Digital Signal Processing Reference
In-Depth Information
Hence, the denominator polynomial of
H(s)
is the determinant of
(s
I
-
A
);
the
poles of the transfer function are the roots of
det(s
I
-
A
) = 0.
(8.54)
This equation is then the system characteristic equation. Note that the stability is a
function only of the system matrix
A
and is not affected by
B, C,
or
D
. In fact, (8.54)
is the characteristic equation of a multivariable system, which has more than one
input and more than one output. We now consider an example illustrating stability.
Stability from state equations
EXAMPLE 8.11
We consider the second-order system of Figure 8.8, and we wish to find the range of the
parameter
a
for which this system is stable. We write the state equations directly from
Figure 8.8:
x
#
(t) =
-2
-a
B
R
x
(t).
1
-4
We have ignored the input and output terms because stability is independent of these terms.
From (8.54), the characteristic equation is given by
s + 2
a
B
R
= s
2
+ 6s + 8 + a = 0.
det(s
I
-
A
) = det
-1
s + 4
The zeros of this polynomial are given by
-6 ;
2
36 - 4(8 + a)
2
-6 ;
2
4 - 4a
s =
=
=-3 ;
2
1 - a.
2
x
1
(
t
)
x
1
(
t
)
5
2
u
(
t
)
y
(
t
)
a
x
2
(
t
)
x
2
(
t
)
2
6
4
Figure 8.8
System for Example 8.11.
