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I Sistemi Positivi Realizzazione: esistenza a tempo continuo e minimalità

X Scuola Nazionale CIRA di dottorato “Antonio Ruberti” Bertinoro, 10-12 Luglio 2006. I Sistemi Positivi Realizzazione: esistenza a tempo continuo e minimalità. Lorenzo Farina Dipartimento di informatica e sistemistica “A. Ruberti” Università di Roma “La Sapienza”, Italy.

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I Sistemi Positivi Realizzazione: esistenza a tempo continuo e minimalità

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  1. X Scuola Nazionale CIRA di dottorato “Antonio Ruberti” Bertinoro, 10-12 Luglio 2006 I Sistemi PositiviRealizzazione: esistenza a tempo continuo e minimalità Lorenzo FarinaDipartimento di informatica e sistemistica “A. Ruberti”Università di Roma “La Sapienza”, Italy

  2. The positive realization problem for continuous-time systems Spectrum translation property

  3. Existence conditions

  4. Examples - I … not to be!

  5. Examples - II … not to be!

  6. Minimality of Positive Realizations

  7. Does positive factorization suffice? For general systems, the minimal inner dimension of a factorization of the Hankel matrix coincides with the minimal order of a realization. Is that true also for positive systems?

  8. Does positive factorization suffice? No rotational simmetry, no 3rd order positive realization...

  9. Does positive factorization suffice? No! A positive factorization of the Hankel matrix!

  10. A prologue via examples (I)

  11. A prologue via examples (I) (contd.) The spectrum must remain unchanged under a rotation of /2(q+1) radians

  12. A prologue via examples (I) The spectrum must remain unchanged under a rotation of /4 radians

  13. The Karpelevich theorem

  14. The Karpelevich regions n = 4 n = 3

  15. A prologue via examples (II) hidden pole

  16. Example 3

  17. 2 3 Ab Ab b @O Ab cx = 0 2 3 cAx = 0 cAx = 0 cAx = 0

  18. Minimality of Positive SystemsNSC for 3rd order systems

  19. Minimality of Positive SystemsNSC for 3rd order systems (contd.) r3 r2 {1 {1

  20. Minimality of Positive SystemsNSC for 3rd order systems (contd.)

  21. Minimality of Positive SystemsNSC for 3rd order systems (contd.)

  22. Minimality of Positive SystemsNSC for 3rd order systems (contd.)

  23. Minimality for continuous-time positive systems

  24. Generation of all positive realizations

  25. Example 1

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