Mathematical models for lti systems, B.2 mathematical models for lti systems – ElmoMC SimplIQ Digital Servo Drives-Bell Getting Started User Manual
Page 83
The SimplIQ for Steppers Getting Started & Tuning and Commissioning Guide
MAN-BELGS (Ver. 1.1)
83
B.2 Mathematical Models for LTI Systems
LTI systems, like any other system, are modeled by differential or algebraic
equations. The basic one is the differential equation, which means that the
relation between the output,
y
, and input,
u
, should obey the differential
equation
( )
( )
( )
( )
(
)
( )
u
b
u
b
u
b
u
y
a
y
a
y
a
y
a
m
n
m
n
n
n
n
0
1
1
1
1
0
1
1
1
1
+
+
+
+
=
+
+
+
+
−
−
−
−
Λ
Λ
(2)
A simple example is a DC motor in current mode, described by the differential
equation
ku
y
=
&
&
(3)
where
u
is the current and
y
the shaft angle. Let us now assume that the input
to the system (2) is
( )
t
u
ω
sin
=
. It can be confirmed by substitution that
( )
( )
(
)
ω
ϕ
+
ω
⋅
ω
=
t
sin
a
y
(4)
solves (2), that is it is the system output, where
( )
ω
a
and
( )
ω
ϕ
are the absolute
value and phase, respectively. The dependence of
α
and
ϕ
on the frequency
ω
is
called a Transfer function. For the system of (2), the transfer function is:
( )
( )
( )
( )
( )
0
1
0
1
1
1
a
j
a
j
b
j
b
j
b
j
b
m
n
n
n
n
+
⋅
+
+
+
⋅
+
+
⋅
+
⋅
−
−
ω
ω
ω
ω
ω
Λ
Λ
(5)
Note that the expression in (5) yields a complex number. The magnitude of this
number is
)
(
ω
α
and its phase is
)
(
ω
ϕ
.
A major property of an LTI system is that its output,
)
(
t
y
, for an input of the
form
( )
t
u
ω
sin
=
is
( )
( )
(
)
ω
ϕ
+
ω
⋅
ω
=
t
sin
a
y
, hence, the output is the same as
the input apart from an amplification factor
( )
ω
a
and time delay
( )
ω
ω
ϕ
−
/
. The
parameter
ω
is called the frequency of the signal
u
(and
y
),
( )
ω
a
the
amplitude of
y
and
( )
ω
ϕ
its phase.
The transfer function is the basic engineering description of a linear system. It
directly describes the frequency response - the way the system responds to a
sinusoidal signal of any frequency. The transfer function is closely related to the
Laplace transform of the system. In fact, the Laplace transform of the system is
obtained by replacing in (2) the expression
ω
j
with the Laplace variable
s
.
The
Laplace transform of the system described by the differential equation (2) is:
0
1
0
1
1
1
a
s
a
s
b
s
b
s
b
s
b
m
n
n
n
n
+
+
+
+
+
+
+
−
−
Λ
Λ
(6)