Stability of fractional positive nonlinear systems
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1 Archives of Cotrol Scieces Volume 5(LXI), 15 No. 4, pages Stability of fractioal positive oliear systems TADEUSZ KACZOREK The coditios for positivity ad stability of a class of fractioal oliear cotiuous-time systems are established. It is assumed that the oliear vector fuctio is cotiuous, satisfies the Lipschitz coditio ad the liear part is described by a Metzler matri. The stability coditios are established by the use of a etesio of the Lyapuov method to fractioal positive oliear systems. Key words: positive, fractioal, oliear, system, stability. 1. Itroductio I positive systems iputs, state variables ad outputs take oly o-egative values. Eamples of positive systems are idustrial processes ivolvig chemical reactors, heat echagers ad distillatio colums, storage systems, compartmetal systems, water ad atmospheric pollutio models. A variety of models havig positive liear behavior ca be foud i egieerig, maagemet sciece, ecoomics, social scieces, biology ad medicie, etc. Positive liear systems are defied o coes ad ot o liear spaces. Therefore, the theory of positive systems is more complicated ad less advaced. A overview of state of the art i positive systems theory is give i the moographs [5, 8]. Positive liear systems with differet fractioal orders have bee addressed i [1, 14 17]. Stability of positive liear systems has bee ivestigated i [1, 5, 8] ad of fractioal liear systems i [ 4]. The problem of preservatio of positivity by approimatio the cotiuous-time liear systems by correspodig discrete-time liear systems has bee addressed i [6, 7]. The approimatio of positive stable cotiuoustime liear systems by positive stable discrete-time liear systems has bee cosidered i [6]. Descriptor (sigular) fractioal liear systems have bee aalyzed i [11 13]. The stability of a class of oliear fractioal-order systems has bee aalyzed i [18]. I this paper the positivity ad asymptotic stability of a class of fractioal oliear T. Kaczorek is with Bialystok Uiversity of Techology, Faculty of Electrical Egieerig, Wiejska 45D, Bialystok. kaczorek@isep.pw.edu.pl Received This work was supported by Natioal Sciece Cetre i Polad uder work No. 14/13/B/ST7/3467. Uautheticated Dowload Date 6/7/18 5:41 PM
2 49 T. KACZOREK systems will be addressed. Necessary ad sufficiet coditios for the positivity ad sufficiet coditios for the asymptotic stability will be established. The paper is orgaized as follows. I sectio ecessary ad sufficiet coditios for a class of fractioal oliear cotiuous-time systems are established. The asymptotic stability of the oliear fractioal system by the use of a etesio of the Lyapuov method is addressed i sectio 3. Cocludig remarks are give i sectio 4. m The followig otatio will be used: R the set of real umbers, R the set of 1 m m real matrices ad R = R, R the set of m matrices with oegative 1 etries ad R = R, M the set of Metzler matrices (with oegative offdiagoal etries), I - the idetity matri.. Positivity of a class of fractioal oliear systems Cosider the fractioal oliear cotiuous-time system where a a d t () Dt t () = = At () f( t ()), < a < 1 (1) a dt t a 1 ( ) ( ), (1 ) τ d( τ ) D = t t Γ ( dτ a ( τ ) = () a t τ ) dτ is the Caputo fractioal derivative of the order a of the state vector ( R ad z 1 t z t e dt Γ ( ) =, Re(z) > (3) is the Euler gamma fuctio, A R ad f ( ( R is the cotiuous vector fuctio of (. It is assumed that the oliear equatio (1) has a solutio ( ad the Lipschitz coditio fi ( t, ',..., ') fi ( t, 1",..., ") < c 1 1 { ' "... ' "}, t, i 1,,..., 1 = (4) is satisfied for some costat c ad k ', k" for k = 1,,..., are some values of the state vector k (. Defiitio 1 The fractioal oliear system (1) is called (iterally) positive if ( R for all iitial coditios = () R. Theorem 1. The fractioal oliear system (1) is positive if A M ad f ( R ( for R ( (5) Uautheticated Dowload Date 6/7/18 5:41 PM
3 STABILITY OF FRACTIONAL POSITIVE NONLINEAR SYSTEMS 493 Proof It is well-kow [14] that if f ( ( = tha ( R, t if ad oly if A M ad R. By assumptio the equatio (1) has a solutio ad the coditio (4) is satisfied. Usig the Picard method it ca be show that the equatio has a solutio ( R if the coditio (5) are met. 6. Stability of cotiuous-time oliear systems Cosider the positive cotiuous-time oliear system where = ( R, A M, ad f ( ) =. = A f ( ), (6) f ) R ( is a cotiuous ad bouded vector fuctio Defiitio The positive cotiuous-time oliear system (6) is called asymptotically stable i the regio R if ( R, t ad D lim ( = for ay fiite t D R. (7) To test the asymptotic stability of the positive system (6) the Lyapuov method will be used. As a cadidate of Lyapuov fuctio we choose T V ( ) = c > for (, t (8) = R where is a vector with strictly positive compoets c > for k = 1,...,. c R k Usig (8) ad (6) we obtai for T T V ( ) = c = c [ A f ( )] < (9) A f ( ) < for, t (1) D R sice c R is strictly positive vector. Therefore, the followig theorem has bee proved. Theorem The positive cotiuous-time oliear system (6) is asymptotically stable i the regio R if the coditio (1) is satisfied. D Eample 1 Cosider the oliear system (6) with =, A =, ( ) =. 1 3 f (11) Uautheticated Dowload Date 6/7/18 5:41 PM
4 494 T. KACZOREK The oliear system (6) with (11) is positive sice A M ad f ( ) R for all R, t. I this case the coditio (9) is satisfied i the regio D defied by 1 1 D : = { 1, } = <. (1) 1 3 From (1) we have 1 ( ) > > ad 1 < (3 ). (13) The regio D is show o the Fig. 1. Fig. 1. Stability regio (iside the curved lie). By Theorem the positive oliear system (6) with (11) is asymptotically stable i the regio (1). 4. Cocludig remarks The positivity ad stability of a class of fractioal oliear cotiuous-time systems described by the equatio (1) has bee addressed. Necessary ad sufficiet coditios for the positivity (Theorem 1) ad sufficiet coditios for the asymptotic stability (Theorem ) have bee established. The cosideratios have bee illustrated by umerical eample of oliear fractioal system. A ope problem is a etesio of the results to descriptor fractioal cotiuous-time oliear systems. Uautheticated Dowload Date 6/7/18 5:41 PM
5 STABILITY OF FRACTIONAL POSITIVE NONLINEAR SYSTEMS 495 Refereces [1] A. Berma ad R.J. Plemmos: Noegative Matrices i the Mathematical Scieces. SIAM, [] M. Busłowicz: Stability of liear cotiuous-time fractioal order systems with delays of the retarded type. Bulleti of the Polish Academy of Scieces: Techical Scieces, 56(4), (8), [3] M. Busłowicz: Stability aalysis of cotiuous-time liear systems cosistig of subsystems with differet fractioal orders. Bulleti of the Polish Academy of Scieces: Techical Scieces, 6(), (1), [4] M. Busłowicz ad T. Kaczorek: Simple coditios for practical stability of positive fractioal discrete-time liear systems, It. J. of Applied Mathematics ad Computer Sciece, 19(), (9), [5] L. Faria ad S. Rialdi: Positive Liear Systems; Theory ad Applicatios. J. Wiley, New York,. [6] T. Kaczorek: Approimatio of positive stable cotiuous-time liear systems by positive stable discrete-time systems. Pomiary Automatyka Robotyka,, (13), [7] T. Kaczorek: Compariso of approimatio methods of positive stable cotiuuous-time liear systems by positive stable discrete-time systems. Computer Applicatios i Electrical Egeeerig, Pub. House of Poza Uiversity of Tehology, Pozań, 13, 1-1. [8] T. Kaczorek: Positive 1D ad D Systems. Spriger Verlag, Lodo,. [9] T. Kaczorek: Liear Cotrol Systems. 1, Research Studies Press J. Wiley, New York, 199. [1] T. Kaczorek: Positive liear systems with differet fractioal orders. Bulleti of the Polish Academy of Scieces: Techical Scieces, 58(3), (1), [11] T. Kaczorek: Applicatio of Drazi iverse to aalysis of descriptor fractioal discrete-time liear systems with regular pecils. It. J. of Applied Mathematics ad Computer Sciece, 3(1), (13), [1] T. Kaczorek: Reductio ad decompositio of sigular fractioal discrete-time liear systems. Acta Mechaica et Automatica, 5(4), (11), [13] T. Kaczorek: Sigular fractioal discrete-time liear systems. Cotrol ad Cyberetics, 4(3), (11), Uautheticated Dowload Date 6/7/18 5:41 PM
6 496 T. KACZOREK [14] T. Kaczorek: Selected Problems of Fractioal System Theory, Spriger Verlag, Berli, 11. [15] T. Kaczorek: Fractioal positive cotiuous-time liear systems ad their reachability. It. J. of Applied Mathematics ad Computer Sciece, 18(), (8), 3-8. [16] T. Kaczorek: Positive liear systems cosistig of subsystems with differet fractioal orders. IEEE Tras. o Circuits ad Systems, 58(7), (11), [17] T. Kaczorek: Positive fractioal cotiuous-time liear systems with sigular pecils. Bulleti of the Polish Academy of Scieces: Techical Scieces, 6(1), (1), 9-1. [18] W. Xiag-Ju, W. Zheg-Mao ad L. Ju-Guo: Stability aalysis of a class of oliear fractioal-order systems. IEEE Tras. Circuits ad Systems-II, Epress Briefs, 55(11), (8), Uautheticated Dowload Date 6/7/18 5:41 PM
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