Fractional Order PID Controller Tuning by Frequency Loop-Shaping: Analysis and Applications

Transcription

1 Fractonal Order PID ontroller Tunng by Frequency oop-shapng: Analy and Applcaton hald Saleh 1, Mohammad T. Haweel,* Department of Electrcal Engneerng, Shaqra Unverty, P.O , Dawadm, Ar Ryadh, SA. orrepondng author Abtract The purpoe of th paper to compute the parameter of the fractonal Proportonal Integral Dervatve (PID) controller to acheve a cloed loop entvty bandwdth approxmately equal to a dered bandwdth ung frequency loop hapng technque. Th work provde the nfluence of the order of a fractonal ntegrator whch ued a a target on loop hapng, tablty and robutne performance. A comparon between clacal PID controller and fractonal PID controller preented. Some advantage of fractonal PID controller over clacal PID controller have been dcued. The propoed model ued n th work are to compute the clacal PID parameter. The novelty n th work to develop a generalzed mathematcal form for the fractonal PID. eyword: Mttag-lffler functon, the wrght and manard functon, fractonal dervatve, fractonal pd controller, complementary entvty functon. I. INTRODUTION It known that the ue of Proportonal Integral Dervatve (PID) controller began n the thrte of the lat century. The degn of thee controller were baed on mathematcal and phycal background. Scentfc development n cence and engneerng ha helped to develop the degn of thee controller. PID controller are commonly ued n proce control ndutre becaue of ther performance and advantageou cot. The frt mathematcal veron of the fractonal PID controller wa ntroduced by the profeor Igor Podlubny, twenty year ago, pecfcally n 1994 [1-3]. The dea of developng clac PID controller of three parameter nto fractonal PID controller of fve parameter began n the nnete of the lat century. It known that the parameter of a clac PID controller are proportonal, ntegral, and dervatve gan.the addtonal parameter n the fractonal PID controller repreent the fractonal ntegral order and the fractonal dfferental order[4-6]. Many optmzaton technque have been ued to fnd the bet parameter n the degn. A lot of tude were made to mprove tunng technque for fractonal PID controller. oop hapng technque ha been developed n degnng to allow PID controller to acheve a cloed loop entvty bandwdth approxmately equal to the dered bandwdth. The dea of th technque to fnd the parameter of the controller by aumng that the PID controller and the plant n a feedback ytem behave a a certan target n the frequency doman. In other word, obtanng the hape of the open loop n the frequency doman mlar to that of the target an objectve [5, 7]. Therefore, the magntude of the target n the frequency doman can be condered a the hape of the open loop whch cont of the controller and plant together. Fndng the controller parameter to hape the loop and meet performance pecfcaton and acheve derable robutne properte are the objectve of degnng the controller. The parameter of the fractonal PID can be obtaned ung frequency loop hapng technque by mnmzng the fttng error functon for dfferent value of the fractonal ntegral order and the fractonal dfferental order [3-5, 8, 9]. There are ome many phycal phenomena that can be repreented by fractonal dfferental equaton. For example, the relatonhp between the current and voltage of a emnfnte loy R tranmon lne gven by a fractonal dfferental equaton. Another example, the relatonhp between the temperature and heat flux n a em-nfnte compote body can be decrbed by a fractonal dfferental equaton. The fractonal calculu theory ha been appled to control theore to mprove control ytem performance. The fractonal calculu appeared n the year 1659, a a reult of a queton raed by bntz n a letter to Hoptal, generalzng dervatve wth non-nteger order cannot gve the meanng of dervatve wth nteger order [10-1]. A queton raed and repled by Hoptal: f the order wa half what wll be?. ebntz anwered n a htorc way, a paradox wll happen whch wll lead to good reult. In the dertaton, the frequency loop hapng technque appled to three dfferent fractonal ytem. The frt ytem the heatng furnace ytem whoe tranfer functon can be obtaned a a fractonal tranfer functon after ung the Grnwald-etnkov defnton. The econd ytem decrbe the moton of a rgd thn plate mmered n a Newtonan vcou flud [13]. The fractonal tranfer functon of the ytem gven by the Bagely-Torvk equaton. The lat ytem Buck converter whch a D to D power converter. The relatonhp between the nput voltage and output voltage can be gven by a fractonal tranfer functon [14, 15]. In th work, Fractonal PID controller wll be appled to the three model n order to obtan good reult and compare them clacal controller. Reult have hown great mprovement compared regardng to entvty and model executon tme. 33

2 II. FRATIONA ORDER SYSTEMS MATHEMATIA MODES Th ecton contan the mathematcal model of the three propoed fractonal PID. The frt furnace temperature control and the econd the moton of an mmered plate n a vcou Newtonan flud. The thrd the Buck converter crcut. All mathematcal equaton wth ther fractonal tranfer functon are found n the lted reference. A. Heatng Furnace Fuel uppled n the form of coal and ron ore contnuouly from the top of the furnace. The ar whch powered by extra oxygen blown nto the bottom of the furnace, o that the chemcal reacton occur nde the furnace a hown n Fgure B. Moton of an Immered plate The moton of a rgd plate mmered n a vcou Newtonan flud can be decrbed by a fractonal dfferental equaton. The ytem cont of a thn rgd plate of ma (M) n a Newtonan flud wth denty (ρ) and vcoelatc contant (µ) connected by a prng of tffne contant () a hown n Fgure.. (5) Fgure. : A rgd plate mmered n a vcou Newtonan flud Fgure 1: Mechanm of heatng furnace ytem The heatng furnace can be expreed mathematcally a the followng dfferental equaton F m x b x kx The equaton above how that the total force the um of ndvdual force caued by ma (m), dampng (b) and prng (k) element. The dfferental equaton of the heatng furnace ytem gvng by [16] F x 4893 x1. 93x It can be expreed n the aplace doman a the followng tranfer functon G The fractonal order model of heatng furnace ytem can be evaluated ung Grunwald-etnkov defnton a (1) () (3) G 1 FOM (4) Where G FOM() repreent the tranfer functon of the plant n the fractonal order ytem. The fractonal order controller wa obtaned ung Nelder-Mead optmzaton technque a [16] The dplacement x(t) of the plate due to an external force f (t) n a Newtonan flud ytem can be decrbed by the followng fractonal dfferental equaton [17-0] 3 M x t 0 t f t xt D xt By takng the aplace tranform of equaton 6, we have [1-4] F X X M X 0 3 (6) (7) Therefore, the tranfer functon of the ytem gven by G M (8) omparng between equaton 8 and the tranfer functon gven wth unty tffne contant [4-9] G n It can be deduced that 1 n m 1 1 n (10) M (9) 34

3 0 n 3 m for (11) The analytcal oluton x(t) for the nhomogeneou Bagley- Torvk equaton n (6) can be calculated a [4, 9] where h and E t x 1 M t t ht f n0 0 d (1) n n 1 k n1 n n! t M E 1 3n, M n 3, 1, n t n m r n t (13) r n! t m (14) 0,1,, r! r n. The Buck onverter The Buck crcut a D to D converter. It ued to tep down the nput voltage to a lower output voltage [30-33]. The buck network contan a voltage ource( v n), wtch (S), fractonal nductor ( α), fractonal capactor( β), Dode (D), and the load (R) whch ha the output voltage(v o)a hown n Fgure 3. Fgure 3: Buck converter crcut r The voltage and current n the fractonal order nductor ( α), are related a [34, 35] v t d dt Takng aplace tranform, v t (15) (16) Smlarly, the fractonal order capactor can be repreented by a chan of retor and a chan of capactor n parallel combnaton at each node a hown n Fgure. 5. Fgure 5: Fractonal order capactor The current acro a fractonal order capactor ( β), gven by [36-38] Takng aplace tranform, t dvo t (17) t dt vo (18) The tranfer functon from the nput voltage to the output voltage(v o/v n) of the crcut can be obtaned by replacng the wtch and the dode by a current-dependent current ource and voltage-dependent current ource a hown n Fgure. 6 whch repreent the crcut averaged model of the Buck crcut [38-41]. The fractonal order nductor contan a chan of retor and a chan of capactor n parallel combnaton at each node connected wth a ere retor a hown n Fgure. 4. Fgure. 6: rcut averaged model of the Buck converter Fgure. 4: Fractonal order nductor Therefore S v D d d v n (19) (0) where (d) the duty cycle wth a value between 0 and 1. The 35

4 D equvalent crcut of Fgure can be obtaned by replacng the nductor wth a hort crcut and replacng the capactor wth an open crcut a hown n Fgure. 7. The crcut n Fgure.8 can be reduced to the followng crcut n Fgure 9. Fgure 9: The mall-gnal equvalent crcut model of the Buck converter crcut Therefore Fgure 7. The D equvalent crcut Vo DV (1) n D R I V n () Alo, the mall-gnal (A-gnal) equvalent crcut model can be obtaned from the followng relatonhp a v o v n I V o V n v o v n (3) (4) (5) d D d (6) Subttutng equaton (3), (4), (5) and (6) nto (1) and (), yeld S v S DI DI d I dv D Dv (7) (8) Equaton (7) and (8) repreent the ummaton of the D and A gnal. Therefore, the mall-gnal equvalent crcut model can be degned a hown n Fgure 8. The relatonhp between the nput voltage and the output voltage can be determned ung rchhoff current and voltage law a Dv v n o (9) vo (30) vo R vo The tranfer functon v can be ealy obtaned by n olvng the prevou two fractonal order equaton. Therefore, v o v n 1 D R (31) whch repreent the tranfer functon of the open loop Buck converter. The followng parameter are ued a n [7]. 100 F 3 mh R 0 D 0.6 III. RESUTS AND DISUSSION In th ecton the analy for the three propoed model are preented and evaluated ung MATAB. Th baed on the mathematcal model dcued n ecton II. Heatng Furnace PID Reult and dcuon Fgure 8: The mall-gnal equvalent crcut 36

5 Fgure 10: The tep repone of the uncompenated ytem Fgure 1: The fttng error at (λ=0.6, µ=0.1) Fgure 10 how the tep repone of the uncompenated ytem. It can be een that the uncompenated ytem table and ha a teady tate repone of (0.4) n repone. The bandwdth of the dered ytem around ( ) rad/ec. A fractonal PID controller can be degned to elmnate the teady tate error and mprove the ytem tranent. The target can be choen to have the dered bandwdth a (3) The obtaned reult would be a follow The dtrbuton of the fttng error all over the range of the fractonal order hown n Fgure 11 From Fgure 1, the bet choce for the order of the fractonal PID controller would be at (λ=0.6, µ=0.1). It atfe the relatonhp mn Ub. p,, d,,, The value of the fractonal PID parameter are U p d b The nformaton above how that the fractonal PID behave a a fractonal PI controller. Fgure 11: The fttng error (U b) wth fractonal order (λ, µ) Fgure 13: The target loop and plant 37

6 The frequency repone and the tep repone of the ytem are hown n Fgure 16 and Fgure 17, repectvely Fgure 14: The approxmaton error (fttng error) The obtaned approxmaton error ( ) at the bandwdth frequency whch le than (0.3). Fgure 14 how that the approxmaton error decreae at hgh frequence. Fgure 17: The tep repone of the ytem The fractonal ntegrator (1/ 0.6) need addng a pure ntegrator and dfferentator at the orgn to elmnate the teady tate error a hown n Fgure 18. Fgure 19 how that dturbance at the plant nput ha been elmnated for the ame reaon. Fgure 15: The entvty and complementary entvty functon Fgure 15 how that the cloed loop tranfer functon ha good noe rejecton at hgh frequence and the dturbance at low frequence wll not affect the output. Alo, t can be een that the bandwdth of the cloed loop entvty approxmately around the dered bandwdth ( ) rad/ec. Fgure 16: The frequency repone of the ytem Fgure 18: The dturbance rejecton repone Therefore, the fnal form of the fractonal PID Or, equvalently (33) (34) hoong (λ=1, µ=1) wth the ame target gve a clacal PID controller wth the followng reult p U d b

7 The nformaton above how that clacal PID controller behave a a PI controller. The fttng error lghtly hgher than the typcal value (0.-0.3). The followng Fgure how a comparon between the fractonal PID controller and the clacal PID controller. Fgure 1 how that the fractonal PID controller ha elmnated the teady tate error fater than the clacal PID controller depte the obtaned overhoot hgher than the overhoot caued by the clacal PID. Fgure how that dturbance rejecton due to the fractonal PID controller fater than the ued clacal PID. Fgure 19: The approxmaton error (fttng error) Fgure : The dturbance rejecton repone The fractonal controller n [16] ha the followng form (35) The compared reult between the fractonal controller obtaned by the frequency loop hapng and the fractonal controller n [16] are hown n the followng fgure Fgure 0: The entvty and complementary entvty functon Fgure 0 how that the fractonal PID and the clacal PID acheved the cloed loop entvty bandwdth approxmately equal to the dered value. Fgure 3: The entvty and complementary entvty functon Fgure 3 how that the fractonal PID and the fractonal PID n Reference [16, 35-40] acheved the cloed loop entvty bandwdth approxmately equal to the dered value. Fgure 1: The tep repone 39

8 It can be een that the fractonal PID ung the frequency loop hapng ha rejected the dturbance fater than one ued n [16]. Smlarly, the followng target can be choen to acheve the dered degn Fgure 4: The tep repone where (α=bw/4) a The obtaned reult would be a follow (36) The dtrbuton of the fttng error all over the range of the fractonal order hown n Fgure 7. Fgure 4 how both of tep repone of the two ytem have no teady tate error. Fgure 5 how that dturbance rejecton due to the fractonal PID ung the frequency loop hapng fater than the ued fractonal PID n Reference [16, 35-40]. The fractonal controller hould contan a pure ntegrator to reject the dturbance at the nput plant. Fgure 7: The fttng error (U b) wth fractonal order (λ, µ) Fgure 5: The dturbance rejecton repone Therefore, an ntegrator and dfferentator are added at the orgn to the fractonal controller [16, 33-36] to elmnate the dturbance a hown n Fgure 6. Fgure 8: The fttng error at (λ=1.1 and µ=0.1) Fgure 6: The dturbance rejecton repone From Fgure 8, the bet choce for the order of the fractonal PID controller would be (λ=1.1 and µ=1.7). It atfe the relatonhp mn Ub. p,, d,,, 40

9 The value of the fractonal PID parameter are p d U b 0.99 The nformaton above how that the fractonal PID behave a a fractonal PD controller a hown n Fgure 9. Fgure 34: The entvty and complementary entvty functon Fgure 34 how that the cloed loop tranfer functon ha good noe rejecton at hgh frequence and the dturbance at low frequence wll not affect the output. Alo, t can be een that the bandwdth of the cloed loop entvty approxmately around the dered bandwdth ( ) rad/ec. Fgure 9: The target loop and plant Fgure 35: The frequency repone of the ytem The frequency repone and the tep repone of the ytem are hown n Fgure 35 and Fgure 36, repectvely Fgure 33: The approxmaton error (fttng error) The obtaned approxmaton error ( 0.99) at the bandwdth frequency whch n between (0.-0.3). Fgure 33 how that the approxmaton error decreae at hgh frequence. Fgure 36: The tep repone of the ytem 41

10 It can be een that the fractonal ntegrator (1/S 1.1 ) whch contan a pure ntegrator (1/) elmnate the teady tate error. Fgure 37 how that dturbance at the plant nput ha been elmnated for the ame reaon. Fgure 37: The dturbance rejecton repone Therefore, the fnal form of the fractonal PID (37) hoong (λ=1, µ=1) wth the ame target gve a clacal PID controller wth the followng reult U p d b The fttng error greater than the typcal value (.-0.3). The followng fgure (fgure 38) how a comparon between the fractonal PID controller and the clacal PID controller. Fgure 39: The entvty and complementary entvty functon Fgure 39 how that reonant peak due to the clacal PID controller hgher than the reonant peak due to the clacal PID. Fgure 40: The tep repone Fgure 40 how that the fractonal PID controller ha mproved the tranent of the ytem wth a reducton of overhoot and ocllaton are obtaned. Fgure 41 how that dturbance rejecton repone are elmnated a tme goe to nfnty. Fgure 38: The approxmaton error (fttng error) Fgure 41: The dturbance rejecton repone 4

11 The compared reult between the fractonal controller obtaned by the frequency loop hapng and the fractonal controller n [16, 35-40] are hown n Fgure 4 Fgure 4: The entvty and complementary entvty functon It expected that fractonal PID ung the frequency loop hapng ha a hgh reonant peak due to the value of the fttng error whch around(0.3). Therefore, an overhoot hould appear n the tep repone a hown n Fgure 43. Fgure 44: The dturbance rejecton repone The tranfer functon of the ytem wth the parameter value ued n [9] 1 G (38) Fgure 45 how the tep repone of the uncompenated ytem. Fgure 43: The tep repone Fgure 43 how both of the tep repone of the two ytem have no teady tate error. Fgure 44 how that dturbance rejecton due to the fractonal PID ung the frequency loop hapng fater than the ued fractonal PID n [16]. The fractonal controller hould contan a pure ntegrator to reject the dturbance at the nput plant. The ame fgure how the effect of addng a pure ntegrator and dfferentator at the orgn to the fractonal controller ued n [16]. Fgure 45: The tep repone of the uncompenated ytem It can be een that the uncompenated ytem ocllatory. The bandwdth of the ytem around (3.553) rad/ec. The fractonal PID controller for the degn expected to reduce the overhoot and reach the tablty fater than the uncompenated ytem. The target can be choen to have the dered bandwdth a 3.37 (39) 1. 4 The obtaned reult would be a follow: The dtrbuton of the fttng error all over the range of the fractonal order hown n Fgure 46 43

12 Fgure 46: The fttng error (U b)wth fractonal order (λ, µ) Fgure 48: The target loop and plant Fgure 47: The fttng error at (λ=1.4, µ=0.5) From Fgure 47, the bet choce for the order of the fractonal PID controller would be(λ=1.4, µ=0.5). It atfe the relatonhp mn Ub. p,, d,,, Fgure 49: The approxmaton error (fttng error) The obtaned approxmaton error ( 0.035) at the bandwdth frequency whch le than(0.-0.3). Fgure 49 how that the approxmaton error decreae at hgh frequence. The value of the fractonal PID controller p d Fgure 50: The entvty and complementary entvty functon 44

13 Fgure 50 how that the cloed loop tranfer functon ha good noe rejecton at hgh frequence and the dturbance at low frequence wll not affect the output. Alo, t can be een that the bandwdth of the cloed loop entvty approxmately around the dered bandwdth (.7144*10 4 ) rad/ec. Fgure 51: The frequency repone of the ytem The frequency repone of and the tep repone of the ytem are hown n Fgure 51 and Fgure 5, repectvely Therefore, the fnal form of the fractonal PID (40) hoong (λ=1, µ=1) wth the ame target gve a clacal PID controller wth the followng reult U p d b The nformaton above how that clacal PID controller behave a a pure ntegrator. The fttng error greater than the typcal value ( ). Fgure 54 how a comparon between the fractonal PID controller and the clacal PID controller. Fgure 5: The tep repone of the ytem It can be een that the fractonal ntegrator (1/S 1.4 ) whch contan a pure ntegrator (1/S) elmnate the teady tate error. Fgure 53 how that dturbance at the plant nput ha been elmnated for the ame reaon. Fgure 54: The approxmaton error (fttng error) Fgure 53: The dturbance rejecton repone Fgure 55: The entvty and complementary entvty functon Fgure 55 how that reonant peak due to the clacal PID 45

14 controller hgher than the reonant peak due to the fractonal PID. Defntely, the fractonal PID meet the dered degn. Therefore, fractonal the tranfer functon of the ytem wth the parameter value ued n [7] would be v o (43) v Fgure 56: The tep repone Fgure 56 how that the fractonal PID controller ha mproved the tranent of the ytem wth a reducton of overhoot and ocllaton are obtaned. Fgure 57 how that dturbance rejecton due to the fractonal PID controller fater than the ued clacal PID. Fgure 58: The tep repone of the uncompenated ytem Fgure 58 how the tep repone of the uncompenated ytem. It can be een that the uncompenated ytem table and ha overhoot. The bandwdth of the ytem around (.7144*104) rad/ec. A fractonal PID controller can be degned to reduce the overhoot and mprove the ytem tranent. The target can be choen to have the dered bandwdth a (44) The obtaned reult would be a follow: The dtrbuton of the fttng error all over the range of the fractonal order hown n Fgure 59. Fgure 57: The dturbance rejecton repone Snce (D) a contant, the fractonal tranfer functon of the ytem can be wrtten a v o v 1 Where v v 1 n R D (41) (4) Fgure 59: The fttng error (U b) wth fractonal order (λ,µ) 46

15 Fgure 6: The approxmaton error (fttng error) Fgure 60: The fttng error at (λ=1.1, µ=1.7). The obtaned approxmaton error ( 0.038) at the bandwdth frequency whch le than (0.-0.3). Fgure 61 and 6 how that the approxmaton error decreae at hgh frequence. From Fgure 60, the bet choce for the order of the fractonal PID controller would be (λ=1.1, µ=1.7). It atfe the relatonhp mn Ub. p,, d,,, The value of the fractonal PID parameter are p d U b The nformaton above how that the fractonal PID behave a a fractonal PI controller. Fgure 63: The entvty and complementary entvty functon Fgure 63 how that the cloed loop tranfer functon ha good noe rejecton at hgh frequence and the dturbance at low frequence wll not affect the output. Alo, t can be een that the bandwdth of the cloed loop entvty approxmately around the dered bandwdth (.7144*10 4 ) rad/ec. Fgure 61: The target loop and plant Fgure 64: The frequency repone of the ytem 47

16 The frequency repone and the tep repone of the ytem are hown n Fgure 64 and Fgure 65, repectvely typcal value (0.-0.3). fgure 67 how a comparon between the fractonal PID controller and the clacal PID controller. Fgure 65: The tep repone of the ytem It can be een that the fractonal ntegrator(1/s 1.1 ) whch contan a pure ntegrator (1/S) elmnate the teady tate error. Fgure 66 how that dturbance at the plant nput ha been elmnated for the ame reaon. Fgure 67: The approxmaton error (fttng error) Fgure 68: The entvty and complementary entvty functon Fgure 66: The dturbance rejecton repone Therefore, the fnal form of the fractonal PID (45) hoong (λ=1 and µ=1) wth the ame target gve a clacal PID controller wth the followng reult U p d b The nformaton above how that clacal PID controller behave a a pure ntegrator. The fttng error greater than the Fgure 68 how that reonant peak due to the clacal PID controller hgher than the reonant peak due to the clacal PID. Fgure 69: The tep repone 48

17 Fgure 69 how that the fractonal PID controller ha mproved the tranent of the ytem wth a reducton of overhoot and ocllaton are obtaned. Fgure 70 how that dturbance rejecton repone are elmnated a tme goe to nfnty. Fgure 70: The dturbance rejecton repone The tep repone of the ytem when the duty cycle (0.6) can be obtaned a hown n Fgure 71. It ha the followng fractonal tranfer a n [7]. v o v n (46) ued n the frequency repone to decrbe fractonal order ytem. Fractonal PID controller ha ome advantage over clacal PID controller lke ncreang tablty a hown n Remann urface plan, mprovng the performance of fractonal ytem and more freedom to tune. Remann urface plan how that the repone of a fractonal order ntegrator whoe order le than one low. Th problem can be olved by addng a pure ntegrator and dfferentator at orgn. The effect of the fractonal order on the hape of fractonal tranfer functon preented. A target can be ued for the tunng to acheve a robut performance. hoong a fractonal ntegrator a a target n the frequency loop hapng technque ha ome advantage uch a rejectng dturbance. The fractonal order play an mportant role on entvty hapng. The entvty functon how how the dturbance goe through the ytem and appear on the output or the error gnal. The complementary entvty functon how how the noe wll be fltered through the ytem. The obtaned reult n th work how gnfcant mprovement n the tranent repone and atfactory reult through the achevement of a cloed loop entvty bandwdth approxmately to a dered value of bandwdth. Indeed, the tudy of Fractonal alculu, epecally mathematcal analy ha been very nteretng. The future work wll be focung on the followng pont: 1. Generatng the target loop hape ung QR oluton.. The mathematcal rule of the root locu n Remann prncpal heet for fractonal order ytem. 3. Studyng the complex order ntegral and dervatve and poblty of extendng fractonal PID to complex fractonal PID. VI. ONUSION Fgure 71: The tep repone Fractonal order calculu provde a good way to tudy the behavor of functon n the pat and future. The fractonal order tell how far the behavor of a functon from preent. Alo, fractonal order calculu can be ued to approxmate functon wthout ung a ub of functon for the ame purpoe. The Outaloup flter can be ued to approxmate fractonal order ytem although the hape around the boundary frequence not deal due the expreon of approxmaton. Intead of havng lope n multple of (±0dB/dec) or zero n the frequency repone of a certan ytem, lope n multple of (±0 α db/dec) or zero can be REFERENES [1] Zhang, X., & hen, Y.Q., Remark on fractonal order control ytem, In: Amercan ontrol onference (A), IEEE, pp , 01. [] aponetto, R., Dongola, G., Fortuna,., and Petra, I., Fractonal order ytem: Modelng and ontrol Applcaton, Sere A -Vol.7, World Scentfc Publhng o. Pte. td., 010. [3] Podlubny, I., Fractonal Dfferental Equaton, Academc Pre, San Dego, [4] Gra, E., & Takal,., PID ontoller Tunng by Frequency oop-shapng: Applcaton to Dffuon Furnace Temperature ontrol, IEEE Tranacton on ontrol Sytem Technology, Vol. 8, NO.5, September 000. [5] Jeu, I.S., & Tenrero Machado, J.A., Fractonal control of heat dffuon ytem, Nonlnear Dynamc (008) 54: [6] Malt, R., Moreau, X., hemane, F., Outaloup, A., Stablty and reonance condton of elementary fractonal tranfer functon, Automatca, Volume 47, Iue 11, pp ,

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