Digital Signal Processing Reference
In-Depth Information
These two equations are solved for the derivative terms
dx
1
(t)
dt
R
L
x
1
(t) -
1
L
x
2
(t) +
1
L
v
i
(t)
=-
and
(8.5)
dx
2
(t)
dt
1
C
x
1
(t).
=
In addition to these coupled first-order differential equations, an equation that re-
lates the system output to the state variables is required. Because the output signal
is the capacitor voltage
v
c
(t),
the output equation is given by
y(t) = v
c
(t) = x
2
(t),
(8.6)
where we denote the system output with the common notation
To simplify the notation, the overdot is used to indicate the first derivative; for
example,
y(t).
x
#
1
(t) = dx
1
(t)
>
dt.
Then (8.5) is expressed as
x
#
1
(t) =-
R
L
x
1
(t) -
1
L
x
2
(t) +
1
L
v
i
(t)
and
(8.7)
1
C
x
1
(t).
x
#
2
(t) =
As a further simplification in notation, the state equations are written in a vector-
matrix format. From (8.6) and (8.7),
R
L
-
1
L
1
L
0
≥
-
x
#
1
(t)
x
#
2
(t)
x
1
(t)
x
2
(t)
B
R
B
R
=
¥
+
C
S
v
i
(t)
1
C
0
and
(8.8)
x
1
(t)
x
2
(t)
B
R
y(t) = [0
1]
.
These then are state equations for the circuit of Figure 8.1. As we discuss in Section 8.6,
this set is not unique; that is, we can choose other variables to be the states of the
system.
The standard form for the state equations of a continuous-time LTI system is
given by
x
#
(t) =
Ax
(t) +
Bu
(t)
and
(8.9)
y
(t) =
Cx
(t) +
Du
(t),
