GENETIC ALGORITHMS APPLIED IN PARAMETERS DETERMINATION OF THE 3D CRANE MODEL UDC D

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1 FACTA UNIVERSITATIS Series: Automati Control and Robotis Vol. 10, N o 1, 2011, pp GENETIC ALGORITHMS APPLIED IN PARAMETERS DETERMINATION OF THE 3D CRANE MODEL UDC D Zoran Jovanović 1, Dragan Antić 1, Zoran Stajić 2, Miloš Milošević 3, Saša Nikolić 1, Staniša Perić 1 1 Universit of Niš, Fault of Eletroni Engineering, Department of Control Sstems, Aleksandra Medvedeva 14, Niš, Serbia zoran.jovanovi@elfak.ni.a.rs, dragan.anti@elfak.ni.a.rs, sasa.s.nikoli@elfak.ni.a.rs, stanisa.peri@elfak.ni.a.rs 2 Universit of Niš, Fault of Eletroni Engineering, Department of Eletroenergetis, Aleksandra Medvedeva 14, Niš, Serbia zoran.staji@elfak.ni.a.rs 3 IRC ALFATEC, Niš, Bulevar Nikole Tesle 63/5, Niš, Serbia milos.milosevi@alfate.rs Abstrat. This paper onsiders the implementation of the geneti algorithms in determine parameters of model 3D Crane sstem, defined b highl nonlinear mathematial model. The validation of the model and parameters are performed b digital simulation and omparison with the eperimental results. Proposed method for determining parameters of the model has man advantages, suh as satisfator ontrol performane under a wide range of operating onditions, reduing flutuations during the positioning of paload art. Ke words: geneti algorithms, 3D rane sstem 1. INTRODUCTION The geneti algorithms are well known evolutionar searh algorithms implemented in man ontrol engineering problems [1-3]. Sine geneti algorithms represent optimization proedures, the an be applied to identifiation problems. There are two inter-related problems eisting in sstem identifiation: the hoie of a suitable model struture and the estimation of model parameters. Tehniques for the seletion of struture and for non-linear in the parameters estimation are still the subjet of Reeived April 29, 2011 Aknowledgement. The work presented here was supported b the Serbian Ministr of Eduation and Siene (projets III43007, III44006 and TR35005).

2 20 Z. JOVANOVIĆ, D. ANTIĆ, Z. STAJIĆ, M. MILOŠEVIĆ, S. NIKOLIĆ, S. PERIĆ ongoing researh [4]. There are man lassial methods for sstem identifiation, suh as least-squares and maimum-likelihood. Most of these are for linear or linear in the parameters non-linear sstems, and the often fails in the searh for a global optimum if the searh spae is not differentiable or linear in parameters. Furthermore, for these methods, initial information on the sstem parameters is needed for onvergene; estimated parameters ma be biased if the noise is orrelated; and the annot easil be applied to nonlinear sstems. The geneti algorithms an be applied to ontinuous and disrete-time sstem, both on-line and off-line and both time domain and frequen domain sstems, and an diretl identif phsial parameters or poles and zeroes of the sstem. In this paper, one geneti algorithm is implemented in the parameters model of 3D rane, whih represents the highl nonlinear plants. 2. 3D CRANE DESCRIPTION The 3D rane [5] shown in Fig. 1 is a non-linear eletromehanial sstem having a omple dnami behavior and reating hallenging ontrol problems. The sstem is ontrolled from a PC. Therefore it is delivered with hardware and software whih an be easil mounted and installed in a laborator. You obtain the mehanial unit with power suppl and interfae to a PC and the dediated A/D, D/A board onfigured in the Xilin tehnolog. The software operates under MS Windows NT using MATLAB and RTWT toolbo pakage. The 3D Crane setup onsists of a paload hanging on a pendulum-like lift-line wound b a motor mounted on a art. The paload is lifted and lowered in the z diretion. Both the rail and the art are apable of horizontal motion in the diretion. The art is apable of horizontal motion along the rail in the diretion. Therefore the paload attahed to the end of the lift-line an move freel in three dimensions. The 3D rane is driven b three DC motors. There are five idential measuring enoders measuring five state variables: the art oordinates on the horizontal plane, the lift-line length, and two deviation angles of the paload. Fig. 1 3D rane sstem b Inteo

3 Geneti Algorithms Applied in Parameters Determination of the 3D Crane Model 21 The enoders measure movements with a high resolution equal to 4096 pulses per rotation (ppr). These enoders together with the speialized mehanial solution reate a unique measurement unit. The deviation of the load is measured with a high aura equal to rad. The power interfae amplifies the ontrol signals whih are transmitted from the PC to the DC motors. It also onverts the enoders pulse signals to the digital 16-bit form to be read b the PC. The PC equipped with the RT-DAC/PCI multipurpose digital I/O board ommuniates with the power interfae board. The whole logi neessar to ativate and read the enoder signals and to generate the appropriate sequene of pulses of PWM to ontrol the DC motors is onfigured in the Xilin hip of the RT-DAC/PCI board. All funtions of the board are aessed from the 3D Crane Toolbo whih operates diretl in the MATLAB /Simulink environment. The user has a rapid aess to all basi funtions of the 3DCrane ontrol sstem from the Main Control Window. It inludes tests, drivers, models and appliation eamples. The main driver is loated in the RTWT Devie Driver olumn. The driver is a software go-between for the real rane MATLAB environment and the RT-DAC/PCI aquisition board. The driver has three PWM inputs (DC motor ontrols) for the X, Y and Z aes. There are 8 outputs of the driver: X position, Y position, Z position, two angles and additionall three safet swithes. Aording to a pre-programmed logi the internal XILINX program of the RT-DAC/PCI board an use the swithes to stop the DC motors. When one wants to build his own appliation one an op this driver to a new model. The shemati diagram of the 3D rane sstem is shown on Fig. 2. Fig. 2 3D rane sstem: oordinates and fores

4 22 Z. JOVANOVIĆ, D. ANTIĆ, Z. STAJIĆ, M. MILOŠEVIĆ, S. NIKOLIĆ, S. PERIĆ Denote also: m - mass of the paload; m w - mass of the art; m v - mass of the moving rail;,, z - oordinates of the paload; S - reation fore in the lift-line ating on the art; F - fore driving the art; F - fore driving the rail with art; F R - fore ontrolling the length of the lift-line; T, T, T R - frition fores. The following smbols are used in the sequel: m µ, µ m 1 2 mw mw ms, (1) F F u u u R 1, 2, 3 mw mw ms m F, (2) T T T T T R 1, 2, 3 mw mw ms m T, (3) N u T, N u T, N u T. (4) The position of the paload is desribed b the equalities: Ros, (5) w Rsin sin, (6) w z Rsin os. (7) The dnami of the rane is given b the equations (Fig. 2): m S m S mz S mg z m F T S w w ( m m ) F T S w s w where S, S and S z are omponents of the fore S: S S S z Sos, (8) Ssin sin. (9) Ssin os 3. GENETIC ALGORITHM The priniples of the geneti algorithms were first published b Holland in 1962 [6]. The mathematial framework was developed in the late 1960 s, and is presented in Holland s book, Adaptation in Natural and Artifiial Sstems published in 1975 [7].

5 Geneti Algorithms Applied in Parameters Determination of the 3D Crane Model 23 Geneti algorithms are optimization tehniques based on simulating the phenomena that takes plae in the evolution of speies and adapting it to an optimization problem. These tehniques impl appling the laws of natural seletion onto the population to ahieve individuals that are better adjusted to their environment. The population is nothing more than a set of points in the searh spae. Eah individual of the population represents a point in that spae b means of his hromosome. The individual s degree of adaptation is given b the objetive funtion. Appling geneti operators to an initial population simulates the evolution mehanism of individuals. Survival of the fittest philosoph is used to speed up the evaluation proess. The geneti algorithms have been used in man diverse areas suh as funtion optimization, image proessing, sstem identifiation, sstem modeling..., and have demonstrated ver good performanes as global optimizers in man tpes of appliations [8-11]. In order to understand more details about how are used geneti algorithms in our work [7, 12]. 4. EXPERIMENTAL RESULTS The art veloities in X and Y diretions are muh alike. The paload moves slower down and up. The angles of the paload are not losed-loop ontrolled and in onsequene the paload osillates freel. That the paload would not flutuate, it is projeted a lassi PI ontroller for all three aes [5], i.e.,,, z. The following simple linear model of the rane dnamis in the X-ais, Y-ais, Z-ais are assumed: K G () s sts ( 1) K G () s. (10) sts ( 1) K z Gz () s sts ( 1) Model parameters values are determined b geneti algorithm and ompared with parameters obtained from the MATLAB Toolbo. Shemati piture of the whole 3D Crane sstem in MATLAB environment is shown in Fig. 3. During the proess of parameter identifiation of the ontroller in our eperiments, as a riteria funtion, mean squared error an be used: 1 J ( ) T z T m 2 i i dt, (11) 0 m where i represent the measured position for the, and z ais of the art and i represent of the simulation results for position for the, and z of the art. Parameters of the ontroller are obtained b adjusting the model in order to minimize the riteria funtion (11) b the geneti algorithm. The omplete blok diagram, whih illustrates the proess of identifiation [9, 13], is given in Fig. 4.

6 24 Z. JOVANOVIĆ, D. ANTIĆ, Z. STAJIĆ, M. MILOŠEVIĆ, S. NIKOLIĆ, S. PERIĆ Fig. 3 Shemati window in MATLAB of the 3D Crane Fig. 4 Priniiple of parameters ontroller identifiation The geneti algorithm used in our eperiments was with the following parameters: initial population of 300, number of generations 200, stohasti uniform seletion, reprodution with 8 elite individuals, Gaussian mutation with shrinking and sattered rossover. The goal of the eperiment was to make a error as small as possible for a hosen input, i.e., to obtain the best parameters of the model of 3D rane sstem. So, we used error (11), as the fitness funtion for the geneti algorithm.

7 Geneti Algorithms Applied in Parameters Determination of the 3D Crane Model 25 Eperimental time was 30 seonds. After the eperiments, following model parameters of 3D Crane sstem were obtained: parameters for X aes K = , T = , parameters for Y aes K = , T = , and for Z aes K z = , T z = when paload is lifted. These results are used in digital simulation in order to validate the proposed solution. The results of the paload positions after optimisation b using geneti algorithm are shown on the following Figures. Fig. 5 Desired position of the art and angle of the paload are traked Fig. 6 Desired position of the art and angle of the paload are traked

8 26 Z. JOVANOVIĆ, D. ANTIĆ, Z. STAJIĆ, M. MILOŠEVIĆ, S. NIKOLIĆ, S. PERIĆ Fig. 7 Desired z position of the art 5. CONCLUSION This paper treats the problem of determining model parameters for alm simultaneousl stabilizes the paload in its hanging down position a art of 3D rane sstem b using geneti algorithm optimisation tehniques. The sstem model is highl nonlinear, so the traditional identifiation tehniques an fail in estimation proess. The proposed geneti algorithm requires nothing more than position measure and do not pose an restritions to the identifiation problem. The obtained parameter values are used in digital simulation and the model verifiation is done b omparing the eperimental data with simulated ones. The proposed geneti algorithm gives good aura in estimation of the model parameters. REFERENCES 1. P. J. Fleming and R. C. Purshouse, "Evolutionar algorithms in ontrol sstems engineering: A surve", Control Engineering Pratie, vol. 10, pp , Q. Wang, P. Spronk and R. Traht, "An overview of geneti algorithms applied to ontrol engineering problems", Proeedings of the Seond International Conferene on Mahine Learning and Cbernetis, Xisan, pp , D. Antić, M. Milojković, Z. Jovanović and S. Nikolić, ''Optimal design of the fuzz sliding mode ontrol for a DC servo drive'', Journal of Mehanial Engineering, vol. 56, no. 7-8, pp , G. J. Gra, D. J. Murra-Smith, Y. Lin, K. C. Sharman and T. Weinbrenner, "Nonlinear model struture identifiation using geneti programming", Control Engineering Pratie, vol. 6, pp , Inteo, 3D Crane Sstem-User s Manual, Available at J. H. Holland, "Outline for a logial theor of adaptive sstems", J. ACM, vol. 3, pp , 1962.; also in A.W. Burks, Ed., Essas on Cellular Automata, Universit Illinois Press, pp , J. H. Holland, Adaptation in Natural and Artifiial Sstems, Universit of Mihigan Press, Ann Arbor, D. Antić and M. Milojković, "Nonlinear sstem ontrol b using geneti algorithms and fuzz sliding mode", TEHNIKA, Elektrotehnika, vol. 56, no. 3, pp. 9-16, 2007.

9 Geneti Algorithms Applied in Parameters Determination of the 3D Crane Model M. Milojković, S. Nikolić, B. Danković, D. Antić and Z. Jovanović, ''Modelling of dnamial sstems based on almost orthogonal polnomials'', Mathematial and Computer Modelling of Dnamial Sstems, vol. 16, no. 2, pp , D. Antić, M. Milojković and S. Nikolić, ''Fuzz sliding mode ontrol with additional fuzz ontrol omponent'', FACTA UNIVERSITATIS Series: Automati Control and Robotis, vol. 8, no. 1, pp , D. Antić, M. Milojković, S. Nikolić and S. Perić, ''Optimal fuzz sliding mode ontrol with a time-varing sliding surfae'', Proeedings of IEEE International Joint Conferenes on Computational Cbernetis and Tehnial Informatis, ICCC-CONTI 2010, Timisoara, Romania, Ma , pp , B. Danković, D. Antić, Z. Jovanović and M. Milojković, ''Geneti algorithms applied in parameter optimization of asaade onneted sstems'', Proeedings of International Sientifi Conferene on Information, Communiation and Energ Sstems and Tehnologies, ICEST 2007, Ohrid, Maedonia, June 24-27, pp , D. Mitić, D. Antić, S. Nikolić and M. Milojković, ''Identifiation of the multitank sstem using geneti algorithm'', Proeedings of the XLV International Sientifi Conferene on Information, Communiation and Energ Sstems and Tehnologies, ICEST 2010, Ohrid, Maedonia, June , 2010, vol. 1, pp , ODREĐIVANJE PARAMETARA MODELA 3D KRANA KORIŠĆENJEM GENETIČKIH ALGORITAMA Zoran Jovanović, Dragan Antić, Zoran Stajić, Miloš Milošević, Saša Nikolić, Staniša Perić U ovom radu razmatra se primena genetičkog algoritma pri određivanju parametara modela 3D krana, definisanog kao visoko nelinearnog sistema. Validnost modela i parametara potvrđena je simulaijom i upoređena je sa eksperimentalnim rezultatima. Predloženi način određivanja parametara modela ima mnoge prednosti, kao što su zadovoljavajuće performanse u širokim uslovima rada, smanjenje osilaija tereta prilikom poziioniranja kolia. Ključne reči: genetički algoritmi, 3D kran sistem