I Sistemi Positivi Realizzazione: esistenza a tempo continuo e minimalità

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

The positive realization problem for continuous-time systems

Im

Re

a

l*1

l*2

l*F

lF alal

alla

A

IA

pAI

IAIp

det

det

Spectrum translation property

if is Metzler, then there exists

0 such that is nonnegative

A

A Ia a

3

Existence conditions

4

Examples - I

2

3 2

4 14

6 11 6

sH s

s s s

1 2 31, 2, 3 p p p

0 1 2 3 4 50

0.5

1

1.5

2

2.5

3

3.5

4

log 5 3 0.51

… not to be!

5

Examples - II

2

2

2 3 3

1 2 2

s sH s

s s s

1 2 31, 1 , 1 p p j p j

2 sin 0th t e t

0 1 2 3 4 50

0.5

1

1.5

2

… 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?

2

2 3

2 3

22

cb cAb cA b

cAb cA b cA bH

cA b cA b

c

cAb Ab A b RS

cA

8

Does positive factorization suffice?

1 0 0 8 1

0 0 1 1 1

0 1 0 1 1

TA b c

1 4 10h k

2 4 8h k 3 4 6h k 4 4 8h k

0

60120

180

240 300

No rotational simmetry, no 3rd order positive

realization...

9

Does positive factorization suffice? No!

11 12

12

5 3 32 1 0 1

4 2 60 1 1 0

3 5 50 0 1 1

4 6 2

ij

H H

H H H

A positive factorization of the Hankel matrix!

10

A prologue via examples (I)

2

1, 1

( 1)( 1)qH z q q

z z

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

= 0

2= 0

2 O

= 0

3= 0 3

18

Minimality of Positive SystemsNSC for 3rd order systems

19

{1{1

(contd.)

2

3

Minimality of Positive SystemsNSC for 3rd order systems

20

(contd.)

Minimality of Positive SystemsNSC for 3rd order systems

21

(contd.)

Minimality of Positive SystemsNSC for 3rd order systems

22

(contd.)

Minimality of Positive SystemsNSC for 3rd order systems

23

Minimality for continuous-time positive systems

Generation of all positive realizations

25

Example 1