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f (oo) = shmo [s f (s)]
s
(5.30)
This theoremis very usefulin determining the steady-state value ofa given function
using its Laplace transform.
6) Initial value theorem. The initial value of a function f (t) is given by
f (O+) = lrnr s f (s)
(5.31)
To prove this theorem, consider the Laplace transform of d f (t)ldt and take the
limit as s -+ oo in Eq. (5.25).
lm /- d~( )]e-s, dt = sl..[s f (s) - f (0)] (5.32)
As s approaches infinity, e-st approaches zero. Hence,
by
f (O) =
slun [s f (s)]
(5.33)
7) Integration theorem. The Laplace transform of the integral of f (t) is given
,[ ff (t)dt] = f(, ) +
where f -1 = f f (t) dt evaluated at t - O.
f -1(0)
(5.34)
LINEAR SYSTEMS, THEORY, AND DESIGN:A BRIEF REVIEW 445
To prove this theorem, we proceed as follows:
,[[f(t)d,] = F [ ffct)d,le-s,dt
(5.35)
=c[f(t)d,].,:+lrf(t)e-srdt (5.36)
= ~ f f (t) dtlr=0 + i, l,oo f (t)e-srdt (5.37)
f ~1(0)
+ f (s)
s
(5.38)
Hence, the theorem is proved.
8) Convolution integral. The integral of the form fo' f~ (t - r ) f2(r) dr is called
the convolution integral and is frequently encountered in the study of control
systems.
The Laplace transform of the convolution integral is given by
,[ [,' f,(t - C) f2(T) d,] = f i (s) f 2(s)
For the proof of this theorem, the reader may refer elsewhere.1.3
5.3 Transfer Function
(5.39)
Let us consider a linear system represented by the following differential equa-
tion:
y +ay + by -. kr(t)
(5.40)
where a is the damping constant, b is frequency parameter, and r(t) is the input
function. We assume y(0) = y(0) : O-
Taking the Laplace transform of both sides and using the initial conditions
y(0) -. y(0) : O, l
so that
s2y(s) + asy(s) + by(s) = kr(s)
y(s)
r~s~ =
S2 +as +b
G(s) = y~ ;
G(s) =
S2 + as +b
..
(5.41)
(5.42)
(5.43)
(5.44)
Here, G(s) is the ratio of the Laplace transform of the output to the Laplace
transform of the input and is called the transfer function of the given system. If
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446' PERFORMANCE, STABILI-FY, DYNAMICS, AND CONTROL
the input is a unit impulse function whose Laplace transform is unity, then the
transfer function is equal to the Laplace transform of the output. In other words,
by measuring the out.put for a unit-impulse function input, one can deduce the
information on the system transfer function.
5.4 System Response
The system response depends on the order ofthe system.The order ofthe system
refers to the order of the differential equation representing the physical system or
the degree of the denominator of the corresponding transfer function. For example,
mx + cx +kx - u(t)
is a second-order differential equation. If
Gi (s) =
G2(S) :
(s +a)
S2 +as + b
(5.45)
(5.46)
(5.47)
then G ] (s) is a first-order system and G2(S) is a sccond-order system.
Generally, the output or response ofa system consists of two parts: 1) the natural
or free response and 2) forced response.ln the following, we discuss the unit-step
response of typical first- and second-order syscems.
Consider a typical first-order system
s+b
G(s) : -
s+a
(5.48)
The response of this system to a unit-step function whose Laplace transform l/s
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