Control Systems
Seminar
October 9, 2023
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Problem 1
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 s-plane and on the jw-axis.
1. s3 + 25s2 + 10s + 50 = 0
2. 2s4 + 10s3 + 5.5s2 + 5.5s + 10 = 0
3. s4 + 2s3 + 10s2 + 20s + 5 = 0
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Problem 2
Given the forward-path transfer function of unity-feedback control systems, apply the Routh-Hurwitz
criterion to determine the stability of the closed-loop system as a function of K, Determine the value
or K that will cause sustained constant-amplitude oscillations in the system. Determine the frequency
of oscillation
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1. G(s) =
K(s+10)(s+20)
s2 (s+2)
2. G(s) =
K(s+1)
s3 +2s2 +3s+1
Problem 3
Reduce the block diagram shown in Fig. below and find the Y / X.
Figure 1: Problem 3
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Problem 4
Find the angles of the asymptotes and the intersect of the asymptotes of the root loci of the following
equations for both K > 0 and K < 0.
1. s3 + 5s2 + (K + 1)s + K = 0
2. s3 + 2s2 + 3s + K(s2 − 1)(s + 3) = 0
3. s4 + 2s2 + 10 + K(s + 5) = 0
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Problem 5
Construct the root-locus diagram for each of the following control systems for which the poles and
zeros of G(s)H(s) are given. The characteristic equation is obtained by equating the numerator of
1 + G(s)H(s) to zero.
1. Poles at 0, -1, -3, -4; no finite zeros
2. Poles at 0, -1+j, -1-j; zero at -2
3. Poles at 0, -1 + j, -1 - j; no finite zeros
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