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

2

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

4

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

9

Irreducible normal form

Each diagonal block is irreducible or it is a 1x1 zero matrix

10

reducible

positive systemsirreducible

cyclic

primitive

classification based only

on the structure of

A!

classification based only

on the structure of

A!

11

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

14

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

15

Example

u 21 # # # # #1 0 1 2

1 0 0 0TA b b A b

not excitable

16

excitable

x(0) 0

u(·) 0

Any positive input

x(0+) 0 continuous-time systems

x(n) 0 discrete-time systems

17

Example

y 21 # # # # #1 0 1 2

1 0 0 1A c c A c

transparent

18

Excitability and/or transparency do not imply

reachability and/or observabilityExample

Excitable and transparent system but neither reachable nor

observable