Control system miscellaneous
- The input x(t) and output y(t) of a system are related as
The system is
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Since y(t) and x(t) are related with some function of time, so they are not time-invariant.
Let x(t) be bounded to some finite value k.
y(t) is also bounded. Thus system is stable.Correct Option: B
Since y(t) and x(t) are related with some function of time, so they are not time-invariant.
Let x(t) be bounded to some finite value k.
y(t) is also bounded. Thus system is stable.
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The matrix [A] = 
2 1 
4 -1
is decomposed into a product of a lower triangular matrix [L] and an upper triangular matrix [U]. The properly decomposed [L] and [U] matrices respectively are
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By Dolittle’s decomposition,
2 1 = 1 0 u11 u12 4 -1 l21 1 0 u22
We have u11 = 2, u12 = 1
l21 u11 = 4
⇒ l21 = 2
and l 21u12 + u22 = – 1
⇒ u22 = – 1 – l 21u12 = – 3then , 2 1 = 1 0 2 1 .......(A) 4 -1 2 1 0 -3
eqn (A) does not signfy any choice
Now, from crout’s decomposition,2 1 = l11 0 1 u12 4 -1 l21 l22 0 1
We get, l11 = 2, l11 u12 = 1u12 = 1 = 0.5 2
and, l 21 = 4 , l21 u12 + l22 = – 1
⇒ l22 = – 3Then , 2 1 = 2 0 1 0.5 4 -1 4 -3 0 1
Hence the choice is (d).Correct Option: D
By Dolittle’s decomposition,
2 1 = 1 0 u11 u12 4 -1 l21 1 0 u22
We have u11 = 2, u12 = 1
l21 u11 = 4
⇒ l21 = 2
and l 21u12 + u22 = – 1
⇒ u22 = – 1 – l 21u12 = – 3then , 2 1 = 1 0 2 1 .......(A) 4 -1 2 1 0 -3
eqn (A) does not signfy any choice
Now, from crout’s decomposition,2 1 = l11 0 1 u12 4 -1 l21 l22 0 1
We get, l11 = 2, l11 u12 = 1u12 = 1 = 0.5 2
and, l 21 = 4 , l21 u12 + l22 = – 1
⇒ l22 = – 3Then , 2 1 = 2 0 1 0.5 4 -1 4 -3 0 1
Hence the choice is (d).
- If the electrical circuit of Fig.(b) is an equivalent of the coupled tank system of Fig.(a), then
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C1 = capacity of tank (1)
C2 = capacity of tank (2)
h1, h2 = heights of water
Q1, Q2 = flow rates
R1, R2 = resistances of flowQ1 = Q2 + C1 dh1 dt Q2 = Q3 + C2 dh2 dt ∴ Q2 = h1 - h2 , Q3 = h2 R1 R2
Equivalent circuit :
∴ A, C = capacitance
B, D = resistances
Correct Option: D
C1 = capacity of tank (1)
C2 = capacity of tank (2)
h1, h2 = heights of water
Q1, Q2 = flow rates
R1, R2 = resistances of flowQ1 = Q2 + C1 dh1 dt Q2 = Q3 + C2 dh2 dt ∴ Q2 = h1 - h2 , Q3 = h2 R1 R2
Equivalent circuit :
∴ A, C = capacitance
B, D = resistances
