ASYMPTOTIC STABILITY OF POSITIVE FRACTIONAL 2D LINEAR SYSTEMS WITH DELAYS

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1 ASYMPTOTIC STABILITY OF POSITIVE FRACTIONAL D LINEAR SYSTEMS WITH DELAYS Tadeusz Kaczorek Faculty of Electrical Egieerig Bialystok Techical Uiversity Wiejska 45D 5-35 Bialystok Polad kaczorek@isep.pw.edu.pl ABSTRACT The asyptotic stability of positive fractioal D liear systes with delays is addressed. Necessary ad sufficiet coditios for the asyptotic stability of the positive fractioal D systes with delays are established. It is show that ) the asyptotic stability of the positive fractioal D systes does ot deped o the uber ad values of the delays but oly o the su of syste atrices ) the checkig of the asyptotic stability of positive fractioal D liear systes with delays ca be reduced to testig the asyptotic stability of correspodig D positive liear systes. Keywords: asyptotic stability fractioal positive D liear syste delay. INTRODUCTION I positive systes iputs state variables ad outputs take oly o-egative values. Eaples of positive systes are idustrial processes ivolvig cheical reactors heat echagers ad distillatio colus storage systes copartetal systes water ad atospheric pollutio odels. A variety of odels havig positive liear behavior ca be foud i egieerig aageet sciece ecooics social scieces biology ad edicie etc. Positive liear systes are defied o coes ad ot o liear spaces. Therefore the theory of positive systes i ore coplicated ad less advaced. A overview of state of the art i positive systes theory is give i the oographs (Faria ad Rialdi Kaczorek ). The ost popular odels of two-diesioal (D) liear systes are the odels itroduced by Roesser (Roesser 975) Forasii-Marchesii (Forasii- Marchesii 976; 978) ad Kurek (Kurek 985). These odels have bee eteded for positive systes i (Kaczorek 966; Valcher 997; Kaczorek 5; Kaczorek ). A overview of positive D systes theory has bee give i the oographs (Faria ad Rialdi ; Kaczorek ). Reachability ad iiu eergy cotrol of stadard ad positive D systes with oe delay i states have bee cosidered i (Kaczorek 5; Klaka 99). The choice of the Lyapuov fuctios for positive D Roesser odel has bee ivestigated i (Kaczorek 8c). The otio of iterally positive D syste (odel) with delays i states ad i iputs has bee itroduced ad ecessary ad sufficiet coditios for the iteral positivity reachability cotrollability observability ad the iiu eergy cotrol proble have bee established i (Kaczorek 6). Stability of positive D ad D liear systes has bee cosidered i (Faria ad Rialdi ; Kaczorek 8b; 8c; Roesser 975; Valcher 997) ad the robust stability i (Buslowicz 8a). The positive fractioal liear systes have bee addressed i (Kaczorek 8e; 8f; 7a; 7b) ad their stability has bee ivestigated i (Buslowicz 8b; Buslowicz ad Kaczorek). LMI approaches to checkig the stability of positive D systes have bee proposed i (Kaczorek 8d; Twardy 7). I this paper ecessary ad sufficiet coditios for the asyptotic stability of the positive fractioal D liear systes will be established ad it will be show that the asyptotic stability of the positive fractioal D liear systes with delays ca be reduced to testig the asyptotic stability of correspodig D positive liear systes.. POSITIVE FRACTINAL D LINEAR SYSTEMS Let R be the set of real atrices. The set atrices with oegative etries will be deoted by R ad R R. The set of oegative itegers will be deoted by Z ad the idetity atri will be deoted by I. A atri A [ a ] R (a vector ) is called strictly positive ad deoted by A > ( > ) if ad oly if a > for i... ; j.... Defiitio. The ( α β ) orders fractioal differece of a D fuctio is defied by the forula Proceedigs of the Europea Modelig ad Siulatio Syposiu EMSS 9 Vol I - ISBN

2 i j α β cαβ ( k l) i k j l (a) < α < < β < ; N {...} where α β α β ad i j for k or/ad l cα β ( k l) k l α( α )...( α k ) β( β )...( β l ) ( ) k! l! for k l > (b) The justificatio of Defiitio is give i (Kaczorek 8g). Cosider the ( α β ) orders fractioal D liear syste with ( q q ) -delays i state vector described by the equatios q q i j A i k j l A i k j l α β A B u B u B u i k j l i j i j (a) y C Du (b) where R u R y R p are the state iput t ad output vectors ad A R k... q ; p l... q R t C R B t p D R. Usig Defiitio we ay write the equatio (a) i the for q q i j i k j l i k j l i k j l A A A c B u B u B u k l D where D D D ad i k j l i j i j \ i j qq {( ) : } D i j Z Z i k j l c c ( k l) αβ (3a) A A I c for k... q ; l... q A A k l A Ick l for k ; l... q A for k... q ; l... q A Ick l for k... q; l A for k... q ; l... q (3b) The boudary coditios for the equatio (3) have the for b R for i... q; j... q (4) where b are give. Defiitio. The syste ()(ad also (3)) is called the (iterally) positive fractioal D syste if ad oly if p R ad R i j Z for ay y oegative boudary coditios (4) ad all iput sequeces R i j Z. Lea. u a) If < α < ad < β < the c > for k...; l 3... (5a) b) If < α < ad < β < the c > for k 3...; l... (5b) Proof is give i (Kaczorek 8g). Theore. The fractioal D syste () for < α < ad < β < (or < α < ad < β < ) is positive if ad oly if A R B R t k k... q l... q t... C R D R p p (6) The proof is siilar to the proof for positive fractioal liear systes without delays give i (Kaczorek 9). Proceedigs of the Europea Modelig ad Siulatio Syposiu EMSS 9 Vol I - ISBN

3 3. ASYMPTOTIC STABILITY Defiitio 3. The positive fractioal D liear syste () is called asyptotically stable if for ay oegative bouded boudary coditios (4) li i j b for R i... q ; j... q (7) I the proof of the ai result of this sectio the followig lea ad theore will be used. Lea. If < α < ad < β < (or < α < ad < β < ) the cαβ ( k l) (8) Proof. It is well-kow (Buslowicz ad Kaczorek) that i α i α( α )...( α i ) ( ) ( ) α > (9) i i! i i Usig (b) ad (9) we obtai αβ k l k l α( α )...( α k ) β( β )...( β l ) ( ) k! l! k l c ( k l) k α( α )...( α k ) ( ) k k! l β( β )...( β l ) ( ) l l! () Theore. The positive D geeral odel with q delays p q ( ) A A A i j i k j l i k j l i k j l for i j Z () t where R is the state vector ad A R k p l q t is asyptotically stable if ad oly if the positive D syste A A A p q ( ) i i k l is asyptotically stable. i i Z R () Proof is give i (Kaczorek 9; Kaczorek 8b). Theore 3. The positive fractioal D syste () is asyptotically stable if ad oly if the positive D syste ˆ ( ) A I ˆ A A A A i i i q q R i Z (3) is asyptotically stable. Proof. Fro (3) for B B B ad () we have q q i j i k j l i k j l A A i j i k j l c i k j l k l > A (4) By Theore the positive D syste with delays is asyptotically stable if ad oly if the positive D syste Aˆ c I i i k l k l> i i Z R (5) is asyptotically stable. Fro (b) we have c ( ) ad fro () we obtai c αβ c I I (6) k l > Substitutio of (6) ito (5) yields (3). Applyig to the positive D syste the well-kow theore (Kaczorek 8b) we obtai the followig theore. Theore 4. The positive fractioal D syste () is asyptotically stable if ad oly if oe of the followig equivalet coditios holds: ) Eigevalues z... z of the atri Aˆ I have oduli less tha ) All coefficiets of the characteristic polyoial of the atri  are positive 3) All leadig pricipal iors of the atri  are positive Proceedigs of the Europea Modelig ad Siulatio Syposiu EMSS 9 Vol I - ISBN

4 Theore 5. The positive fractioal D syste () is ustable if at least oe diagoal etry of the atri  is positive. Proof. If at least oe diagoal etry of the atri  is positive the at least oe diagoal etry of the atri Aˆ I is greater tha ad it is well-kow (Buslowicz ad Kaczorek; Galkowski ; Kaczorek 6) that the syste (5) is ustable. Eaple. Usig Theore 4 check the asyptotic stability of the positive fractioal D syste () for α.3 β. ad q q with the atrices A A A A A A A A A A A A.. (7) The fractioal syste is positive sice the atrices.4. A A Ic.74 A A Ic A A Ic.6.6. A A Ic.6.. A A Ic..8. A A Ic.. A A.3.. A A.... A A Ic A A..5.4 A A Ic.5. A A have oegative etries. I this case Aˆ ( A A A ) The first coditio of Theore 4 is satisfied sice the atri ˆ.69.5 A I.75 has the eigevalues z.69 z.75 whose oduli are less tha. The secod coditio of Theore 4 is also satisfied sice the characteristic polyoial ˆ z.3.5 I z A z z z.5 det[ ] has positive coefficiets. The leadig priciple iors of the atri ˆ.3.5 A.5 i.e Therefore all three coditios of Theore 4 are satisfied ad the positive fractioal D syste with (7) is asyptotically stable. 4. CONCLUDING REMARKS Necessary ad sufficiet coditios for the asyptotic stability of the positive fractioal D systes with delays have bee established. Siple sufficiet coditios for istability of the syste have bee also give. It has bee show that ) the asyptotic stability of the positive fractioal D systes does ot deped o the uber ad values of the delays but oly o the su of syste atrices ) checkig the asyptotic stability of the positive fractioal D liear systes with delays ca be reduced to testig the stability of correspodig D positive liear systes. Assuig i (a) B B we obtai the first Forasii-Marchesisi odel ad assuig A ad B we obtai the secod Forasii-Marchesisi odel.therefore the cosideratios for the first ad secod F-M odels are particular case of the preseted Proceedigs of the Europea Modelig ad Siulatio Syposiu EMSS 9 Vol I - ISBN

5 geeral D odel. The cosideratios preseted for the geeral odel ca be easily eteded for the Roesser odel. Etesio of those cosideratios for the hybrid positive D liear systes ad for D cotiuous-tie liear systes are ope probles. ACKNOWLEDGMENTS This work was supported by Miistry of Sciece ad Higher Educatio i Polad uder work No NN REFERENCES Bose N.K. 985 Multidiesioal Systes Theory Progress Directios ad Ope Probles D. Reidel Publishig Co. Buslowicz M. 8a. Stability of liear cotiuoustie fractioal order systes with delays of the retarded type. Bull. Pol. Acad. Tech. Sci. vol. 56 o Buslowicz M. 8b. Siple stability coditios for liear discrete-tie systes with delays. Bull. Pol. Acad. Tech. Sci. vol. 56 o Buslowicz M. Kaczorek T. Siple coditios for practical stability of positive fractioal discretetie liear systes. (i Press). Faria L. Rialdi S.. Positive Liear Systes; Theory ad Applicatios. J. Wiley New York. Forasii E. Marchesii G State-space realizatio theory of two-diesioal filters. IEEE Tras. Auto. Cotr. Vol. AC Forasii E. Marchesii G Double ideed dyaical systes Math. Sys. Theory Galkowski K.. State-space Realizatios of Liear -D Systes with Etesios to the Geeral D (>) Case. Spriger Lodo. Kaczorek T.. Positive D ad D Systes Spriger-Verlag Lodo. Kaczorek T. 8a. Asyptotic stability of positive D liear systes. Proc of 3 th Scietific Cof. Coputer Applicatios i Electrical Eg. April 4-6 Poza Polad. Kaczorek T. 8b. Asyptotic stability of positive D ad D liear systes. Recet Advaces i Cotrol ad Autoatio Acad. Publ. House EXIT 4-5. Kaczorek T. 8c. Practical stability of positive fractioal discrete-tie liear systes Bull. Pol. Acad. Tech. Sci. vol. 56 No Kaczorek T. 8d. LMI approach to stability of D positive systes with delays. Multidiesioal Systes ad Sigal Processig vol. 8 o. 3 (i Press) Kaczorek T Reachability ad cotrollability of o-egative D Roesser type odels Bull. Acad. Pol. Sci. Ser. Sci. Tech. vol. 44 o Kaczorek T. 5. Reachability ad iiu eergy cotrol of positive D systes with delays Cotrol ad Cyberetics vol. 34 o Kaczorek T. 8e. Reachability ad cotrollability to zero tests for stadard ad positive fractioal discrete-tie systes Joural of Autoatio ad Syste Egieerig. (i Press). Kaczorek T. 7a. Reachability ad cotrollability to zero of positive fractioal discrete-tie systes. Machie Itelligece ad Robotic Cotrol vol. 6 o. 4. Kaczorek T. 7b. Reachability ad cotrollability to zero of coe fractioal liear systes Archives of Cotrol Scieces vol. 7 o Kaczorek T. 8f. Fractioal positive cotiuous-tie liear systes ad their reachability. It. J. Appl. Math. Coput. Sci. vol. 8 o. (i Press). Kaczorek T. 6. Positive D systes with delays MMAR 6 th IEEE IFAC Iteratioal Coferece o Methods i Autoatio ad Robotics Polad. Kaczorek T. Practical stability of positive fractioal D liear systes Bull. Acad. Pol. Sci. Ser. Sci.Tech. vol. 56 o Kaczorek T. 9b. Idepedece of the asyptotic stability of the D liear systes with delays of their delays. It. J. Appl. Math. Coput. Sci. vol. 9 (i Press). Kaczorek T. 8g. Positive differet orders fractioal D liear systes Acta Mechaica et Autoatica vol. o Klaka J. 99. Cotrollability of Dyaical Systes. Kluwer Acadeic Publ. Dordrecht. Kurek J The geeral state-space odel for a two-diesioal liear digital systes IEEE Tras. Auto. Cotr. AC Roesser R. P A discrete state-space odel for liear iage processig. IEEE Tras. O Auto. Cotr. AC- -. Twardy M. 7. A LMI approach to checkig stability of D positive systes Bull. Pol. Acad. Tech. Sci. vol. 55 o Valcher M. E O the iitial stability ad asyptotic behavior of D positive systes IEEE Tras. o Circuits ad Systes I vol. 44 o AUTHORS BIOGRAPHY TADEUSZ KACZOREK received the MSc. PhD ad DSc degrees fro Electrical Egieerig of Warsaw Uiversity of Techology i ad 964 respectively. I the period he has the dea of Electrical Egieerig Faculty ad i the period he was the prorector of Warsaw Uiversity of Techology. Sice 97 he has bee professor ad sice 974 full professor at Warsaw Uiversity of Techology. I 986 he was elected a corresp. eber ad i 996 full eber of Polish Acadey of Scieces. I the period he was the director of the Research Cetre of Polish Acadey of Scieces i Roe. I May 4 he was elected the hoorary eber of the Hugariar Acadey of Scieces. He was awarded by the title doctor hooris causa by the several Proceedigs of the Europea Modelig ad Siulatio Syposiu EMSS 9 Vol I - ISBN

6 uiversities. His research iterests cover the theory of systes ad the autoatic cotrol systes theory specially sigular ultidiesioal systes positive ultidiesioal systes ad sigular positive D ad D systes. He has iitiated the research i the field of sigular D ad positive D systes. He has published books (si i Eglish) ad over 85 scietific papers. He supervised 66 PhD theses. He is Editor-i-Chief of Bulleti of Polish Acadey of Scieces Techology Scieces ad Editorial Meber of about te iteratioal jourals. Proceedigs of the Europea Modelig ad Siulatio Syposiu EMSS 9 Vol I - ISBN