Modern Control Engineering

Finally, we present two frequently used theorems, together with Laplace transforms

of the pulse function and impulse function.

866 Appendix A / Laplace Transform Tables

Initial value theorem

Final value theorem

Pulse function

Impulse function

=

As

s

=A

=tlim

0 S^0

d

dt 0

[A( 1 - e-st^0 )]

d

dt 0

(t 0 s)

=0, for t 6 0,t 06 t

lCg(t)D=tlim

0 S^0

c

A

t 0 s

g(t)=tlim ( 1 - e-st^0 )d

0 S^0

A

t 0

, for 0 6 t 6 t 0

lCf(t)D=

A

t 0 s

-

A

t 0 s

f(t)= e-st^0

A

t 0

1 (t)-

A

t 0

1 (t-t 0 )

f(q) =tlimSqf(t)=limsS 0 sF(s)

f( 0 +)=tlimS 0 +f(t)=slimSqsF(s)

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