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Assignment_4_2007_Solutions

1 s 12 c s s 05 s 3 k and 00 4 3 39

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Unformatted text preview: 12) 3s 3 s + 5 2 + 9 + 12 s s Using a P controller, i.e., g c ( = k c , we obtain the characteristic equat ion as: s) 3 + 12 s æ öæ 1 ö 1 + g p g c g m = 0 = 1 + ç 3 ÷ç ÷k c 2 . s s s è s + 5 + 9 + 12 øè 0 5 + 1 ø o r 0 = ( s 3 + 5 2 + 9 + 12)( . + 1) + ( s + 12 c s s 0 5 s 3 ) k and 0 = 0. 4 + 3. 3 9. 2 + (15 + 3 c ) s + (12 + 12 c ) 5s 5 s 5 s k k Figures 1 and 2 show the Root­Locus diagram for this system with the squares indicating different values of the controller gain. It can be seen fro m the figures that, as the gain are increased, the system moves towards the unstable regio n. The crit ical value for the controller gain (fro m Figure 7) is 2.35 since at this po int the Root­Locus crosses the imaginary axis. Figure 1: Root­locus diagram (squares indicate the location of locus for k c = 1.14 ). Figure 2: Root­locus diagram (squares indicate location of locus for k c = 2.4 ). The Nyquist plot for the system under P­controller for a value of the controller gain equal to 2.35...
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