Image Processing Reference
In-Depth Information
1, and u
(
t
) ¼
e
t
; derive expressions for x
1
(
t
)
.
(a) Find a nonsingular transformation matrix T such that the change in variables
z
¼
Tx will lead to a decoupled system of differential equations.
(b) Solve the decoupled system for the new variables z
1
(
t
)
Let x
1
(
0
) ¼
1, x
2
(
0
) ¼
and x
2
(
t
)
, and
demonstrate that your results are consistent with the results obtained for
x
1
(
t
)
and z
2
(
t
)
and x
2
(
t
)
.
4.9
A discrete LTI system is described by the following state equations:
x
1
(
k
)
x
2
(
k
)
u
(
k
)
x
1
(
k
þ
1
)
0
1
1
1
¼
þ
x
2
(
k
þ
1
)
0
:
51
:
5
k
, derive expressions for x
1
(
k
)
Let x
1
(
) ¼
1, x
2
(
) ¼
1, and u
(
k
) ¼ (
:
)
0
0
0
5
and x
2
(
k
)
.
(a) Find a nonsingular transformation matrix T such that the change in vari-
ables z
(
k
) ¼
Tx
(
k
)
will lead to a decoupled system of equations.
(b) Solve the decoupled system for the new variables
, and
demonstrate that your results are consistent with the results obtained for
x
1
(
k
)
z
1
(
k
)
and z
2
(
k
)
and x
2
(
k
)
.
4.10
Consider the continuous transfer function
3s
þ
4
H
(
s
) ¼
s
2
þ
6s
þ
8
Obtain the state-space representation of this system in controllable and observ-
able canonical forms.
4.11
Consider the following transfer function
3z
2
þ
2z
þ
1
H
(
z
) ¼
z
2
þ
0
:
7z
þ
0
:
12
Obtain the state-space representation of this system in controllable and observ-
able canonical forms.
4.12
Consider the system de
ned by
x
1
(
t
)
x
2
(
t
)
u
(
t
)
x
1
(
t
)
x
2
(
t
)
12
6
1
¼
þ
3
5
x
1
(
t
)
x
2
(
t
)
y
(
t
) ¼
½
23
(a) Transform the system equations into the controllable canonical form.
(b) Transform the system equations into the observable canonical form.
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