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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 Fall '08
 Hjortso,M
 WCO, Closedloop response

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