i sistemi positivi grafi dinfluenza: irriducibilità, eccitabilità e trasparenza lorenzo farina...
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I Sistemi Positivi Grafi d’influenza: irriducibilità, eccitabilità e trasparenza
Lorenzo FarinaDipartimento di informatica e sistemistica “A.
Ruberti”Università di Roma “La Sapienza”, Italy
X Scuola Nazionale CIRA di dottorato “Antonio Ruberti”
Bertinoro, 10-12 Luglio 2006
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Influence graph
Given a continuous-time system
tx,,txhty
n,,,itu,tx,,txftx
n
nii
1
1 21
or discrete-time
tx,,txhty
n,,,itu,tx,,txftx
n
nii
1
1 211
the corresponding influence (oriented) graph is
denoted by G (Guxy): an arc represents the
direct influences among variables
3
An influence graph is described by a triple (A#,b#,cT# ) with elements in [0,1].
#
#
#
1 , in
0 otherwise
1 0, in
0 otherwise
1 , 1 in
0 otherwise
ij
i
i
j i Ga
i Gb
i n Gc
note index inversion
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Example (pendulum)
1
21122
21
sen
sencos1
xLy
hxxmgLxuLmL
x
xx
0
1
1
0
11
10 ### cbA
5
For linear systems, the influence graph can be easily obtained from the triple (A,b,cT ) because each arc of G corresponds to a nonzero element
of A, b and cT .
Therefore, the matrices A#, b# and cT# are simply the matrices AT , b and
cT where the nonzero entries are replaced by ones.
Example
0
2
1
0
12
10cbA
0
1
1
0
11
10 ### cbA
6
# # #
1 1 0 1 1
0 0 1 0 0
1 0 0 0 1
A b c
4 3 4 3# # # # # # # #
3 2 1 2 4
1 1 1 1 2 3
1 2 1 1 3
4
T T TA A b A c c A b
k
Example
u
y
21 3
7
2221
11 0
AA
AA
22
11
0
0
A
AA
Examples
1 2
n1n2
1
n1
2
n2
+
+
8
1, , reducible : is block triangularA P PAP
P is a permutation matrix (P-1=PT )
Example
***
****
***
**
***
*
A
1 2 3 4 5 6
7
8
12
3
C5
0
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Irreducible normal form
Each diagonal block is irreducible or it is a 1x1 zero matrix
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reducible
positive systemsirreducible
cyclic
primitive
classification based only
on the structure of
A!
classification based only
on the structure of
A!
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Example
1
23
4
5
6
7
21
3
12
Sufficient conditions for primitivity
i Gx primitive
i
h
j Gx primitive
13
222 nnmmin
Wielandt formula
In this case n=4, mminm=10
Example
# #9
0 0 1 1 1 3 3 1
1 0 0 0 1 1 3 2
0 1 0 0 2 1 1 1
0 0 1 0 1 2 1 0
A A
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More examples
(a) is irreducible (Gx connected) with r 6(b) is irreducible (Gx connected) with r 2
(c) and (d) are reducible (Gx not connected)
(a)(b)
(c) (d)
C1 C2C1 C2 C3
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Example
u 21 # # # # #1 0 1 2
1 0 0 0TA b b A b
not excitable
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excitable
x(0) 0
u(·) 0
Any positive input
x(0+) 0 continuous-time systems
x(n) 0 discrete-time systems
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Example
y 21 # # # # #1 0 1 2
1 0 0 1A c c A c
transparent
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Excitability and/or transparency do not imply
reachability and/or observabilityExample
Excitable and transparent system but neither reachable nor
observable