Digital Signal Processing Reference
In-Depth Information
and the first property is satisfied. The second property, (8.41), yields
2e
-t
1
- e
-2t
1
e
-t
1
- e
-2t
1
2e
-t
2
- e
-2t
2
e
-t
2
- e
-2t
2
≥(t
1
)≥(t
2
) =
B
R
*
B
R
.
-2e
-t
1
+ 2e
-2t
1
-e
-t
1
+ 2e
-2t
1
-2e
-t
2
+ 2e
-2t
2
-e
-t
2
+ 2e
-2t
2
The (1, 1) element of the product matrix is given by
(1, 1) element = [2e
-t
1
- e
-2t
1
][2e
-t
2
- e
-2t
2
]
+ [e
-t
1
- e
-2t
1
][-2e
-t
2
+ 2e
-2t
2
] = [4e
-(t
1
+ t
2
)
- 2e
-(2t
1
+ t
2
)
- 2e
-(t
1
+ 2t
2
)
+ e
-2(t
1
+ t
2
)
] + [-2e
-(t
1
+ t
2
)
+ 2e
-(2t
1
+ t
2
)
+ 2e
-(t
1
+ 2t
2
)
- 2e
-2(t
1
+ t
2
)
].
Combining these terms yields
(1, 1) element = 2e
-(t
1
+ t
2
)
- e
-2(t
1
+ t
2
)
,
(8.44)
which is the (1, 1) element of
≥(t
1
+ t
2
).
The other three elements of the product matrix can
be verified in a like manner.
To illustrate the third property, (8.43), we assume that the property is true. Hence,
2e
-t
- e
-2t
e
-t
- e
-2t
2e
t
- e
2t
e
t
- e
2t
B
R
B
R
≥(t)≥(-t) =
*
=
I
.
-2e
-t
+ 2e
-2t
-e
-t
+ 2e
-2t
-2e
t
+ 2e
2t
-e
t
+ 2e
2t
As with the last property, we test only the (1, 1) element of the product. This product is given
in (8.44); in this equation, we let
t
1
= t
and
t
2
=-t,
with the result
(1, 1) element = 2e
-(t - t)
- e
-2(t - t)
= 1,
as expected. In a like manner, the other three elements of the product matrix can be verified.
■
In this section, three properties of the state-transition matrix are developed.
Property (8.40),
≥(0) =
I
,
is easily applied as a check of the calculation of a state-
transition matrix.
8.5
TRANSFER FUNCTIONS
A procedure was given in Section 8.2 for writing the state equations of a system
from the system transfer function. In this section, we investigate the calculation of
the transfer function from state equations.
The standard form of the state equations is given by
x
#
(t) =
Ax
(t) +
B
u(t)
and
y(t) =
Cx
(t) + Du(t)
(8.45)
