Homework 3 - MAE143B Spring 2010: due Thursday April 22
Question 1: Initial state calculation
Consider the state-variable realization of the system described by the
n
th
-order ordinary differential
equation
y
(
n
)
(
t
) +
a
1
y
(
n
-
1)
(
t
) +
· · ·
+
a
n
-
1
˙
y
(
t
) +
a
n
y
(
t
) =
b
1
u
(
n
-
1)
(
t
) +
· · ·
+
b
n
-
1
˙
u
(
t
) +
b
n
u
(
t
)
,
with initial conditions
y
(0)
,
˙
y
(0)
, . . . ,
y
(
n
)
(0)
;
˙
x
(
t
)
=
Ax
(
t
) +
Bu
(
t
)
,
y
(
t
)
=
Cx
(
t
) +
Du
(
t
)
.
Our aim is to transfer the known initial conditions on output
y
(
t
)
to initial conditions on the state
vector
x
(
t
)
.
For this, assume that the initial conditions on
u
(
t
)
are all zero.
[The argument for
this is that
u
(
t
)
is ours to control and we typically do things like step, impulse and sine-wave test,
which are preceded by zero input. If the initial conditions of
u
were important, then – according to
our definition of state – we would have to incorporate them into the state and we could do this by
redefining the initial conditions on
y
.]
Using the state equations, show that
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- Winter '10
- Bitmead
- initial conditions, various initial conditions
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