MEEN 651
Control System Design
Homework 10: State Feedback and Observers
Solutions
Assigned:
Tuesday, 4 Nov. 2008
Due:
Tuesday, 11 Nov. 2008, 5:00 pm
Problem 1)
For the following problems do the following (by hand):
a)
determine stability (eigenvalues)
b)
determine controllability using the controllability matrix
c)
determine controllability using the Hautus-Rosenbrock test
d)
if appropriate, determine controllability using the grammian; if this is not appropriate,
explain why.
⎥
⎦
⎤
⎢
⎣
⎡
=
⎥
⎦
⎤
⎢
⎣
⎡
−
−
=
1
1
4
0
1
5
B
A
,
and
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
=
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
=
1
0
0
0
0
0
1
0
0
0
1
0
B
A
Solution:
For System 1:
Stable with eigenvalues of -5 and -4.
Not controllable (one controllable
mode, one uncontrollable mode).
This can be determined from the controllability matrix:
which has rank 1, or from the Hautus Rosenbrock test, which is rank
deficient for
.
Since the system is stable, we can
determine the controllability grammian as
, which is also rank deficient.
For System 2:
Marginally stable with eigenvalues of {0, 0, 0}.
Controllable, determined
from the controllability matrix:
which has rank 3, or from the Hautus
Rosenbrock test, which is full rank for all
.
Since the system is not stable, it is not
appropriate to compute the controllability grammian.

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