Active Sway Control of a Single Pendulum Gantry Crane System using Output-Delayed Feedback Control Technique

Transcription

1 21 11th. Int. Conf. Contro, Automation, Robotic and Viion Singapore, 5-8th December 21 Active Sway Contro of a Singe Penduum Gantry Crane Sytem uing Output-Deayed Feedback Contro echnique Rajeeb Dey, Nihant Sinha, Priyanka Chaubey Dept. of Eectrica & Eectronic Engineering, Sikkim Manipa Univerity, Sikkim, India. E-mai: rajeeb_de@ieee.org, ee: Abtract hi paper invetigate the impementation of output-deayed feedback contro (ODFC) technique for controing the way ange of inge penduum gantry crane (SPGC) ytem. Linearized mathematica mode of the SPGC in tate pace form i conidered for the invetigation. he deigned ODFC ha undergone compete tabiity anayi for a given controer gain. Keyword Anti-Sway contro, Output-deayed feedback contro (ODFC), LQR Contro, Singe penduum Gantry Crane (SPGC). I. INRODUCION A SPGC ytem i a crane carrying the cart with a movabe or fixed hoiting mechanim, they are required in modern indutria environment to tranport heavy payoad from one poition to another a fat and a accuratey a poibe with out coiion with other equipment. he baic motion of SPGC ytem invove crane travering, oad hoiting and oad owering. We conider an SPGC which i of fixed hoit mode. In cae of fat crane travering, a arge way of the hoiting mechanim take pace. he objective of thi work i to deign ODFC for controing way ange of the hoiting mechanim. hi contro probem i imiar to vibration contro probem deat in [1,4,6,9]. Finding contro method that wi eiminate vibration or ociation from wide range of phyica ytem i of interet for pat few decade [1], and one uch appication of vibration contro of indutria ignificance i way contro of gantry crane. he active vibration contro trategie for controing vibration in phyica tructure or ytem [1,2,4,6,8,1,9] i the principe ued here, thu caing it a active way contro. In [4,1] haped input contro method have been ued, thi method ha the effect of pacing zero at the ocation of the fexibe poe of the origina ytem, but being a feedforward contro trategy contro i not robut to externa diturbance. In [1,6] and reference there in, it i found that time-deay contro (DC) i another approach for active vibration (ociation) contro. he inuion of time-deay in the ytem dynamic make the ytem an infinite dimeniona [3,7,13-15] thu direct computation of the characteritic root and conequenty deciding about tabiity i a difficut tak. S. Ghoh and G. Ray Dept. of Eectrica Engineering, Nationa Intitute of echnoogy, Rourkea Oria, India. Dept. of Eectrica Engg., Indian Intitute of echnoogy, Kharagpur, Wet Benga, India. A detaied review of the reearch on time-deay tabiity and tabiization iue uing both frequency domain technique and time domain method can be found in [3,7,15]. he former technique for aeing the tabiity of DS can be found in the iterature [1,5,6,7,9,11,12], thi technique provide exact tabiity anayi for time-invariant deay and hence DC ha been ued in many contro appication [1,6,9] and reference there in to uppre vibration or ociation of the ytem. he ater technique can treat both the nature of deay, time-variant and time-invariant, a numericay tractabe agorithm exit to ove the probem, but provide conervative anayi compared to the former technique [3,7,13,15]. In thi paper, to contro thi under-damped ytem a igna i derived from the poition enor which i then combined with the deayed output igna from the ame enor and fed back to the ytem, thu caing it output-deayed feedback controer (ODFC). hi deign invove priori knowedge of the controer gain for which the time-deay i treated a deign parameter. he frequency domain technique of [11] i adopted for thi deign to compute the deay time for a pre-eected gain vaue. II. DYNAMIC MODEL OF SPGC he two-dimeniona inge penduum gantry crane ytem with it payoad conidered in thi work i hown in the Fig.1. he payoad i upended from the point of upenion S, which denote the centre of gravity of the cart. he downward vertica poition of the payoad i taken a reference poition. he centre point G denote the centre of gravity of the payoad and the direction of the veocity of the payoad with it component in X and Y Carteian coordinate are repreented in the Fig.1. F x repreent the force cauing tranationa motion of the crane. he nomenature aong with the vaue of the phyica parameter are given in abe 1. he dynamic mode of SPGC can be found in [2,8]. Foowing impification appy in thi mode, (i) Mode doe not inude hoiting drive, thu rod ength i fixed (ii) troey or payoad i aumed to be point ma (iii) troey and payoad aumed to move in X-Y pane and (iv) force on troey due to penduum wing i negected /1/$ IEEE ICARCV21 532

2 FIG.1: SPGC Mode he non-inear dynamic mode of the gantry crane uing Euer-Lagrange formuation with above impification yied 2 Fx Beqx = ( M + m) x+ m[ θ coθ θ in θ] + 2m θcoθ + m (1) inθ B pθ = θ + 2 θ + xcoθ + ginθ (2) Auming ma way ange θ, an approximate inear dynamic mode for (1)-(2) can be repreented in tate pace form a Xt () = AXt () + But () (3) yt () = Cxt () 41 where Xt () R i the tate vector, ut ( ) R i the contro input, yt R ( ) i the output of the ytem, Xt () = [ x θ x θ ] and the matrice A and B are given by 1 1 mg Beq mb p A = M M M ( M + m) g Beq ( M + m) B p M M M B 1 = M 1 M In order to ue deayed feedback contro method, we choe to feedback the way ange θ of the rod thu we choe C = [ 1 ]. he pair ( A, B) i found to be controabe. III. DESIGN OF OUPU-DELAYED FEEDBACK CONROLLER (ODFC) In thi ection, we expain the controer deign foowing the technique in [11]. he contro aw adopted for thi controer i mathematicay given by ut () = K[ yt () yt ( )] (4) Uing (4) the oed oop ytem dynamic of (3) can be written a Xt () = AXt () + AXt 1 ( ) (5) where, A = A+ BKC and A1 = BKC. he characteritic equation of (5) i a trancedenta equation and one can write it a I A A1e = (6) Equation (6) have infinite number of characteritic root due to preence of the deay term in (5). o carry out the tabiity anayi of uch time-deay ytem evera appraoche have evoved in pat a found in the iterature. Frequency domain technique give exact tabiity anayi and invove finding root of (6) and are dicued in [1,9,11,12], wherea time-doamin technique do not invove actuay computing root of (6) and hence provide conervative tabiity reut [3,7,15]. We adopt the exact tabiity anayi of [11] for the deign of ODFC. he impementation of the technique for the tabiity anayi of SPGC under deayed feedback contro i preented tructuray in the form of agorithmic tep. A. Agorithm: he characteritic equation in (6) i written in the genereic form a n Δ (, ) = p () e (7) = Finding the compete root croing tructure for (7) 1 uing the Rekaiu ubtitution for e = term to 1 + convert it into reuting poynomia without trancedentaity, which take the form 2n b = (8) = where, b = b (, p, b ), p, b,1 i, j n being the ij ij ij ij eement of A and B matrice. We appy Routh-Hurwitz criterion on (8), we determine et of vaue i.e, { c }, by equating the eement of 1 row of Routh array to zero, and for each vaue when the auxiiary equation (which i formed by the row preceeding 1 in the Routh array) i oved it give either a pair of imaginary root ( i ) or rea and equa root with oppoite ign. A, we are intereted ony in the imaginary croing frequencie o we oveook thoe vaue that give rea root. he computed vaue of and are paced in abe

3 For each (or correponding ) there are infinitey many time deay { }, which one can obtain uing 2 [tan 1 ( ) k ], k,1, 2... = π = So, for the ytem in (5) one can get finite number of purey imaginary root { }, = 1, 2,3... m but infinitey many time deay i.e, { k} with k =,1, he computed deay vaue are paced in abe 3 for K = 1 and K = 2. he characteritic root of ytem in (5) croe the imaginary axi at { } vaue for infinitey many timedeay { k}, that i computed a decribed above. he tabiity region or witche are found by computing the root tendencie (R) (or root croing direction) at the point of correponding deay k by the foowing equation from R = n ' j pe j j= = gn Im n = k j jp je j= = i = k (9) he vaue of R wi be +1 or -1, if it come out to be +1 NU increae by 2, if it i -1 it decreae by 2. he computation of NU can be done uing equtaion (2) in [11]. Now for finding the deay range (or tabiity witche) for the conidered ytem, we rearrange the computed k vaue in an acending order and check for NU by ooking at the R correponding to deay vaue. he compete information about the tabiity witche for K = 1 and K = 2 are paced in abe 4. he neceary condition to be fufied i that at = the ytem mut be Hurwitz. Remark 1: he advantage of thi technique i that it can dea with ytem having commenurate deay ao, but method expained in [1] cannot hande cae of commenuarte deay. IV. SIMULAION RESULS he oed oop imuation for the ytem in (3) i carried uing MALAB for evera vaue of time-deay and for a given gain. he initia condition for a the imuation i taken to be x () = [ 1.5 ]. he open oop imuation for the way ange i hown in Fig.2. Fig.3 how the output repone for the SPGC under ODFC with K = 1. hree different deay vaue are conidered in the firt tabiity region i.e, < <.498, (i) =.3 [,.489) for which the ytem i tabe (ii) =.489, the ytem i marginay tabe and (iii) =.52 [.49,1.46), the ytem i untabe. Fig.4 how the repone of the ytem for the tabiity region between 1.46< < Fig.5 how the comparion of the output repone between LQR and ODFC. he Q matrix for the LQR controer i choen to be I 4 4and R = 1, the LQR optima gain obtained for the ytem i K = [ ]. For ODFC deign, the controer parameter i choen to be K = 1, =.3ec, thu the contro aw i ut ( ) = 1[ θ( t) θ( t.3)]. ABLE 1: PHYSICAL PARAMEERS OF SPGC Ma of the cart M = 1.731Kg Ma of payoad m =.23Kg Acceeration due to gravity 2 g = 9.81 m/ Vicou damping coefficient, Bp =.24 Nm / rad een from penduum axi Equivaent vicou damping Beq = 5.4 Nm / rad coefficient of the crane aong X-axi Penduum ength from the =.332meter point of upenion to CG ABLE 2: AND VALUES FOR VARIOUS GAINS. K = 1 K = 2 =.169 = =.81 = =.1739 = =.1889 = = = ABLE 3: DELAY VALUES FOR GIVEN AND K = 1 K = 2 = = = = = = : : : : : : : :

4 Remark 2: here i no croing frequency correponding to c3 =.2265 for K = 1 and c3 =.2237 for K = 2, a it yied rea and equa root of oppoite ign and hence negected. Remark 3: We oberve from abe 4 the ytem i tabe in the range and again for 1.46< < After =1.95 the ytem become untabe and the tabiity i never regained. Simiary from abe 5 the ytem i tabe in the range < <.3543, at =.3543 the ytem become marginay tabe and for >.3543 the ytem become untabe. ABLE 4: SABILIY SWICHES (OR REGIONS) K=1 FIG. 2: OPEN LOOP SIMULAION OF SPGC SYSEM Critica ime Deay (ec) k Imaginary Root ( =± i), ck Root endency R Number of Untabe Root (NU) S-NU= S-NU= FIG. 3: SWAY ANGLE FOR K = 1, (I) =.3 [,.498), (II) =.498 AND (III) =.52 >.498. ABLE 5: SABILIY SWICHES (OR REGIONS) K=2 Critica ime Deay Imaginary Root ( =± i), Root endency R Number of Untabe Root (NU) S-NU= FIG. 4: SWAY ANGLE FOR K = 1, (I) = 1.14 [1.46,1.224), (II) = AND (III) = 1.26 >

5 FIG. 5: COMPARISON BEWEEN LQR & ODFC WIH K = 1 AND =.3ec. V. CONCLUSIONS In thi work, impementation of the ODFC deign for controing the way ange of the inearized SPGC ytem uing the exact tabiity anayi in [11] i preented for the firt time. hi anayi aowed u to find out the tabiity region for different deay range with a pre-eected vaue of controer gain. he method of finding tabiity witche for time-deay ytem uing thi method i much more convenient and tructured than that preented in [1] and reference there in. One can oberve that the tabiity region determined uing the anayi for a given gain matche exacty with the imuation reut a hown in FIG.3 and 4. he tabiity region reduce graduay a the vaue of gain i increaed, thi fact can be oberved by comparing the region obtained for K = 1 and K = 2 in abe 4 and 5 repectivey. he ODFC deign i compared with the LQR and found that the former one i uperior in term of quaity of tranient repone. he impementation of thi ODFC i much imper a we need to feed back ony one tate information, whie in cae of LQR we are to feed back a the four tate thu needing four enor. [3] J.P Richard, ime-deay ytem: An overview of ome recent probem, Automatica, Vo 39, pp , 23. [4] J.M. Hyde,W.P.Seering, Uing input command pre-haping to upre mutipe mode vibration, IEEE Int. Conf. on Robotic and Automation, Sacramento, CA, Vo 3, pp , [5] J.E Marha, Contro of ime deay ytem. Stevenage, U.K: Peter Peregrinu, [6] K. V. Singh, B.N. Datta, M. ayagi, Zero aignment in vibration: with or without time-deay, Proceeding of ASME IDEC/CIE, La Vega, NV, Sep. 27. [7] K. Gu, V.L Kharitonov, J.Chen, Stabiity of time-deayed ytem, Boton, Birkhauer, 2. [8] Md. A. Ahmad, Active way uppreion technique of gantry crane ytem, European Journa of Scientific Reearch, Vo7, No. 4, pp , 29. [9] M. Rameh, S. Narayana, Controing Chaotic motion ina two dimeniona airfoi uing time-deay feedback, J. Sound Vibration, Vo. 239, No. 5, pp , 21. [1] M. Bodon, Experimenta comparion of two input haping method for contro of reonant ytem, IFAC word congre, San Francico, CA, [11] N. Ogac, R. Sipahi, An eaxct method for tabiity anayi of timedeayed inear time-invariant ytem, IEEE ranc. On Automatic Contro, Vo. 47, No. 5, pp , May 22. [12] N. Ogac, R. Sipahi, A new practica tabiity anayi method for timedeayed LI ytem, 3 rd IFAC workhop on DS (DS 21), Santa Fe, December 21. [13] R.Dey, G.Ray, S.Ghoh, A. Rakhit, Stabiity anayi for continou ytem with additive time-varying deay: A e conervative reut, App. Mathem. Comput., Vo. 215, pp , 21. [14] R. Dey, S.Ghoh, G.Ray, Deay-dependent tabiity anayi of inear ytem with mutipe tate deay, 2 nd IEEE,ICIIS, Srianka, pp , 27. [15] S. Xu, J. Lam, A urvey of Linear matrix inequaity technique in tabiity anayi of deayed ytem, Int. J. of Sytem Science, Vo. 39, No. 12, , Dec 28. ACKNOWLEDGMEN hi project work i upported by AICE. Govt. of India under reearch promotion cheme, vide grant no. 823/BOR/RID/RPS-229/ REFERENCES [1] A. Jnifene, Active vibrationa contro of fexibe tructure uing deayed poition feedback, Sytem and contro etter, Vo 56, pp , 27. [2] D.Liu,J.Yi,D.Zhao,W.Wang, Swing free tranporting of twodimeniona overhead crane uing iding mode Fuzzy contro, Proceeding of American Contro Conf., Boton, ,