Edwar Yazid 1. Indonesian Institute of Sciences Abstract

Transcription

1 Otimasi Gain engenali D Menggunakan Artificial Bee Colony untuk engenalian Sistem Kren Gantry Kaku Otimizing the Gains of D Controller Using Artificial Bee Colony for Controlling the Rigi Gantry Crane System Ewar Yazi 1 1 Research Center for Electrical ower an Mechatronics Inonesian Institute of Sciences ewar.utra@gmail.com Abstract Control osition an reuction of swinging of the ayloa of a rigi gantry crane system is a challenging work because of uner-actuate system. his aer aresses challenges by roosing the artificial bee colony (ABC) algorithm to otimize the gains of the D controller to form what the so-calle the artificial bee colony (ABC)-D controller. he effectiveness of the roose control algorithm is teste uner constant ste functions an comare with Ziegler-Nichols (ZN)-D controller. Simulation results show that the roose controller rouces slower rise time an eak time, but faster settling time than the ZN-D controller as well as no overshoot uner the reefine traectories. Keywors: Gantry crane system, swing angle, D gains, ABC Abstrak Kontrol osisi an reuksi ayunan muatan ari sistem kren gantry kaku meruakan sebuah ermasalahan yang menantang karena meruakan sistem yang uner-actuate. ulisan ini menyelesaikan tantangan tersebut engan mengusulkan algoritma artificial bee colony (ABC) untuk mengotimalkan gain engenali D untuk membentuk yang inamakan engenali D berbasis metoe ABC. Keefektifan engenali yang iusulkan iui lewat fungsi ste konstan an ibaningkan engan engenali D berbasis metoe Ziegler-Nichols (ZN). Hasil-hasil simulasi menunukkan bahwa engenali yang iusulkan menghasilkan waktu naik an waktu uncak yang lebih lambat tetai waktu turun yang lebih ceat uga ketiaaan lonakan ibaningkan engenali D berbasis ZN melalui trayektori yang iefinisikan. Kata kunci: Sistem kren gantry, suut ayunan, gain D, ABC 1. Introuction 1 Gantry crane system, a non-slewing-luffing crane system is most wiely use in several work laces such as orts, factories, construction sites. his tye of crane is esigne for reeat motions such as hoisting, transorting which inclues longituinal, transverse motion, an lowering heavy ayloa, as well as combination of each motion. Schematic of gantry crane system is shown in Figure 1. Gantry crane system can be ivie into two subsystems, namely gantry crane an stationary crane framework. Gantry crane incororates interaction among trolley, wire roe as hoist cable an ayloa which is maniulate by trolley an hoist mechanism. he ayloa is grabbe using hook system, which is then hoiste from trolley by means of cable. he function of cable rum is to win u or unwin the cable, raises or lowers the ayloa attache to the hook. he trolley an hoist work simultaneously to erform the task of gantry crane. Framework rolley Cable Hook o beam ayloa Receive: 14 March 17; Revise: 17 May 17; Accete: 6 May 17 Wheels Rails Figure 1. Gantry crane system [1]

2 In general, the task erforme by a gantry crane is to ick the ayloa, raise it, move it to target osition an lower it own on the crane framework. Because the traverse motion of trolley uring transort oerations, the ayloa has the tenency to swing naturally ue to traverse motion of trolley. he swinging motion reuces the see, accuracy an safety requirements of crane oerations. It lowers the see of crane oerations because the ayloa swing must be avoie before the ayloa can be safely lowere into secifie osition. he swings make it ifficult to erform alignment, fine osition, or other accuracy riven task. Swing effect also causes safety roblems to the crane framework. hat s why control systems are neee to suress the effects. If the cable an stationary crane framework are consiere as rigi boies, then the gantry crane system is regare as a rigi system. Much attention has been lace on the moeling of ynamics an control of gantry crane system. Khali et al. [] have controlle the brige an gantry cranes by roosing the D an inut shaing controllers an succeee to achieve goo ositioning accuracy an significant sway reuction. Hazriq et al. [3]-[4] have stuie numerically the ynamic behaviour of a nonlinear gantry crane system by varying the inut voltage, cable length, ayloa mass an trolley mass an controlle the system. hey moele the ynamics using Lagrange s aroach an simulate the system through the simulink of Matlab. he rest can be referre to references [5]-[1]. he maor contribution of this stuy is to imrove the control caacity of the D controller (DC) from the aforementione revious works. here are several D tuning methos in the literature (e.g. Ziegler-Nichols (ZN), Cohen-Coon, lamba tuning. However, they o not always lea to an accetable erformance an they may also suffer from some from robustness issues [11]. An otimizer, namely artificial bee colony (ABC) algorithm is roose to otimize the gains of DC. hat is because this algorithm is a owerful otimization technique, goo numerical convergence an multi-imensional search sace comare to other soft comuting techniques such as genetic algorithm an article swarm otimization [1] as well as fuzzy logic an neural network. he combination between the ABC algorithm an DC is calle the artificial bee colony (ABC)-D controller. his roose controller is alie to control the osition an to reuce the swinging of the ayloa of rigi gantry crane system. Ziegler-Nichols (ZN) base D controller is chosen as a benchmark because of its common ractice. he remainer of this aer is organize as follows. Section erives the moeling of rigi gantry crane system. Section 3 escribes a tyical structure of a DC with roose otimizer. Section 4 resents the concet of otimization of DC via ABC algorithm. Section 5 iscusses the control an otimization. Finally, Section 6 conclues this aer.. Moeling of Rigi Gantry Crane System For simlicity of the characteristics of the hysical gantry crane system, several assumtions are ut forwar to the ynamical moel. Mass of trolley m an ayloa m are moele as lume mass which is connecte by inetensible hoist cable. ayloa an its cable behave as enulum moel as eicte in Fig.. Z Y X m m Figure. Moel of rigi gantry crane system he ayloa has one swing angle with resect to the inference frame: is enote as angle between the -ais an y -lane. he ayloa swings either small or large swing angles. Friction between trolley an the to beam of crane framework an ynamics of hoist cable an rum in hoist system an hoist rive mechanism are not consiere. he structure is treate as a rigi boy. he equations of motion of rigi gantry crane system can be erive by Lagrange s equations, with the following form, t L L F fi,q q q q All terms of q, f, q,., (1) are efine as the generalize coorinates to escribe the trolley an ayloa motion. General velocity of the system is q,. he Lagrangian L is efine as L K, where K is kinetics energy an is otential energy of system. Generalize force is

3 enote as f i, where they are f, f y an f z alie inut force for the, y an z motions resectively. otal kinetics energy of the system K in terms of generalize coorinates an velocities are the kinetics energy of the trolley an ayloa, K K K K K 1 m 1 m m r m r, cos. 1 1, () otal otential energy of the system is the otential energy of the trolley an the otential energy of the ayloa, m g cos. (3) he energy of aming is eresse as, 1 F c. (5) By ifferentiating the Lagrangian oerator L an F with resect to the generalize coorinates q an q, t t L L F f L L F., (6) (7) he nonlinear equation of motion of gantry crane system can be written in Eqs. (8)-(9), m m c m cos sin f, (8) g cos sin. (9) Equations (8)-(9) are ynamics of gantry crane coule with ynamics of crane framework an call for some remarks. 1. Equation (8) resents ynamics of trolley motion with the inut force, while Eq. (9) is ynamics of ayloa.. erm f is inut force or riving force for the trolley motion while m m is mass total from trolley an ayloa. 3. erm is the acceleration of trolley which aears as the forcing term in the ayloa ynamics if the inut force for gantry crane is set u to be zero. he inut force f is generate from the torque of trolley motor. Dynamic of trolley motor can be written as, f K z K z z u. (1) r R r R r Equations (8) an (1) are combine to yiel, m m c m cos sin K z u R r K z R r. (11) erms in Eq. (11) can be elaine as: K,R, u are torque constant, motor resistance an inut voltage, resectively while z,r are gear ratio an raius of motor ulley. e n e n n n u n n Rigi e e n DC Motor Dynamics u Gantry Crane System n n ABC Figure 3. Diagram block of D controller for rigi gantry crane system

4 3. Close-Loo Control System with D Controller (DC) A tyical structure of a DC for controlling the rigi gantry crane system is shown in Fig. 3. It is seen that the system is classifie as a ouble inut single outut system (uner-actuate system). he first inut for the DC is error an error erivative e ( n ), e ( n ) of trolley osition while the secon inut is error an error erivative e ( n ), e ( n ) of ayloa swing. he errors an error erivatives are multilie by the gains of DC, namely roortional K an erivative K an then combine to form the control signal of DC as follows, u n K e ( n ) e ( n ) K e ( n ) e ( n ). (1) erms in Eq. (1) are as follows: e ( n ),e ( n ), e ( n ), e ( n ) are errors of trolley osition an ayloa swing as well as their error erivatives, resectively. he value of u n is limite by using saturation function before it is sent to the motor. he control action u n is then alie to the gantry crane system. Uner any tyes of system inut, the gains in Eq. (1) significantly affect the close-loo resonse. Subotimal gains lea to the system becomes unstable, high overshoot an large steay-state error. he gains must be esigne to match the system inut (set-oint) an the system outut by giving the corrective action in terms of control action. Hence, ABC algorithm is emloye to otimize those gains. In aition, Eq. (1) also reflects that gain for the errors of trolley osition an swinging of the ayloa are set u similarly an so o the gain of error erivatives. Both gains can be set u searately, but this is intene to make the otimization rocess efficient. It is worthy to note that integral action is not require ue to the resence of integral (reset) winu leaing to significant amount of overshoots. Hence, D controller is an aroriate choice for controlling the system. 4. Otimization of D Controller via ABC Algorithm arameters K, K are moele as foos in artificial bee colony (ABC) algorithm. For the sake of clarity, otimizer for the arameters of DC using ABC algorithm is calle ABC-D. DC arameters recalle as K, K are initialize ranomly in ABC-D controller. he maor hases of this algorithm has been outline [1], but it is revisite here for a new cost function, 4.1 Initialization he initial caniate solutions for are rouce for emloye bees by using Eq. (1), i, min ma min,i 1,, S, 1,, D. (1) erms in Eq. (81 can be elaine as follows: is -th imension of i-th i, min ma emloye bee, an are lower an uer bouns of -th arameters, resectively, is a ranom number in range of,1, S is the number of foo sources an D is the number of gains of DC. he cost function values of i-th the initial solution are calculate using Eq. (13), SSE N n 1 ^ n n. (13) erms in Eq. (13) are as follows: N is the number of ata, n is the actual trolley ^ n osition while is the calculate trolley osition. he fitness value of Eq. (13) is calculate by using following formula, f i 1 if f i 1 CFi. (14) 1 CFi otherwise 4. Emloye bee hase In this hase, emloye bees uate the initial caniate solutions using Eq. (15), g i, i, i, i, k, i,k 1,, S, 1,, D., (15) erms in Eq. (15) can be elaine as follows: g is -th imension of i-th uate caniate i, solution, is -th imension of i-th i, emloye bee, is -th imension of k-th k, emloye bee, term is a ranom number in range of 1,1. Also, an k inices are ranomly selecte among initial solutions so that i k to assure that the initial caniate solutions can be uate. After the solutions are uate, the cost function an fitness values of i-th uate caniate solutions are calculate using Eqs. (14)-(15). If the fitness value of uate caniate solutions is better than fitness value of initial caniate

5 solutions, then the initial caniate solutions are relace with uate caniate solutions. 4.3 Onlooker bee hase In this hase, the emloye bees communicate with the onlooker bees by calculating the robability of each fitness value as follows, i S f i1 i f i. (16) Base on the robability value, the onlooker bees also ranomly imrove the caniate solutions in the emloye bee hase by using Eq. (16), as well as calculate the cost function an fitness values of i-th uate caniate solutions using Eqs. (15)-(16). If the fitness value of the uate caniate solutions foun in the onlooker bee hase is better than fitness value of the solutions in the emloye bee hase, the onlooker bee is relace with the emloye bee. Scout bee hase If the caniate solutions uring the emloye bee hase cannot be uate until a reefine number of iterations reaches to the limit, the emloye bee becomes scout bee an new caniate solutions are rouce by using Eq. (1). Ste by ste of otimization rocess above is eicte in Fig. 4. arameters of ABC algorithm are shown in able 1 an able, resectively. Interval for the searching sace is 1 for K, K D. Since this aer works on moel base control, then Eqs. (9) an (11) are solve using fourth-orer Rung- Kutta with samling time of.1 s. Control an otimization rocesses are erforme simultaneously using Matlab. able 1. Gantry crane arameters arameters rolley mass, ayloa mass, m m 5 kg kg Cable length, 1 m Gravitational acceleration, g 9.81 m / s Initial conitions, o,, o,, able. arameters of ABC algorithm ABC Number of iterations 5 Onlooker number 5% Number of foos 5 Emloye bee number 5% Number of otimize Scout number 1 arameters ime omain resonses obtaine from ZN-D an ABC-D controllers are comare one to another. Control erformances in time omain are then assesse in terms of rise time, settling time, overshoot an eak time. Foos Cost Function, Eq. (13) e K e K D Emloye Bee hase Onlooker Bee hase Gains Scout Bee hase New Figure 4. Gains otimization rocess in ABC-D controller 5. Results an Discussions In this section, erformance of roose controller is investigate uner constant ste function. Other functions such as varying ste function an sinusoial functions are not elaborate in this stuy. Basic arameters of gantry crane system an Crane is commane to track a osition in 1m by giving a constant ste function. he system resonse for uration 18 s is shown in Fig. 5. he figure eicts that the crane is able to track the commane osition an both controllers have successfully stabilize it with resect to time. his

6 ayloa swing angle, θ (eg) rolley osition error, e (m) ayloa swing angle, θ (eg) rolley osition, (m) is confirme by Fig. 6 where the error of both controllers ecays to zero. However, each controller has ifferent erformances uring the tracking rocess. ZN-D controller has faster rise time an eak time than the ABC-D controller. However, fast rise time an eak time lea to overshoot until it reaches its settling time. In the other sie, slow rise time an eak time of ABC-D controller lea to no overshoot so that settling time can be achieve faster than ZN-D controller. Details are elaborate in able 3. realizations, it is foun that the otimize K an K of ABC-D controller are lower than the otimize gains of ZN-D controller. It elains why the ABC-D controller has the slow rise time an eak time, fast settling time as well as no overshoot as the function of gain K is to increase the rise time of the system resonse an the function of K is to reuce the oscillation. able 4. Otimal gains of D controller uner ste function ZN-D ABC-D Gains ZN-D Otimize ABC-D Otimize K K ime (s) Figure 5. rolley osition uner ste function 5 ZN-D ABC-D ZN-D SO-D ime (s) (a) 4 5 ZN-D ABC-D ime (s) Figure 6. Error of trolley osition uner ste function able 3. Controller s erformances uner ste function erformance ZN-D ABC-D Rise time Settling time Overshoot 4.58 eak time Otimal gains of D controller are tabulate in able 4. It is worthy to note that the initial gains are set ranomly an it will lea to the change of the otimize gains accoringly. After ime (s) (b) Figure 7. ayloa swing angle uner for ste function (a) time winow -18 s (b) time winow -5 s Control erformance in Figs. 5-6 is elaborate by Fig. 7. he figure shows the consequent of fast rise time an eak time of ZN-D controller. Faster the crane reaches the target osition, bigger the swing angle of ayloa occurs. Large swing angle of ayloa of ZN-D controller in Fig. 7 is the consequent of using full nonlinear ynamic

7 rolley osition, (m) Cost function (SSE) moel in Eq. 9 an Eq. 11. At this oint, control esigner can choose whether the trolley moves fast with large swing angle as eense or reasonable see of trolley with no overshoot. he latter is favorable since it is require for safety in crane oeration. Hence, all results confirm that the ABC- D controller outerforms the ZN-D controller. In otimizing the gains, D controller otimize by ABC algorithm rouces cost function as shown in Fig. 8. It islays the cost function with resect to the number of iterations. As observe, the cost ehibits a graual convergence an seems like a laer function as the number of iteration increases. However, the cost function starts to converge after the-31 st iteration an is steay to a certain value Number of iterations Figure 8. Cost function for ste function with resect to number of iterations kg 3 kg 4 kg ime (s) Figure 9. rolley osition uner ayloa mass variation o test the effectiveness the roose controller for a more challenging conition, ayloa mass is varie from kg to 4 kg. Uner this conition, the commane osition 4 is fie an the result is shown in Fig. 9. he crane, as eecte, is still able to track the commane osition. However, the figure shows that as the ayloa mass increases, the rise time an settling time also increases with resect to time. With increase ayloa mass, erformance of ABC-D controller will eteriorates further. 6. Conclusions In this aer, a controller namely the ABC-D is roose for controlling the rigi gantry crane system. Simulation results show that the roose controller can imrove the erformance of closeloo control system uner constant ste function. Imortant conclusions an suggestions from this work are erive below. - he ABC-D controller outerforms the ZN- D controller in terms of fast settling time an no overshoot. - Control an otimization are still erforme uner simulation since the ABC-D controller is comutationally eensive for real-time imlementation. - he generate cost function seems like a laer function. - he roose controllers can easily be alie to ID controller, where the gain K i is inclue. - he roose controllers can be alie to control other ynamic systems. Acknowlegments Research facility for this work from research center for electrical ower an mechatronics, Inonesian Institute of Sciences is gratefully areciate. s [1] SANCO. (1998). Installation an arts manual for SANCO F series gantry cranes, Available: htt:// cranes.html [] Khali, L.S., William, S. an Stehen, D., DA Controller Enabling recise ositioning an Sway Reuction in Brige an Gantry Cranes, Control Engineering ractices Vol. 15, 7, [3] Hazriq, et al., Dynamic Behaviour of a Nonlinear Gantry Crane System, roceeings of the ICEEI Vol. 11, 13, [4] Hazriq, et al., erformance analysis for a Nonlinear Gantry Crane System using rioritybase Fitness Scheme in Binary SO Algorithm, Avance Materials Research Vol. 93, 14, [5] Die, D.V. an Khoa, V.V., ID-Controllers uning Otimization with SO Algorithm for Nonlinear Gantry Crane System, International Journal of Engineering an Comuter Science Vol. 3 No. 6, 14,

8 [6] Sanor, I., et al., Stabilizing Moel reictive Control of a Gantry Crane Base on Fleible Set- Membershi Constraints, roceeings of the IFAC Vol. 48-3, 15, [7] Ahma, B.A., et al., Otimal Analysis an Control of D Nonlinear Gantry Crane System, roceeings of the ICSSA, 15, [8] Hussien, S.Y.S., Ghazali, R., Jaafar, H.I., an Soon C.C., An Eerimental of 3D Gantry Crane System in Motion Control by ID an D controller via FSO, Journal of elecommunication, Electronic an Comuter Engineering Vol. 8 No. 7, 16, [9] Hussien, S.Y.S., Ghazali, R., Jaafar, H.I., an Soon C.C., Analysis of 3D Gantry Crane System by ID an VSC for ositioning rolley an Oscillation Reuction, Journal of elecommunication, Electronic an Comuter Engineering Vol. 8 No. 7, 16, [1] Wei, H. an Shuzi, S.G., Cooerative Control of a Nonuniform Gantry Crane System with Constraine ension, Automatica Vol. 66, 16, [11] Skoczowski, S., Domek, S., ietrusewicz. K. an Broel-later, B., A metho for imroving the robustness of ID control, IEEE ransactions on Inustrial Electronics Vol. 5, 5, [1] Yazi, E. Liew, M.S., arman, S., an Kurian, V.J., Imroving the moelling caacity of Volterra moel using evolutionary comuting methos base on Kalman Smoother aative filter, Journal of Alie Soft Comuting, Vol. 35, 15,