1
GATE EE 2000
Subjective
+5
-0
A unity feedback system has open loop transfer function $$G\left( s \right) = {{K\left( {s + 5} \right)} \over {s\left( {s + 2} \right)}};K \ge 0$$
(a) Draw a rough sketch of the root locus plot; given that the complex roots ofthe characteristic equation move along a circle.
(b) As K increases, does the system become less stable? Justify your answer.
(c) Find the value of $$K$$ (if it exists) so that the damping $$\xi$$ of the complex closed loop poles is $$0.3.$$
2
GATE EE 2000
Subjective
+5
-0
Open-loop transfer function of a unity - feedback system is $$G\left( s \right) = {G_1}\left( s \right).{e^{ - s{\tau _D}}} = {{{e^{ - s{\tau _D}}}} \over {s\left( {s + 1} \right)\left( {s + 2} \right)}}$$\$
Given : $$\,\left| {{G_1}\left( {j\omega } \right)} \right| \approx 1$$ when $$\omega = 0.446$$

(a) Determine the phase margin when $${\tau _D} = 0$$
(b) Comment in one sentence on the effect of dead time on the stability of the system.
(c) Determine the maximum value of dead time $${\tau _D}$$ for the closed-loop system to be stable.

3
GATE EE 2000
+2
-0.6
$$D\left( s \right) = {{\left( {0.5s + 1} \right)} \over {\left( {0.05s + 1} \right)}}$$ Maximum phase lead of the compensator is
A
$$52$$ deg at 4 rad/sec
B
$$52$$ deg at $$10$$ rad/sec
C
$$55$$ deg at $$12$$ rad/sec
D
none of the answers is correct
4
GATE EE 2000
Subjective
+5
-0
Consider the state equation $$\mathop X\limits^ \bullet \left( t \right) = Ax\left( t \right)$$
Given : $${e^{AT}} = \left[ {\matrix{ {{e^{ - t}} + t{e^{ - t}}} & {t{e^{ - t}}} \cr { - t{e^{ - t}}} & {{e^{ - t}} - t{e^{ - t}}} \cr } } \right]$$

(a) Find a set of states $${x_1}\left( 1 \right)$$ and $${x_2}\left( 1 \right)$$ such that $${x_1}\left( 2 \right) = 2.$$
(b) Show that $$\,{\left( {s{\rm I} - A} \right)^{ - t}} = \Phi \left( s \right) = {1 \over \Delta }\left[ {\matrix{ {s + 2} & 1 \cr { - 1} & s \cr } } \right];$$ $$\Delta = {\left( {s + 1} \right)^2}$$
(c) From $$\Phi \left( s \right),$$ find the matrix $$A$$.

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