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ACT3701 ASSESSMENT 3
Assessment 3
Using the Routh-Hurwitz criterion, determine the stability of the closed-loop system that has the following characteristic equations. Determine the number of roots of each equation that are in the right-half-plane and on the j
\omega
-axis.
(a)
s^(3) 25s^(2) 10s 450=0
(5)
(b)
2s^(4) 10s^(3) 5.5s^(2) 10=0
(6)
[11 marks]
Consider a unity feedback system with the closed-loop transfer function
(Y(s))/(x(s))=(\omega _(n)^(2))/(s^(2) 2\omega _(n)s \omega _(n)^(2))
For a unit step input, calculate the following performance indices
a)
IAE=\int_0^(\infty ) |e(t)|dt
.
(4)
b)
ITAE=\int_0^(\infty ) |e(t)|dt
.
(5)
c) Calculate the values of
\omega _(n)
which minimize IAE and ITAE respectively. Are these values of
\omega _(n)
practical? If not, then choose suitable values of
\omega _(n)
and determine the corresponding performance indices.
(10)
Hint:
(i) Since damping factor
\zeta =1
, the system does not overshoot. Therefore
|e(t)|=e(t)
(ii)
\int_0^(\infty ) |e(t)|dt=\lim_(s->0)E(s),
and
\int_0^(\infty ) |e(t)|dt=\lim_(s->0)[-(d)/(ds)E(s)]
[19 marks]
[30]
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