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Statespace systems
Representations:
From transfer function to statespace.
Consider a linear, shift invariant, system:
P
(
z
)
±±
u
(
k
)
y
(
k
)
We can express this as a transfer function,
y
(
z
)=
P
(
z
)
u
(
z
b
(
z
)
a
(
z
)
u
(
z
)
where
a
(
z
) and
b
(
z
) are polynomials, so,
P
(
z
b
(
z
)
a
(
z
)
=
b
0
z
m
+
b
1
z
m

1
+
···
+
b
m
z
n
+
a
1
z
n

1
+
+
a
n
.
For causal systems the order of
b
(
z
) is less than or equal to the order of
a
(
z
). So
m
≤
n
above.
Assume for now that
m < n
,
Roy Smith: ECE 147b
8
:2
Statespace systems
Approach:
•
Represent the plant,
P
(
z
) (or
P
(
s
)) as an
n
th order di±erential equation.
•
Represent
n
th order di±erential equation as a 1st order matrix di±erential equation with
dimension
n
.
•
Design methods now involve linear algebra.
•
Easy to handle large systems (with
Matlab
).
•
Easy to handle systems with multiple inputs and outputs.
•
Easy to simulate systems.
x
(
k
+ 1) =
Ax
(
k
)+
Bu
(
k
)
,
y
(
k
Cx
(
k
Du
(
k
)
.
A
,
B
,
C
and
D
can be matrices.
x
(
k
) is a vector (state vector).
Roy Smith: ECE 147b
8
:1
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View Full DocumentStatespace systems
Chain of delays:
Consider the frst equation:
ζ
(
z
)=
1
a
(
z
)
u
(
z
)
We want to develop a chain oF delay model to get
ζ
(
k
):
First step:
Write the expression For
n
delays:
ζ
(
k
z

1
ζ
(
k
+ 1)
.
.
.
.
.
.
ζ
(
k
+
n

1) =
z

1
ζ
(
k
+
n
)
In pictures .
..
z

1
z

1
z

1
±
±
±
±
±
ζ
(
k
+
n
)
ζ
(
k
+ 1)
ζ
(
k
+
n

2)
ζ
(
k
+
n

1)
ζ
(
k
)
Second step:
Express
ζ
(
k
+
n
) in terms oF
ζ
(
k
), .
. . ,
ζ
(
k
+
n

1) and
u
(
k
).
Roy Smith: ECE 147b
8
:4
Statespace systems
Outline:
1. Draw the system as an interconnected “chain oF delays”,
2. Relabel the signals in the system,
3. Rewrite the input/output equations in terms oF the new signals,
4. Abbreviate the equations to a matrix Form (statespace).
Drawing a digital system block diagram in terms oF delays is exactly the same as drawing a
continuous system block diagram in terms oF integrators.
Split the system,
b
(
z
)
1
a
(
z
)
±
±
±
u
(
z
)
ζ
(
z
)
y
(
z
)
ζ
(
z
1
a
(
z
)
u
(
z
)
and
y
(
z
b
(
z
)
ζ
(
z
)
.
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 Spring '07
 RODWELL

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