G1 (dB) 0 -5 -10 -15 -20 -25 -2 -1 0 1 10 10 10 10 When K c is increased to 4 mag. plot goes through 0dB -0 Gain Margin = 4 = 12dB -50 Phase -100 -150 -200 -250 -2 -1 0 1 ω pc 10 10 10 10 13/27 Process Control Margins

Method used to get gain margin ◮ Locate ω c , where ∠ G(j ω c ) = − 180 ◦ ◮ Find | G(j ω c ) | at that point ◮ Can increase gain of the system by K c until K c | G(j ω c ) | = 1 ◮ Can verify that we can increase K c until 4 ◮ This method was followed in the previous slide 14/27 Process Control Margins

Phase margin Bode plot 20 15 10 5 G1 (dB) 0 -5 -10 -15 -20 -25 ω gc -2 -1 0 1 10 10 10 10 Phase Margin = 56.8 deg -0 -50 -100 -150 Phase -200 -250 -300 -350 -400 -2 -1 0 1 10 10 10 10 15/27 Process Control Margins

Method used to get phase margin ◮ When φ ( ω g ) = − 123 . 2 ◦ , | G(j ω g ) | = 1 ◮ ω g = 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 ω gc or ω g 16/27 Process Control Margins

3. Margins using Nyquist plots 17/27 Process Control Margins

Nomenclature used in Nyquist Plot ◮ C 1 contour covers RHP in the s-plane ◮ C 2 contour is a plot of 1 + K c G(s) evaluated along C 1 and plotted on G plane ◮ C 3 is C 2 − 1 ◮ Stability in terms of C 3 contour: ◮ For stability, C 3 contour should encircle − 1 point − P times ◮ P is the number of open loop unstable poles (or poles of G within C 1 ) ◮ 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 C 3 18/27 Process Control Margins

C 3 with K c = 1 or Nyquist Plot 30 G(s) = (s + 1)(s + 2)(s + 3) Im(G(s)) C 43 R 4 R 1 (5 , 0) R 3 R 2 ( − 0 . 5 , 0) Re(G(s)) √ at ω = 11 C 41 At ω = 1 , (0 , − 3) Gain margin is 2. What about phase margin? 19/27 Process Control Margins

Nyquist Plot Im(G) Re(G) This is only an approximate plot. The real plot will be a smooth one. 20/27 Process Control Margins

GM: Nyquist Plot and Unit Circle Im(G) α Re(G) The gain margin is 1. α 2. 1 / alpha 3. -1 / alpha 4. Inadequate information is given Answer: 2 21/27 Process Control Margins

PM: Nyquist Plot and Unit Circle The Phase margin is 1. φ 2. − φ Im(G) 3. 180 − φ 4. Insufficient information φ Re(G) Answer: 1 22/27 Process Control Margins

Nyquist Plot: Gain and Phase Margin Im(G) α φ Re(G) 23/27 Process Control Margins

Efficacy of GM, PM Metric ◮ Have to use GM and PM together ◮ One alone is insufficient ◮ Examples ◮ Modulus/vector margin 24/27 Process Control Margins

Can we draw Nyquist plot from Bode plot? 25/27 Process Control Margins

What we learnt today ◮ Margins ◮ Relation between Nyquist and Bode plots 26/27 Process Control Margins

Thank you 27/27 Process Control Margins