Digital Signal Processing Reference
In-Depth Information
where the system input was the engine thrust and the system output was the angle of the
rocket relative to the vertical. The system modes are and the latter mode is
obviously unstable. A control system was added to the rocket, such that the overall system
was stable and responded in an acceptable manner.
e
-0.172t
e
0.172t
;
■
We restate the definition of the inverse of a system from Section 2.7 in terms of
transfer functions.
Inverse of a System
The inverse of an LTI system
H(s)
is a second system
H
i
(s)
that, when cascaded with
H(s),
yields the identity system.
Thus,
H
i
(s)
is defined by the equation
1
H(s)
.
H(s)H
i
(s) = 1 QH
i
(s) =
(7.64)
These systems are illustrated in Figure 7.14.
We now consider the characteristics of the inverse system, assuming that the
transfer function
H(s)
of a causal system can be expressed as in (7.49):
+
Á
+ b
0
b
n
s
n
+ b
n- 1
s
n- 1
H(s) =
.
(7.65)
Á
a
n
s
n
+ a
n- 1
s
n- 1
+
+ a
0
Hence, the inverse system has the transfer function
+
Á
+ a
0
a
n
s
n
+ a
n- 1
s
n- 1
H
i
(s) =
.
(7.66)
Á
b
n
s
n
+ b
n- 1
s
n- 1
+
+ b
0
This inverse system is also causal because (7.66) is a unilateral transfer function.
Note that the differential equation of the inverse system can easily be written from
(7.66), since the coefficients of the transfer function are also the coefficients of the
system differential equation. (See Section 7.6.)
Next, we investigate the stability of the inverse system. For the system of
(7.65) to be stable, the poles of the transfer function must lie in the left half of
the
s
-plane. For the inverse system of (7.66) to be stable, the poles of
H(s)
H
i
(s)
[the
Inverse
system
System
X
(
s
)
Y
(
s
)
X
(
s
)
H
(
s
)
H
i
(
s
)
H
i
(
s
)
1/
H
(
s
)
Figure 7.14
System with its inverse.
