1
GATE EE 2005
MCQ (Single Correct Answer)
+2
-0.6
A state variable system
$$\mathop X\limits^ \bullet \left( t \right) = \left( {\matrix{ 0 & 1 \cr 0 & { - 3} \cr } } \right)X\left( t \right) + \left( {\matrix{ 1 \cr 0 \cr } } \right)u\left( t \right)$$ with the initial condition $$X\left( 0 \right) = {\left[ { - 1\,\,3} \right]^T}$$ and the unit step input $$u(t)$$ has

The state transition equation

A
$$X\left( t \right) = \left( {\matrix{ {t - {e^{ - t}}} \cr {{e^{ - t}}} \cr } } \right)$$
B
$$X\left( t \right) = \left( {\matrix{ {t - {e^{ - t}}} \cr {3{e^{ - 3t}}} \cr } } \right)$$
C
$$X\left( t \right) = \left( {\matrix{ {t - {e^{ - 3t}}} \cr {3{e^{ - 3t}}} \cr } } \right)$$
D
$$X\left( t \right) = \left( {\matrix{ {t - {e^{ - 3t}}} \cr {{e^{ - t}}} \cr } } \right)$$
2
GATE EE 2004
MCQ (Single Correct Answer)
+2
-0.6
The state variable description of a linear autonomous system is, $$\mathop X\limits^ \bullet = AX,\,\,$$ where $$X$$ is the two dimensional state vector and $$A$$ is the system matrix given by $$A = \left[ {\matrix{ 0 & 2 \cr 2 & 0 \cr } } \right].$$ The roots of the characteristic equation are
A
$$-2$$ and $$+2$$
B
$$-j2$$ and $$+j2$$
C
$$-2$$ and $$-2$$
D
$$+2$$ and $$+2$$
3
GATE EE 2003
MCQ (Single Correct Answer)
+2
-0.6
The following equation defines a separately exited $$dc$$ motor in the form of a differential equation $${{{d^2}\omega } \over {d{t^2}}} + {{B\,d\omega } \over {j\,\,dt}} + {{{K^2}} \over {LJ}}\omega = {K \over {LJ}}{V_a}$$

The above equation may be organized in the state space form as follows
$$\left( {\matrix{ {{{{d^2}\omega } \over {d{t^2}}}} \cr {{{d\omega } \over {dt}}} \cr } } \right) = P\left( {\matrix{ {{{d\omega } \over {dt}}} \cr \omega \cr } } \right) + Q{V_a}$$

where the $$P$$ matrix is given by

A
$$\left( {\matrix{ { - {B \over J}} & { - {{{K^2}} \over {LJ}}} \cr 1 & 0 \cr } } \right)$$
B
$$\left( {\matrix{ { - {{{K^2}} \over {LJ}}} & { - {B \over J}} \cr 0 & 1 \cr } } \right)$$
C
$$\left( {\matrix{ 0 & 1 \cr { - {{{K^2}} \over {LJ}}} & { - {B \over J}} \cr } } \right)$$
D
$$\left( {\matrix{ 1 & 0 \cr { - {B \over J}} & { - {{{K^2}} \over {LJ}}} \cr } } \right)$$
4
GATE EE 2002
MCQ (Single Correct Answer)
+2
-0.6
For the system $$X = \left[ {\matrix{ 2 & 3 \cr 0 & 5 \cr } } \right]X + \left[ {\matrix{ 1 \cr 0 \cr } } \right]u,$$ Which of the following statement is true?
A
The system is controllable but unstable
B
The system is uncontrollable and unstable
C
The system is controllable and stable
D
The system is uncontrollable and stable
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