Part A | 10.1
230 Part A Fundamentals
2. Differentiation with respect to time
L fx.t/g D X.s/ ) L fP x.t/g D sX.s/ x.t D 0/ :
(10.4)
3. Integration with respect to time
L fx.t/g D X.s/
y.t/ D
Z
x../d
9
=
;
) L
8
<
:
t
Z
0
x../d
9
=
;
D
X.s/
s
and L
8
<
:
t
Z
1
x../d
9
=
;
D
X.s/
s
C
x.t D 0/
s
:
(10.5)
4. Initial and final value theorems
L fx.t/g D X.s/
)
8
<
:
lim
t!0
.x.t// D lim
s!1
.sX.s//
lim
t!1
.x.t// D lim
s!0
.sX.s// :
(10.6)
5. Theorem of convolution
X 1 .s/ D L fx 1 .t/g
X 2 .s/ D L fx 2 .t/g
x.t/ D x 1 .t/ x 2 .t/
,
C1 Z
1
x 1 .v /x 2 .t v /dv
9
> > > > > > > =
> > > > > > > ;
)
X.s/ D L fx.t/g D X 1 .s/X 2 .s/ :
(10.7)
The signal x.t/ resulting from a convolution operation must becausal; it may not depend on the values of
signals x 1 .t 1 / and x 2 .t 2 / that will occur at time instants
t 1 and t 2 , respectively, later than t. In effect, the limits
of the convolutional integral in (10.7) must be bounded
as follows
t 1 D v Ä t
t 2 D t v Ä t ) v 0
)
x 1 .t/ x 2 .t/ D
t
Z
0
x 1 .v /x 2 .t v / dv :
Furthermore, the convolution operation, for which symbol ˝ is also used in some texts, obtains the commutative, associative, and distributive with respect to
addition properties. The identity element of convolution
is the Dirac delta signal defined in the next section.
The proof of the above Laplace transform properties
as well as various other lemmas of the transform can be
sought after in either ordinary differential equations or
introductory control systems literature [10.1, 2]. Some
important Laplace transform pairs, used throughout this
text, are given in Table 10.1.
10.1.3 Linear Time-Invariant Systems
Definition of LTI Systems
A category of dynamic systems of high importance in
feedback control theory is linear time-invariant (LTI)
systems. Such systems are described by linear differential equations with constant coefficients. The general
form of such differential equations is given below for
a single-input, single-output (SISO) system
Â
a n
d
n
dt n C a n1
d
n1
dt n1 C C C C C a 1
d
dt
C a 0
Ã
y.t/
D
Â
b m
d
m
dt m C b m1
d
m1
dt m1 C C C C C b 1
d
dt
C b 0
Ã
u.t/ :
(10.8)
In the above, y.t/ and u.t/ are the input and output
signal, respectively, of the system. The initial conditions required are those for the up to the (n 1)-th
derivative with respect to time of the output signal
y.t/ at some initial time instant t 0 (commonly t 0 is
set to zero by an appropriate translation of the time
axis).
d v
dt v y.t/
ˇ
ˇ
ˇ
ˇ
tDt0
D y
.v /
0 ;
v D 0; 1; : : : ; .n 1/
(10.9)
In practically realizable systems, that is, systems that
can be implemented by interconnecting electrical (induction coils, capacitors, resistors, etc.), or mechanical
(masses, springs, dampers, etc.), elements, it holds that
m Ä n; this assumption will be observed in the following of this text as well.
Order of an LTI System
The order of any LTI system is integer n, that is, the
order of its differential equation.
Transfer Function of an LTI System
Considering a SISO–LTI system as in (10.8) with zero
initial conditions in (10.9), by consecutively applying
Laplace transform property (10.3) and then property
(10.4), the following algebraic relationship between the
Laplace transformations of the output and input signal