1
GATE ECE 2009
MCQ (More than One Correct Answer)
+1
-0.3
Consider the system $${{dx} \over {dt}} = Ax + Bu$$ with $${\rm A} = \left[ {\matrix{ 1 & 0 \cr 0 & 1 \cr } } \right]\,\,\,and\,\,\,{\rm B} = \left[ {\matrix{ p \cr q \cr } } \right],$$

where p and q are arbitrary real numbers. Which of the following statesments about the controllability of the system is true?

A
The system is completely state controllable for any nonzero values of p and q.
B
Only p =0 and q=0 result in controllability.
C
The system is uncontrollable for all values of p and q.
D
We cannot conclude about controllability from the given data.
2
GATE ECE 2005
+1
-0.3
A linear system is equivalently represented by two sets of state equations. $$\mathop x\limits^ \bullet = \,\,{\rm A}X\,\, + BU$$ and $$\mathop W\limits^ \bullet = \,\,CW\,\, + DU.$$ The eigen values of the representations are also computed as $$\left[ \lambda \right]\,\,\,\,and\,\,\,\,\left[ \mu \right].$$ Which one of the following statements is true?
A
$$\left[ \lambda \right]\, = \,\left[ \mu \right]\,\,and\,X\, = W$$
B
$$\left[ \lambda \right]\, = \,\left[ \mu \right]\,\,and\,X\, \ne W$$
C
$$\left[ \lambda \right]\, \ne \,\left[ \mu \right]\,\,and\,X\, = W$$
D
$$\left[ \lambda \right]\, \ne \,\left[ \mu \right]\,\,and\,X\, \ne W$$
3
GATE ECE 2002
+1
-0.3
The transfer function Y(s)/U(s) of a system described by the state equations
$$\mathop x\limits^ \bullet$$(t) = -2x(t)+2u(t)
y(t) = 0.5x(t) is
A
$${{0.5} \over {\left( {s - 2} \right)}}$$
B
$${1 \over {\left( {s - 2} \right)}}$$
C
$${{0.5} \over {s + 2}}$$
D
$${1 \over {s + 2}}$$
4
GATE ECE 1999
+1
-0.3
The system mode described by the state equations $$X = \left( {\matrix{ 0 & 1 \cr 2 & { - 3} \cr } } \right)x + \left( {\matrix{ 0 \cr 1 \cr } } \right)u,y = \left[ {\matrix{ 1 & 1 \cr } } \right]x$$\$
A
controllable and observable
B
controllable, but not observable
C
observable, but not controllable
D
neither controllable nor observable
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