TimeVarying Systems and Computations
Lecture 3
Klaus Diepold
1. November 2012
Linear TimeVarying Systems
StateSpace System Model
We aim to derive the matrix containing the timevarying impulse responses by inspection of a generic
causal timevarying system. Recall the state equations
x
k
+1
=
A
k
·
x
k
+
B
k
·
u
k
y
k
=
C
k
·
x
k
+
D
k
·
u
k
,
(1)
or in matrix form,
±
x
k
+1
y
k
²
=
±
A
k
B
k
C
k
D
k
²
·
±
x
k
u
k
²
,
(2)
These statespace equations are similar to the equations for the timeinvariant case except that now the
entries
{
A
k
,B
k
,C
k
,D
k
}
of the realization matrix are depending on the time index
k
, which means that
these matrices can change from time step to time step, incorporating the timevariation that we are
looking for. A generic causal timevarying system is drawn as shown in Figure 1.
Z
Z
u
−
1
B
−
1
D
−
1
y
−
1
x
0
Z
u
0
A
0
C
0
D
0
B
0
y
0
x
1
A
1
C
1
D
1
B
1
u
1
y
1
x
2
Z
Z
Z
C
2
A
2
D
2
B
2
u
2
y
2
x
3
Z
Z
C
3
D
3
u
3
y
3
Abbildung 1: A timevarying system
Besides the timevarying system parameters
{
A
k
k
k
k
}
we can see that the dimension of the
statespace, i.e. the number of delay elements may change from time step to time step. Similarly, the
dimension of the input signal vector
u
k
and the dimension of the output signal vector
y
k
may change
with time. However, we will not dwell in more detail on the the latter point.
1
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2
TimeVarying ImpulseResponse
We seek to determine the I/O operator, i.e. the Toeplitz operator for this timevarying causal system.
We proceed in a similar fashion as with the timeinvariant case, by applying an impulse at time instant
k
and noting the output of the system (only in timevarying case, the response depends on
k
as well).
Taking the impulse responses as the columns of the matrix
T
we arrive at
T
=
k
= 0
k
= 1
k
= 2
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
0
.
.
.
.
.
.
.
.
.
D
0
0
.
.
.
.
.
.
C
1
B
0
D
1
0
.
.
.
C
2
A
1
B
0
C
2
B
1
D
2
.
.
.
.
.
.
C
3
A
2
A
1
B
0
C
3
A
2
B
1
C
3
B
2
.
.
.
C
4
A
3
A
2
A
1
B
0
C
4
A
3
A
2
B
1
C
4
A
3
B
2
.
.
.
.
.
.
C
5
A
4
A
3
A
2
B
1
C
5
A
4
A
3
B
2
.
.
.
.
.
.
.
.
.
C
6
A
5
A
4
A
3
B
2
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
(3)
We can note from the above that
•
i
th
column represents the impulse response of the system for an impulse input at
k
=
i
, i.e.
u
i
=
±
1
i
=
k,
0
else
•
T
has the same functionality as Toeplitz matrix of the timeinvariant case. In fact, if the system is
timeinvariant, all the matrices
A
k
,B
k
,C
k
,D
k
are identical, hence
T
will exhibit Toeplitz structure.
•
We can identify the system to be causal by lowertriangular structure of
T
.
•
The clear diﬀerence is that
T
has no Toeplitz structure.
•
Because of its timevarying nature, the dimension of the matrix
T
may be ﬁnite.
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 Winter '12
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 Matrices, UK, realization, Transfer Operator

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