Stability, Observability, and Controllability
Kai Borre
Copyright
c
2011 by Kai Borre
Stability of Discrete Filter
Remember the update equation:
O
x
k
j
k
D
O
x
k
j
k
1
C
K
k
.
b
k
A
k
O
x
k
j
k
1
/:
(1)
Prediction of the state equation is
O
x
k
j
k
1
D
F
k
1
O
x
k
1
j
k
1
. Substitute to get
O
x
k
j
k
D
.
F
k
1
K
k
A
k
F
k
1
/
O
x
k
1
j
k
1
C
K
k
b
k
:
(2)
We now take the
z
-transform on both sides. Note that retarding one step in the
time domain (k to k
1) is the equivalent of multiplying by
z
1
in the
z
-domain.
Omitting most indices the
z
-transform becomes
O
X
.
z
/
D
.
F
K
k
AF
/
z
1
O
X
.
z
/
C
K
k
B
.
z
/
Copyright
c
2011 by Kai Borre
or
.
z
I
F
C
K
k
AF
/
O
X
.
z
/
D
z
K
k
B
.
z
/
. The characteristic polynomial is
det
.
z
I
F
C
K AF
/
D
0
:
(3)
The roots of this polynomial tell the story about the stability of the filter. The
transfer function is the reciprocal
z
I
F
C
K
k
AF
1
. In this particular case
equation (3) becomes
z
I
1
C
K
1
1
D
0 or
z
D
1
K
:
In order to determine K we run the M-file k_updatf
[x, P, K, inn] = k_updatf(1,1,1,1.5,0.1,1.0,1); % x,P,A,b,
6
e
,
6
,F
[x, P, K, inn] = k_updatf(x,P,1,1.5,0.1,1.0,1);
[x, P, K, inn] = k_updatf(x,P,1,1.5,0.1,1.0,1);
. . .
Copyright
c
2011 by Kai Borre
The actual computation runs like this (notice P twice between updates of k):
P
1
j
0
D
P
0
j
0
C
1
D
1
C
1
D
2
K
1
D
P
1
j
0
.
P
1
j
0
C
0
:
1
/
1
D
2
=
2
:
1
D
0
:
952 380 95
P
1
j
1
D
.
1
K
1
/
P
1
j
0
D
0
:
095 238 095
P
2
j
1
D
P
1
j
1
C
1
D
1
:
095 238 095
K
2
D
1
:
095 238 095
=.
1
:
095 238 095
C
0
:
1
/
D
0
:
916 334 66
: : :
or
K
D
0
:
95 238
0
:
91 633
0
:
91 608
0
:
91 608.
Copyright
c
2011 by Kai Borre
Hence
z
D
1
K
D
0
:
08 392 which is well within the unit circle in the
z
-plane.
We realize the filter is highly stable.
Even though the input b is non-stationary, the filter itself is intrinsically stable.
The location of the filter zero (pole of the transfer function) tells that any small
perturbation (roundoff errors) from the steady-state condition will damp out
quickly. Any perturbation will be attenuated by a factor of 0
:
084 with each step
so the effect rapidly fades out.
We conclude that we can gain considerable insight into the filter operation just by
looking at its characteristic poles in the steady-state condition, provided, of
course, the steady-state condition exists.
We have just dealt with stability. Next we introduce the twin concepts of system
observability, and system controllability.
Copyright
c
2011 by Kai Borre