Digital Signal Processing Reference
In-Depth Information
Then, for the first property, (8.74),
`
s
-1
= s
2
+ 3s + 2;
det(s
I
-
A
) =
2
s + 3
`
s + 2
0
= s
2
+ 3s + 2.
det(s
I
-
A
v
) =
3
s + 1
Next, the eigenvalues are found. From
det(s
I
-
A
) = s
2
+ 3s + 2 = (s + 1)
(s + 2),
we obtain
For the second property, (8.75), the determinants of the two matrices are given by
l
1
=-1, l
2
=-2.
`
01
-2
-20
-3
ƒ
A
ƒ =
= 2;
ƒ
A
v
ƒ =
= 2,
-3
-1
and both determinants are equal to the product of the eigenvalues.
For the third property, (8.76), the traces of the two matrices are the sums of the diagonal
elements
tr
A
= 0 + (-3) =-3;
tr
A
v
=-2 + (-1) =-3.
Thus, the traces are equal to the sum of the eigenvalues.
■
The eigenvalues of an matrix can be found with the MATLAB program
in Example 8.11. The following MATLAB program also calculates eigenvalues:
A=[0 1;-2 -3], Av=[-2 0;-3 -1]
'Compare the coeficients of the characteristic polynomials'
poly(A), poly(Av)
'Compare the eigenvalues of the two matrices'
eig(A), eig(Av)
'Compare the traces of the two matrices'
trace(A), trace(Av)
n * n
In this section, we develop similarity transformations for state equations. It is
shown that any system has an unbounded number of state models. However, all
state models for a given system have the same transfer function. As a final topic,
four properties of similarity transformations for state equations are derived.
In earlier chapters, we specified the model of a continuous-time LTI system by a
differential equation or a transfer function. In both cases, the system input-output
characteristics are given. In this chapter, a third model, the state-variable model, is
developed. This model is a set of coupled first-order differential equations. The
state model can be specified either by state equations or by a simulation diagram.
