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Find the value of the gain k so that the closed-loop system is stable and is operat,ing
with a damping ratio of 0.7.
5.5 For the following unity feedback systems, sketch the Nyquist plot.
~
25
. (a) G(s) = ~s+ ) s+3~
k(s + 2)
(b) G(S)=~S 3~s 4.)
LINEAR SYSTEMS, THEORY, AND DESIGN: A BRIEF REVIEW 533
For the system in (b), determine the \ralue of the gain for which the closed loop
system becomes unstable.
5.6 Using the Nyquist plot, determine the gain and phase margins of the system
given by
G(s)=k(,+~~( 3~
)(s + 9)
5.7 For the system in Exercise 5.6, use Bode plots to obtain the gain and phase
margins. '
5.8 For a unity feedback system with
rc
G(s) = (+2;(s +4)
design a PI controller to reduce the steady-state error to zero for a unit-step input.
Assume that the system is operating with a damping ratio of 0.6. Plot the unit~step
response to verify your design. Compare the values of Ts and Tp for the basic and
compensated systems.
5.9 For the system given in Example 5.8, design alag compensator to reduce the
steady-state error by a factor of 15.
5.10 For the unity feedback system given by
k
G(s) = ~s+ ) s+2~s+3~
(a) Determine the value of the gain k for 15% overshoot. Determine the corre-
sponding values of Ts and Tp.
(b) Design a PD controller for reducing Tp by a factor of 2 and Ts by 50o/o, while
operating at 15o/o overshoot in both cases.
5.11 Given the unity feedback system with
rc
G(s) = ~+3k~+5~
determine the value ofgain k for the system to operate with a damping ratio of0.51.
Find the corresponding locations of closed-loop poles. If a lead compensator is to
be designed to reduce the time for peak amplitude Tp by 50qo, with compensator
zero placed at -2.5, find the compensator polelocation. How does the performance
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