State-Variable Representation of Discrete-Time Systems
Summary: State-variable, or state-space, representations provide a general description of all linear, time-invariant (LTI) systems that is useful both for their analysis and for generating alternate forms with more convenient implementation or with less sensitivity to quantization.
State and the State-Variable Representation
Definition 1:
State
Essentially, the state of a system is the information held in the
delay registers in a filter structure or signal flow graph.
the minimum additional information
at time n n
fact: Any LTI (linear, time-invariant) system of finite order
M M
x n + 1 = A x n + B u n
x
n
1
A
x
n
B
u
n
y n = C x n + D u n
y
n
C
x
n
D
u
n
x x
M x 1
x
M
1
u n
u
n
n n
y n
y
n
n n A A
M x M
x
M
M
B B
M x 1
x
M
1
C C
1 x M
x
1
M
D D
1 x 1
x
1
1
One can always obtain a state-variable description of a signal
flow graph.
Example 1: 3rd-Order IIR
y n = -
a
1
y n - 1 -
a
2
y n - 2 -
a
3
y n - 3 +
b
0
x n +
b
1
x n - 1 +
b
2
x n - 2 +
b
3
x n - 3
y
n
a
1
y
n
1
a
2
y
n
2
a
3
y
n
3
b
0
x
n
b
1
x
n
1
b
2
x
n
2
b
3
x
n
3
x
1
n + 1
x
2
n + 1
x
3
n + 1 = 0 1 0 0 0 1 -
a
3
-
a
2
-
a
1
x
1
n
x
2
n
x
3
n + 0 0 1 u n
x
1
n
1
x
2
n
1
x
3
n
1
0
1
0
0
0
1
a
3
a
2
a
1
x
1
n
x
2
n
x
3
n
0
0
1
u
n
y n = -
a
3
b
0
-
a
2
b
0
-
a
1
b
0
x
1
n
x
2
n
x
3
n +
b
0
u n
y
n
a
3
b
0
a
2
b
0
a
1
b
0
x
1
n
x
2
n
x
3
n
b
0
u
n
 Figure 1 |
Problem 1
Is the state-variable description of a filter
H z
H
z
Problem 2
Does the state-variable description fully describe the
signal flow graph?
State-Variable Transformation
Suppose we wish to define a new set of state variables, related
to the old set by a linear transformation:
q n = T x n
q
n
T
x
n
T T
M x M
x
M
M
q n
q
n
x n = T -1 q n
x
n
T
q
n
x n + 1 = A x n + B u n ⇒ T -1 q n = A T -1 q n + B u n ⇒ q n = T A T -1 q n + T B u n
⇒
x
n
1
A
x
n
B
u
n
T
q
n
A
T
q
n
B
u
n
q
n
T
A
T
q
n
T
B
u
n
y n = C x n + D u n ⇒ y n = C T -1 q n + D u n
⇒
y
n
C
x
n
D
u
n
y
n
C
T
q
n
D
u
n
q n =
A
^
q n +
B
^
u n
q
n
A
^
q
n
B
^
u
n
y n =
C
^
q n +
D
^
u n
y
n
C
^
q
n
D
^
u
n
A
^
= T A T -1
A
^
T
A
T
B
^
= T B
B
^
T
B
C
^
= C T -1
C
^
C
T
D
^
= D
D
^
D
These transformations can be used to generate a wide
variety of alternative stuctures or implementations of a filter.
Transfer Function and the State-Variable Description
Taking the z z
Z x n + 1 = Z A x n + B u n
Z
x
n
1
Z
A
x
n
B
u
n
Z y n = Z C x n + D u n
Z
y
n
Z
C
x
n
D
u
n
⇓
⇓
z X z = A X z + B U z
z
X
z
A
X
z
B
U
z
Y z = C X n + D U n
Y
z
C
X
n
D
U
n
z I - A X z = B U z ⇒ X z = z I - A -1 B U z
⇒
z
I
A
X
z
B
U
z
X
z
z
I
A
B
U
z
Y z = C z I - A -1 B U z + D U z = C - z I -1 B + D U z
Y
z
C
z
I
A
B
U
z
D
U
z
C
z
I
B
D
U
z
H z = C z I - A -1 B + D
H
z
C
z
I
A
B
D
z I - A -1 = ± det
(
z
I
-
A
)
red
T det z I - A
z
I
A
±
(
z
I
-
A
)
red
z
I
A
M M z z
D z = det z I - A
D
z
z
I
A
M M
det z I - A
z
I
A
Note:
X z
X
z
z z
X z T =
X
1
z
X
2
z …
X
z
X
1
z
X
2
z
…
Consider the transformed state system with
A
^
= T A T -1
A
^
T
A
T
B
^
= T B
B
^
T
B
C
^
= C T -1
C
^
C
T
D
^
= D
D
^
D
H z =
C
^
z I -
A
^
-1
B
^
+
D
^
= C T -1 z I - T A T -1 -1 T B + D = C T -1 T z I - A T -1 -1 T B + D = C T -1 T -1 -1 z I - A -1 T -1 T B + D = C z I - A -1 B + D
H
z
C
^
z
I
A
^
B
^
D
^
C
T
z
I
T
A
T
T
B
D
C
T
T
z
I
A
T
T
B
D
C
T
T
z
I
A
T
T
B
D
C
z
I
A
B
D
State-variable descriptions of systems are useful because they
provide a fairly general tool for analyzing all systems; they
provide a more detailed description of a signal flow graph than does the
transfer function (although not a full description); and they suggest
a large class of alternative implementations. They are even more
useful in control theory, which is largely based on state descriptions
of systems.
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Last edited by Douglas L. Jones on Dec 27, 2004 1:06 pm US/Central.