Question

Question 2 System Stability in the s-Domain and in the Frequency Domain: Bode Plots, Root Locus Plots and Routh- Hurwitz Criterion ofStability A unit feedback control system is to be stabilized using a Proportional Controller, as shown in Figure Q2.1. Proportional Controller Process The process transfer function is described as follows: Y(s) G(s) s2 +4s 100 s3 +4s2 5s 2 Figure Q2.1 Your task is to investigate the stability of the closed loop system using s-domain analysis by finding: a) the value of the Proportional Controller Gain or Gains at which the closed loop system becomes marginally stable (Kp - Kcrit) and the corresponding frequency or frequencies of marginally stable oscillations (wosc), and b) the range or ranges of safe operating gains for the Proportional Controller. 1) (8 marks) Root Locus sketch for the system in question is provided in Figure Q2.2. Use it to read off the frequency or frequencies (wosc) corresponding to the frequency of marginally stable oscillations. Next, use the Root Locus Magnitude Criterion to determine the corresponding values of critical gains (Kcrit). Finally, use the Root Locus sketch to decide on the range or ranges of safe operating gains 2) (6 marks) The frequency response (Bode plots) for the process G(s) are shown in Figure Q2.3. Use them to verify the above calculations of the critical value (or values) of the gain, Kcrit , and the corresponding frequency (or frequencies) of marginally stable oscillations 3) (6 marks) Finally, apply the Routh-Hurwitz Criterion of Stability to verify the above calculations of the critical value (or values) of the gain, Kcrit , and the corresponding frequency (or frequencies) of marginally stable oscillations wosc, as well as to establish the range or ranges of safe operating gains for the Proportional Controller.

Answer 1

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