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 X Scuola Nazionale CIRA di dottorato Antonio Ruberti Bertinoro, Luglio 2006

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 3 rd order positive realization...

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

10 A prologue via examples (I)

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

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

13 The Karpelevich theorem

14 The Karpelevich regions n = 3 n = 4

15 hidden pole A prologue via examples (II)

16 Example 3

17 cA x = 0 2 Ab b A bA b 2 c x = 0 3 cA x = 0 A bA b 3

18 Minimality of Positive Systems NSC for 3 rd order systems

19 {1 (contd.) r2r2 r3r3 Minimality of Positive Systems NSC for 3 rd order systems

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

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

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

23 Minimality for continuous-time positive systems

Generation of all positive realizations

25 Example 1