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Modern Control
GUC Faculty of Engineering and Material Science
Department of Mechatronics
Tutorial #8
Controllability and Observability II
and Minimal Realization
Modern Control MCTR 702
Dr. Ayman Ali El-Badawy
GUC Faculty of Engineering and Material Science
Department of Mechatronics
Modern Control MCTR 702
Dr. Ayman Ali El-Badawy
Kalman Decomposition (controllability):
When the system is NOT completely controllable, we may want to decompose it to know
which modes are uncontrollable. And then see if the Eigen values of these modes are
already stable, then the system is stabilizable. This is by constructing a similarity transformation matrix from any rank(P) linearly
independent columns (basis of P) appended to them n-rank(P) other independent columns
such that the resulting matrix is non singular.
GUC Faculty of Engineering and Material Science
Department of Mechatronics
Modern Control MCTR 702
Dr. Ayman Ali El-Badawy
Kalman Decomposition (observability):
When the system is NOT completely observable, we may want to decompose it to know
which modes are unobservable. And then see if the Eigen values of these modes are already
stable, then the system is Detectable.
GUC Faculty of Engineering and Material Science
Department of Mechatronics
Modern Control MCTR 702
Dr. Ayman Ali El-Badawy
For the system Given in Problem two, consider the second input only and find
which modes are uncontrollable. Is the system stabilizable?
Solution:
0
3660
000
601
0
0
1
,
606
010
100
2
PBAABBP
BA
Problem 1:
|P| = 0 because the 3rd column is linearly dependent on the 1st and 2nd columns
(-6x1st -6x2nd). If it’s not noticeable, we perform row operations to find the basis
vectors of the P matrix. The decomposition is by constructing a transformation
matrix constituting of these basis vectors along with added independent vectors.
GUC Faculty of Engineering and Material Science
Department of Mechatronics Modern Control MCTR 702
Dr. Ayman Ali El-Badawy
u
w
w
w
w
w
w
BVBAVVA
V
0
0
1
100
061
060
0
0
1
ˆ,
100
061
060
ˆ
060
100
001
3
2
1
3
2
1
11
I have only one uncontrollable mode (W3). Its Eigen value is 1 (unstable). Then
this system is NOT stabilizable.
GUC Faculty of Engineering and Material Science
Department of Mechatronics
Modern Control MCTR 702
Dr. Ayman Ali El-Badawy
State space realization
The transfer function associated with this realization is given by;
The question that arises now is “can we reduce this realization ?”.i.e. is there
any state that could be cancelled?
Minimal Realization
GUC Faculty of Engineering and Material Science
Department of Mechatronics
Modern Control MCTR 702
Dr. Ayman Ali El-Badawy
Problem 1:
Compute a minimal realization for each of the following state equations.
Solution :
)1)(2(
1 + s
23s
1 + s
)(
)()(
1
0
32
10
0
0]11[)(
2
1
ss
ssU
sXsH
s
ssH
Due to pole-zero cancellation,
it is clear that the new H(s) is
given as
GUC Faculty of Engineering and Material Science
Department of Mechatronics
Modern Control MCTR 702
Dr. Ayman Ali El-Badawy
]1[],1[],2[
2
2
2
)2(
1
)(
)()(
111
1
CBA
uxx
uxx
uxx
L
ssU
sXsH
One state is cancelled, and the
irreducible realization (minimal
realization) is
GUC Faculty of Engineering and Material Science
Department of Mechatronics
Modern Control MCTR 702
Dr. Ayman Ali El-Badawy
Important Note:
If we apply controllability and observability Kalman decomposition to a
system (in either order)(the second being applied to separate SUB-
systems only), we get
uB
B
X
X
X
X
AA
A
AAAA
AA
X
X
X
X
oc
co
oc
oc
oc
co
oc
oc
oc
co
oc
oc
oc
co
0
0
00
000
00
43
242321
13
Du
X
X
X
X
CCy
oc
oc
oc
co
occo
]00[
GUC Faculty of Engineering and Material Science
Department of Mechatronics
Modern Control MCTR 702
Dr. Ayman Ali El-Badawy
The most systematic way, to get the minimal (irreducible) realization, is to perform
controllability and observability Kalman decomposition (in either order), the second being
applied to separate SUB-systems only, to get Aco , Bco, and Cco
Solution :
By first performing (controllability) Kalman decomposition, we find the controllability matrix
as
3271
710
100
][ 2BAABBP
Which is full row rank. This means that the
system is completely controllable. It is clear
that the system is already in the controllable
as it is given in CCF
GUC Faculty of Engineering and Material Science
Department of Mechatronics
Modern Control MCTR 702
Dr. Ayman Ali El-Badawy
Now, we apply observability Kalman decomposition
41715
130
013
2CA
CA
C
Q
rank = 2
The 3rd row depends on the first 2 rows
100
0.33-0.330
0.110.11-0.33
100
130
0131
U
u
z
z
z
z
z
z
1
0
0
100
0.33-0.330
0.110.11-0.33
100
0.33-0.330
0.110.11-0.33
71715
100
010
100
0.33-0.330
0.110.11-0.331
3
2
1
1
3
2
1
GUC Faculty of Engineering and Material Science
Department of Mechatronics
Modern Control MCTR 702
Dr. Ayman Ali El-Badawy
001ˆ C
u
z
z
z
z
z
z
1
1
0
3-45
045
010
3
2
1
3
2
1
coA coB
coC
01coC
Then, the minimal realization is given by the following matrices
1
0coB
45
10coA
GUC Faculty of Engineering and Material Science
Department of Mechatronics
Modern Control MCTR 702
Dr. Ayman Ali El-Badawy
And the transfer function is given by
54s
1)()(
2
1
sBAsICsH cococo
In this problem, it is clear that one state is cancelled due to pole-zero cancellation