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1

### GATE EE 2003

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
$$\left( {\matrix{ 1 & 0 \cr { - {B \over J}} & { - {{{K^2}} \over {LJ}}} \cr } } \right)$$
2

### GATE EE 2002

For the system $$\mathop X\limits^ \bullet = \left[ {\matrix{ 2 & 0 \cr 0 & 4 \cr } } \right]X + \left[ {\matrix{ 1 \cr 1 \cr } } \right]u;\,\,\,y = \left[ {\matrix{ 4 & 0 \cr } } \right]X,\,$$ with u as unit impulse and with zero initial state, the output, $$y$$, becomes
A
$$2{e^{2t}}$$
B
$$4{e^{2t}}$$
C
$$2{e^{4t}}$$
D
$$4{e^{4t}}$$
3

### GATE EE 2002

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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