EE221A Linear System Theory
Problem Set 7
Professor C. Tomlin
Department of Electrical Engineering and Computer Sciences, UC Berkeley
Fall 2007
Issued 11/15; Due 11/27
Problem 1.
Show that
braceleftbiggbracketleftbigg
A
0
c
0
bracketrightbigg
,
bracketleftbigg
b
0
bracketrightbiggbracerightbigg
where
A
∈
R
n
×
n
,
b
∈
R
n
,
c
T
∈
R
n
is completely controllable iff (i)
{
A, b
}
is completely controllable and (ii)
the matrix
bracketleftbigg
A
b
c
0
bracketrightbigg
is full rank.
Problem 2: Grammians under Similarity Transforms.
Consider the controllability and observability grammians
W
c
, W
o
of a linear timeinvariant system (
A, B, C
)
over the time period [0
,
Δ]. Determine what happens to them under similarity transformations of the state
space. That is, determine the controllability and observability grammians of (
TAT

1
, TB, CT

1
). Prove that
the eigenvalues of the product
W
c
W
o
are constant under similarity transformations.
Problem 3: RLRC circuit example.
Consider the network shown in Figure 1 with voltage
v
(
t
) as input
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 Fall '10
 ClaireTomlin
 Electrical Engineering, Professor C. Tomlin, Echo Canceller, product Wc Wo

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