The University of Texas at Austin
Department of Electrical and Computer Engineering
EE362K: Introduction to Automatic Control—Fall 2009
Problem Set Four Solutions
C. Caramanis
Not Due.
This problem set should serve as a start for your review for the test. It covers the concepts
introduced in Chapter 5. This includes homogeneous and particular solutions to LTI systems (CT
and DT); The matrix exponential; Jordan Canonical Form and linear algebra; and stability of LTI
systems. Also, while it also covers some of the earlier concepts discussed, it is not exhaustive. We
had a lot of practice with linearization in class and in previous problem sets, so while that is an
important concept, it is not emphasized in this one. It is de±nitely important for the midterm.
1. Consider the matrix:
A
=
1111 2
0
−
30 2 8
000
−
2
−
5
0004
−
7
0000 6
(a) If you can, write down the Jordan Canonical Form (JCF) of this matrix and explain
which results you are using in order to arrive to your answer. If you cannot, explain why
there is ambiguity.
(b) Is the system ˙
x
=
Ax
stable, unstable, or stable i.s.L., for
A
the above matrix?
(c) Is the system
x
[
k
+1]=
Ax
[
k
] stable, unstable, or stable i.s.L., for
A
the above matrix?
Solution:
(a) The matrix is upper triangular, and hence the eigenvalues are precisely the values on
the diagonal. Thus the eigenvalues are:
λ
=1
,
−
3
,
0
,
4
,
6. Since these values are distinct,
the JCF of the matrix must be composed of 1
×
1 blocks, i.e., it is diagonal, with the
above values along the diagonal.
(b) The continuous time system is unstable, since there is an eigenvalue (several, in fact)
with strictly positive real part.
(c) The discrete time system is also unstable, since there is an eigenvalue (several, in fact)
strictly outside the unit circle.
2. Suppose
A
is a 7
×
7 symmetric matrix with characteristic polynomial:
p
A
(
λ
)=3
·
λ
·
(
λ
+2)
2
·
(
λ
+1)
3
·
(
λ
+ 12)
.
(a) If you can, write down the Jordan Canonical Form (JCF) of this matrix and explain
which results you are using in order to arrive to your answer. If you cannot, explain why
there is ambiguity.
1
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View Full Document(b) Is the system ˙
x
=
Ax
stable, unstable, or stable i.s.L., for
A
the above matrix?
(c) Is the system
x
[
k
+1]=
Ax
[
k
] stable, unstable, or stable i.s.L., for
A
the above matrix?
Solution
:
(a) The eigenvalues are not distinct. They are:
λ
=0
,
−
2
,
−
2
,
−
1
,
−
1
,
−
1
,
−
12. However,
since the matrix is symmetric, we know it is diagonalizable, and hence its JCF must be
the diagonal matrix with the values above in its diagonal.
(b) The continuous time system is neutrally aka marginally stable. This is because there
are no eigenvalues strictly in the right half plane, and the one with real part equal to
zero is in a 1
×
1Jordanb
lock
.
(c) The discrete time system is unstable, since there is an eigenvalue (several, in fact) strictly
outside the unit circle.
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 Fall '08
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