I Sistemi Positivi Grafi dinfluenza: irriducibilità, eccitabilità e trasparenza Lorenzo Farina Dipartimento di informatica e sistemistica A. Ruberti Università di Roma La Sapienza, Italy X Scuola Nazionale CIRA di dottorato Antonio Ruberti Bertinoro, Luglio 2006

2 Influence graph Given a continuous-time system or discrete-time the corresponding influence (oriented) graph is denoted by G ( G uxy ): an arc represents the direct influences among variables

3 An influence graph is described by a triple ( A #,b #,c T# ) with elements in [ 0, 1 ]. note index inversion

4 Example (pendulum)

5 For linear systems, the influence graph can be easily obtained from the triple ( A,b,c T ) because each arc of G corresponds to a nonzero element of A, b and c T. Therefore, the matrices A #, b # and c T# are simply the matrices A T, b and c T where the nonzero entries are replaced by ones. Example

6

7 Examples 1 2 n1n1 n2n2 1 n1n1 2 n2n2 + +

8 P is a permutation matrix ( P -1 =P T ) Example C 1 C 2 C 3 C 4 C 5 0

9 Irreducible normal form Each diagonal block is irreducible or it is a 1x1 zero matrix

10 classification based only on the structure of A !

11 Example C C C 2 1 3

12 Sufficient conditions for primitivity G x primitive

13 Wielandt formula In this case n=4, m min m=10 Example

14 More examples ( a ) is irreducible ( G x connected) with r 6 ( b ) is irreducible ( G x connected) with r 2 ( c ) and ( d ) are reducible ( G x not connected) (a) (b) (c) (d) C1C1 C2C2 C1C1 C2C2 C3C3

15 Example 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 transparent

18 Excitability and/or transparency do not imply reachability and/or observability Example Excitable and transparent system but neither reachable nor observable