ZOH
equivalence in statespace
ZOH
Equivalence
P
(
s
)
ZOH
a
a
T
A
A
A
A
u
(
k
)
u
(
t
)
y
(
t
)
y
(
k
)
P
(
s
)
dx
(
t
)
dt
=
Ax
(
t
)
+
B u
(
t
)
y
(
t
) =
C x
(
t
) +
D u
(
t
)
.
We would like to get a description of the form,
P
(
z
)
x
(
k
+ 1) =
A
d
x
(
k
) +
B
d
u
(
k
)
y
(
k
) =
C
d
x
(
k
) +
D
d
u
(
k
)
.
Approach:
Solve the state equation over one sample period.
x
(
t
) = e
At
x
(0) +
i
t
0
e
A
(
t

τ
)
Bu
(
τ
)
dτ,
And over a single sample period (
kT
to
kT
+
T
) this is,
x
(
kT
+
T
) = e
AT
x
(
kT
) +
i
kT
+
T
kT
e
A
(
kT
+
T

τ
)
Bu
(
τ
)
dτ,
Roy Smith: ECE 147b
10
: 2
Discretetime equivalents
Continuous to discrete transforms in statespace
We have several ways of calculating a discretetime transfer function from a continuoustime
one, depending on the application.
ZOH
Equivalence
P
(
s
)
ZOH
a
a
T
A
A
A
A
u
(
k
)
u
(
t
)
y
(
t
)
y
(
k
)
P
(
z
) =
(
1

z

1
)
Z
b
P
(
s
)
s
B
.
Controller approximation
z

1
a
a
x
(
k
+ 1)
x
(
k
)
1
s
A
A
dx
(
t
)
dt
x
(
t
)
Forward di±erence:
C
(
z
) =
C
(
s
)

s
=
z

1
T
Backward di±erence:
C
(
z
) =
C
(
s
)

s
=
z

1
Tz
Tustin/bilinear:
C
(
z
) =
C
(
s
)

s
=
2
T
(
z

1)
(
z
+1)
Roy Smith: ECE 147b
10
: 1
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentZOH
equivalence in statespace
ZOH
equivalent
So far we have calculated
x
(
k
+ 1) as a linear function of
x
(
k
) and
u
(
k
). What about
y
(
k
)?
y
(
kT
) =
C x
(
kT
) +
D u
(
kT
)
.
By deFnition,
y
(
k
) =
y
(
kT
), and as
u
(
t
) is constant over the sample period,
u
(
k
) =
u
(
kT
).
y
(
k
) =
C x
(
k
) +
D u
(
k
)
.
Clearly then,
C
d
=
C
and
D
d
=
D
.
b
A
B
C
D
B
ZOH
=
⇒
e
AT
i
T
0
e
Aη
B dη
C
D
A
d
and
B
d
are calculated via
Matlab
commands
c2d
or
zohequiv
.
Roy Smith: ECE 147b
10
: 4
ZOH
equivalence in statespace
Key observation
The integration involves
u
(
τ
) from
τ
=
kT
to
τ
=
kT
+
T
.
But
u
(
τ
) is
constant
over this time period. It is the output of a
ZOH
.
So,
u
(
τ
) =
u
(
k
) for
kT
≤
τ < kT
+
T
.
Therefore,
x
(
kT
+
T
) = e
AT
x
(
kT
) +
bi
kT
+
T
kT
e
A
(
kT
+
T

τ
)
B dτ,
B
u
(
k
)
.
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '07
 RODWELL
 Tom DeLay, Roy Smith

Click to edit the document details