Closed-Loop Control System - IES Academy

India’s No 1
Control System
IES Academy Chapter 1
Theorem 3: Differentiation
dF(t)
i. L = SF(S)-F(0)
dt
[ ]
2
dF(t) 2 1
ii. L = ⎡S F(S)-F(0)-f (0)
2
dt
1 dF(0)
where, F (0) = dt
In general, for higher order derivatives or F(t)
n
⎡d F()
t ⎤
L ⎢ ⎥ = sFS
n
− s f O − s f − f
⎣ dt ⎦
⎣
n n−1 n−2 (1) ( n−1)
( ) ( ) (0) (0)
Where, F 1 (0) denotes the i th order derivative of f(t) with respect to t1,
Theorem 4: Integration
⎡
i. L∫
F(t) = ⎢ +
⎣ S
1
F(S) F (0)
⎡
ii. L∫∫
F(t) = ⎢ + +
⎣ S S S
S
⎤
⎥
⎦
F(S)
1
F (0)
2
F (0)
2 2
Theorem 5: Shift in time
The laplace transform of F(t) delayed by time T is equal to the laplace transform F(t)
multiplied by e –ST that is
-ST
L [ F(t – T)u
s(t – T) ] = e F(S)
Where US(t–T) denotes the unit step function that is shifted in time to the right by T.
Theorem 6: Complex shifting
The laplace transform of F(t) multiplied by
transform F(S), with S replaced by ( S ± α )
L ⎡
⎣e
∓αt
Theorem 7: Initial-value theorem
If the laplace transform of F(t) is F(S), then
t→0
αt
e ∓
⎤
⎦
⎤
⎥
⎦
, where α is a constant is equal to the laplace
that is
F(t) ⎤
⎦=F(S±α)
lim F( t) = lim SF( S )
S→∞
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