Explanation Given: The equation is G p ( s ) = 8 ( 2 s + 2 ) ( s + 2 ) ( 4 s + 12 ) . Concept Used: Ultimate cycle technique of root locus method. Calculation: G p ( s ) = 8 ( 2 s + 2 ) ( s + 2 ) ( 4 s + 12 ) G p ( s ) = 8 2 × 4 ( s + 1 ) ( s + 2 ) ( s + 3 ) G p ( s ) = 1 ( s + 1 ) ( s + 2 ) ( s + 3 ) G p ( s ) = 1 s 3 + 6 s 2 + 11 s + 6 The ultimate cycle method starts by using proportional action only. The transfer function for P action with this plant is C ( s ) R ( s ) = K p s 3 + 6 s 2 + 11 s + 6 + K p The characteristic equation is s 3 + 6 s 2 + 11 s + 6 + K p = 0 To apply the ultimate cycle method. We must find the ultimate period P u and the associated gain is K P u . To do this analytically we note that sustained oscillations occur when a pair of roots is purely imaginary, and the rest of the roots have negative real parts. To calculate this put s = j ω u where ω u is unknown frequency of oscillation. Replacing the values in the above characteristic equation we get, ( j ω u ) 3 + 6 ( j ω u ) 2 + 11 ( j ω u ) + 6 + K p u = 0 Separating the real and the imaginary parts we get, ( j ω u ) 3 + 6 ( j ω u ) 2 + 11 ( j ω u ) + 6 + K p u = To determine (b) The plot for unit step response for the design and if PID response is unsatisfactory tuning the gains to improve its performance by using the root locus plot.

3rd Edition

ISBN: 9780073398068

Author: III William J. Palm

Publisher: MCG

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Question

Chapter 11, Problem 11.26P

To determine

(a)

The control gains for PID controller by ultimate cycle method.

To determine

(b)

The plot for unit step response for the design and if PID response is unsatisfactory tuning the gains to improve its performance by using the root locus plot.

Expert Solution & Answer

Chapter 11 Solutions

System Dynamics

Ch. 11 - Prob. 11.1P Ch. 11 - Prob. 11.2P Ch. 11 - Prob. 11.3P Ch. 11 - Sketch the root locus plot of ms~ 4- 12s + 10 = 0...Ch. 11 - Prob. 11.5P Ch. 11 - Prob. 11.6P Ch. 11 - Prob. 11.7P Ch. 11 - PID control action is applied to the plant GPU) =...Ch. 11 - Consider the following equation where the...Ch. 11 - 11.10 In the following equation. K > 0. j2(j+9) +...

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