Assignment 5 Solutions 2009

5 as the value of the controller gain is increased

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Unformatted text preview: beco mes ) e -0. 1s k c 5 + 1 s (5 + 1) + e - 0. 1 s k c = 0 s This is an irrational transfer funct ion, so in order to use the Routh’s Criterion, we need to transform the po lyno mial into a rational one. This is acco mplished by a first­order Pade approximat ion. 1 - 0.1s e 0.1 s » 1 + 0.1s Consequent ly, the characterist ic equat ion beco mes, (1 - 0.1s ) k = 0 ( 5s + 1) + c (1 + 0.1s ) or ( 5s + 1) (1 + 0.1s ) + (1 - 0.1s )kc = 0 1 + g p g c = 0 = 1 + This is now a polyno mial equat ion. Expressing this equation in the standard form yields, 0.5 s 2 + (5.1 - 0.1k c ) s + (1 + k c ) = 0 For the Routh’s criterion, all coefficients in the characterist ic po lyno mial need to be posit ive. Since k is posit ive, the condition for stabilit y, then, beco mes c ( 5.1 - 0.1kc ) > 0 Or, we can show that, kc < 5.1 / 0.1 = 51 Figures 26 and 27 illustrate the Root­Locus diagrams for two values of the controller gain, 10 and 50.5. As the value of the controller gain is increased, the locat ion o f the roots (indicated by the squares in the plot) moves towards the imaginary axis. When the value of k is around 51, it c is exact ly at the crossing point reaching the crit ical stabilit y value, which is consistent with the results provided by the Routh’s Criterion. Figure 26: Root­Locus for the blending process, squares show the root location when k c = 10 . Figure 27: Root­Locus for the blending process, squares show the root location when k c = 50.5 . The stabilit y analysis for this process was studied before and the limit ing value for the controller gain was obtained to be 51 using Routh’s criterio n and Root­Locus. We now validate these results by performing the cont inuous cycling to obtain the values of the crit ical controller gain under P­controller. Figures 28 and 29 illustrate the closed­loop response for a set­point change. As the gain is increased, the system rea...
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