DumitrescuBogdanUPB PCE 2007 1 - schur.pub.ro fileIn context mai larg, pozitivitatea este o...

PROIECTE DE CERCETARE EXPLORATORIE PCE -

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PN–II–ID–PCE–2007–1

CERERE DE FINANTAREPENTRU PROIECTE

DE CERCETARE EXPLORATORIE

1. Date personale ale directorului de proiect :

1.1. Nume: DUMITRESCU

1.2. Prenume: BOGDAN

1.3. An nastere: 1962

1.4. Titlu didactic si/sau stiintific : Profesor (Selectati)

1.5. Doctor din anul: 1994 1.6 Conducator doctorat: NU (Selectati)

1.7 Numar doctoranzi: 0

2. Institutia gazda a proiectului:

2.1. Codul Institutiei :

28 [A se vedea ANEXA 1]

2.2. Denumire Institutie: Universitatea POLITEHNICA Bucuresti [completati denumirea institutiei]

2.3. Facultate/ Department: Facultatea de Automatica si Calculatoare

2.4. Functie: profesor

2.5. Adresa: spl. Independentei nr. 313, 060042 Bucuresti

2.6. Telefon: 021 4029100 (021 4029167 catedra)

2.7. Fax: 021 3181001

2.8. E-Mail: [email protected] ([email protected])

3. Titlul proiectului in limba romana: (Max 200 caractere)

Pozitivitatea in analiza si sinteza sistemelor multidimensionale

4. Titlul proiectului in limba engleza: (Max 200 caractere)

Positivity in the analysis and synthesis of multidimensional systems

ROMANIA

Ministerul Educatiei, Cercetarii si Tineretului

Autoritatea Nationala pentru Cercetare Stiintifica


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5. Termeni cheie (max 5 termeni ):

1 sisteme multidimensionale 2 polinoame pozitive

3 programare semidefinita

4 optimizare

5 prelucrarea semnalelor 6. Incadrarea proiectului in domeniile de expertiza:

COD COMISIE COD SUBCOMISIE COD DOMENIU

2 (selectati) 2J 75

[ Pentru cod: Comisie/Sucomisie/ Domeniu - A se vedea ANEXA 2 ] 7. Durata proiectului ( 3 ani ) : 8. Rezumatul proiectului in limba romana: (Max. 2000 caractere)

Sistemele dinamice multidimensionale apar curent in prelucrarea semnalelor si automatica. In proiectul de fata, ne propunem studiul unor probleme de analiza si sinteza a acestor sisteme folosind teoria recenta a optimizarii cu polinoame trigonometrice pozitive. In context mai larg, pozitivitatea este o proprietate de baza a sistemelor, dar aici ne referim cu precadere la transformarea unor probleme de optimizare evidentiind prezenta (deseori ascunsa) a pozitivitatii. Diversele parametrizari ale polinoamelor trigonometrice multivariabile pozitive descoperite in ultimii cativa ani permit legatura cu o categorie distincta a optimizarii convexe, anume programarea semidefinita. Obiectivele proiectului sunt de doua tipuri : de implementare si stiintifice. Primele vizeaza realizarea unor biblioteci de programe (inexistente la ora actuala) dedicate manipularii facile a polinoamelor trigonometrice pozitive in contextul programarii semidefinite, precum si a unei colectii de programe rezolvand probleme care fac apel la polinoamele pozitive. Obiectivele de implementare vor fi realizate de cercetatorii tineri din echipa, cu scopul acomodarii lor la subiectul proiectului. Obiectivele stiintifice constau in studierea unor probleme de baza legate de pozitivitate, in contextul sistemelor multidimensionale: parametrizarea polinoamelor mixte (cu variabile reale si complexe), o lema Kalman-Yakubovich-Popov pentru pozitivitate pe anumite domenii, caracterizarea pozitivitatii sistemelor interconectate spatial. De asemenea, avem in vedere obiective cu caracter derivat, precum studiul conditiilor de stabilitate (eventual robusta) a sistemelor multidimensionale, calculul razei de controlabilitate, proiectarea bancurilor de filtre 2D ortogonale cu doua canale. Toate problemele enumerate mai sus sunt fie noi, fie tratate nesatisfacator in literatura de specialitate, de aceea rezolvarea lor (fie si partiala) va conduce la publicarea unor articole in reviste cotate ISI.

9. Rezumatul proiectului in limba engleza:

(Max. 2000 caractere) Multidimensional dynamic systems are used currently in signal processing and control. In this project, we will study some multidimensional system analysis and synthesis problems, using the recent theory of optimization with positive trigonometric polynomials. In a wider context, positivity is a basic system property, but here we deal especially with the transformation of some optimization problems by revealing the presence (often hidden) of positivity. The parameterizations of positive trigonometric polynomials discovered in the latest few years allow the connection with a distinct category of convex optimization, namely semidefinite programming. The objectives of the project are of two kinds: implementation and scientific. The first aim to writing a library (not existent at this moment) dedicated to the manipulation of positive trigonometric polynomials in the context of semidefinite programming, and also a collection of programs solving problems appealing to positive polynomials. These implementation objectives will be accomplished by the young researchers of the team, with the goal of getting them familiarised with the main subjects of the project. The scientific objectives are the study of a number of basic positivity problems, in the context of multidimensional systems : a parameterization of hybrid polynomials (with real and complex variables), a Kalman-Yakubovich-Popov lemma for

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positivity on certain domains, the positivity characterization of spatially interconnected systems. Also, we consider derived scientific objectives, such as the study of (possibly robust) stability conditions for multidimensional systems, the computation of the controllability radius, the design of 2D orthogonal two-channel filter banks. All the problems mentioned above are either new, or not satisfactorily treated in the literature, and so their solution (even partial) will lead to the publication of articles in ISI journals.

10. Prezentarea proiectului in limba romana:

[Va rugam sa completati max. 10 pag. in ANEXA 3]

11.Prezentarea proiectului in limba engleza: [Va rugam sa completati max. 10 pag. in ANEXA 4]

12. Modul de organizare a proiectului (managementul proiectului):

[Va rugam sa completati ANEXA 5]

13. Structura bugetului:

NR. CRT

DENUMIRE CAPITOL BUGET

VALOARE 2007 (lei)

VALOARE 2008 (lei)

VALOARE 2009 (lei)

VALOARE 2010 (lei)

VALOARE TOTALA

(lei)

1. CHELTUIELI DE PERSONAL - max. 60% 32100 128400 128400 96300 385200

2. CHELTUIELI INDIRECTE (regie) 8025 32100 32100 24075 96300

3 MOBILITATI (se asigura participarea la stagii de documentare-cercetare in strainatate)

7000 32000 32000 25000 96000

4.

CHELTUIELI DE LOGISTICA pentru derularea proiectului (infrastructura de cercetare, cheltuieli materiale, diseminare etc.)

2000 54500 18000 18000 92500

TOTAL 49125 247000 210500 163375 670000

14. Conformare la cerintele programului „Cercetare de Excelenta” -CEEX: Directorul de proiect a fost /este director de proiect in cadrul programului CEEX ?

[Daca da va rugam sa completati ANEXA 6]

15. Directorul de proiect are activitatea profesionala de baza (Cartea de Munca) in institutia care propune proiectul (criteriu de eligibilitate) : (Selectati)

DA


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PRIN ACEASTA SE CERTIFICA LEGALITATEA SI CORECTITUDINEA DATELOR CUPRINSE IN PREZENTA CERERE DE FINANTARE

CODUL INSTITUTIEI : Codul trebuie sa fie identic cu cel de la punctul 2.1 (vezi ANEXA 1) DATA: 26.06.2007 RECTOR/DIRECTOR, Nume, prenume:prof.dr.ing. Ecaterina Andronescu Semnatura: DIRECTOR EC./CONTABIL SEF Stampila Nume, prenume:ec. Dorina Adamescu Semnatura:

DIRECTOR DE PROIECT, Nume, prenume:prof.dr.ing. Bogdan Dumitrescu Semnatura:

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ANEXA 3 10. Prezentarea proiectului in limba romana: (Max. 10 pagini)

10.1. Importanta si relevanta continutului stiintific

1. Introducere Sistemele dinamice multidimensionale apar in mai multe domenii, modeland in mod natural fenomene sau procese. La loc de frunte este prelucrarea semnalelor, unde filtrele 2D si 3D (recent chiar si 4D) au utilizari diverse, de exemplu in prelucrarea imaginilor, reducerea zgomotului (denoising), compresie, prelucrarea semalelor de tip radar. Variabilele independente au aici semnificatie spatiala. In automatica, sistemele repetitive (laminarea metalelor, fabricarea hartiei) pot fi modelate prin sisteme 2D. Un alt exemplu, mai recent, este cel al sistemelor interconectate spatial [DD03], in care subsisteme identice interactioneaza, fiecare doar cu vecinii sai. In aceste sisteme, o variabila este timpul, iar celelalte reprezinta coordonate spatiale. In proiectul de fata suntem interesati in special de sistemele discrete (digitale). Ele pot fi modelate fie prin functia de transfer (eventual cu coeficienti matriceali), fie prin modele de stare ca Roesser si Fornasini-Marchesini. Pozitivitatea reprezinta atat o proprietate fundamentala, cat si un instrument. Sistemele pasive (real pozitive) formeaza o clasa importanta in studiul stabilitatii. Ele sunt caracterizate fie prin exprimarea efectiva a pozitivitatii, in modele polinomiale, fie prin lema Kalman-Yakubovich-Popov, in modele de stare. Referindu-ne la pozitivitate ca instrument, in contextul sistemelor in timp discret, ne gandim la polinoame trigonometrice pozitive, cu ajutorul carora se pot modela natural o serie de probleme de optimizare in context sistemic, mentionate in cele ce urmeaza. In continuare, vom prezenta pe scurt modul de lucru cu polinoame trigonometrice pozitive in optimizare, apoi vom enumera aplicatii recente ale acestora in rezolvare unor probleme din teoria semnalelor si sistemelor. In final, vom discuta importanta si relevanta studiului lor aprofundat, propus in acest proiect. 2. Polinoame trigonometrice pozitive

Un polinom trigonometric are forma generala ( ) ,n

kk

k nR z r z−

=−

= ∑ unde coeficientii sunt in general

complecsi si simetrici, i.e. *k kr r− = , iar n este gradul polinomului. Pe cercul unitate, unde z=ejω,

polinomul are valori reale. Numim pozitiv un polinom pentru care ( ) 0,jR e ω ω≥ ∀ . Interesul recent pentru polinoamele pozitive a provenit din combinatia a doua rezultate. Primul, bine cunoscut, spune ca daca putem optimiza patratul magnitudinii unui polinom H(z) (in general nesimetric), adica lucram cu 2( ) | ( ) |j jR e H eω ω= , care este un polinom trigonometric pozitiv, atunci


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putem calcula H(z) prin factorizare spectrala. (“Ridicarea la patrat” este un mod de convexificare a unor astfel de probleme de optimizare.) Al doilea rezultat a fost propus acum mai putin de o decada si constutuie o parametrizare a multimii polinoamelor pozitive: conditia ca R(z) sa fie pozitiv este echivalenta cu existenta unei matrice pozitiv semidefinite Q astfel incat [ ]k kr trace Q= Θ , unde kΘ sunt matrice constante (cu o

structura Toeplitz elementara) [AV02, DTS01, GHNSVX02, GHNV03]. Aceasta parametrizare deschide calea utilizarii programarii semidefinite (SDP) in probleme de optimizare cu polinoame trigonometrice, permitand rezolvarea lor rapida si sigura folosind biblioteci disponibile online ca [SeDuMi]. Acest rezultat a fost generalizat in mai multe feluri, de exemplu pentru polinoame cu coeficienti matriceali [GHNV03] sau pentru polinoame pozitive pe un interval (i.e. nu global) [AV02, DLS02], si utilizat in diverse aplicatii care vor fi mentionate mai jos. De interes special pentru proiectul de fata este generalizarea parametrizarii la polinoame trigonometrice cu mai multe variabile. Notand d numarul variabilelor, aceste polinoame au forma

( ) ,R r −

=−

= ∑n

kk

k n

z z unde variabilele scrise cu caractere ingrosate sunt d-dimensionale, de exemplu

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dkkdz z=kz … , iar suma este luata pentru toate valorile dZ∈k pentru care

− ≤ ≤n k n (inegalitatile sunt intelese la nivel de element). Din nou polinomul este simetric, i.e. *r r− =k k , deci pe d-cercul (torul) unitate, polinomul ia valori reale.

Un polinom trigonometric este suma-de-patrate daca poate fi scris in forma

2( ) | ( ) |j ji

iR e H e=∑ω ω . Parametrizarea capata urmatoarea forma: polinomul R(z) este suma-de-

patrate daca si numai daca exista o matrice pozitiv semidefinita Q astfel incat [ ]r trace Q= Θk k ,

unde Θk sunt matrice constante (produse Kronecker de matrice Toeplitz elementare)

[MW02,Du06a]. Se observa ca, in cazul multidimensional, pozitivitatea este inlocuita de suma-de-patrate (aceasta restrictie provine din rezultatul cunoscut inca de la Hilbert, in legatura cu a 17-a sa problema, ca nu toate polinoamele pozitive, de variabila reala, sunt sume-de-patrate). Practic, pentru a putea aplica SDP, se fixeaza gradul polinoamelor Hi ale sumei-de-patrate si, prin aceasta, dimensiunea matricei Q. Multimea sumelor-de-patrate astfel obtinute este inclusa in multimea polinoamelor pozitive (si diferenta este nevida !). Asadar, problemele de optimizare cu polinoame pozitive se rezolva intr-un cadru oarecum restrictiv, impus de parametrizarea descrisa mai sus. Generalizarea parametrizarii la cazul polinoamelor pozitive pe anumite domenii (analogul pozitivitatii pe un interval, din cazul monodimensional) a fost prezentata in [Du06b]. O piedica in calea optimizarii cu polinoame pozitive cu mai multe variabile este inexistenta unui


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rezultat de tip “factorizare spectrala”. In unele probleme, aceasta poate fi inlocuita cu o inegalitate de tip Bounded Real Lemma, care in [Du06b] a fost exprimata sub forma unei inegalitati matriceale liniare (LMI). In general, in cazul multidimensional, rezultatele matematice sunt mai slabe decat in cel monovariabil. Cu toate acestea, exista parametrizarea de baza descrisa mai sus care a permis rezolvarea unor probleme, deocamdata foarte putine ca numar. 3. Aplicatii si alte dezvoltari In cazul monodimensional, utilizarea parametrizarii polinoamelor pozitive a condus la rezolvarea unor probleme de optimizare referitoare (printre altele) la :

• proiectarea filtrelor FIR [AV02, DLS02] si IIR (optimizare doar in magnitudine) [AV02] • proiectarea bancurilor de filtre ortogonale cu doua canale [DP02] • proiectarea regulatoarelor robuste de ordin redus [HSB03] • proiectarea bancurilor de filtre ortogonale supraesantionate [WDR04] • proiectarea filtrelor IIR [DN04] • proiectarea bancurilor de filtre ortogonale simetrice [KTVN06] • optimizarea unei constructii de tip dual-tree wavelet (cercetare in curs B.Dumitrescu

impreuna cu I.Selesnick si I.Bayram, articol trimis in iunie 2007 la IEEE Signal Proc. Letters)

Alte dezvoltari ale subiectului se refera la:

• Algoritmi rapizi de rezolvare a problemelor SDP rezultate din utilizarea polinoamelor trigonometrice pozitive. Dupa unele tentative [AV02] cu viteza buna, dar stabilitate numerica redusa (interzicand rezolvarea unor probleme de dimensiuni mari), rezultate mai bune par a fi obtinute in [RV06].

• Generalizarea lemei Kalman-Yakubovich-Popov (sursa permanenta de rezultate legate de pozitivitate) la cazul sistemelor pozitive pe intervale [IH05], conducand tot la SDP.

• Margini de robustete pentru sisteme complexe [GHNSVX02] In cazul multivariabil, aplicatiile sunt mai recente si deci mai putine

• Descrierea unor domenii convexe de stabilitate utilizate in proiectarea filtrelor IIR 2D [Du05]

• Test de stabilitate a sistemelor multidimensionale [Du06a] • Proiectarea filtrelor FIR 2D [Du06b]


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Un prima varianta de algoritm rapid pentru cazul 2D a fost prezentata in [RDV07]. In toata discutia de mai sus am ignorat voluntar eforturile matematicienilor. In ultimii 15 ani au fost produse mai multe rezultate importante referitoare la polinoamele suma-de-patrate. O parte dintre ele sunt prezentate in [PD01], dar suntem la curent si cu articolele mai recente. Investigarea potentialului lor aplicativ (in general nu foarte evident si nu neaparat cautat de autori) este parte a lucrului la proiectul propus. 4. Concluzii – importanta si relevanta Interesul recent acordat optimizarii cu polinoame pozitive se justifica prin posibilitatea rezolvarii, practic exact sau cu aproximatie buna, a unor probleme pentru care anterior se foloseau abordari conducand la formulari neconvexe. In cazul monovariabil se pare ca intelegerea fundamentelor parametrizarii polinoamelor pozitive este completa; mai raman eventual unele aplicatii in care acest instrument poate fi folosit, dar ele nu pot constitui obiectul unei cercetari sistematice, ci pot fi eventual produse derivate ale unor cercetari in alte directii. In schimb, studiul pozitivitatii in contextul sistemelor multidimensionale este inca plin de zone neexplorate. O cauza primara este neidentitatea intre “pozitiv” si “suma-de-patrate”, care conduce la aproximatii care trebuie studiate de la o problema la alta. O alta este dificultatea in sine a problemelor, modelele complicate si programele aferente a caror scriere necesita, in afara intelegerii complete a problemei, multa atentie si minutie. Tema propusa se afla la granita intre optimizare, teoria sistemelor si matematica, o zona in general accesibila cercetatorilor in automatica (teoria sistemelor), dar poate de interes mai mare in prelucrarea semnalelor. Acesta poate fi motivul pentru care explorarea decurge mai incet, dar fereastra de oportunitate nu va dura probabil prea mult. Directorul de proiect este bine plasat in cursa pentru studierea temei (dovada articolele sale recente), dar momentul este propice pentru o abordare cu forte mai mari, ale unui intreg grup. Bibliografie [AV02] B. Alkire, L. Vandenberghe - Convex optimization problems involving finite autocorrelation sequences, Math. Progr. Ser. A, vol. 93, no. 3, pp. 331–359, 2002 [CD06] R. S. Chandra, R.D’Andrea - A Scaled Small Gain Theorem With Applications to Spatially Interconnected Systems, IEEE Trans. Automatic Control, vol.51, no.3, pp.465-469, Mar. 2006. [DD03] R.D’Andrea, G.E.Dullerud - Distributed control design for spatially interconnected systems, IEEE Trans. Automatic Control, vol.48, no.9, pp.1478-1495, Sept. 2003.


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[DLS02] T. N. Davidson, Z. Q. Luo, and J. F. Sturm - Linear matrix inequality formulation of spectral mask constraints with applications to FIR filter design, IEEE Trans. Signal Process., vol. 50, no. 11, pp. 2702–2715, Nov. 2002. [DTS01] B.Dumitrescu, I.Tabus, P.Stoica - On the Parameterization of Positive Real Sequences and MA Parameter Estimation, IEEE Trans. Signal Processing, vol.49, no.11, pp.2630-2639, Nov. 2001 [DP02] B. Dumitrescu, C. Popeea - Accurate Computation of Compaction Filters with High Regularity, IEEE Signal Proc. Letters, vol.9, no.9, pp.278–281, Sept. 2002. [DN04] B.Dumitrescu, R.Niemistö - Multistage IIR Filter Design Using Convex Stability Domains Defined by Positive Realness, IEEE Trans. Signal Processing, vol.52, no.4, pp.962-974, April 2004. [Du05] B.Dumitrescu - Optimization of 2-D IIR Filters with Nonseparable and Separable Denominator, IEEE Trans. Signal Processing, vol.53, no.5, pp.1768-1777, May 2005 [Du06a] B.Dumitrescu - Stability Test of Multidimensional Discrete-Time Systems via Sum-of-Squares Decomposition, IEEE Trans. Circuits & Systems I, vol.53, no.4, pp.928-936, April 2006.

[Du06b] B.Dumitrescu - Trigonometric Polynomials Positive on Frequency Domains and Applications to 2-D FIR Filter Design, IEEE Trans. Signal Processing, vol.54, no.11, pp.4282-4292, Nov. 2006 [Du07] B.Dumitrescu – Positive Trigonometric Polynomials and Signal Processing Applications, Springer, 2007. [Du07a] B.Dumitrescu - Positivstellensatz for Trigonometric Polynomials and Multidimensional Stability Tests, IEEE Trans. Circuits & Systems II, vol.54, no.4, pp.353-356, 2007. [GHNSVX02] Y. Genin, Y. Hachez, Yu. Nesterov, R. Stefan, P. Van Dooren, S. Xu - Positivity and linear matrix inequalities, Euro. J. Control, vol. 8, pp.275–298, 2002. [GHNV03] Y. Genin, Y. Hachez, Y. Nesterov, P. Van Dooren – Optimization problems over positive pseudopolynomial matrices, SIAM J. MatrixAnal. Appl., vol. 25, no. 1, pp. 57–79, 2003. [HSB03] D. Henrion, M. Sebek, V. Kucera - Positive Polynomials and Robust Stabilization With Fixed-Order Controllers, IEEE Trans. Auto. Control, vol.48, no.7, pp.1178–1186, July 2003. [IH05] T.Iwasaki, S.Hara - Generalized KYP Lemma: Unified Frequency Domain Inequalities With Design Applications, IEEE Trans. Automatic Control, vol.50, no.1, pp.41-59, Jan. 2005. [KTVN06] H.H. Kha, H.D. Tuan, B.Vo, T.Q. Nguyen - Symmetric Orthogonal Complex-Valued


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Filter Bank Design by Semidefinite Programming, ICASSP, vol.3, pp.221–224, Toulouse, France, 2006. [MW02] J. W. McLean and H. J. Woerdeman, “Spectral factorizations and sums of squares representations via semidefinite programming,” SIAM J. Matrix Anal. Appl., vol. 23, no. 3, pp. 646–655, 2002. [PD01] A. Prestel and C.N. Delzell. Positive Polynomials: From Hilbert’s 17th Problem to Real

Algebra. Springer Monographs in Mathematics, Berlin, 2001. [RDV07] T.Roh, B.Dumitrescu, L.Vandenberghe - Interior-Point Algorithms for Sum-of-Squares Optimization of Multidimensional Trigonometric Polynomials, ICASSP, Honolulu, Hawaii, April 2007. [RV06] T. Roh, L. Vandenberghe - Discrete transforms, semidefinite programming and sum-of-squares representations of nonnegative polynomials,” SIAM J. on Optimization, vol. 16, no. 4, pp. 939–964, 2006 [SeDuMi] J. F. Sturm, “Using SeDuMi: A Matlab toolbox for optimization over symmetric cones,” Optim. Meth. Softw., vol. 11–12, pp. 625–653, 1999. Available online: http://sedumi.mcmaster.ca [WDR04] M.R. Wilbur, T.N. Davidson, J.P. Reilly - Efficient Design of Oversampled NPR GDFT Filterbanks. IEEE Trans. Signal Proc., vol.52, no.7, pp.1947–1963, July 2004. [YXZ05] R. Yang, L. Xie, and C. Zhang - Kalman–Yakubovich–Popov lemma for two-dimensional systems,” presented at the IFAC World Congr., Prague, Czech Rep., Jul. 2005.

10.2. Obiectivele proiectului

Obiectivele proiectului pot fi structurate pe trei nivele. N1. Obiective practice (O1 si O2 de mai jos), care vizeaza scrierea unor programe si biblioteci de programe cu scopul facilitarii rezolvarii de probleme de optimizare cu polinoame trigonometrice pozitive, utilizand rezultate de baza deja cunoscute. Programele vor fi facute publice pe internet, iar realizarea lor rapida poate asigura intaietatea in acest domeniu. In momentul actual exista doar facilitati pentru lucrul cu polinoame de variabila reala, nu si cu polinoame trigonometrice. Aceste obiective sunt pe termen scurt (un an fiind probabil suficient) si urmaresc in secundar sudarea grupului. Alte programe pot fi adaugate ulterior, pe masura ce lucrul avanseaza in celelalte directii propuse. N2. Obiectivele stiintifice sunt cele mai importante. Pe acestea le putem imparti in obiective fundamentale (O3-O5), care se refera la proprietati de baza ale sistemelor multidimensionale, si obiective aplicative (O6-O8), prin care se vizeaza rezolvarea unor probleme de analiza sau proiectare, utilizand intrumentele teoretice si practice produse de indeplinirea celorlalte obiective. N3. Obiective strategice. Pe termen lung, scopul acestui proiect este coagularea unui grup de cercetare stabil, a carui forta sa constea in special in domeniul teoretic, in fundamentele stiintei sistemelor. (La ora


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actuala, exista in Facultatea de automatica si calculatoare mai multe grupuri cu preocupari mai degraba ingineresti, rezolvand probleme cu implementabilitate imediata.) Obiectivele formale ale proiectului sunt urmatoarele: O1. Scrierea unei biblioteci de programe dedicata manipularii facile a polinoamelor trigonometrice pozitive in contextul programarii semidefinite. Din punct de vedere utilizator, polinoamele apar ca un tip de variabila, cu ajutorul careia se pot descrie operatii si restrictii intr-un mod facil. Biblioteca poate fi scrisa intr-o prima faza intr-un context independent (utilizand direct SeDuMi, de exemplu), iar ulterior prin intermediul unor biblioteci de nivel superior ca CVX (http://www.stanford.edu/~boyd/cvx) sau Yalmip (http://control.ee.ethz.ch/~joloef/yalmip.php). O sursa de inspiratie poate fi Gloptipoly (http://www.laas.fr/~henrion/software/gloptipoly3), o biblioteca pentru polinoame de variabila reala. Biblioteca propusa ar fi prima de acest gen pe plan mondial. O2. Scrierea unei colectii unitare de programe rezolvand probleme standard de optimizare cu polinoame trigonometrice pozitive. Punctul de pornire sunt programele rezolvand problemele prezentate in monografia [Du07]. Prima etapa vizeaza programarea directa in SeDuMi (exista deja o serie de programe, gandite in majoritate independent) si, pentru simplitate, in CVX. A doua etapa va consta in utilizarea bibliotecii de la O1, programele realizate insotind biblioteca pe post de exemple. O3. Studierea parametrizarii polinoamelor mixte (de exemplu cu o variabila reala si mai multe complexe, partea complexa formand un polinom trigonometric). Aceste polinoame pot fi utile in abordarea problemelor de stabilitate robusta in care coeficientii sistemului depind polinomial de un numar redus de parametri. De asemenea, pot fi utile in studiul sistemelor hibride (cu variabile atat pentru timpul continuu cat si pentru timpul discret). Nu avem la cunostinta ca ar exista o astfel de parametrizare. O4. Gasirea unei forme a lemei Kalman-Yakubovich-Popov, pentru unul din modele de stare ale sistemelor 2D, care sa caracterizeze un sistem pozitiv pe domenii definite de pozitivitatea unor polinoame. Acest rezultat ar fi o generalizare a lemei KYP pentru pozitivitate pe intervale [IH05], utilizand cadrul din [Du06b]. O astfel de lema a fost propusa in [YXZ05], dar numai pentru domenii dreptunghiulare, intr-o forma care nu pare foarte eficienta (distanta dintre suficienta si necesitate fiind mare, i.e. formularea pare mult prea conservatoare). O5. Caracterizarea pozitivitatii sistemelor interconectate spatial [DD03], in vederea studierii conditiilor de pasivitate, cu extensie in studiul stabilitatii. O teorema a amplificarilor mici (small gain) este prezentata in [CD06], utilizand lema KYP, dar consideram ca o abordare bazata pe utilizarea polinoamelor pozitive poate fi mai eficienta. O6. Studiul stabilitatatii robuste a sistemelor, in mai multe abordari: a) sisteme monovariabile, ale caror coeficienti depind de niste parametri; b) sisteme multidimensionale, in model de stare, cu matricele modelului luand valori intr-un politop cunoscut. Un astfel de studiu este [Du07a], dar el se ocupa de o problema particulara de stabilitate, pentru sisteme unidimensionale in timp discret. Exista potential de a rezolva probleme mai generale, in special in modele de stare (Fornasini-Marchesini, de exemplu). O7. Proiectarea bancuri de filtre 2D ortogonale, cu doua canale. Aceasta este o problema inselator de simplu formulata, pentru care exista singurele metode populare de rezolvare se bazeaza pe transformarea McClellan (de la 1D la 2D, Tay & Kingsbury 1993) sau pe metoda lifting (care asigura reconstructia perfecta). Ambele metode sunt neoptimale. Pentru problema 1D, rezolvarea este simpla [DP02], dar ea nu poate fi generalizata direct intrucat se bazeaza pe factorizare spectrala. Astfel de bancuri de filtre se pot utiliza de exemplu pentru compresia imaginilor. O8. Ca aplicatie a teoriei pozitivitatii in teoria sistemelor, ne propunem investigarea unei proceduri de calcul al asa-numitei „raze” de controlabilitate, adică distanţa de la o pereche controlabilă (A,B) la mulţimea tuturor perechilor necontrolabile de aceeaşi dimensiune. Problema calculului acestei distante se poate reformula ca o problema de optimizare convexa peste multimea polinoamelor matriceale pozitive de doua variabile. O metoda eficienta de calcul este data de aproximarea de tip suma-de-patrate a clasei de polinoame pozitive implicate in problema. In acest context, dorim studiul comparativ al aproximarilor folosind polinoame cu variabila reala, respectiv folosind polinoame mixte. De asemenea, este interesant studiul conditiilor asupra perechii (A,B) care fac ca polinomul matriceal in doua variabile mentionat mai sus sa se poată scrie ca suma de patrate.


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O9. Alte aplicatii. Din experienta directa a autorului, o parte dintre articolele publicate au rezultat aproape imprevizibil ca urmare a doi factori: i) detinerea unor metode emergente, cunoscute doar in cercuri restranse (cum sunt acum diverse rezultate referitoare la parametrizarea polinoamelor trigonometrice multivariabile pozitive) si ii) descoperirea (prin lecturi sistematice sau intamplatoare) unor probleme care se preteaza la rezolvari cu metodele i). Ne asteptam ca acest proiect sa produca si astfel de realizari, greu de prevazut, dar pentru care masa critica de cunostinte este aproape acumulata. Impact estimat. Ne asteptam ca realizarea obiectivelor O1 si O2 sa produca cel putin lucrari de conferinte (eventual articole de revista in conjunctie cu descrierea rezolvarii unor aplicatii), precum si un numar de utilizatori in afara grupului, care sa citeze bibliotecile realizate. De asemenea, realizarea bibliotecilor propuse va favoriza colaborarea cu alte grupuri de cercetatori care se ocupa de probleme similare. Realizarea obiectivelor O3-O9 poate produce lucrari publicabile in reviste cotate ISI. Probabil ca 7 articole ISI (cate unul pentru fiecare obiectiv) este un tel prea indraznet, dar apreciam ca publicarea unui numar de 3-4 articole este perfect realizabila. In plus, din realizare obiectivelor vor realiza in mod cert si cateva lucrari de conferinta (avem in vedere numai conferinte internationale de prima mana, gen ICASSP, EUSIPCO, MTNS). In plus, materialul acumulat va permite realizarea a doua teze de doctorat de calitate. O parte dintre obiective pot fi atacate impreuna cu cercetatori din alte universitati. Datorita timpului scurt de pregatire a proiectului, nu avem angajamente ferme in acest sens, dar mizam pe bunele relatii cu coautori mai vechi sau mai noi (vezi “resurse umane”).

10.3. Metodologia cercetarii

Metodologia generala dupa care ne ghidam este una obisnuita: • Documentarea completa prin studierea articolelor celor mai recente din literatura de

specialitate si a rapoartelor tehnice disponibile prin Internet (care contin uneori rezultate recente de mare interes). Parte din articole vor fi obtinute prin legaturile noastre cu cercetatori din universitati straine. Activitatea de recenzent asigura de asemenea informatii de actualitate.

• Gasirea unor idei noi, care sa imbunatateasca abordarile precedente, sau sa produca abordari noi. Este etapa cea mai dificila si cea mai greu predictibila in activitatea de cercetare.

• Testarea ideilor prin realizarea de programe (Matlab) si compararea rezultatelor obtinute de noi cu cele obtinute de alti cercetatori.

• In cazul ideilor bune: incercarea de generalizare, gasirea unor demonstratii complete, fundamentarea tuturor argumentelor.

• Scrierea unor lucrari care sa prezinte aceste idei si rezultatele testelor. Reteta de publicare a unor articole in reviste bine cotate este in principiu simpla : o inovatie teoretica + rezultate experimentale convingatoare + redactare ingrijita + gasirea revistei potrivite.

10.4.Resurse necesare:

10.4.1 Resursa umana 10.4.1.1. Directorul de proiect 10.4.1.1.1 Competenta stiintifica a directorului de proiect

Domenii de competenta: prelucrarea semnalelor, optimizare convexa, programare semidefinita, algoritmi paraleli. Lucrari stiintifice: 19 articole in reviste cotate ISI, dintre care 10 in ultimii 5 ani, in special in reviste de prelucrarea semnalelor (de exemplu, 4 articole in IEEE Trans. on Signal Processing, cea mai prestigioasa revista in domeniu). Peste 40 de lucrari la conferinte internationale, dintre care 12 apar in baza de date IEEE. O lista completa a lucrarilor se gaseste la http://www.cs.tut.fi/~bogdand/BD_PublicationList.html.


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Carti: 9, dintre care cea mai importanta este Positive trigonometric polynomials and signal processing applications, aparuta in primavara 2007 la Springer. Rezultate semnificative: cele mai semnificative rezultate au fost obtinute prin aplicarea unor metode de optimizare, in special programare semidefinita, in probleme specifice prelucrarii semnalelor. Intre problemele abordate se numara proiectarea filtrelor FIR si IIR (inclusiv in cazul 2D) sau proiectarea bancurilor de filtre. Un alt subiect cu rezultate recente este stabilitatea sistemelor, inclusiv a celor multidimensionale. Colaborari internationale: mai multe stagii de cercetare la Tampere University of Technology, Finlanda (peste 4 ani, in total, in ultimii 9 ani). Colaborari directe cu cercetatori de la Nokia Research Center (Tampere, Finlanda), Drexel University (Philadelphia, SUA), University of California (Los Angeles), Brooklyn University (New York). Conducere de doctorat: dosarul de conducere de doctorat a fost depus in aprilie 2007. Altele: Recenzent la mai mult de 10 reviste in diverse domenii (prelucrarea semnalelor, circuite si sisteme, automatica, dar si la Mathematical Reviews). Membru in comitetul tehnic de program la mai multe conferinte europene (EUSIPCO 2007, Poznan, Polonia) si IEEE (ISSPA 2007, Sharjah, UAE, NSIP 2007, Bucuresti). Cele mai recente 10 articole (toate in reviste cotate ISI):

• B.Dumitrescu - Positivstellensatz for Trigonometric Polynomials and Multidimensional Stability Tests, IEEE Trans. Circuits & Systems II, vol.54, no.4, pp.353-356, 2007.

• B.Dumitrescu - Trigonometric Polynomials Positive on Frequency Domains and Applications to 2-D FIR Filter Design, IEEE Trans. Signal Processing, vol.54, no.11, pp.4282-4292, Nov. 2006.

• B.Dumitrescu, B.C.Chang - Robust Schur Stability with Polynomial Parameters, IEEE Trans. Circuits & Systems II, vol.53, no.7, pp.935-937, July 2006.

• B.Dumitrescu - Stability Test of Multidimensional Discrete-Time Systems via Sum-of-Squares Decomposition, IEEE Trans. Circuits & Systems I, vol.53, no.4, pp.928-936, April 2006.

• B.Dumitrescu, R.Bregovic, T.Saramäki - Simplified Design of Low-Delay Oversampled NPR GDFT Filterbanks, EURASIP Journal on Applied Signal Processing, vol. 2006, Article ID 42961, 11 pages, 2006.

• B.Dumitrescu - Bounded Real Lemma for FIR MIMO Systems, IEEE Signal Processing Letters, vol.12, no.7, pp.496-499, July 2005.

• B.Dumitrescu - Optimization of 2-D IIR Filters with Nonseparable and Separable Denominator, IEEE Trans. Signal Processing, vol.53, no.5, pp.1768-1777, May 2005.

• B.Dumitrescu, R.Niemistö - Multistage IIR Filter Design Using Convex Stability Domains Defined by Positive Realness, IEEE Trans. Signal Processing, vol.52, no.4, pp.962-974, April 2004.

• R.Niemistö, B.Dumitrescu - Simplified Procedures for Quasi-Equiripple IIR Filter Design, IEEE Signal Processing Letters, vol.11, no.3, pp.308-311, March 2004.

• C.Popeea, B.Dumitrescu, B.Jora - Efficient State-Space Approach for FIR Filter Bank Completion, Signal Processing, vol.83, no.9, pp.1973-1983, Sept. 2003.

10.4.1.1.2. Competenta manageriala a directorului de proiect

B.Dumitrescu a fost director la urmatoarele proiecte de cercetare (codirector, la al treilea): 1. Proiectarea filtrelor de semnal utilizand programarea semidefinita, contract CNCSIS, cod 406, 2002


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(nr.33784/23.07.2002). (55 000 000 ROL) 2. Noi aplicatii ale programarii semidefinite in proiectarea filtrelor digitale, contract CNCSIS, cod 153, 2003 (nr. 40528/5.11.2003). (91 500 000 ROL) 3. Filterbanks for speech enhancement, contract al Universitatii Tehnice din Tampere, Finlanda, cu Nokia Research Center, 2004 (cu prelungire in 2005). (80 000 EUR) Toate proiectele au fost duse la bun sfarsit, producand printre altele si articole in reviste cotate ISI.

10.4.1.2. Echipa de cercetare

Lista membrilor echipei de cercetare: (Fara directorul de proiect)

Nr. crt. Nume si prenume Anul

nasterii Titlul didactic

stiintific * Doctorat

* * Semnatura

1 RADU STEFAN 1969 Conferentiar DA 2 CRISTIAN FLUTUR 1982 Cercetator NU 3 X – va fi selectat

ulterior (toamna 2007) NU

* La “Titlu didactic/stiintific” completati cu una din variantele: Profesor / Conferentiar / Lector / Asistent / CS I / CS II / CS III / Cercetator ** La “Doctorat” completati cu una din variantele: DA /NU / Doctorand

10.4.1.2.1. Cercetatori cu experienta

Radu Stefan s-a nascut in 1969. Din 1993 este cadru didactic (conferentiar, din 2005) la Facultatea de Automatica si Calculatoare a Universitatii Politehnica din Bucuresti. A obtinut titlul de doctor inginer in 1998. Intre feb. 1999 si august 2001 a fost cercetator asociat la Faculte des Sciences Appliquees, Universite Catholique de Louvain, Belgia, facand parte dintr-unul din grupurile care au initiat cercetarile curente in utilizarea polinoamelor pozitive in optimizare, vezi mai sus lucrarea [GHNSVX02]. Domenii de competenta : teoria sistemelor, stabilitate, optimizare, sisteme neliniare Experienţa acumulata in programe naţionale/internaţionale:

• coordonator proiect ``Masuri de robustete pentru sisteme complexe'' in colaborare cu Universitatea catolica din Louvain, 2002-2003.

• conducator proiect "Advanced topics in stability radius theory", SSTC Fellowship al guvernului Belgiei, 1999-2000.

• conducator proiect "Positivity, dissipativity and robustness of control systems: theory and algorithms", NATO Fellowship, 2000-2001.

• participare la alte 10 granturi/contracte nationale/internationale ca membru al echipei de cercetare (CNCSIS, Banca Mondiala, MCT).

Lucrari semnificative (in ultimii 5 ani): 1. Vl. Rasvan si R. Stefan, Systemes Nonlineaires: Theorie et Applications, Hermes Science, 2007. 2. R. Stefan, Disturbance attenuation in the chain-scattering formalism, Mathematical Reports, 5(55), 4, pp. 371-387, 2003. 3. V. Rasvan si R. Stefan, Discussion on Passification of nonsquare nonlinear systems, European Journal of Control, 9, 6, pp. 587-588, 2003. 4. V. Ionescu si R. Stefan, Generalized time-varying Riccati theory: A Popov operator based approach, Integral Equations and Operator Theory, 48, pp. 159-212, 2004. 5. Y. Hachez si R. Stefan, Computing the distance to uncontrollability: A convex optimization approach, IFAC CAO06, Paris, France , 2006.

10.4.1.2.2. Cercetatori in formare


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Cristian Flutur s-a nascut in 1982. In 2006 a terminat Facultatea de Automatica si Calculatoare a Universitatii Politehnica Bucuresti, unde este cercetator din ianuarie 2007, in cadrul centrului de cercetare ACPC. In toamna 2007 se va inscrie la doctorat. Un posibil titlu al tezei de doctorat este “Optimizare convexa in analiza si sinteza sistemelor”, cu planul de a acorda cea mai mare parte a studiului optimizarii cu polinoame trigonometrice pozitive. In cadrul proiectului prima sarcina a lui C.Flutur va fi realizarea obiectivului O1 (vezi 10.2), de implementare a unei biblioteci de programe dedicata manipularii facile a polinoamelor trigonometrice pozitive in contextul programarii semidefinite. Acest obiectiv necesita cunostinte standard de programare si informatii despre polinoamele trigonometrice pe care le va capata prin documentare, pe parcursul lucrului. Dupa terminarea bibliotecii (sau a unei prime versiuni, cvasi-complete), C.Flutur va trece la nivelul urmator, cel de participare la realizarea unor obiective stiintifice (vezi 12.1); in prima faza, participarea se va materializa in implementare de programe dupa idei primite de la cercetatorii cu experienta si la teste si comparatii folosind aceste programe. In mod ideal, va urma faza emiterii unor idei inovative proprii, validate corespunzator. Un scop al participarii la proiect este finalizarea tezei de doctorat, chiar daca sustinerea va avea loc efectiv doar dupa terminarea proiectului. Al patrulea membru al echipei, notat X, nu a fost identificat. Ar putea fi unul dintre absolventii din 2007 ai Facultatii de Automatica si Calculatoare. X va fi selctionat in toamna 2007, procesul de cautare incepand de acum si intensificandu-se imediat ce proiectul va fi fost aprobat. Este posibil ca X sa nu se inscrie la doctorat in 2007, inscrierea urmand sa aiba loc in 2008. X se va ocupa de realizarea obiectivului O2, care are caracter de implementare, deci este potrivit pentru introducerea activa in domeniul optimizarii cu polinoame multivariabile. Dupa terminare, urmeaza implicarea tot mai activa in obiectivele stiintifice, asa cum este explicat mai sus pentru C.Flutur. Asa cum este mentionat la 12.1, deocamdata nu ne propunem o separare clara a obiectivelor O3-O8 intre C.Flutur si X. In orice caz, de un obiectiv se va ocupa cel mult unul dintre ei, iar numarul de obiective in care va fi implicat fiecare va depinde de capacitatea lor de progres. Alegerea obiectivelor pentru fiecare dintre ei se va face printr-o abordare treptata, in functie de afinitati si evolutia lucrului.

10.4.2 Alte resurse

10.4.2.1. Resurse financiare

1. Cheltuieli de personal – salarii incluzand contributiile platite de angajator B.Dumitrescu, jumatate de norma: 3000 RON/luna R.Stefan, jumatate de norma: 2500 RON/luna 2 tineri cercetatori, norma intreaga: 2600 RON/luna Total: (3000 + 2500 + 2600 + 2600) * 36 = 385200 RON 2. Regie 25 %: 96 300 RON 3. Mobilitati

• Pentru cercetatorii cu experienta, cate 3 deplasari de 15-30 de zile. In afara documentarii si discutiile cu cercetatorii de acolo, beneficiarii deplasarilor vor face prezentari stiintifice in universitatile gazda. Suma pentru fiecare deplasare: 7000 RON.

• Pentru tinerii cercetatori, cate 3 deplasari de 2 luni. Suma pentru fiecare deplasare: 9000 RON. Total: (7000 + 9000)*6 = 96 000 RON Deplasarile vor fi efectuate la universitati unde exista grupuri cu preocupari apropiate si cercetatori cu care suntem in contact (chiar daca nu am avut colaborari directe), anume:

• Tampere University of Technology (Finlanda): I.Tabus, J.Astola, T.Saramaki • Universite Catholique de Louvain (Belgia): P.Van Dooren, Y,Nesterov • Lab. d’Analyse et d’Architecture de Systemes (CNRS), Toulouse (Franta): D.Henrion • INRIA Futurs, Orsay (Franta): L.Grigori • eventual, universitati din SUA, in conjunctie cu deplasari la conferinte (pentru reducerea

costurilor): L.Vandenberghe, I.Selesnick, B.C.Chang


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Deplasarile vor fi (pe cat posibil) uniform distribuite in cei 3 ani ai proiectului. 4. Cheltuieli de logistica a. Infrastructura

• 2 calculatoare bune (pentru tinerii cercetatori): 10000 RON • 1 imprimanta (pentru uz curent al grupului): 1500 RON • Matlab, licenta academica pentru un grup de aprox. 10 utilizatori: 30000 RON

b. Diseminare rezultate • cheltuieli pentru articole publicate (de exemplu paginile suplimentare in revistele IEEE—peste 8—

costa 150-200 USD/pagina): 6000 RON • sprijin pentru publicarea unei carti in limba romana: 5000 RON • intretinere si actualizare site web www.schur.pub.ro: 3000 RON

c. Documentare

• Cumparare carti: 3000 RON • Taxa IEEE + abonamente electronice la cateva reviste: 3000 RON

d. Invitati straini:

• 3 invitati straini la Bucuresti, pentru o saptamana fiecare, pentru discutii si o prezentare stiintifica, de exemplu in seminarul stiintific al catedrei. Cel putin unul dintre invitati va tine o prelegere la traditionala scoala de vara organizata in Facultatea de automatica si calculatoare, in ultima decada a lunii mai. Vor fi invitati cercetatori cu preocupari similare celor ale grupului, de preferat dintre cei cu care exista relatii de colaborare. Vor fi suportate cheltuielile de transport si cazare, eventual o indemnizatie. Suma pentru fiecare invitat: 6000 RON.

e. Altele

• Consumabile: 8000 RON • Neprevazute: 5000 RON

Total: 41500 + 14000 + 6000 + 18000 + 13000 = 92500 RON Distribuire cheltuieli pe ani:

2007 2008 2009 2010 Calculatoare 10000 Imprimanta 1500

Matlab 30000 Articole 1000 2000 3000

Carte 5000 Site 1000 1000 1000

Cumparare carti 1000 2000 IEEE 1000 1000 1000

Invitati 6000 6000 6000 Consumabile 1000 2000 3000 1000 Neprevazute 1000 1000 2000 1000

Total 2000 54500 18000 18000

10.4.2.2. Infrastructura disponibila (calitatea infrastructurii de cercetare existente)

Pentru cercetarea propusa in acest proiect sunt suficiente calculatoare cu o viteza de calcul rezonabila (produse in ultimii 2-3 ani) si conexiune la internet. In afara programului Matlab (pentru care avem o singura licenta), restul programelor necesare sunt disponibile liber pe internet. Infrastructura existenta: Retea de 6 calculatoare (dintre care 3 sunt utilizate, iar unul este server), in sala ED206, unde lucreaza B.Dumitrescu (si Grupul de calcul numeric). Patru dintre calculatoare au mai putin de trei ani vechime. O imprimanta, destul de veche. De asemenea, acces la laboratorul de Sisteme si Optimizari, unde lucreaza R.Stefan.


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Infrastructura necesara: 2 calculatoare PC de buna calitate, pentru cei doi tineri cercetatori. O imprimanta de birou pentru grup mic de lucru. Licenta Matlab pentru intreg grupul.


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ANEXA 4

11. Prezentarea proiectului in limba engleza: (Max. 10 pagini)

11.1. Importance and relevance of the scientific contents 11.1.1. Introduction Multidimensional dynamic systems are used in many domains, modelling naturally phenomena and processes. In prominent position is signal processing, where 2D and 3D (recently even 4D) filters are employed diversely, for example in image processing, denoising, compression, radar signal processing. The independent variables have here spatial significance. In control, repetitive systems (metal rolling, paper making) can be modelled via 2D systems. A more recent example is that of spatially interconnected systems [DD03], in which identical subsystems interact, each only with its neighbours. In these systems, a variable is time, the others represent spatial coordinates. In this project, we are interested especially by discrete-time (digital) systems. They can be modelled either by a transfer function (matrix), or by state-space models like Roesser or Fornasini-Marchesini. Positivity represents a fundamental property as well as an instrument. Passive (positive real) systems make an important class in stability treatment. They are characterized either by the direct expression of positivity, in polynomial models, or by the Kalman-Yakubovich-Popov, in state-space models. Talking of positivity as an instrument, in the context of discrete-time systems, we are referring to positive trigonometric polynomials, which model naturally a number of optimization problems, mentioned below, in a systemic context. In the sequel, we will briefly present the basics of the use of positive trigonometric polynomials in optimization, we will then enumerate some of their recent applications in solving problems in signals and systems theory. Finally, we will discuss the relevance of their deeper study, as proposed by this project. 11.1.2. Positive trigonometric polynomials

A trigonometric polynomial has the form ( ) ,n

kk

k n

R z r z−=−

= ∑ where the coefficients are generally

complex and symmetric, i.e. *k kr r− = , and n is the degree of the polynomial. On the unit circle,

where z=ejω, the polynomial has real values. We name positive a polynomial for which ( ) 0,jR e ω ω≥ ∀ . The recent interest for positive polynomials comes from the combination of

two results. The first, well known, says that if we can optimize the squared magnitude of a


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polynomial H(z) (in general not symmetric), i.e. we work with 2( ) | ( ) |j jR e H eω ω= , which is a positive trigonometric polynomial, then we can compute H(z) through spectral factorization. (“Squaring” is a standard way to convexify such optimization problems.) The second result was proposed less than a decade ago and represents a parameterization of the set of positive trigonometric polynomials: the condition that R(z) is positive is equivalent to the existence of a positive semidefinite matrix Q such that [ ]k kr trace Q= Θ , where kΘ are

constant matrices (with an elementary Toeplitz structure) [AV02, DTS01, GHNSVX02, GHNV03]. This parameterization opens the way to the use of semidefinite programming (SDP) in optimization problems with trigonometric polynomials, allowing their fast and reliable solution with libraries available online, like [SeDuMi]. This result has been generalized in several ways, for example for polynomials with matrix coefficients [GHNV03] or for polynomials positive on an interval [AV02, DLS02], and used in several applications as mentioned later. Of special interest for the current project is the generalization of the parameterization to multivariate trigonometric polynomials. Denoting d the number of independent variables, these

polynomials have the form ( ) ,R r −

=−

= ∑n

kk

k nz z where the bold characters denote d-dimensionale

variables, e.g 11

dkkdz z=kz … , and the sum is taken for all the values dZ∈k for which

− ≤ ≤n k n (the inequalities are understood elementwise). Again, the polynomial is symmetric, i.e. *r r− =k k , and so the polynomial takes real values on the unit d-circle (torus).

A trigonometric polynomial is sum-of-squares if it can be written in the form

2( ) | ( ) |j ji

i

R e H e=∑ω ω . The parameterization gets the following formulation: the polynomial

R(z) is sum-of-squares if and only if there exists a positive semidefinite matrix Q such that [ ]r trace Q= Θk k , where Θk are constant matrices (Kronecker products of elementary Toeplitz

matrices) [MW02,Du06a]. We note that, in the multidimensional case, positivity is replaced by sum-of-squares (this restriction comes from a result known already from Hilbert, in connection with his 17th problem, that not all real multivariate polynomials are sum-of-squares). Practically, to be able to apply SDP, the degree of the polynomials Hi appearing in the sum-of-squares is bounded, and so the size of the matrix Q is fixed. The set of sum-of-squares thus obtained is a subset of the set of positive polynomials (and the difference is not empty). So, optimization problems with positive polynomials cand be solved in a somewhat restrictive setup, imposed by the above parameterization. The generalization of the parameterization in the case of polynomials positive on certain domains (the analogue of positivity on an interval for the monovariate case) has been


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presented in [Du06b]. An impediment to optimization with positive multivariate polynomials is the inexistence of a “spectral factorization” result. In some problems, it can be replaced by a Bounded Real Lemma style inequality, which in [Du06b] was expressed as a linear matrix inequality (LMI). In general, in the multivariate case, the mathematical results are weaker than in the univariate one. However, the basic parameterization described above has allowed the solution of some problems, only a few for the moment. 11.1.3. Application and other developments In the univariate case, the use of the parameterization of positive trigonometric polynomials has led to the solution of optimization problems related (among others) to:

• the design of FIR [AV02, DLS02] and IIR (only in magnitudine) filters [AV02] • the design of orthogonal two-channel filter banks [DP02] • the design of robust low-order controllers [HSB03] • the design of orthogonal oversampled filter banks [WDR04] • the design of IIR filters [DN04] • the design of orthogonal symmetric filter banks [KTVN06] • the optimization of a dual-tree wavelet construction (current research of B.Dumitrescu,

together with I.Selesnick and I.Bayram, article submitted in June 2007 to IEEE Signal Proc. Letters)

Other developments of the subject are related to:

• Fast algorithms for solving SDP problems result from the utilization of positive trigonometric polynomials. After some relatively fast tentatives [AV02], with low numerical stability (prohibitting the solution of large size problems), better results seem to have been obtained in [RV06].

• A generalization of the Kalman-Yakubovich-Popov lemma to systems that are positive on intervals [IH05], leading also to SDP.

In the multivariate case, the applications are more recent and thus lesser:

• The description of convex stability domains used in the design of 2D IIR filters [Du05] • Multidimensional system stability test [Du06a] • Design of FIR 2D filters [Du06b]


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A first version of fast algorithm for the 2D case has been presented in [RDV07]. In the above discussion we have voluntarily ignored the efforts of mathematicians. In the latest 15 years, several important results related to sum-of-squares polynomials have been produced. Some of them are presented in [PD01], but we are aware of the most recent results. The investigation of their applicative potential (in general not obvious and not explicitly searched by the authors) is part of the work in this project. 11.1.4. Conclusions – importance and relevance The recent interest devoted to positive polynomials is motivated by the possibility of solving, practically exactly or with very good approximation, some problems for which previously nonconvex approaches were used. In the univariate case, it seems that the understanding of the fundaments of positive polynomial parameterization is complete; there may be still applications in which this instrument is used, but they cannot constitute the object of a systematic research—they may be derived products of research oriented to other goals. In contrast, the study of positivity in the context of multidimensional systems is still full of unexplored regions. A primary cause is the non-identity between “positive” and “sum-of-squares”, leading to approximations that have to be studied on problem basis. Another cause is the intrinsic difficulty of the problems, the complicated models and the corresponding programs whose development requires, besides a complete understanding of the problem, considerable attention and skill. The proposed research theme is on the border of optimization, system theory and mathematics, a region generally accessible to researchers in control, but maybe of greater intereset to the signal processing community. This may be one of the reasons for which the results appear rather slowly, but this opportunity window will not last too long. The principal investigator of this project is well placed in studying the subject (a proof being his recent articles), but the timing is favorable to an attack with wider forces, those of a whole group. 11.1.5. Bibliography [AV02] B. Alkire, L. Vandenberghe - Convex optimization problems involving finite autocorrelation sequences, Math. Progr. Ser. A, vol. 93, no. 3, pp. 331–359, 2002 [CD06] R. S. Chandra, R.D’Andrea - A Scaled Small Gain Theorem With Applications to Spatially Interconnected Systems, IEEE Trans. Automatic Control, vol.51, no.3, pp.465-469, Mar. 2006.


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[DD03] R.D’Andrea, G.E.Dullerud - Distributed control design for spatially interconnected systems, IEEE Trans. Automatic Control, vol.48, no.9, pp.1478-1495, Sept. 2003. [DLS02] T. N. Davidson, Z. Q. Luo, and J. F. Sturm - Linear matrix inequality formulation of spectral mask constraints with applications to FIR filter design, IEEE Trans. Signal Process., vol. 50, no. 11, pp. 2702–2715, Nov. 2002. [DTS01] B.Dumitrescu, I.Tabus, P.Stoica - On the Parameterization of Positive Real Sequences and MA Parameter Estimation, IEEE Trans. Signal Processing, vol.49, no.11, pp.2630-2639, Nov. 2001 [DP02] B. Dumitrescu, C. Popeea - Accurate Computation of Compaction Filters with High Regularity, IEEE Signal Proc. Letters, vol.9, no.9, pp.278–281, Sept. 2002. [DN04] B.Dumitrescu, R.Niemistö - Multistage IIR Filter Design Using Convex Stability Domains Defined by Positive Realness, IEEE Trans. Signal Processing, vol.52, no.4, pp.962-974, April 2004. [Du05] B.Dumitrescu - Optimization of 2-D IIR Filters with Nonseparable and Separable Denominator, IEEE Trans. Signal Processing, vol.53, no.5, pp.1768-1777, May 2005 [Du06a] B.Dumitrescu - Stability Test of Multidimensional Discrete-Time Systems via Sum-of-Squares Decomposition, IEEE Trans. Circuits & Systems I, vol.53, no.4, pp.928-936, April 2006.

[Du06b] B.Dumitrescu - Trigonometric Polynomials Positive on Frequency Domains and Applications to 2-D FIR Filter Design, IEEE Trans. Signal Processing, vol.54, no.11, pp.4282-4292, Nov. 2006 [Du07] B.Dumitrescu – Positive Trigonometric Polynomials and Signal Processing Applications, Springer, 2007. [Du07a] B.Dumitrescu - Positivstellensatz for Trigonometric Polynomials and Multidimensional Stability Tests, IEEE Trans. Circuits & Systems II, vol.54, no.4, pp.353-356, 2007. [GHNSVX02] Y. Genin, Y. Hachez, Yu. Nesterov, R. Stefan, P. Van Dooren, S. Xu - Positivity and linear matrix inequalities, Euro. J. Control, vol. 8, pp.275–298, 2002. [GHNV03] Y. Genin, Y. Hachez, Y. Nesterov, P. Van Dooren – Optimization problems over positive pseudopolynomial matrices, SIAM J. MatrixAnal. Appl., vol. 25, no. 1, pp. 57–79, 2003. [HSB03] D. Henrion, M. Sebek, V. Kucera - Positive Polynomials and Robust Stabilization With Fixed-Order Controllers, IEEE Trans. Auto. Control, vol.48, no.7, pp.1178–1186, July 2003. [IH05] T.Iwasaki, S.Hara - Generalized KYP Lemma: Unified Frequency Domain Inequalities


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With Design Applications, IEEE Trans. Automatic Control, vol.50, no.1, pp.41-59, Jan. 2005. [KTVN06] H.H. Kha, H.D. Tuan, B.Vo, T.Q. Nguyen - Symmetric Orthogonal Complex-Valued Filter Bank Design by Semidefinite Programming, ICASSP, vol.3, pp.221–224, Toulouse, France, 2006. [MW02] J. W. McLean and H. J. Woerdeman, “Spectral factorizations and sums of squares representations via semidefinite programming,” SIAM J. Matrix Anal. Appl., vol. 23, no. 3, pp. 646–655, 2002. [PD01] A. Prestel and C.N. Delzell. Positive Polynomials: From Hilbert’s 17th Problem to Real

Algebra. Springer Monographs in Mathematics, Berlin, 2001. [RDV07] T.Roh, B.Dumitrescu, L.Vandenberghe - Interior-Point Algorithms for Sum-of-Squares Optimization of Multidimensional Trigonometric Polynomials, ICASSP, Honolulu, Hawaii, April 2007. [RV06] T. Roh, L. Vandenberghe - Discrete transforms, semidefinite programming and sum-of-squares representations of nonnegative polynomials,” SIAM J. on Optimization, vol. 16, no. 4, pp. 939–964, 2006 [SeDuMi] J. F. Sturm, “Using SeDuMi: A Matlab toolbox for optimization over symmetric cones,” Optim. Meth. Softw., vol. 11–12, pp. 625–653, 1999. Available online: http://sedumi.mcmaster.ca [WDR04] M.R. Wilbur, T.N. Davidson, J.P. Reilly - Efficient Design of Oversampled NPR GDFT Filterbanks. IEEE Trans. Signal Proc., vol.52, no.7, pp.1947–1963, July 2004. [YXZ05] R. Yang, L. Xie, and C. Zhang - Kalman–Yakubovich–Popov lemma for two-dimensional systems,” presented at the IFAC World Congr., Prague, Czech Rep., Jul. 2005. 11.2. Project objectives The objectives of the projects can be structured on three levels. L1. The practical objectives (O1 and O2 below), aiming at writing programs and libraries with the purpose of simplifying the manipulation of positive trigonometric polynomials in the context of solving optimization problems, using alread known basic results. The programs will be made available on the internet and their fast realization will assure our first position in the field; as of this moment, there are only libraries for working with polynomials of real variable, not with trigonometric polynomials. These are short term objectives (a year being probably enough) and have the secondary goal of uniting the group. Other programs can be added later, as the work advances in the other proposed directions. L2. The scientific objectives are the most important. We can split them in fundamental objectives (O3-O5), referring to basic properties of multidimensional systems, and applied objectives (O6-O8), with the


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goal of solving analysis or design problems by using the theoretical and practical instruments produced by the realization of the other objectives. L3. Strategic objectives. On a long term, the purpose of this project is the formation of a stable research group whose force would rely especially in the theoretical domain, in the fundaments of system theory. (At this moment, there are, in the Faculty of Automatic Control and Computers, several groups with engineering interests, solving problems with immediate implementability.) The formal objectives of the project are the following: O1. To write a library of programs dedicated to the easy manipulation of positive trigonometric polynomials in the context of semidefinite programming. From the user viewpoint, the polynomials appear as a variable type, used to describe operations and constraints in a simple way. In a first stage, the library can be written as an independent entity (calling directly only SDP libraries, as SeDuMi); later, the library can call upper level optimization libraries such as CVX (http://www.stanford.edu/~boyd/cvx) or Yalmip (http://control.ee.ethz.ch/~joloef/yalmip.php). An inspiration source can be Gloptipoly (http://www.laas.fr/~henrion/software/gloptipoly3), a library for polynomials of real variable. Such a library would be the first of its kind. O2. To write a unitary collection of programs for solving standard optimization problems using positive trigonometric polynomials. The starting set are the programs solving problems presented in the monography [Du07]. The first stage will consists in writing programs calling directly SeDuMi (some programs already exist, but they were written individually, not as a collection) and, for simplicity, CVX. The second stage will consist of using the library from O1, the programs becoming part of the library as examples of use. O3. To study the parameterization of hybrid polynomials (for example, with a real variable and several complex ones, the complex part being a symmetric polynomial). These polynomials could be useful in the treatment of robust stability problems in which the coefficients of the system depend polynomially on a small number of parameters. Also, they can be useful in the study of hybrid systems (mixed continuous-discrete-time). We are not aware of the existence of such a parameterization. O4. To find a form of the Kalman-Yakubovich-Popov lemma, for one of the state-space models of 2D systems, characterizing a system whose response is positive on domains defined by the positivity of some given polynomials. This result would be a generalization of the KYP lemma for positivity on intervals [IH05], using the framework from [Du06b]. Such a lemma was proposed in [YXZ05], but only for rectangular domains, in a form that does not seem efficient (the distance between sufficiency and necessity is too large, i.e. the formulation is much too conservative). O5. To characterize the positivity of spatially interconnected systems [DD03], aiming at a study of passivity conditions, with extensions in the study of stability. A small gain theorem is presented in [CD06], using the KYP lemma, but we consider that an approach based on positive polynomials can be more efficient. O6. To study the robust stability of systems, in several approaches: a) 1D systems whose coefficients depend on some parameters; b) multidimensional systems, in state-space model, the matrices of the model taking values in a given polytope. Such a study is [Du07a], but it deals with a particular stability problem for 1D discrete-time systems. There is a potential to solve more general problems, especially in state-space models (e.g. Fornasini-Marchesini). O7. To find algorithms for designing 2D two-channel orthogonal filter banks. This is a deceivingly simple problem, for which the only existing methods are based on the McClellan transformation (from 1D to 2D, Tay & Kingsbury 1993) or on the lifting method (ensuring perfect reconstruction). Both methods are suboptimal. For the 1D problem, the solution is simple [DP02], but it cannot be generalized as it is based on spectral factorization. These filter banks can be used for example in image compression. O8. As an application of positivity theory in system theory, we propose to investigate a procedure to compute the so called controllability radius, i.e. the distance from a controllable pair (A,B) to the set of uncontrollable pairs of the same size. The problem of computing this distance can be reformulated as a convex optimization problem on the set of bivariate matrix polynomials of degree 2. An efficient computation method is based on an a sum-of-squares approximation. In this context, we want to perform


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a comparative study of the approximation using real polynomials and hybrid polynomials. Also, it is interesting to study the conditions on the pair (A,B) that make sum-of-squares the above bivariate polynomial. O9. Other applications. The direct experience of the author tells that some of his published articles have resulted almost unpredictably as a result of two factors: i) the possession of some emergent methods, known only in narrow circles (as are now the diverse results dealing with the parameterization of positive multivariate trigonometric polynomials), and ii) the discovery (through systematic or random reading) of problems that can be solved with these methods. We expect that this project will produce also this kind of result, hard to forecast, but for which the critical mass of knowledge is already accumulated. Estimated impact. We expect that the realization of the objectives O1 and O2 will produce at least conference papers (maybe journal articles, in conjunction with the solution of some applicative problems), as well as a number of users outside the group, who will cite our libraries. Also, working at these libraries will favorize the connections with researchers who are studying similar problems. The finalization of objectives O3-O9 can produce papers publishable in ISI journals. Maybe a number of 7 articles (one for each objective) is a too daring purpose, but we appreciate that publishing 3-4 ISI articles is perfectly possible. Additionally, a number of conference papers will certainly result (we aim only at first class international conferences, such as ICASSP, EUSIPCO, MTNS). Also, the accumulated material will be sufficient for two quality doctoral theses. Some of the objectives can be attacked together with researchers from other universities. Due to the short time available for preparing the project, we have no formal agreements to this purpose, but we count on the good relations with old or new coauthors (see “human resources”). 11.3 Research methodology The general methodology guiding us is a usual one:

• Bibliographic study of the most recent articles in the litereature and also of technical reports available on the internet (often containing recent results of great interest). Part of the articles will be obtained through our connections with researchers from other universities. The reviewing activity is also a source of timely information.

• Find new ideas, that improve on the existing ones or produce new approaches. This is the most difficult and less predictable activitity in research.

• Test the ideas through programs and compare the results with those obtained with other approaches.

• For good ideas: try to generalize, find complete proofs, order all arguments on a logical basis.

• Write papers that present the ideas and the results of the tests. The recipe for producing an article that can be published in a good journal is simple, in principle: theoretical innovation + convincing experimental results + careful writing + finding the appropriate journal.

11.4 Necessary resources

11.4.1 Human resources 11.4.1.1. Project director

11.4.1.1.1 Scientific competence of project director Competence fields: signal processing, convex optimization, semidefinte programming, parallel algorithms. Scientific activity: 19 articles in ISI journals, out of which 10 in the latest 5 years, especially in signal processing journals (for example, 4 articles in IEEE Trans. on Signal Processing, the most prestigious journal in the field). More than 40 papers at international conferences, out of which 12 appear in the IEEE databasis. A complete publication list is http://www.cs.tut.fi/~bogdand/BD_PublicationList.html.


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Books: 9, the most important being Positive trigonometric polynomials and signal processing applications, published by Springer, in 2007. Significant results: the most significant results have been obtained through the application of some optimization methods, especially semidefinite programming, to signal processing problems. Among these problems are the design of FIR and IIR filters (including the 2D case) or the design of filter banks. Another subject with recent results is system stability, including the multidimensional case. International cooperation: several research stages at Tampere University of Technology, Finland (more than 4 years in the latest 9 years). Direct cooperation with researchers from Nokia Research Center (Tampere, Finland), Drexel University (Philadelphia, USA), University of California (Los Angeles), Brooklyn University (New York). Docentship: the file for obtaining the docentship was submitted in april 2007. Other: Reviewer for more than 10 journals in several fields (signal processing, circuits & systems, control, but also for Mathematical Reviews). Member in the technical program committee at several conferences, european (EUSIPCO 2007, Poznan, Poland) or IEEE (ISSPA 2007, Sharjah, UAE, NSIP 2007, Bucarest). Most recent 10 articles (all in ISI journals):

• B.Dumitrescu - Positivstellensatz for Trigonometric Polynomials and Multidimensional Stability Tests, IEEE Trans. Circuits & Systems II, vol.54, no.4, pp.353-356, 2007.

• B.Dumitrescu - Trigonometric Polynomials Positive on Frequency Domains and Applications to 2-D FIR Filter Design, IEEE Trans. Signal Processing, vol.54, no.11, pp.4282-4292, Nov. 2006.

• B.Dumitrescu, B.C.Chang - Robust Schur Stability with Polynomial Parameters, IEEE Trans. Circuits & Systems II, vol.53, no.7, pp.935-937, July 2006.

• B.Dumitrescu - Stability Test of Multidimensional Discrete-Time Systems via Sum-of-Squares Decomposition, IEEE Trans. Circuits & Systems I, vol.53, no.4, pp.928-936, April 2006.

• B.Dumitrescu, R.Bregovic, T.Saramäki - Simplified Design of Low-Delay Oversampled NPR GDFT Filterbanks, EURASIP Journal on Applied Signal Processing, vol. 2006, Article ID 42961, 11 pages, 2006.

• B.Dumitrescu - Bounded Real Lemma for FIR MIMO Systems, IEEE Signal Processing Letters, vol.12, no.7, pp.496-499, July 2005.

• B.Dumitrescu - Optimization of 2-D IIR Filters with Nonseparable and Separable Denominator, IEEE Trans. Signal Processing, vol.53, no.5, pp.1768-1777, May 2005.

• B.Dumitrescu, R.Niemistö - Multistage IIR Filter Design Using Convex Stability Domains Defined by Positive Realness, IEEE Trans. Signal Processing, vol.52, no.4, pp.962-974, April 2004.

• R.Niemistö, B.Dumitrescu - Simplified Procedures for Quasi-Equiripple IIR Filter Design, IEEE Signal Processing Letters, vol.11, no.3, pp.308-311, March 2004.

• C.Popeea, B.Dumitrescu, B.Jora - Efficient State-Space Approach for FIR Filter Bank Completion, Signal Processing, vol.83, no.9, pp.1973-1983, Sept. 2003.

11.4.1.1.2. Managerial competence of project director B.Dumitrescu was the director of the following projects (codirector, for the third one): 1. Filter design using semidefinite programming, CNCSIS contract, code 406, 2002 (no.


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33784/23.07.2002). (55 000 000 ROL) 2. New applications of semidefinite programming in the design of digital filters, CNCSIS contract, code 153, 2003 (nr. 40528/5.11.2003). (91 500 000 ROL) 3. Filterbanks for speech enhancement, contract of Tampere University of Technology with Nokia Research Center, 2004-2005. (80 000 EUR) All the project have been finalized, producing articles in ISI journals, among other results.

11.4.1.2. Research team (excluding the project director)

1. Radu Stefan (born 1969, associate professor, Ph.D.) 2. Cristian Flutur (born 1982, researcher) 3. X – to be selected in autumn 2007

11.4.1.2.1. Senior researchers Radu Stefan was born in 1969. Since 1993 he is with the Faculty of Automatic Control and Computers of the Politehnica University of Bucharest (associated professor since 2005). He obtained the Ph.D. title in 1998. Between February 1999 and August 2001 he was an associated researcher at Faculte des Sciences Appliquees, Universite Catholique de Louvain, Belgium, being member of one of the groups that initiated the current research in the optimization use of positive polynomials, see above [GHNSVX02]. Competence fields: system theory, stability, optimization, nonlinear systems Experience in national/international research programs :

• Project coordinator ``Robustness measures in complex systems'', in cooperation with Universite Catholique de Louvain, 2002-2003.

• Project coordinator "Advanced topics in stability radius theory", SSTC Fellowship of Belgium government, 1999-2000.

• Project coordinator "Positivity, dissipativity and robustness of control systems: theory and algorithms", NATO Fellowship, 2000-2001.

• Participation at other about national/international 10 grants/contracts as a member of the research team (CNCSIS, World Bank, MCT).

Significant papers (in the latest 5 years): 1. Vl. Rasvan si R. Stefan, Systemes Nonlineaires: Theorie et Applications, Hermes Science, 2007. 2. R. Stefan, Disturbance attenuation in the chain-scattering formalism, Mathematical Reports, 5(55), 4, pp. 371-387, 2003. 3. V. Rasvan si R. Stefan, Discussion on Passification of nonsquare nonlinear systems, European Journal of Control, 9, 6, pp. 587-588, 2003. 4. V. Ionescu si R. Stefan, Generalized time-varying Riccati theory: A Popov operator based approach, Integral Equations and Operator Theory, 48, pp. 159-212, 2004. 5. Y. Hachez si R. Stefan, Computing the distance to uncontrollability: A convex optimization approach, IFAC CAO06, Paris, France , 2006. 11.4.1.2.2. Young researchers Cristian Flutur was born in 1982. In 2006, he graduated the Faculty of Automatic Control and Computers of the Politehnica University of Bucharest, where he is now a researcher in the ACPC research center. In fall 2007, he will enroll in a Ph.D. program. A possible title of his thesis is “Convex optimization in system analysis and synthesis”, with the goal of dedicating the largest part of the study to the optimization with positive trigonometric polynomials. In this project, the first task of C.Flutur will be the realization of objective O1 (see 11.2), namely to implement a library dedicated to the easy manipulation positive trigonometric polynomials in a semidefinite programming context. This objective requires standard programming skills and some knowledge on trigonometric polynomials that he will get by working at this project. After finishing the


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library (or at least a first version with full features), C.Flutur will go on at a superior level, that of participating at the scientific objectives (see 12.1); in a first stage, his participation will consist of implementing ideas generated by the senior researchers, test them and compare results using the programs. Ideally, in a second stage he will start himself producing and validating ideas. A purpose of his participation at this project is the writing of his Ph.D. thesis, even if the defence will take place after the project is finished. The fourth member of the team, denoted X, has not been identified yet. He could be one of the students graduating in 2007 at the Faculty of Automatic Control and Computers. X will be selected in fall 2007, the search process starting now and intensifying after this project is approved. It is possible that X will not enroll to the Ph.D. program in 2007; this will happen in 2008. X will deal with objective O2, which has an implementation character, and so it is appropriated to the introduction to optimization with multivariate polynomials. After finalizing O2, X will become more involved in the scientific tasks, as explained above for C.Flutur. As mentioned in 12.1, for the moment we do not plan a clear split of the objectives O3-O8 between C.Flutur and X. In any case, a certain objective will be the task of a single one of them, and the number of objectives in which one of them will be involved will depend on their progress capacity. The choice of the objectives for each of them will be made gradually, function of their affinities and evolutions.

11.4.2 Other resources 11.4.2.1. Financial resources

1. Salaries – including contributions payd by the employer B.Dumitrescu, half salary: 3000 RON/month R.Stefan, half salary: 2500 RON/month 2 young researchers, full salary: 2600 RON/month Total: (3000 + 2500 + 2600 + 2600) * 36 = 385200 RON 2. University overhead 25 %: 96 300 RON 3. Mobilities

• For senior researchers, 3 mobilities of 15-30 days. Besides documentation and discussions with the host researchers, the researchers will make scientific presentations. Amount for each mobility: 7000 RON.

• For the young researchers, 3 mobilities, 2 months each. Amount for each mobility: 9000 RON. Total: (7000 + 9000)*6 = 96 000 RON The mobilities will be made at universities where there are groups with similar interests and researchers with which we are in contact:

• Tampere University of Technology (Finland): I.Tabus, J.Astola, T.Saramaki • Universite Catholique de Louvain (Belgium): P.Van Dooren, Y,Nesterov • Lab. d’Analyse et d’Architecture de Systemes (CNRS), Toulouse (France): D.Henrion • INRIA Futurs, Orsay (France): L.Grigori • Possibly, universities in the USA, in conjunction with trips at conferences (for cutting the costs):

L.Vandenberghe, I.Selesnick, B.C.Chang The mobilities will be spread uniformly on the 3 years of the project. 4. Logistic expenses a. Infrastructure

• 2 performant computers (for the young researchers): 10000 RON • 1 printer (for current use of the group): 1500 RON • Matlab, academic licence for a group of about 10 users: 30000 RON


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b. Result dissemination • Expenses for the published articles (for example, extra pages in IEEE journals—above 8

pages—cost 150-200 USD/page): 6000 RON • Support for the publication of a book in Romanian: 5000 RON • Support and update the web site www.schur.pub.ro: 3000 RON

c. Documentation

• Books: 3000 RON • IEEE tax + online access at some journals: 3000 RON

d. Foreign guests

• 3 foreign guests at Bucarest, each for about a week, for discussions and a scientific presentation. At least one of the guests will participate with a lecture at the traditional summer school in control, organized in the last decade of May at the Faculty of Automatic Control and Computers. The guests will be researchers with similar interests, preferrably among those cooperating with us. Amount for each guest: 6000 RON.

e. Other

• Consumables: 8000 RON • Unplanned: 5000 RON

Total: 41500 + 14000 + 6000 + 18000 + 13000 = 92500 RON

11.4.2.2. Infrastructure For the research proposed in this project we need only computers with a reasonable power (produced in the latest 2-3 years) and connection to the internet. Besides Matlab (for which we have a single licence), the other programs are freely available on the internet. Existing infrastructure: network of 6 computers (of which 3 are used and one is the server), in ED206, where is the office of B.Dumitrescu (and the Numerical computation group). Four computers are less than 3 years old. There is also a rather old printer. Also, we have access to the System and optimization lab, where R.Stefan works. Necessary infrastructure: 2 PCs, good quality, for the two young researchers. A desk printer for small workgroups. Matlab licence for the whole group.


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ANEXA 5 12. Modul de organizare a proiectului (managementul proiectului):

12.1. Planul de lucru. Obiective si activitati Mai jos, BD, RS, CF, X reprezinta membrii grupului (B.Dumitrescu, R.Stefan, C.Flutur si al patrulea

membru, ce urmeaza a fi selectionat ulterior). Prin O1-O8 sunt notate obiectivele expuse la 10.2. Prin natura lui, O9 este un obiectiv permanent, de aceea nu e trecut in planificarea de mai jos. De altfel, intre obiectivele O3-O8 pot aparea permutari, in functie de viteza de avans, rezultate aparute intre timp, inspiratia cercetatorilor.

An *

Obiective (Denumirea obiectivului)

Activitati asociate

2007

1 Selectionare X (responsabil BD) Anunturi pe diverse grupuri, interviuri, decizie de selectie

2 O1 documentare (CF + BD) Studiu comparativ al posibilitatilor de realizare a

bibliotecii. Alegere modalitate de implementare.

3 O2 documentare (X) Studiu probleme de baza si parametrizari polinoame trigonometrice pozitive.

4 O3, O8 (BD+RS) incercari Derivare conditii de baza, implementare programe

2008

1 Achizitii infrastructura (BD)

2 O1 implementare (CF)

Scriere efectiva biblioteca, teste, elaborare documentatie, publicare pe internet (eventual in versiuni succesive), elaborare lucrare conferinta

3 O2 implementare (X)

Alegerea continut biblioteca (impreuna cu BD), implementare, documentatie. Conectare cu O1 si definitivare pachet.

4 Inscriere doctorat X (daca nu s-a inscris in 2007)

5 O3 (BD+RS) finalizare O8 (RS+BD) finalizare

Teste, comparatii, redactare lucrari, supunere spre publicare.

6 O4 (BD+RS) start O5 (RS+BD) start

Documentare, incercari, ipoteze etc.

2009

1 O6, O7 (toata echipa) start

Documentare, incercari, ipoteze etc. CF si X vor lucra fiecare la un singur obiectiv (alegerea va fi facuta in functie de afinitati etc.)

2 O4, O5 (toata echipa) finalizare

CF si X vor participa la partea de testare, eventual la implementarea programelor. Din nou, fiecare va lucra la un singur obiectiv.

3 BD+RS – pregatire cuprins carte, adunare materiale primare

Documentare, selectionare materiale si idei

2010

1 O6, O7 (toata echipa) finalizare Vezi mai sus

2 Redactare carte (BD+RS, cu participarea CF si X)

Scriere efectiva a unei carti de aproximativ 200 de pagini, cautare editor, angajare publicare

3 Redactare teza (CF, eventual X, daca s-a inscris in 2007)

Scriere efectiva teza, cu scopul de a sustine doctoratul cel tarziu in 2010

12.2. Fezabilitatea proiectului Obiectivele practice O1 si O2 au caracter de implementare, deci nu ar trebui sa fie obstacole semnificative in calea realizarii lor. In cel mai rau caz poate aparea o intarziere in finalizarea lor, dar in nici un nici un caz nu se poate depasi durata acestui proiect. Obiectivele stiintifice O3-O9 nu sunt nici evident tangibile, nici utopice, dar ne apar realizabile. Desigur, este dificil de estimat daca vom obtine rezultate demne de publicat in toate cazurile. De asemenea, este imposibil de prevazut daca vom avea intaietate sau daca nu cumva alti cercetatori vor obtine primii


32

rezultate similare. Deocamdata ne situam intr-o pozitie de atac favorabila. Mizam pe indeplinirea obiectivului O9, cel al rezultatelor colaterale ; factorul ‘neprevazut’ tine de obicei cu cei activi in domeniul respectiv. Obiectivul strategic al formarii, pe termen lung, a unui grup, tine foarte mult de factorul uman, deci e mai greu de prevazut succesul lui. Crearea unui pol de atractie, stiintific si organizatoric, este insa fezabila, cel putin pentru durata acestui proiect.

12.3. Planul de diseminare a rezultatelor

Diseminarea rezultatelor se va face prin • publicarea bibliotecilor de programe realizate pe un site al facultatii de Automatica si calculatoare

(de exemplu pe www.schur.pub.ro, al grupului din care face parte B.Dumitrescu). • publicare de articole in reviste cotate ISI (prim obiectiv: revistele IEEE) • comunicarea de lucrari la conferinte internationale de prestigiu • comunicari periodice in seminarul stiintific al catedrei de Automatica si Ingineria sistemelor • prezentari stiintifice ale membrilor echipei (in special ale celor cu experienta) in institutiile pe care le

vor vizita (obligatoriu atunci cand sunt folosite fonduri din acest proiect pentru deplasare) 12.4. Masurile prevazute pentru respectarea normelor deontologice ale cercetarii

Se vor cita corespunzator rezultatele altor cercetatori utilizate in articolele scrise de membrii echipei. Nu se vor face duble submiteri (trimiterea unor versiuni similare ale unor articole la reviste diferite). In general, se vor respecta normele etice (vezi de exemplu normele IEEE) in privinta publicatiilor stiintifice.Relatiile dintre membrii grupului vor fi de colegialitate, iar cercetatorii cu experienta se vor stradui sa asigure o comunicare directa cu cei tineri, cu scopul de a crea o atmosfera propice cercetarii.


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ANEXA 6

14. Directorul de proiect a fost /este director de proiect in cadrul programului CEEX

NU

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