Stability and Stabilization of Fractional Order Time Delay Systems

Transcription

1 Scieific Techical Review, 2,Vol.6,No. 3 UDK: : :656.4 OSTI: 4-7, 2- Sabiliy ad Sabilizaio of Fracioal Order Tie Delay Syses Mihailo Lazarević I his paper, soe basic resls of he sabiliy crieria of fracioal order syse wih ie delay as well as free delay are preseed. lso, we obaied ad preseed sfficie codiios for fiie ie sabiliy ad sabilizaio for (oliear (ohoogeeos as well as perrbed fracioal order ie delay syses. Several sabiliy crieria for his class of fracioal order syses are proposed sig a recely sggesed geeralized Growall ieqaliy as well as classical Bella-Growall ieqaliy. Soe coclsios for sabiliy are siilar o hose of classical iegerorder differeial eqaios. Fially, a erical eaple is give o illsrae he validiy of he proposed procedre Key words: oliear syse, syse sabiliy, syse sabilizaio, syse wih delay, ie delay, perrbaio, fracioal order syse. Irodcio HE qesio of sabiliy is of ai ieres i he T corol heory. lso, he proble of ivesigaio of ie delay syses has bee eploied over ay years. Delay is very ofe ecoered i differe echical syses, sch as elecric, peaic ad hydralic eworks, cheical processes, log rasissio lies, ec.,[]. Delays are ihere i ay physical ad egieerig syses. I pariclar, pre delays are ofe sed o ideally represe he effecs of rasissio, rasporaio, ad ierial pheoea. This is becase hese syses have oly liied ie o receive iforaio ad reac accordigly. Sch a syse cao be described by prely differeial eqaios, b has o be reaed wih differeial differece eqaios or he so-called differeial eqaios wih differece variables. Delay differeial eqaios (DDEs cosie basic aheaical odels for real pheoea, for isace i egieerig, echaics, ad ecooics, [2]. The basic heory cocerig he sabiliy of syses described by eqaios of his ype was developed by Poryagi i 942. lso, ipora works have bee wrie by Bella ad ooke i 963, [3]. The presece of ie delays i a feedback corol syse leads o a closed-loop characerisic eqaio which ivolves he epoeial ype rascedeal ers. The epoeial rascedealiy brigs ifiiely ay isolaed roos, ad hece i akes he sabiliy aalysis of ie-delay syses a challegig ask. I is well recogized ha here is o siple ad iversally applicable pracical algebraic crierio, like he Roh Hrwiz crierio for sabiliy of delay-free syses, for assessig he sabiliy of liear ieivaria ie-delayed (LTI-TD syses. O he oher side, he eisece of pre ie delay, regardless if i is prese i he corol or/ad sae, ay case a desirable syse rasie respose, or geerally, eve a isabiliy. Neros repors have bee pblished o his aer, wih a pariclar ephasis o he applicaio of Lyapov`s secod ehod, or o sig he idea of ari easre[4-7]. The aalysis of ie-delay syses ca be classified sch ha he sabiliy or sabilizaio crieria ivolve he delay elee or o. I oher words, delay idepede crieria garaee global asypoic sabiliy for ay ie-delay ha ay chage fro zero o ifiiy. s here is o pper lii o ie-delay, ofe delay idepede resls ca be regarded as coservaive i pracice, where boded ie-delays are o so realisic. I pracice, oe is o oly ieresed i syse sabiliy (e.g. i he sese of Lyapov, b also i he bods of syse raecories. syse cold be sable b sill copleely seless becase i possesses desirable rasie perforaces. Ths, i ay be sefl o cosider he sabiliy of sch syses wih respec o cerai sbses of sae-space which are defied a priori i a give proble. Besides ha, i is of pariclar sigificace o cosider he behavior of dyaical syses oly over a fiie ie ierval. These bodedess properies of syse resposes, i.e. he solio of syse odels, are very ipora fro he egieerig poi of view. Realizig his fac, eros defiiios of he so-called echical ad pracical sabiliy were irodced. Roghly speakig, hese defiiios are esseially based o he predefied bodaries for he perrbaio of iiial codiios ad he allowable perrbaio of a syse respose. Ths, he aalysis of hese pariclar bodedess properies of solios is a ipora sep, which precedes he desig of corol sigals, whe fiie ie or pracical sabiliy corol is cosidered. Moivaed by a brief discssio o pracical sabiliy i he oograph of LaSalle ad Lefsche,[8] ad Weiss ad Ifae,[9] have irodced varios oaios of sabiliy over a fiie ie ierval for coios-ie syses ad cosa se raecory bods. ore geeral ype of sabiliy ( pracical sabiliy wih selig ie, pracical epoeial sabiliy, ec. which icldes ay previos defiiios of fiie sabiliy was irodced ad cosidered by Grić,[,]. cocep of fiie-ie sabiliy, called Uiversiy of Belgrade, Facly of Mechaical Egieerig, Depare of Mechaics; Kralice Mrie 6, 2 Belgrade, SERBI

2 32 LZREVIĆ.M.: STBILITY ND STBILIZTION OF FRTIONL ORDER TIME DELY SYSTEMS fial sabiliy, was irodced by Lashirer ad Sory, [2] ad a frher develope of hese resls was de o La ad Weiss,[3]. Recely, fiie-ie corol/sabilizaio, ad ehods for sabiliy evalaio of liear syses o fiie ie horizo have bee proposed by ao e al., [4,5], respecively. lso, a aalysis of liear ie-delay syses i he coe of fiie ad pracical sabiliy was irodced ad cosidered i [6-8] ad as well as fiieie sabiliy ad sabilizaio [9]. Recely here have bee soe advaces i he corol heory of fracioal (o-ieger order dyaical syses for sabiliy qesios sch as robs sabiliy, boded ip boded op sabiliy, ieral sabiliy, fiie ie sabiliy, pracical sabiliy, roo-locs, robs corollabiliy, robs observabiliy, ec. For eaple, regardig liear fracioal differeial syses of fiie diesios i a sae-space for, boh ieral ad eeral sabiliies are ivesigaed by Maigo,[2]. Soe properies ad (robs sabiliy resls for liear, coios, (cerai fracioal order sae-space syses are preseed ad discssed [2,2]. However, we cao direcly se algebraic ools, e.g. Roh-Hrwiz crieria, for he fracioal order syse becase we do o have a characerisic polyoial b a psedopolyoial wih a raioal power-livaled fcio. aalyical approach was sggesed by he ad Moore,[22], who cosidered he aalyical sabiliy bod sig Laber fcio W. Frher, aalysis ad sabilizaio of fracioal (epoeial delay syses of rearded/eral ype are cosidered [23,24], as well as BIBO sabiliy [25]. Whereas Lyapov ehods have bee developed for he sabiliy aalysis ad he corol law syhesis of ieger liear syses ad have bee eeded o sabiliy of fracioal syses, oly few sdies deal wih o-lyapov sabiliy of fracioal syses. Recely, for he firs ie, he fiieie sabiliy aalysis of fracioal ie delay syses has bee preseed ad repored i papers [26,27]. Here, a Bella-Growall`s approach is proposed, sig a classical Bella-Growall ieqaliy as well as a recely obaied geeralized Growall ieqaliy repored i [28] as a sarig poi. The proble of sfficie codiios ha eable syse raecories o say wihi he a priori give ses for a pariclar class of (oliear (oaooos fracioal order ie-delay syses has bee eaied. Fdaeals of he fracioal calcls Fracioal calcls (F as a eesio of ordiary calcls has 3 years old hisory. F was iiiaed by Leibiz ad L`Hospial as a resl of a correspodece which lased several ohs i 695. Boh Leibiz ad L`Hospial, aware of ordiary calcls, raised he qesio of a oieger differeiaio (order /2 for siple fcios. Fracioal derivaives were sbseqely eioed, i oe coe or he oher, by (for eaple Eler i 73, Lagrage i 772, Laplace i 82, Lacroi i 89, Forier i 822, Riea i 847, Gree i 859, Holgre i 865, Grwald i 867, Leikov i 868, Soii i 869, Lare i 884, Nekrassov i 888, Krg i 89, ad Weyl i 99, ec. [29]. I ha way, he heory of he fracioal-order derivaive was developed aily i he 9h cery. Sice 9h cery, as a fodaio of fracioal geoery ad fracioal dyaics, he heory of FO, he heory of F ad FDEs ad applicaio research i pariclar, have bee developed rapidly i he world. The oder epoch sared i 974 whe a cosise foralis of he fracioal calcls has bee developed by Oldha ad Spaier,[4], ad laer Podlby,[6]. pplicaios of F are very wide owadays, i rheology, viscoelasiciy, acosics, opics, cheical physics, roboics, corol heory of dyaical syses, elecrical egieerig, bioegieerig, ec. [4-2]. I fac, real world processes geerally or os likely are fracioal order syses. The ai reaso for he sccess of F applicaios is ha hese ew fracioal-order odels are ore accrae ha iegerorder odels, i.e. here are ore degrees of freedo i he fracioal order odel. Frherore, fracioal derivaives provide a ecelle isre for he descripio of eory ad herediary properies of varios aerials ad processes de o he eisece of a eory er i a odel. This eory er isres he hisory ad is ipac o he prese ad fre. ypical eaple of a oieger (fracioal order syse is he volage-crre relaio of a sei-ifiie lossy rasissio lie [7] or diffsio of he hea hrogh a sei-ifiie solid, where hea flow is eqal o he half-derivaive of he eperare [6]. I his 7 page-log book o alcls, 89 Lacroi [3] developed he forla for he -h derivaive of y, is a posiive ieger,! D (! where ( is a ieger. Replacig he facorial sybol by he Gaa fcio, he frher obaied he forla for he fracioal derivaive β Γ ( β + β D ( Γ ( β + where ad β are fracioal bers ad he Gaa fcio Γ ( z is defied for z > by he so-called Eler iegral of he secod kid: z Γ ( z e d, Γ ( z+ zγ( z (2 O he oher had, Lioville ( forally eeded he forla for he derivaive of iegral order a a a a De ae De ae, arbirary order (3 Usig he series epasio of a fcio, he derived he forla kow as Lioville`s firs forla for fracioal derivaive, where ay be raioal, irraioal or cople. where a D f c a e (4 f c ep( a, Re a >. However, i ca be oly sed for fcios of he previos for. lso, i was J.B.J.Forier [3] who derived he fcioal represeaio of he fcio f ( f cos π d d ( ζ ( ξ ( ζ ζ ξ, (5 2 R R where he also forally irodced he fracioal derivaive versio. I 823, bel cosidered a echaical proble, aely bel s echaical proble [32]. I he absece of fricio, he proble is redced o a iegral eqaio

3 LZREVIĆ.M.: STBILITY ND STBILIZTION OF FRTIONL ORDER TIME DELY SYSTEMS 33 y /2 ( y z ( z dz 2 g f( y, y [, H ], (6 where ( z + φ 2 ( z, φ( z is a icreasig fcio, g is he cosa dowward acceleraio, f ( y is a prescribed fcio. The bel solved (6 i [33]. lso a bel rasfor of a sfficiely well behaved fcio was geeralized o ( ( ( d, a< < b Γ, (7 where a b, (, a < ad Γ (. is he well kow Eler's gaa fcio. Here, i is assed he solio of classical bel iegral eqaio eiss ad he fracioal derivaive wih order (, eiss i L ( a, b, [34], so we have followig resls: Lea.osider, for (,, a < b, he classical bel iegral eqaio Γ a ( d f, a < < b, (8 The here eiss a os oe solio of eqaio (8 i L ( a, b. Moreover, if he fcio f is absolely coios o [a, b], he eqaio (8 has a solio i L ( a, b, give by (9 d, Γ( d a If a ad f ( a are fiie, he ( f d f ( a < < b, (9 f( a( a + ( f ( d,, Γ( a ( a< < b If a is fiie ad f is eeded by o he lef of a, he Γ( a ( df, a < < b, ( If a is fiie ad li f he Γ( ( df, < < b, (2 Fro he viewpoi of fracioal calcls, we ca see ha (9 (2 are s soe oher fors of fracioal derivaives, wih order (,, der soe differe hypoheses o f. Fracioal derivaives are ypically reaed as a pariclar case of psedo-differeial operaors. Sice hey are olocal ad have weakly siglar kerels, he sdy of fracioal differeial eqaios sees o be ore difficl ad fewer heories have bee esablished ha for classical differeial eqaios. I a series of papers by Lioville [35,36] repored he earlies for of he fracioal iegral, hogh o qie rigorosly fro he aheaical poi of view. The forla was ake as follows p D ϕ + d, p ( Γ( p < <, p > p ϕ, (3 Tha is ow called he Lioville for of fracioal iegral wih he facor ( p beig oied. Ne sigifica work was doe by Riea [37]. lhogh he wroe his paper i 847 whe he was s a sde, i was o pblished il 876, e years afer his deah. Riea arrived a he epressio ( φτ ( τ D RL φ( d τ, > Γ (4 for fracioal iegraio. Frherore, we have he os sefl fors of lef-had ad righ-had Riea- Lioville (RL derivaives defied as follows RL a, d D f ( τ f ( τ dτ, Γ( d a (5 b d D f ( τ f ( τ dτ, Γ d ( ( RL, b where <, a, b are he erial pois of he ab, which ca also be,. The defiiio ierval [, ] (5 of he fracioal differeiaio of Riea-Lioville ype leads a coflic bewee he well-esablished ad polished aheaical heory ad proper eeds, sch as he iiial proble of he fracioal differeial eqaio, ad he ozero proble relaed o he Riea-Lioville derivaive of a cosa, ec. cerai solio o his coflic was proposed by apo firs i his paper [38] (967.apo s defiiios ca be wrie as ( D a, f τ f ( τ dτ, Γ( a (6 b d D f ( τ f ( τ dτ, Γ d ( (, b + where <. Obviosly, he apo derivaive is sricer ha he Riea-Lioville derivaive, oe reaso is ha he -h order derivaive is reqired o eis. The apo ad Riea-Lioville forlaios coicide whe he iiial codiios are zero. Besides, he RL derivaive is eaigfl der weaker soohess reqirees. I addiio, he RL derivaive ca be preseed as: RL D f D Da, f,,, (7 ad he apo derivaive

4 34 LZREVIĆ.M.: STBILITY ND STBILIZTION OF FRTIONL ORDER TIME DELY SYSTEMS a, a, D f D D f,,, (8 where Z +, D is he classical -order derivaive. Moreover, previos epressios show ha he fracioalorder operaors are global operaors havig a eory of all pas eves, akig he adeqae for odelig herediary ad eory effecs i os aerials ad syses. I addiio, for he RL derivaive, we have li RL d + a D ( d ad RL d ( li a D ( (9 d However, for he apo derivaive, we have d li D ( D ( a ( a + ( d ad d ( li a D ( (2 d Obviosly, RL Da, (, + varies coiosly wih, b he apo derivaive cao do his. O he oher had, he iiial codiios of fracioal differeial eqaios wih he apo derivaive have a clear physical eaig ad he apo derivaive is eesively sed i real applicaios. O he oher had, Grwald [39] (i 867 ad Leikov [4] (i 868 developed a approach o fracioal differeiaio based o he defiiio ( Δh f GL li, h h < D f Δ f h ( f ( h, h >, (2 which is he lef Grwald-Leilov (GL derivaive as a lii of a fracioal order backward differece. Siilarly, we have he righ oe as ( Δ h f GL h h D f li, Δ f h ( f ( + h, h<, < (22 Therefore, oe ca defie a ew for of he Grwald- Leikov derivaive as follows GL ( Δ h f +Δ h f D f li, h π 2cos h 2 (23 which is called he Grwald-Leikov-Riesz derivaive. s idicaed above, he previos defiiio of GL is valid for > (fracioal derivaive ad for < (fracioal iegral ad, cooly, hese wo oios are groped io oe sigle operaor called differiegral. The GL derivaive ad RL derivaive are eqivale if he fcios hey ac o are sfficiely sooh. For erical calclaio of he fracioal order differ-iegral operaor, oe ca se a relaio derived fro he GL defiiio. (( L N ± ( ± D f( h b f( h (24 where L is he "eory legh", h is he sep size of he calclaio, L N ( i,, h h [ ] is he ieger par of ad b ( coefficie give by ( ± ± b, b b ( ± ( ± (25 ± is he bioial (26 For coveiece, he Laplace doai is sally sed o describe he fracioal iegro-differeial operaio for solvig egieerig probles. The forla for he Laplace rasfor of he RL fracioal derivaive has he for: s RL k, e D f d s F( s k k RL, s D f (27 Where for < (i.e., for he case of a fracioal iegral he s i he righ-had side s be oied. lso, he Laplace rasfor of he apo fracioal derivaive is: s k ( k k e D f( d s F( s s f (, (28 < < which iplies ha all he iiial vales of he cosidered eqaio are preseed by a se of oly classical iegerorder derivaives. Besides ha, a geoeric ad physical ierpreaio of fracioal iegraio ad fracioal differeiaio ca be fod i Podlby s work [4]. Preliiaries o ieger ie-delay syses liear, livariable ie-delay syse ca be represeed by a differeial eqaio: d( ( + ( τ (29 d ad wih he associaed fcio of he iiial sae: ( ψ (, τ, (3 Eqaio (29 is referred o as a hoogeos sae eqaio. lso, a ore geeral, liear, livariable iedelay syse ca be represeed by he followig differeial eqaio: d( ( + ( τ + B ( + B ( τ, (3 d

5 LZREVIĆ.M.: STBILITY ND STBILIZTION OF FRTIONL ORDER TIME DELY SYSTEMS 35 ad wih he associaed fcio of he iiial sae ad corol: ( ψ (, τ, (32 ( ψ (, Eqaio (3 is referred o as a ohoogeos or forced sae eqaio, ( is a sae vecor, ( is a corol vecor,,, B ad B are cosa syse arices of appropriae diesios, ad τ is pre ie delay, τ cos. (τ >. Moreover, a class of a o-liear syse wih ie delay, cosidered here, is described by he sae space eqaio: d( ( + ( τ + B( + B( τ + d (33 + f ( + f ( τ, i i wih he iiial fcios (32 of he syse. The vecor fcios fi, f, i,,, prese oliear paraeer perrbaios of he syse i respec o ( ad ( τ, respecively. I addiio, he e asspio ha: f i( ( c i (, i, [, (34 f ( (- τ c (- τ,,,, is irodced, where ci, c R + are kow real posiive bers. Moreover, a liear livariable ie-varyig delay syse ca be represeed by he differeial eqaio d( ( + ( τ ( + B (, (35 d ad wih he associaed fcio of he iiial sae ( ψ (, τ. (36 where τ ( is a kow ie varyig paraeer which saisfies τ ( τ, J, J [, + T], J R (37 M M Moreover, here is cosidered a class of perrbed oliear syse wih ie delay described by he sae space eqaio d( ( +Δ ( + ( +Δ ( τ ( + B( + d (38 + f (, ( τ (, wih he give iiial fcios of he syse ad he vecor fcio f.the vecor fcio f preses oliear paraeer perrbaios of he syse i respec o ( ad ( τ, respecively, ad he arices Δ, Δ prese perrbaios of he syse, oo. lso, i is assed ha he e asspio is re. f(, - ( τ c + c - ( τ, (39,, [ where c, c R + are kow real posiive bers. The [ dyaical behavior of syse (29, (3 or (33 wih he iiial fcios (3 or (32 is defied over he ie ierval J {, + T}, where he qaiy T ay be eiher a posiive real ber or he sybol +, so fiie ie sabiliy ad pracical sabiliy ca be reaed silaeosly. I is obvios ha J R. Tie ivaria ses, sed as he bods of syse raecories, are assed o be ope, coeced ad boded. Le he ide " ε " sads for he se of all allowable saes of he syse ad he ide " δ " for he se of all iiial saes of he syse, sch ha he se S S. I geeral, oe ay wrie: δ ε { : ( 2 }, [, Q ] Sρ < ρ ρ δ ε, (4 where Q will be assed o be a syeric, posiive defiie, ad real ari. S deoes he se of he all allowable corol acios. Le be ay vecor or (. (e.g.,.,2, ad (. he ari or idced by his vecor. The ari easre has bee widely sed i he lierare whe dealig wih sabiliy of ie delay syses. The ari easre μ for ay ari is defied as follows: li I + ω μ (4 ω ω The ari easre defied i (36 ca be sbdefied i hree differe ways, depedig o he or ilized i is defiiios [42]. ad μ a Re akk + a k μ a Re 2 akk + a k ik, (42 i i k ik, (43 i i k μ ( a Re( aii aki i + (44 i i k Epressio (32 ca be wrie i is geeral for as: [ ] [ ] ( + θ ψ( θ, τ θ, ψ( θ τ,, (45 ( + θ ψ ( θ, τ θ, ψ ( θ τ, where is he iiial ie of observaio of he syse (29 ad -τ [,] is he Baach space of coios fcios over a ie ierval of he legh τ, appig he ierval [ τ,] io R wih he or defied i he followig aer: ψ a ψθ, (46 τ θ I is assed ha he sal soohess codiio is prese so here is o difficly wih qesios of eisece, iqeess, ad coiiy of solios wih respec o iiial daa.

6 36 LZREVIĆ.M.: STBILITY ND STBILIZTION OF FRTIONL ORDER TIME DELY SYSTEMS Soe previos resls relaed o ieger ie-delay syses The eisig ehods developed so far for sabiliy check are aily for ieger-order syses. Defiiio : The syse give by (3 wih ( -τ,, saisfyig iiial codiio (4 is fiie ς(, ε,, τ, J, μ, if d oly if: sable w.r. { ( } ad iply: [ τ ] ψ S,, (47 δ (, S J (48 [ ] (;, S,, T (49 The illsraio of he previos defiiio is give i Fig.. ε Defiiio 2: The syse give by (3 saisfyig iiial codiio (32 is fiie sable w.r. δε,,,, τ, J, μ( if ad oly if: { ψ } ad iply: [ τ ] ψ Sδ,, (5 [ τ ] ψ S,, (5 (, S J (52 (;,, ( S, J (53 Theore. The syse give by (3, wih iiial fcio (32 is fiie ie sable w.r. { δε,, ψ,, τ, J, μ( } if he followig codiio is saisfied,[43]: where: μ ε ( ε δ σ 2( 2 ( e μ / < (54 μ ( ( 2 τ μ 2 μ σ a a + e c + e c (55 2 2, c2 γ b b c b γ γψ (56 γ / ε, γ / ε, a, b B a b B a ψ ψ /, / (57 The resls ha will be preseed i he seqel eable checkig he fiie ie sabiliy of he oaooos syse o be cosidered (29,(3 or (33 ad (3,(32 wiho fidig he fdaeal ari or he correspodig ari easre. Defiiio 3: The syse give by (3 saisfyig iiial codiio (32 is fiie sable w.r. { δε,,,, o, J, }, δ < ε if ad oly if: iply: ψ < δ, ψ <, (58 ( <, J (59 ( < ε, J (6 Theore 2. The oaooos syse give by (3 saisfyig iiial codiio (33 is fiie ie sable w.r.. { δε,,,,, J, }, δ < ε if he followig codiio is saisfied,[44]: a σa ( * * σ ( e γ γ τ ε δ + + ( + /, J. where * * γ γ δ γ γ δ γ ( b + b γ ( (6 /, /,,, (62 b Preliiaries o he sabiliy of fracioal order syses icldig ie-delays I he field of fracioal-order corol syses, here are ay challegig ad solved probles relaed o he sabiliy heory sch as robs sabiliy, boded ip boded op sabiliy, ieral sabiliy, roo-locs, robs corollabiliy, robs observabiliy, ec. I egieerig, he fracioal order is ofe less ha, so we resric (, as sal. Eve if >, we ca raslae he fracioal syses io syses wih he sae, provided soe siable fracioal order which lies i codiios are saisfied [45]. I order o deosrae he advaage of fracioal calcls i characerizig a syse behavior (here, sabiliy properies, le s cosider he followig illsraive eaple, [46]. Eaple : opare he followig wo syses wih he iiial codiio ( for < <, d ( ν, D, ( ν ν ν, < <. (63 d The aalyical solios of he previos syses are ν ν+ νγ( ν + ( ad + (, respecively. Oe ay Γ ( ν + coclde ha he ieger-order syse is sable for ay ν (,. However, he secod give fracioal dyaic syse is sable as < ν <, which iplies ha he fracioal-order syse ay have a addiioal aracive feare over he ieger-order syse. lso, i [47], Tarasov proposed ha sabiliy is coeced o oio chages a fracioal chages of variables where syses which are

7 LZREVIĆ.M.: STBILITY ND STBILIZTION OF FRTIONL ORDER TIME DELY SYSTEMS 37 sable i sese of Lyapov ' ca be sable wih respec o fracioal variaios. I 996, Maigo [48] sdied he followig fracioal differeial syse ivolvig he apo derivaive d, (, (, (, D (64 d T 2 where (,,..., wih he iiial vale T 2,,...,, R. The sabiliy of he eqilibri of syse (64 was firs defied ad esablished by Maigo as follows. Defiiio 4. The aooos fracioal order syse (64 is said o be a sable if for ay, here eiss ε > sch ha b asypoically sable if ε for (65 ( li (66 lso, Maigo [48] proposed a defiiio of he BIBO sabiliy for he fracioal differeial syse. Defiiio 5. ip/op liear fracioal syse (67 R, y d B, ( d + y (67 p R is eerally sable or boded-ip + boded-op (BIBO if L ( R, R + p y h L ( R, R + p h L ( R, R., which is eqivale o: lso, i [49], he ahors give wo defiiios of he sabiliy for differeial syses wih he apo derivaive ad he Riea-Lioville derivaive, respecively. Besides, he asypoical sabiliy of higher-diesioal liear fracioal differeial syses wih he Riea- Lioville fracioal order ad he apo fracioal order were sdied where he asypoical sabiliy heores were also derived. Defiiio 6. The zero solio of he followig differeial syse wih he -h order apo derivaive i which < <, D X X (68 is said o be: (i Sable, if ε >, δ >, whe X δ, he solio X ( o (68 wih he iiial codiio ( X X saisfies X( ε for ay. (69 (ii sypoically sable, if he zero solio o (68 is sable, ad i is locally aracive, i.e., here eiss a δ sch ha X δ iplies ha li X (7 + Defiiio 7. The zero solio of he followig differeial syse wih he -h order Riea- Lioville derivaive i which < < RL, is said o be: (iiisable, if ε >, δ >, whe X D X X (7 δ, he solio X ( o (7 wih he iiial codiio D X ( X RL, saisfies X ( ε for ay. (72 (ivsypoically sable, if he zero solio o (7 is sable, ad i is locally aracive, i.e., here eiss a δ sch ha X δ iplies ha li X (73 + Ne, oe ay sdy he sabiliy of fracioal differeial syses i wo spaial diesios, ad he sdy he fracioal differeial syses wih higher diesios. Now, he fracioal differeial syse wih he apo derivaive is sdied, * D, X X,,, R (74 where he fracioal derivaive * D, (.. D, (.. or RLD, (... They sdied he fracioal differeial syse wih he apo derivaive, as follows: D, X X,,, R (75 Theore 3. If he real pars of all he eigevales of are egaive, he he zero solio o syse (75 is asypoically sable. lso for he fracioal differeial syse wih he Riea-Lioville derivaive RL D, X X,,, R (76 hey saed he followig heore. Theore 4. If he real pars of all he eigevales of are egaive, he he zero solio o syse (76 is asypoically sable. fracioal-order liear ie ivaria syse ca be represeed i he followig psedosae space for: d ( ( + B ( d y ( ( (77 where he oaio d / d idicaes he apo fracioal derivaive of he fracioal coesrae order, p R, R ad y R are psedo-sae, ip, ad op vecors of he syse, respecively, ad p R, B R, R. I is worh eioig ha he sae space for Eq.(77 is a psedo-represeaio becase he kowledge of he vecor a he ie ad he ip vecor ( for are o eirely sfficie o kow he behavior of syse (77 for >. fracioal-order odel is i fac ifiie diesioal;

8 38 LZREVIĆ.M.: STBILITY ND STBILIZTION OF FRTIONL ORDER TIME DELY SYSTEMS herefore, is re sae vecor shold also be ifiie diesioal. Theore 5[48]: The followig aooos syse, (64 d ( (,, < (78 d R, ad is a ari, is asypoically sable if ad oly if arg ( λ > π /2 is saisfied for all eigevales ( λ of he ari. I his case, each copoe of he saes decays owards sch as. lso, his syse is sable if ad oly if arg ( λ > π /2 is saisfied for all eigevales ( λ of he ari wih hose criical eigevales saisfyig arg ( λ π /2 have a geoeric lipliciy of oe. The deosraio of his heore is based o he copaio of he sae vecor of syse ( < N, >, >. Respose o o-zero iiial codiios. However, his resl reais valid whaever he defiiio sed, give ha for a liear syse wiho delay, a aooos syse wih he o-zero iiial codiios ca be rasfored io a o-aooos syse wih he ll iiial codiio. lso, he sable ad sable regios for < are show i Fig.2 ad hey deoe he sable ad sable regios for < by ad +, respecively. Figre 2. Sabiliy regio of a fracioal-order liear ie ivaria syse wih order < For a ii realizaio of (77, Maigo has also deosraed he followig heore,[48]. Theore 6. I [48], cosider a syse give by he followig liear psedosae space for wih he ier diesio : d ( ( + B (, ( d y ( ( (79 where <. lso, asse ha he riple (,B, is iial. Syse (79 is boded-ip boded-op arg λ > π / 2 is saisfied (BIBO sable if ad oly if for all eigevales λ of he ari. Whe syse (79 is eerally sable, each copoe of is iplse respose behaves like a ifiiy. Epoeial sabiliy hs cao be sed o characerize asypoic sabiliy of fracioal syses. ew defiiio is irodced. Defiiio 8. γ sabiliy The raecory ( of syse d ( / d f (, ( (forced syse is γ asypoically sable if he ifor asypoic sabiliy codiio is e ad if here is a posiive real γ sch ha: γ (, ( ( o c Q sch ha (, Q γ (8 sabiliy will hs be sed o refer o he asypoic sabiliy of fracioal syses. s he copoes of he sae ( slowly decay owards followig γ, fracioal syses are soeies called log eory syses. Sabiliy of fracioal delay syses I spie of iesive research, he sabiliy of fracioal order (ie delay syses reais a ope proble. s for liear ie ivaria ieger order syses, i is ow well kow ha he sabiliy of a liear fracioal order syse depeds o he locaio of he syse poles i he cople plae. However, he poles locaio aalysis reais a difficl ask i he geeral case. For coesraig fracioal order syses, powerfl crieria have bee proposed. The os well-kow is Maigo's sabiliy heore [48]. I peris s o check he syse sabiliy hrogh he locaio i he cople plae of he dyaic ari eigevales of he sae space like syse represeaio. Maigo's heore is i fac he sarig poi of several resls i he field. s we kow, de o he e τ s presece of he epoeial fcio, his eqaio has a ifiie ber of roos, which akes he aalyical sabiliy aalysis of a ie-delay syse ereely difficl. I he lierare few heores are available for sabiliy esig of fracioal-delay syses. los all of hese heores are based o he locaios of he rasfer fcio poles [24, 5] ad sice here is o iversally applicable aalyical ehod for solvig fracioal-delay eqaios i s doai, he erical approach is cooly sed. I he field of ifiiediesioal fracioal-delay syses os sdies are cocered abo he sabiliy of a class of disribed syses whose rasfer fcios ivolve s ad/or, [5]. May eaples of fracioal differeial syses wih delay ca be fod i he lierare. Siple eaples sch as Gs ( ep( a s/ s, a> arisig i he heory of rasissio lies [52], or oe ca fid i [53] fracioal delay syses wih he rasfer fcio of liked o he hea eqaio which leads o rasfer fcios Gs ( sch as or cosh( s Gs (,( (8 ssih( s a s Gs ( 2e 2a s b( e e s (82 For eaple, Hozel [54] preseed he sabiliy codiios for fracioal-delay syses wih he

9 LZREVIĆ.M.: STBILITY ND STBILIZTION OF FRTIONL ORDER TIME DELY SYSTEMS 39 ρs characerisic eqaio ( as b ( cs d e he ad Moore [22] aalyzed he sabiliy of a class of fracioal-delay syses whose characerisic fcio ca be represeed as he prodc of facors of he for r cs as b e d + + where he paraeers abcd,ad,,, r are all real bers. I fac, hey coped he characerisic roos of he syse sig he Laber W fcio, which has becoe a sadard library fcio of ch aheaical sofware. I oher words, hey go a sabiliy codiio of (83, give by a rascede ieqaliy via he Laber fcio [22, 55]. They cosidered he followig delayed fracioal eqaio q d y( Kp y( τ (83 q d where q ad Kp are real bers ad < q <, ie delay τ is a posiive cosa ad all he iiial vales are zeros. We are ieresed i ellig wheher he syse ( is sable or o for a give se of cobiaio of he hree paraeers: q, Kp ad τ. The sabiliy codiio is ha for all possible q, r ad Kp τ ( / q q W Kp (84 τ r I ieqaliy, W(. deoes he Laber fcio sch ha W e W. However, sch a bod reais aalyic ad is difficl o se i pracice. I paper [55], he applicaio of he Laber W fcio o he sabiliy aalysis of ie-delay syses is re-eaied hrogh acally cosrcig he roo disribios of a rascedeal characerisic eqaio s (TE of soe chose orders. I is fod ha he righos roo of he origial TE is o ecessarily a pricipal brach Laber W fcio solio, ad ha a derived TE obaied by akig he h power of he origial TE irodces sperflos roos o he syse. Frher, Maigo's heore has bee sed i [56] o ivesigae fracioal differeial syses wih liple delays sabiliy. The proposed sabiliy codiios are based o he roo locs of he syse characerisic ari deeria b he proposed codiios are hs difficl o se i pracice. hors sed fracioal derivaive apo defiiio of derivaive where, by sig he Laplace rasfor, a characerisic eqaio for he above syse wih liple ie delays is irodced. They discovered ha if all roos of he characerisic eqaio have egaive pars, he he eqilibri of he above liear syse wih fracioal order is Lyapov globally asypoical sable. If he eqilibri eiss, i is alos he sae as ha of classical differeial eqaios. Naely, he followig - diesioal liear fracioal differeial syse wih liple ie delays: q q d ( d a ( τ + a ( τ a ( τ, q2 2 q2 d ( d (85 a2 ( τ2 + a222 ( τ a2( τ2,... q d ( q d a ( τ + a ( τ a ( τ, where q i is real ad lies i (,, he iiial vales i( φi( are give for ai, τ i τa ad i, 2,...,. I his syse, he ie-delay ari + T τ i R, he coefficie ari sae variables i, i( τ i R, φ ( [ τ ] a i, he ad he iiial vales i a,. Is fracioal order is defied as q ( q, q2,..., q. If q i q ad τ i, i,, 2,...,, he syse (85 is acally he oe cosidered i [56]. ( s Δ q sτ sτ2 sτ s ae a2 e... a e sτ2 q2 sτ22 sτ a e s a e a2e M M O M sτ sτ2 q sτ ae a2e... s ae (86 where Δ ( s deoes a characerisic ari of syse ( ad de ( Δ ( s a characerisic polyoial of (86. The disribio of de ( Δ ( s s eigevales oally deeries he sabiliy of syse (86. Theore 7. If all he roos of he characerisic eqaio de ( Δ ( s have egaive real pars, he he zero solio of syse ( is Lyapov globally asypoically sable. If, he (86 is redced o he syse sdied i [56]. Boe ad Parigo [23,24] aalyze he BIBO sabiliy of fracioal epoeial delay syses which are of rearded or eral ype. The codiios esrig sabiliy are give ad hese codiios ca be epressed i ers of he locaio of he poles of he syse. I view of cosrcig robs BIBO sabilizig corollers, eplici epressios of coprie ad B ezo facors of hese syses are deeried. I addiio, hey have hadled he robs sabilizaio of fracioal epoeial delay syses of rearded ype. The deeriaio of coprie ad B ezo facors i he case of eral syses is der sdy i boh cases. However, all hese coribios do o provide iversally accepable pracical effecive algebraic crieria or algorihs for esig he sabiliy of a give geeral fracioal delay syse. lhogh he sabiliy of he give geeral characerisic eqaio ca be checked wih he Nyqis crierio or he Mikhailov crierio, i becoes sfficiely difficl whe a coper is sed sice oe shold fid a agle of r of he freqecy respose plo for a ifiie variaio of he freqecy ω. visal coclsio o sabiliy wih respec o he cosrced par of he plo is o pracically reliable, sice, alog wih a ifiie spiral, he delay geeraes loops he ber of which is ifiie. s evide fro he lierare, he lack of iversally accepable algebraic algorihs for esig he sabiliy of he characerisic eqaio has hidered he advace of corol syse desig for fracioal delay syses. This is pariclarly re i he case of desigig a fied-srcre fracioal-order coroller, e.g., PI D β. O he oher had, Hwag ad heg [57] proposed a erical algorih which ses he ehods based o he achy iegral heore ad sggesed he odified cople iegral i he for of J k i f ( s ( + + k ds (87 s h ih f( ih i 2 2

10 4 LZREVIĆ.M.: STBILITY ND STBILIZTION OF FRTIONL ORDER TIME DELY SYSTEMS where h > ad h 2 are radoly chose real cosas lyig i a specified ierval ad k is a posiive ieger. The radoess of he paraeers h ad h2 akes he probabiliy of he zero s of he resides of all poles of he iegrad beig pracically zero. Hece, he sabiliy of a give fracioal-delay syse ca be achieved by evalaig he iegral J k ad coparig is vale wih zero. lso, he proposed algorih provides o idea abo he ber ad he locaio of sable poles. I paper [58], a effecive erical algorih for deeriig he locaio of poles ad zeros o he firs Riea shee is preseed. The proposed ehod is based o he Roche s heore ad ca be applied o all li-valed rasfer fcios defied o a Riea srface wih a fiie ber of Riea shees where he origi is a brach poi. This covers all pracical (fiie-diesioal fracioal-order rasfer fcios ad fracioal-delay syses. Fiie ie sabiliy ad sabilizaio of fracioal order ie delay syses s we kow, he bodedess properies of syse resposes are very ipora fro he egieerig poi of view. This eables syse raecories o say wihi a priori give ses for he fracioal order ie-delay syses i he sae-space for, i.e., syse sabiliy fro he o- Lyapov poi of view is cosidered. Fro his fac ad or he bes kowledge, we firsly irodced ad defied fiie-ie sabiliy for fracioal order ie delay syses [26-27, 6, 62-63]. We also eed he followig defiiios o aalyze he case of fracioal order syses wih iedelay fro o-lyapov poi of view. Firs, we irodce he sae order fracioal differeial syse wih ie-delay (88 as well as liple ie delays (9 represeed by he followig differeial eqaios: d ( * Do, ( ( + ( τ + B(, d (88 < <, wih he associaed fcio of he iiial sae: ψ [ τ ] τ ( +,,. (89 Moreover, i is show i [26] ha fracioal-order ie delay sae space odel of PD corol of Newcasle robo ca be preseed by (88 i he sae space for. Here, * Do, (. deoes eiher he apo fracioal derivaive Do, (. or he Riea-Lioville fracioal derivaive RL D o, (.. lso, a fracioal differeial syse wih liple ie delays ca be preseed as follows: d ( * o, + i τi + d i D ( ( ( B (, < <, τ < τ <... < τ <... < τ Δ 2 i wih he associaed fcio of he iiial sae: [ ] (9 ( + ψ Δ,, τ. (9 ad where i ( i,,...,, B are cosa syse arices of appropriae diesios, ad τ i > ( i, 2,..., are pre ie delays. Defiiio 9.[59] The syse give by (88, ( saisfyig iiial codiio (89 is fiie sable w.r. {, J, δ, ετ, }, δ< ε if ad oly if: o iplies: ψ < δ, (92 ( < ε, J, (93 Defiiio.[59] The syse give by (9, ( saisfyig iiial codiio (9 is fiie sable w.r. {, J, δ, ε, Δ }, δ < ε if ad oly if: o iplies: ψ < δ, J, J [ Δ,] R, (94 Δ Δ ( < ε, J, (95 Defiiio.[27,62] The syse give by (9 saisfyig iiial codiio (9 is fiie sable w.r. δ, ε,, Δ,, J,, δ< ε if ad oly if: { } ad iply: o [ ] ψ < δ, JΔ, JΔ Δ, R (96 ( <, J, > (97 ( < ε, J (98 lso, a oliear fracioal differeial syse wih ie delay i sae ad corol ca be preseed as follows: d ( * Do, ( ( + ( τ + B( + d (99 B( τ + f + f ( τ, < <, i i ad wih he associaed fcio of he iiial sae ad corol: ( ψ (, ( ψ (, τ ( Eqaio (99 is referred o as a oliear ohoogeos sae eqaio,,, B ad B are he cosa syse arices of appropriae diesios, ad he vecor fcios fi, f, i,,, prese oliear paraeer perrbaios of he syse i respec o ( ad ( τ respecively. Defiiio 2: The syse give by (99 saisfyig iiial codiio ( is fiie sable w.r. δ, ε,,,, J,, δ< ε if ad oly if: { } iply: o ψ < δ, ψ <, ( ( <, J (2

11 LZREVIĆ.M.: STBILITY ND STBILIZTION OF FRTIONL ORDER TIME DELY SYSTEMS 4 ( < ε, J (3 We he irodce he sfficie codiios o fiieie sabiliy. I [59], we cosidered he fracioal iedelay syses (88,(9 i he case of (. Theore 8.( The aooos syse give by (88 saisfyig iiial codiio (89 is fiie ie sable w.r.. { δ, ετ,, o, J, }, δ< ε, if he followig codiio is saisfied: a ( σ a Γ ( + ( ( σ + e ε / δ, J. (4 Γ + where σ a (. beig he larges siglar vale of he ari (., aely: σ σ ( σ ( +, (5 a a a ad Γ (. is he Eler's gaa fcio. B The aooos syse give by (9 saisfyig iiial codiio (9 is fiie ie sable w.r.. { δ, ε, Δ, o, J, }, δ < ε, if he followig codiio is saisfied: a ( σσ Σ a Γ ( + ( ( σ + e ε / δ, J. (6 Γ + where σ σ ( Σ a (. i i of he arices i i, i,,2,...,. where σ a (. beig he larges siglar vale of he ari i, i,, 2,...,. The above sabiliy resls for liear ie-delay fracioal differeial syses are derived sig Bella - Growall s ieqaliy. I ha way, oe ca check syse sabiliy over a fiie ie ierval. Reark [6]: If, case, oe ca obai he sae codiios which relaed o ieger order ie delay syses ( as follows: ( σa ( a σ + e ε / δ, J, Γ (2 (7 For he oaooos case, Zhag [6] also cosidered he followig iiial vale proble RL (, D ( + ( τ + f(, ( φ( [ τ ] ;,, (8 where < <, φ is a give coios fcio o [-τ, ], ad are he cosa syse arices of appropriae diesios, ad τ is a cosa wih τ >. The syse is defied over he ie ierval J [, T ], where T is a posiive ber, f( is a give coios fcio o [, T]. Siilarly, he sfficie codiios of fiie-ie sabiliy were derived by applyig Bella- Growall s ieqaliy. Theore 9. The syse give by (8 saisfyig he iiial codiio φ, [ τ,] is fiie-ie sable w.r. {, J, δ, ε, τ}, δ <ε, if he followig codiio is saisfied: μ a ( Γ ( + ( M + μ ( ( Γ + Where M f / φ φ sp τ θ, ad (. fcio, φ( θ e ε / δ, J, (9 Γ is he Eler s gaa μ μ ( μ (, μ μ ( +. a a a a I paper [62], we cosidered a class of fracioal oliear perrbed aooos syses wih ie delay described by he sae space eqaio: d ( d ( +Δ ( + +Δ ( + (, ( ( τ f ( wih he iiial fcios (89of he syse ad he vecor fcios f saisfied (34. Theore. The oliear perrbed aooos syse give by ( saisfyig iiial codiio (89 ad (34 is fiie ie sable w.r.. { δ, ε, o, J, }, δ < ε, if he followig codiio is saisfied: μ p ( Γ ( + μ p ( + e ε / δ, J, ( Γ ( + where Γ (. Eler's gaa fcio, ad μoco σ o+ γ Δ + c, σ Δ σ + γ Δ, μ p μoco + σ Δ, σδo γδo, σ Δ γ Δ Reark 2: If we have o perrbed syse Δ, Δ, f( oe ca obai he sae codiios which relaed o Theore 7. Frher, paper [63] preses a aral eesio of or paper [59] where ew sabiliy crieria for oaooos fracioal order ie delay syses are obaied (88. Theore. The oaooos syse give by (88 saisfyig iiial codiio (89 is fiie ie sable w.r.. { δ, ε,,, o, J, }, δ< ε, if he followig codiio is saisfied: a ( σ a Γ ( + ( ( ( σ + e + γ ε / δ, Γ + Γ ( + (2 J. where γ b / δ, B b. Reark 3. If, oe ca obai he sae codiios which relaed o ieger order ie delay syses (3, B as follows, [8]:

12 42 LZREVIĆ.M.: STBILITY ND STBILIZTION OF FRTIONL ORDER TIME DELY SYSTEMS ( a ( a σ σ + e + γ ε / δ, J, Γ (2 (3 Moreover, he sae paper [63] proposes fiie ie sabiliy crieria for a class of fracioal o-liear oaooos syses wih ie delay i sae ad i corol as follows: d ( ( + ( τ + B( + d (4 + B( τ + f + f ( τ, wih he iiial fcios (99 of he syse ad he vecor fcios f, f saisfied (34. Theore 2: The oliear oaooos syse give by (4 saisfyig iiial codiio (99 is fiie ie sable w.r.. { δ, ε,,, o, J, }, δ< ε, if he followig codiio is saisfied: σ ( a c σ a c( Γ ( + γ ( e Γ ( + Γ ( + γ ( τ γ ( τ ε / δ, J Γ ( + Γ ( (5 where γ b / δ, γ b / δ, γ b / δ. Recely, a fiie-ie sabiliy aalysis of liear fracioal order sigle ie delay syses has bee carried o ad repored i [27]. The Bella-Growall`s approach is proposed here, sig as he sarig poi a recely obaied geeralized Growall ieqaliy repored i [28]. Theore 3. The liear oaooos syse give by (88 saisfyig iiial codiio ( ψ (, τ is fiie ie sable w.r.. { δ, ε,, J, }, δ< ε, if he followig codiio is saisfied: Γ + Γ + σ a γ + E σ + ε / δ, ( a ( ( {, } J T, (6 where γ b / δ, ad σ a (. beig he larges siglar vale of he ari (., where: σa σa ( + σa ( ad E (. deoes he Miag-Leffler fcio (see ppedi. Reark 4. If, oe ca obai he sae codiios which relaed o ieger order ie delay syses (3, B as follows [8]: ( a ( a σ ( σ + e + γ ε / δ, J, Γ (2, E ( z e z (7 Theore 4. The liear aooos syse give by Eq.(88 B, saisfyig iiial codiio ( ψ (, τ is fiie ie sable w.r.. { δ, ε, J,}, δ ε <, if he followig codiio is saisfied: σ a + E ( σ ε / δ, J Γ ( + a, (8 Reark 5. I he sae aer, oe ay coclde ha if, see (2, he sae codiios follow [6], Eq.(7 which relae o ieger order ie delay syses (29. Here, we are ieresed i fiie ie sabilizaio of he liear perrbed fracioal order ie- delay syse scalar case as follows d ( ( a +Δ a ( + ( a+ a ( τ + b(, (9 d Theore 5: (Fiie ie sabilizaio Syse (9 corolled by he followig liear feedback ( k ( (2 i.e closed loop syse is fiie ie sable J wr { δ, ε,t }, here eiss he scalar k sch ha he followig codiio is saisfied ( μabo ( k + μσ + E (( μabo ( k + μσ ε / δ (2 Γ ( + Proof: Usig (2 ad applyig he or (. we obai a solio i he for of he eqivale Volerra iegral eqaio Le + ( ( a + bk + Δa ( s + ( s ds Γ ( + ( a + Δa ( s τ( s μ ( k a + b k, μ Δ a, μ a + Δa μ μ μ μ μ μ abo o Δo Σ Σ abo ( k + Δ +. o Σ abo k + Σo Takig io acco (23 ad (22, i follows (24 μσ ( μσ ψ + ( s sp ( ds. ( + Γ + Γ( τ M (22 (23 (24 Fially, applyig he geeralizaio of Bella- Growall lea ad he codiio of Theore 5, (2 we coplee he proof of he heore. illsraive eaple Usig a Tie-Delay PD copesaor o a liear syse of eqaios wih respec o he sall perrbaio e ( y ( y (, oe ay obai: d ( e &( + ωe ( K e ( τ + K de ( τ/ d + (, (25 P D where: /2, ω 2 K p 3, KD 4, -feedforward corol. lso, all iiial vales are zeros. Irodcig /2 /2 ( e(, 2( d e(/ d,oe ay wrie (25 i T (, :, ( he sae-space for, 2 /2 ( D τ ( + + ( ( τ, (26 wih a associaed fcio of he iiial sae: ( ψ (,

13 LZREVIĆ.M.: STBILITY ND STBILIZTION OF FRTIONL ORDER TIME DELY SYSTEMS 43 τ. Now, we ca check he fiie ie sabiliy wr, J {, 2 }, δ., ε, τ., where { } (, [.,] ψ. Fro he iiial daa ad Eq.(26 i yields: ψ ( <., σa ( 2, (27 σ 5, σ 7 a a, pplyig he codiio of Theore 3 (6, oe ca ge:.5 7Te ( E Te +.5 Te + /. Te. s..886 T e beig esiaed ie of fiie ie sabiliy. (28 oclsio I his paper, soe basic resls of he sabiliy crieria of fracioal order syses wih ie delay as well as free delay are preseed. We have eployed he classical ad he geeralizaio of Growall Bela lea o obai fiie ie sabiliy ad sabilizaio crieria for a proposed class of ie delay syses. I addiio, we preseed soe basic resls cocerig he sabiliy of fracioal order ie delay syses as well as free delay syses. Fially, a erical eaple is give o illsrae he validiy of he proposed procedre. ckowledgee. This work is parially sppored by he EUREK progra- E!493 ad he Miisry of Sciece ad Eviroeal Proecio of he Repblic of Serbia as Fdaeal Research Proec 46 ad 356. Miag-Leffler Fcio ppedi Siilar o he epoeial fcio freqely sed i he solios of ieger-order syses, a fcio freqely sed i he solios of fracioal-order syses is he Miag-Leffler fcio defied as E ( z k z, ( Γ + k ( k where > ad z. The Miag-Leffler fcio wih wo paraeers appears os freqely ad has he followig for E β, ( z k z, (2 Γ + k ( k β where >, β >, ad z. For β we obai, ad E z E z E z e, Lea (Growall Ieqaliy. Sppose ha g( ad ϕ are coios i,, g, λ ad r are wo cosas. If [ ] ( z he g s s r ds (3 ϕ( λ+ [ ϕ + ] ϕ( ( λ+ r( ep [ g( s ] ds, (4 Theore ([28] Geeralized Growall ieqaliy Sppose (, a( are oegaive ad local iegrable o < T, soe T +, ad g( is a oegaive, odecreasig coios fcio defied o < T, g M cos, > wih (5 ( a( + g( s ( s ds o his ierval. The ( g ( Γ ( a( + ( s a( s ds, Γ (6 < T orollary 2. of (Theore [28] Uder he hypohesis of Theore 2.2, le a ( be a odecreasig fcio o (,T. The holds: ( ( ( ( a E g Γ (7 Refereces [] ZVREI,M., JMSHIDI,M.: (987, Tie-Delay Syses: alysis, Opiizaio ad pplicaios, Norh-Hollad, serda. [2] GOREKI,H., FUKS,S., GRBOWSKI,P., KORYTOWSKI,.: (989, alysis ad Syhesis of Tie Delay Syses, Joh Wiley ad Sos, PWN-Polish Scieific Pblishers-Warszawa. [3] BELLMN,R., OOKE,K.L.: Differeial-Differece Eqaios, cadeic Press, New York, 963. [4] LEE,T.N., DINTT.S.: (98, Sabiliy of Tie Delay Syses, IEEE Tras. oa. o. 3( [5] MORI,T.: (985, rieria for sypoic Sabiliy of Liear Tie Delay Syses, IEEE Tras. oa. orol, -3, [6] HMMED,.: (986, O he Sabiliy of Tie Delay Syses: New Resls, I. J. orol 43, (, [7] HEN,J., XU,D., SHFI,B.: (995, O Sfficie odiios for Sabiliy Idepede of Delay, IEEE Tras. oa orol - 4 ( [8] L SLLE, LEFSHET,S.: Sabiliy by Lyapov s Direc Mehod, cadeic Press, New York, 96. [9] WEISS,L., INFNTE,F.: (965, O he Sabiliy of Syses Defied over Fiie Tie Ierval. Proc. Naioal cad. Sciece 54( [] GRUJIĆ,L.T.: (975a, No-Lyapov Sabiliy alysis of Large- Scale Syses o Tie-Varyig Ses. I. J. orol 2(3,4-45. [] GRUJIĆ,L.T.: (975b, Pracical Sabiliy wih Selig Tie o oposie Syses, oaika, T.P. 9,. [2] LSHIRER,.M., STORY..: (972, Fial-Sabiliy wih pplicaios, J. Is. Mah. ppl., 9, 379-4,972. [3] LM,L., WEISS,L.: (974, Fiie Tie Sabiliy wih Respec o Tie-Varyig Ses, J. Frakli Is., 9, [4] MTO,F., RIOL,M., DORTO,P.: Fiie-ie corol of liear sbec o paraeric ceraiies ad disrbaces,

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