Problem 1. i) Sketch the root locus for the feedback system in Fig. 1 with the given transfer functions. ii)
Find all information about the root locus (such as asymptotes, break-in and away points, angle of
departure and arrival), if applicable. iii) Find a critical gain if it exists.
Compensator
Plant
$R(s)$
$E(s)$
$M(s)$
$C(s)$
$G_c(s)$
$G_p(s)$
Sensor
$H(s)$
Fig. 1.
$s+5$
a) $G_p(s) = \frac{s+5}{s(s+2)(s+4)}$
$G_c(s) = k$,
$H(s) = 1$
1
b) $G_p(s) = \frac{1}{s(s+2)(s+4)}$
$G_c(s) = k$,
$H(s) = s + 3$
$s+2$
c) $G_p(s) = \frac{(s+2)}{(s+3)(s^2+2s+2)}$
$G_c(s) = k$,
$H(s) = 1$
$s^2-4s+20$
d) $G_p(s) = \frac{s^2-4s+20}{(s+2)(s+4)}$
$G_c(s) = k$,
$H(s) = 1$
1
e) $G_p(s) = \frac{1}{s(s+5)(s+6)(s^2+2s+2)}$
$G_c(s) = k$,
$H(s) = s + 3$
Note. Sketch the root locus for positive gain.
Note. Compute angle of departure or arrival only at complex conjugate poles or zeros.
