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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, 10-12 Luglio 2006

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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

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

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4 Example (pendulum)

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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

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

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8 P is a permutation matrix ( P -1 =P T ) Example 12 3 4 56 7 8 C 1 C 2 C 3 C 4 C 5 0

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

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10 classification based only on the structure of A !

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11 Example 1 2 3 4 5 6 7 C C C 2 1 3

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12 Sufficient conditions for primitivity G x primitive

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

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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

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15 Example not excitable

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16 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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17 Example transparent

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

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MULTIPLICATION EQUATIONS 1. SOLVE FOR X 3. WHAT EVER YOU DO TO ONE SIDE YOU HAVE TO DO TO THE OTHER 2. DIVIDE BY THE NUMBER IN FRONT OF THE VARIABLE.

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