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the desired handling qualities. It was not possible to go into all the details oflinear
systems theory. Readers interested in getting more detailed information on control
systems may refer elsewhere.l-3
References
IOgata, K., Modem Conrrol Engineering, Prentice-Hall, Englewood Cliffs, NJ, 1986.
2Norman, S., and Nisc, N. S,, Control System Engineering, Benjamin/Cummings, I 992.
3Kuo, B., Automatic Control Systems, 4th ed., Prentice-Hall, Englewood Cliffs, NJ,
1982.
4Pro_MATIAB for Sun Workstations, The MathWorks, Natick, MA, Jan. 1990
5
g,
g
o
= [-,/
-6is Og5]
532 PERFORMANCE, STABILITY, DYNAMICS, AND CONTROL
Problems
5.1 Sketch the Bode plot for the open-loop systems with
50
(a) G(s) = s~s+5~
25(s + 2)
(b) G(s) = ~s+3~s+5~s+7)
5.2 Using Routh's criterion, examine the stability ofthe closed-loop system with
a characteristic polynomial grven by
(a) S4+5S3+3S2+s+2 = O
(b) S4 +2S3 +O.OOIS2 +3s +4 = 0
(C) S4 +4S3 +7s +2 = O
[Answer: (a) Two sign changes, unstable; (b) Two sign changes, unstable; and
(c) Two sign changes, unstable.]
5.3 Sketch the root-locus for a unity feedback system with
k(s + 3)
(a) G(s) = s~s+2~s+4)
k(s + 1)
(b) G(s) = s~s+2~s+3 (s+7)
Determine the value of the gain k when the closed-loop system in (b) becomes
unstable.
5-4 Sketch the root-locus for a unity feedback system with
;)(s + 7)
G(s) = k~+~3~ 1
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