Identification of Helicopter Dynamics based on Flight Data using Nature Inspired Techniques

Transcription

1 Identfcaton of Helcopter Dynamcs based on Flght Data usng Nature Inspred Technques S.N. Omkar 1, Dheevatsa Mudgere 2, Senthlnath J 1, M. Vaya Kumar 3 1 Department of Aerospace Engneerng, Indan Insttute of Scence-Bangalore, , KA, Inda 2 Lehrsthul für Scentfc Computng n Computer Scence, Informatk Department, TUM, München Rotary Wng Research and Desgn Center, HAL, Bangalore Emal:omkar@aero.sc.ernet.n Abstract The complexty of helcopter flght dynamcs makes modelng and helcopter system dentfcaton a very dffcult task. Most of the tradtonal technques requre a model structure to be defned a pror and n case of helcopter dynamcs, ths s dffcult due to ts complexty and the nterplay between varous subsystems. To overcome ths dffculty, non-parametrc approaches are commonly adopted for helcopter system dentfcaton. Artfcal Neural Network are a wdely used class of algorthms for non-parametrc system dentfcaton, among them, the Nonlnear Auto Regressve exogeneous nput network (NARX) model s very popular, but t also necesstates some n-depth knowledge regardng the system beng modelled. There have been many approaches proposed to crcumvent ths and yet stll retan the advantageous characterstcs. In ths paper we carry out an extensve study of one such newly proposed approach - usng a modfed NARX model wth a two-tered, externally drven recurrent neural network archtecture. Ths s coupled wth an outer optmzaton routne for evolvng the order of the system. Ths generc archtecture s comprehensvely explored to ascertan ts usablty and crtcally asses t s potental. Dfferent nstantatons of ths archtecture, based on nature nspred computatonal technques (Artfcal Bee Colony, Artfcal Immune System and Partcle Swarm Optmzaton) are evaluated and crtcally compared n ths paper. Smulatons have been carred out for dentfyng the longtudnally uncoupled dynamcs. Results of dentfcaton ndcate a qute close correlaton between the actual and the predcted response of the helcopter for all the models. Keywords: System dentfcaton, Helcopter dynamcs, Nonlnear Auto Regressve exogenous model, Artfcal Bee Colony, Artfcal Immune System, Partcle Swarm Optmzaton. I. INTRODUCTION Helcopter system dentfcaton s the extracton of system characterstcs/dynamcs from measured flght test data (Mane, R. E., & Ilff, K. W., 1985; Anon, 1991). The complexty of helcopter flght dynamcs makes, modellng and helcopter system dentfcaton a very challengng task. Unlke fxed-wng arcrafts, the helcopters exhbts a hgh degree of nter-axs couplng, hghly unstable, non-mnmum phase dynamc characterstcs and large response varatons wth flght condton. These characterstcs of the helcopter make t a hghly non-lnear and a complex dynamcal system. Further, the predcton of aeromechancal forces, loads on the rotor system and man rotor wake nterferences wth the empennage and tal rotor requre wnd tunnel experments and flght tests. But the wnd tunnel expermental data suffers from scale effects and model defcences. Therefore, a key tool for helcopter flght/ground test correlaton s provded by system dentfcaton usng flght data. Identfcaton of a system requres pckng a functon (or model) so as to approxmate the nputoutput behavour of the system (the helcopter n ths case) n the best possble manner. There has been consderable amount of work carred out n ths regard, explorng the varous methods avalable for dentfcaton of dynamcal systems (Mller, W. T., Sutton, R. S., & Werbos, P.J. 1990;

2 Narendra K. S. & Parthasarathy, K., 1989; Narendra K. S. & Parthasarathy, K., 1990; Narendra K. S. & Parthasarathy, K., 1991; Ichkawa. Y & Sawa. T, 1992; Sastry, P. S., Santharam G. & Unnkrshnan K. P., 1994; Chen, S., Bllngs, S. A., & Grant, P. M., 1990; Hoskns, D. A., Hwang, J. N., & Vagners J., 1992). Identfcaton of nonlnear physcal models contnues to be a challenge snce both the structure and parameters of the physcal model must be determned. Many exstng system dentfcaton methods are based on parametrc dentfcaton. Structure determnaton s based on expert knowledge of the underlyng physcs, lackng whch often tral and error approach to test canddate model structures s employed. Possble structures are deduced from engneerng knowledge of the system and the parameters of these models are estmated. But n the case of a helcopter, defnng an a pror model s dffcult due to nteracton between the varous subsystems lke the rotor, fuselage, power plant, tal rotor and transmsson systems (Tschler, M. B., 1996) the dynamcs are of relatvely hgher order, and t s dffcult to know how many states to nclude and whch states are mportant. Also, ncrease n the nonlnearty, uncertanty and complextes of the model together wth the strngent specfcatons of accuracy lmts to be mantaned renders modellng helcopter systems a dauntng task. Ths ntated an nterest among researchers to dentfy the system characterstcs usng nonparametrc methods. Artfcal Neural Network (ANN) have found wdespread applcaton n nonlnear dynamc system dentfcaton as unversal approxmators (Mller, W. T., Sutton, R. S., & Werbos, P.J. 1990; Narendra K. S. & Parthasarathy, K., 1989; Narendra K. S. & Parthasarathy, K., 1990; Narendra K. S. & Parthasarathy, K., 1991; Ichkawa. Y & Sawa. T, 1992; Sastry, P. S., Santharam G. & Unnkrshnan K. P., 1994; Chen, S., Bllngs, S. A., & Grant, P. M., 1990; Hoskns, D. A., Hwang, J. N., & Vagners J., 1992). ANNs have also been used for helcopter system dentfcaton, Vaya Kumar et.al. (2003) have explored the dfferent Recurrent Neural Networks (RNN) for the dentfcaton of helcopter dynamcs and based on the results, the practcal utlty, advantages and lmtatons, the models have been crtcally apprased. The authors after a comprehensve study of the three popular RNN archtecture Nonlnear Auto Regressve exogenous (NARX) nput model, Memory Neuron Network (MNN) model and Recurrent Mult-Layer Perceptron (RMLP) model, concluded that the NARX model s most sutable for the dentfcaton of helcopter dynamcs (Vay Kumar, M. et. al. 2003). The NARX model, proposed by Narendra (1989, 1990 & 1991) uses Tapped-Delay-Lnes (TDL) and the Mult-Layer Perceptron neural network archtecture (MLP) for non-lnear system dentfcaton. The NARX model s traned usng the Back Propagaton (BP) algorthm. Although NARX model s popular and wdely used, t has certan drawbacks. NARX model requres the necessary past nputs and outputs of the system beng modelled, to be fed as explct nputs to the network; ths necesstates a farly n-depth knowledge of the system. But, as prevously dscussed, n case of helcopters ths s challengng ask. Whle, n prncple, we can always feed a suffcent number of past values to the network, n practce, all reported applcatons assume that the exact order s known. Both from aesthetcs and practcalty, learnng transformatons n ths manner may not lead to versatle dynamcal models (Sastry, P. S., Santharam G. & Unnkrshnan K. P., 1994). Ths can be overcome by usng alternatves such as the memory neuron networks (MNN) (Sastry, P. S., Santharam G. & Unnkrshnan K. P., 1994) etc... But The NARX model has been proven to have far superor stablty characterstcs when compared

3 to MNN and the other models (Vay Kumar, M. et. al. 2003). Ths makes the NARX model a very popular choce for dentfcaton and adaptve control of dynamcal systems even wth ts set of lmtatons. Mudgere, D., et.al (2008) have proposed a PSO (partcular the partcle swarm optmzaton) (Eberchart, R., & Kennedy, J., 1995) Drven Recurrent Neural Network (PSO- NARX) Model, for crcumventng ths draw-back of the conventonal NARX model. Ths model employs a PSO based algorthm for evolvng the order (and relatve degree) of the system, coupled wth the (NARX-lke) mult-layer perceptron recurrent neural network wth tapped delay lnes, traned by a PSO based learnng algorthm, for the dentfcaton of the dynamcal system. Ths model employs a generc II-Ter archtecture, whch offers a great deal of flexblty and allows for usng a host of dfferent algorthm combnatons for evolvng the order and tranng the system. The obectve of the current work s to further develop ths concept by comprehensvely explorng ths generc II-Ter archtecture wth dfferent combnatons of nature nspred technques for evolvng the order and tranng the network. In order to explot the nhert flexblty of ths archtecture to develop an effcent and accurate model for dentfcaton of non-lnear dynamcal systems. In ths paper we consder algorthms nspred from nature to acheve the above obectve n partcular the partcle swarm optmzaton (PSO) (Eberchart, R., & Kennedy, J., 1995), Artfcal Bee Colony (ABC) (Karaboga, D., & Basturk, B., 2008; Karaboga, D., Akay, B., & Ozturk, C., 2007) and Artfcal Immune System (AIS) (De Castro, L. N., & Von Zuben, F. J., 2000; Dasgupta D., 1999) are consdered. Ths results n three dfferent varants of ths model for nonlnear dynamcal system dentfcaton, whch do not necesstate any apror nformaton about the system and the favourable characterstcs (stablty) of the popular NARX model. Further the proposed models are successfully employed for the dentfcaton of helcopter dynamcs usng flght test data. These dfferent models are crtcally apprased and compared. II. NATURE INSPIRED TECHNIQUES Nature nspred technque s the feld of research that works wth computatonal technques nspred n part by nature and natural systems. These nature nspred technques provde a more robust and effcent approach for solvng complex real-world problems (Bäck, T., & Schwefel, H. P., 1993). Many nature nspred technques such as Artfcal Bee Colony (ABC) (Karaboga, D., & Basturk, B., 2008), Artfcal Immune System (AIS) (De Castro, L. N., & Von Zuben, F. J., 2000), Artfcal Neural Network (Haykn, S., 1994), Genetc Algorthm (GA) (Goldberg, D. E., 1989), Partcle Swarm Optmzaton (PSO) (Eberchart, R., & Kennedy, J., 1995) etc.. have been proposed. Snce they are heurstc and stochastc n nature, they are less lkely to get stuck n local mnmum, and they are based on populatons made up of ndvduals wth a specfed behavor smlar to bologcal phenomenon. These characterstcs led to the development of nature nspred computaton as t s ncreasngly appled n varous domans (Engneerng problems). Presently, t s one of the mportant areas of research. A. Artfcal Bee Colony Artfcal Bee Colony (ABC) (Karaboga, D., & Basturk, B., 2008) s a class of optmzng numercal problem based on swarm ntellgence, nvestgatng the foragng behavour of bees. In

4 ABC algorthm, the colony of artfcal bees contans three groups of bees whch nclude scout bees, employed bees and onlookers. A bee carryng out random search s called a scout. A bee watng on the dance area for makng decson to choose a food source s called an onlooker and a bee gong to the food source vsted by tself prevously s named an employed bee. At the frst step, create a populaton of n artfcal bees placed randomly n the search space representng the food source poston. After ntalzaton, the populaton of the postons (solutons) s subected to repeated teraton of the search processes of the employed bees, the onlooker bees and scout bees. For each soluton x, where = 1, 2...n and s dmensonal vector. The scout bees explore a new food source wth x. Ths operaton can be defned as n (1) x x mn ( x max x mn )* rand( 0,1) (1) The populaton spread s restrcted wthn the search space S.e x S and n the equaton (1) xmn and xmax are the lower and upper lmt respectvely of the search scope along each dmenson; The bees whch have explored the food source are selected as employed bees. Whch results n a modfcaton on the poston (soluton) n those canddate bees memory, as a functon of the local nformaton and tests the nectar amount (ftness value) of the new source. After all the employed bees complete the search process; they communcate the nectar nformaton of the food sources and ther poston nformaton wth the onlooker bees n the dance area. An onlooker bee evaluates the nectar nformaton taken from all employed bees and chooses a food source wth better nectar amount. An artfcal onlooker bee chooses a food source dependng on the new postons, usng the equaton (2). P v, f x, f ( ( f ( x ) f ( x ) f ( v )) f ( v )) (2) In order to select the better nectar poston found by an onlooker, Ob s defned as O b arg mn P f ( P ), 1 n (3) Where P s the best ftness value of the soluton whch s proportonal to the nectar amount of the food source n the poston and n s the number of food sources. In order to produce a canddate food poston from the old one n memory, the ABC uses the followng equaton (4): v x ( x x k ) (4) where k=1, 2,..., n and = 1, 2,...,D are randomly chosen ndces. Although k s determned randomly, t has to be dfferent from. s an adaptvely generated random number whch controls the learnng/adopton rate.

5 B. Artfcal Immune System As descrbed by De Castro, et.al (2002), Artfcal Immune Systems (AIS) are adaptve systems nspred by theoretcal mmunology and observed mmune functons, prncples and models whch are appled to problem solvng. Much of the early work carred out n the development of AIS has been smlar to genetc and evolutonary computaton technques. The man dstncton between the feld of genetc algorthms and AIS s the nature of populaton development. In a genetc algorthm the populaton s evolved usng crossover and mutaton (McCall, J., 2005). However n the AIS, as n evolutonary strateges reproducton s asexual (clonng), each chld produced by a cell s an exact copy of ts parent. Both systems then use mutaton to alter the progeny of the cells and ntroduce genetc varaton. We make use of a varant of AIS called the Clonal selecton algorthm for optmzaton. Clonal Selecton Prncple (De Castro, L. N., & Von Zuben, F. J., 2000) s one of the nsprng methodologes employed n AIS for mult obectve optmzaton problems (Coello Coello, C.A., & Cruz Cortes, N., 2002). Based on the clonal selecton prncple - an algorthm s developed n whch varous mmune system aspects are taken nto account such as: mantenance of the memory cells, selecton and clonng of the most stmulated cells, death of non-stmulated cells and re-selecton of the clones wth hgher affnty and generaton and mantenance of dversty. The clonal selecton algorthm works wth an ntal repertore of antbodes. When an antgen s presented to t, the antbodes that are more effectve n neutralzng the threat are allowed to prolferate. The least effectve ones are elmnated. Among the ones selected to multply, some amount of mutaton s ntroduced to n the hope of fndng antbodes that mght perform better. However the mutaton rate s lesser n antbodes wth better ftness although t has a hgher number of clones. C. Partcle Swarm Optmzaton Partcle Swarm Optmzaton (PSO) s a swarm ntellgence algorthm proposed by Kennedy and Eberhart n the md 1990s. In the proposed algorthm, an agent of the swarm, called a partcle, learns from the best poston that t has occuped and also the best poston that any partcle of the swarm mght have encountered. These postons are retaned n memory of every partcle and constantly updated to drect the swarm to the global best poston. The best poston of a partcle s called cogntve ndex - pbest, and the best of the swarm the socal ndex - gbest. The equatons that govern the change n postons of a partcle are: V ( 1) w V { C p r1 ( pbest X )} { C g r2 ( gbest X )} (5) X ( 1) X V ( 1) (6) Snce each partcle s constantly movng, there s a velocty vector assocated wth each partcle that governs ts moton. Ths s denoted by V and X s ts poston. pbest s the best poston of partcle. gbest s the global best poston of the swarm. Cp s the Cogntve learnng rate and Cg s

6 the Socal learnng rate. The factors r1 and r2 are randomly generated wthn the range (0, 1) and w s the nerta factor. III. HELICOPTER SYSTEM IDENTIFICATION PROBLEM FORMULATION Conventonal system dentfcaton methods can be broadly classfed nto tme-doman and frequency doman methods. In tme-doman system dentfcaton, technques such as least square estmaton (Astrom, K. J., & Eykhoff, P. E., 1987), quas-lnearzaton (Kalaba, R., & Spngarn, K., 1982), and stochastc modellng (Kuzta, B., 1983), have been successfully used. In these methods, the model structure must be defned apror to estmate all the requred system parameters. These methods are extensvely used n helcopter system dentfcaton for lnear models and the approaches can be extended to nonlnear regons as well. Although tme doman methods have been more frequently used, frequency doman approach s also used successfully n rotorcraft dentfcaton (Tschler, M. B., & Cauffman, M. G., 1992; Fu, K.H., & Marchand, M., 1983 & Fu, K. H., & Kaletka, J., 1990). Modelng helcopter dynamcs s a multple nput multple output (MIMO) problem wth a hgh degree of nteracton between all the nputs and outputs. The obectve of system dentfcaton s then to construct a sutable model, such that the nputoutput behavor ( u( k) yˆ( k) ) of the model approxmates, n some sense, the nput-output behavor ( u( k) y( k) ) of the helcopter system,.e. for some specfed small postve constant ( ). y( k) yˆ( k) k 0 (7) Hence, the nput-output model for helcopter dynamcs s descrbed as u( k),.. u( k n 1), y( k 1),.. y( k n 1), 1,...,p y ( k d) F (8) where n1 s the order (or equvalent delay) of the system, d s the relatve degree and the functon F(.) s smooth and contnuous. IV. FLIGHT DATA ACQUISITION Flght test s carred out n calm wnd condtons on a helcopter havng a soft n-plane fourbladed hnge-less man rotor and a four bladed tal rotor wth conventonal mechancal controls. The helcopter s nstrumented to measure ptch (q), roll (p) and yaw (r) rates, longtudnal (ax), lateral (ay) and normal acceleratons (nz), ptch atttude ( ) and roll atttude ( ), ndcated arspeed (V), barometrc alttude (H) and sdeslp ( ). The four-control dsplacements namely, longtudnal cyclc (δlong), lateral cyclc (δlat), collectve (δcol) and pedal nput (δdr) are also measured. However, for the present study, ax, ay, V, H and are not used for smulaton. The helcopter s trmmed n straght and the plots mantan level flght condtons at two dfferent arspeeds of 120 Kmph and 230 Kmph. Inputs are provded manually and no automatc sgnal generators are used. The flght test engneer brefs plots at the start of each flght and before carryng out each test pont. For example, whle gvng an nput manually to δlat, plot has been brefed to adust the other control surfaces (δlong, δdr, δcol) approprately to mantan the flght

7 condton n off-axs. Flght data for the response of the helcopter for 5 to 10% of step/doublet nputs has been recorded for a mnmum of 20 sec duraton. V. METHODOLOGY The model proposed n ths paper, s capable of dentfyng a nonlnear dynamcal system gven ust a set of nput-output pars of the system under study. It does not necesstate any a pror knowledge about the system. Ths model s based on the NARX model. Ths model employs the MLP archtecture n combnaton wth TDLs for dentfyng dynamcal systems. As n the NARX model here also, the past values of outputs and nputs are fed back usng tapped-delay-lnes (TDL). The MLP approxmates the nonlnear functon and TDL ntroduces the dynamcs nto the model. But the dstncton here s that, the order and relatve degree of the system need not be known beforehand, as ths model s developed to evolve by tself the order and relatve degree of the system beng dentfed. The model determnes the necessary past nputs and outputs of the system beng modelled that need to be fed as explct nputs to the network. Further, the network so dentfed s traned usng a stochastc learnng algorthm based nature nspred technques, nstead of backpropagaton as n the conventonal NARX model. Schematc representatons of the models are gven n Fg. 1. Fg.1. Schematc representaton of the proposed modelng approach The helcopter dynamc dentfcaton s handled n a two-ter approach. Frst, the order and the relatve degree of the system has to be evolved.e. the number of hstory values the nputs/outputs that nfluences the current output of the system. Ths determnes the confguraton of network as ths defnes the total number of parameters to be fed as explct nputs (node n the nput layer) to the network. The second step would be to tran the network so dentfed. In ths paper we employ the II-Ter archtecture proposed by Mudgere, D., et.al (2008). In ths paper three dfferent nature nspred methods ABC, AIS and PSO are used. Further 3 model varants, wth dfferent combnatons of these algorthm are proposed - ABC-ABC, wth I-ter ABC for determnng the

8 order of the system and a learnng algorthm based on ABC for tranng the dentfed network and smlarly AIS-PSO and the PSO-NARX model whch used PSO based algorthms for both the ters. These varants use the NARX model as the basc substrate wth an external ABC, PSO and AIS shell for evolvng the order of the system. The generc mplementaton (three of the algorthm(s) used) of the above-descrbed archtecture s descrbed below. A. Ter I Ter I s for evolvng the order of the system. Ths s to determne the number of parameters to be fed as explct nputs to the network. Hence the decson varables n ths case are the number of past values of each nput and number of past values of each output that nfluences the current output of the system. Number of past values of each nput s represented by, U [u1, u2, u3 u] and the number of past values of each output s gven by, Y [y1, y2, y3 y], where and are the number of nputs and outputs of the system under consderaton. Snce the dynamcs of the helcopter s represented by a MIMO system (as dscussed earler), the number of hstory states for each nput and output has to be determned. In ths module the soluton-space s the varous combnatons of [u1, u2, u3 u, y1, y2, y3 y] that could possbly be the order of the system under consderaton. Where u s the number of past values of the nput and y s the number past values of output that nfluences the system under consderaton. The dmensonalty of an algorthm used for the I-Ter, when dentfyng a MIMO system wth m- nputs and n-outputs would be (m + n). A swarm of partcles (PSO), bee partcles (ABC) and antbody populaton (AIS) are employed to search all possble combnaton of [U, Y] and dentfy the order and relatve degree of the system. For each order, the network s traned and the model s checked for how well t conforms to the actual system. The mean square error (MSE) s used as the ftness value to check the extent of conformance of the dfferent combnatons obtaned. The swarm partcles (PSO), bee partcles (ABC) and antbody populaton (AIS) are updated such that ths value s mnmzed. Ths way the dfferent network confguratons are evaluated and fnally the swarm partcles, bee partcles and antbody populaton converge on to the combnaton that results n the least MSE and whch would ndcate the order and relatve degree of the helcopter system under consderaton. For a MIMO dynamcal system wth m-nputs and n-outputs, A possble soluton: [u1, u2, u3 u, y1, y2, y3 y] U [u1, u2, u3 u], u u Z where = [1 m] - number of past hstory states of each nput whch has to be ncluded. Y [y1, y2, y3 y], y y Z where = [1 n] -number of past hstory states of each output whch has to be ncluded.

9 Hence the total number of parameters that has to be gven as explct nputs to the network s gven by m, n ( u 1, 1 y ). m u - order of the dynamcal system and 1 n y s ts relatve degree. 1 Ths forms the frst stage of proposed model wheren the structure of the system s dentfed, determnng all the necessary past nputs and outputs of the system beng modelled needed to be fed as explct nputs to the network. B. Ter II Ths module s to tran the network dentfed by Ter I. Here we use a stochastc tranng algorthm based on PSO/ABC for tranng the MLP network wth TDLs. As ndcated earler these stochastc algorthms have been extensvely used for tranng ANNs and proved to be more effcent than many other gradent based tranng algorthms (Engelbrecht, P., & Ismal, A., 1999). Ths can be manly attrbuted to stochastc nature of the algorthm whch makes t very robust and flexble. In Ter II, the optmum weghts are evolved for each network confguraton determned by Ter I. The network structure determned by the frst level serves as the startng pont for the II-Ter. In ths stage, the varable s the weght matrx of the dentfed network. When appled to Feedforward Neural Network (FNN) tranng, each partcle represents a possble FNN confguraton,.e., ts weghts. Therefore, each vector has a dmenson equal to the number of weghts n the FNN. The unque feature of the current tranng algorthm, whch dstngushes t from the varous other smlar tranng algorthms, s the varable/dynamc dmensonalty. The dmensonalty of the algorthm used for Ter II keeps changng at each teraton, as t depends on the network confguraton determned by the outer-ter. Based on the ftness value used, the weghts are modfed so that the tranng error s mnmzed. Hence, fnally arrvng at a set of weghts, that result n the least error for a gven network confguraton. But from each tme to tme the number of weghts changes as the network confguraton changes. In our study, we use the root mean sum of squared resduals (error) n the tranng data as the ftness values of the Ter II PSO/ABC, ths serves as a qualtatve performance measure of the network learnng. rmse 1 ( y( k) yˆ( k)) N1 N 1 2 (9) where y ( k ) s the tme value of the output, and ŷ ( k ) s the estmated output of the recurrent neural network. N1 s the number of data ponts used n the tranng set. In our case, y ( k ) s the response of helcopter obtaned from the flght test. Our obectve s to mnmze ths ftness value and drve t to zero, whch then ndcates the evolve model to accurately nterpolate the data. The nner loop forms the ftness functon evaluaton n the outer loop. Ths s executed every tme the ftness functon s evaluated n the outer loop. Thus ncreasng the complexty of the model and makng t computatonally ntensve.

10 A. Artfcal Bee Colony Artfcal Bee Colony The model handles helcopter dynamcs dentfcaton n a two ter archtecture usng ABC. The frst ter ABC s used for evolvng the order of the dynamcal system under consderaton. The second ter ABC s used wth the frst ter ABC to evolve the optmum weght matrx for the network dentfed. In two ter archtecture of ABC, the number of bees (n) s ntalzed. At each tme step the randomness ampltude and speed of convergence of each employed and onlooker bees s changed towards ts food source (nectar). The random factor prevents the bees gettng stuck n local mnmae and speed of convergence s used to dentfy the rate at whch employed and onlooker bees explot a soluton. For each of these bees, ftness s evaluated. Our obectve s to mnmze the ftness value. The Ter II ABC s executed every tme the ftness functon s evaluated n Ter I ABC. The selecton of the ABC parameters plays an mportant role n the optmzaton as the performance of the ABC s qute senstve to control parameter choces (Karaboga, D., & Basturk, B., 2008). In the current work, there s two ter ABC algorthms beng employed, each wth dfferent confguratons of parameters. The fnal optmal ABC parameters have been selected after extensve numercal smulatons wth varous combnatons. For both cases, optmal refers to the set of ABC parameters whch results n fastest convergence along wth the most accurate dentfcaton of the consdered dynamcal system. A number of dfferent confguratons of parameters have been expermented wth tryng to acheve a balance between the computatonal tme and the performance. The fnal optmal set of ABC-ABC parameters for both I and II ter algorthms have been lsted n Table 1. ABC Parameters Ter I ABC Ter II ABC Number of Bees N = 30 N = 40 Randomness ampltude = 0.45 = 0.45 Speed of convergence = 0.85 = 0.85 Learnng rate s adaptvely generated for each teraton = [0.5,.,1] = [0.5,,1] Maxmum number of teratons Table 1. The ABC Parameters B. Artfcal Immune System Partcle Swarm Optmzaton In ths model, the order and the relatve degree of the system are evolved usng AIS n the Ter 1. PSO based learnng algorthm s used to tran the dentfed network. The AIS forms the frst stage of the algorthm where n the structure of the system are dentfed. Ths network structure serves as startng pont of the PSO n the second stage of the algorthm. The antbody (Ab) populaton s randomly ntalzed for the AIS. For each of these Ab, ftness s evaluated. The PSO n the Ter II s executed every tme the ftness functon s evaluated n Ter I

11 AIS. The dmensonalty of PSO keeps changng at each teraton, as t depends on the network confguraton determned by the AIS. After the ftness evaluaton of Ab populaton, varous AIS steps of clonng, mutaton and reselecton are done to obtan mproved set of populaton. Ftness of ths mproved populaton s reevaluated and new random Ab partcles are added to ths populaton for subsequent teraton. As wth the PSO, n the case of AIS also, the parameters play an mportant role n the optmzaton as performances of these algorthms are qute senstve to control parameter choces. The fnal optmal parameters have been selected after extensve numercal smulatons wth varous combnatons. A number of dfferent confguratons of parameters have been expermented wth tryng to acheve a balance between the computatonal tme and the performance. The fnal set of parameters used for both AIS and PSO are lsted n Table 2. AIS-PSO Parameters Ter I AIS Ter II PSO Number of clones per antbody n = 4 - Hypermutaton probablty pm = Number of Antbodes P = 4 - Maxmum number of teratons Max_t = 10 - Cogntve Learnng Rate - CP = 2.0 Socal Learnng Rate - Cg = 2.0 Inerta factor - w = 0.9 Number of Swarm Partcles - N = 10 Maxmum number of teratons End Condton teratons C. Partcle Swarm Optmzaton Table 2. The AIS - PSO Parameters Ths varant, employs a PSO based external algorthm for evolvng the order of the system and also a PSO based learnng algorthm for the tranng the evolved network. Ths model was proposed by Mudgere, D., et.al (2008) and ths paper provdes a detaled descrpton of the detals and characterstcs of ths model. Ths s a two-ter PSO archtecture, I level PSO for evolvng the order of the dynamcal system and the II level PSO to evolve the optmum weght matrx for the network dentfed by the I level PSO. Ths model uses the NARX model as the basc substrate wth an external PSO shell for evolvng the order of the system. The dmensonalty of the I-Ter PSO, when dentfyng a MIMO system wth m-nputs and n-outputs would be (m + n). A swarm of partcles are employed to search all possble combnaton of [U, Y] and dentfy the order and relatve degree of the system. In II-Ter PSO, the optmum weghts are evolved for each network confguraton determned by I-Ter PSO. The network structure determned by the frst level PSO serves as the startng pont for the II-Ter

12 PSO. The unque feature of the current tranng algorthm, whch dstngushes t from the varous other PSO-based ANN tranng algorthms, s the varable/dynamc dmensonalty. The dmensonalty of the II-Ter PSO here keeps changng at each teraton, as t depends on the network confguraton determned by the outer-ter PSO. The selecton of these PSO parameters plays an mportant role n the optmzaton as the performance of the PSO s qute senstve to control parameter choces. The fnal optmal PSO parameters have been selected by after extensve numercal smulaton wth varous combnatons. For both cases optmal refers to the set of PSO parameters whch results n fastest convergence along wth the most accurate dentfcaton of the consdered dynamcal system. A number of dfferent confguratons of parameters have been expermented wth tryng to acheve a balance between the computatonal tme and the performance. The fnal optmal set of PSO parameters for both for I and II ter algorthms have been lsted n Table 3. PSO Parameters Ter I PSO Ter I PSO Cogntve Learnng Rate CP = 2 CP = 1.85 Socal Learnng Rate Cg = 2 Cg =1.85 Inerta factor w = 0.9 w = 0.9 Number of Swarm Partcles N = 5 N=8 Maxmum number of teratons End Condton (number of teratons wthout update n the best values) 25 teratons 100 teratons VI. RESULT AND DISCUSSION Table 3. The PSO - PSO Parameters To address the complexty of helcopter dynamcs due to strong nter-axs couplng n ths paper the longtudnal and lateral states of the helcopter are decoupled and ths decoupled model s used for neural network based dentfcaton. For smulaton purposes n ths paper we only consder the Longtudnal Uncoupled Dynamcs. The results of the helcopter dynamcs dentfcaton s shown n Fg.2 depctng both the actual helcopter response and the predcted response of the model usng nature nspred technques (ABC/AIS/PSO). The actual system response and the predcted response of the system evolved nature nspred technques agree qute well over the entre range. The performance of the three nature nspred technques for longtudnal model as shown n Fg. 2. The sold lne represents the flght data and dotted lne represents network output of nature nspred technques (dotted blue ndcates PSO-NARX, dotted red ndcates ABC-NARX and dotted green ndcates AIS-NARX). It can be observed that, for the hghly nonlnear varaton n q and, the ABC and PSO appear to gve better mappng than the AIS.

13 A comparson of the NARX model predcted response and the nature nspred technques predcted response could be seen n Fg.3. It can be seen both ABC and PSO models perform comparatvely over the entre range along NARX model. However AIS doesn t gve better result for q and. Fg.2. Predcted Response and Actual Response for longtudnal cyclc doublet nput. Fg.3. Predcted Response of NARX model V/s predcted response of PSO/ABC/AIS-NARX model. From Table 3, t can be clearly seen that n comparson wth other model ABC-NARX evolved model s more effcent and also the tranng error s sgnfcantly reduced. Ths ndcates qute a close correlaton between the actual and the predcted response of the helcopter. More or less consstently an order of {1,1} (number of past nputs) and a relatve degree of the {1, 2, 0} (number of past outputs) evolved over a number of smulatons. But the evolved order and relatve degree was not very consstent between the models, a varaton of ±1 past value for both the nput and output was observed. Ths could possbly be a short-comng, t has to be further looked nto and further means to control ths nconsstency has to be ntroduced. But ths varaton n not substantal, and more or less the evolved agree well wth the actual physcs of helcopter dynamcs. The evolved confguraton can be accounted for as follows; t s qute evdent that the thrd output - normal acceleraton (az), does not have a bg nfluence on the response of the consdered longtudnally uncoupled helcopter system. For a longtudnally uncoupled helcopter system, among the outputs the greatest nfluence on the system response s due to the ptch rate (q). Ths s clearly dentfed by the PSO/ABC-NARX and reflected n the evolved confguraton. Further, ths confguraton for longtudnally uncoupled helcopter dynamcal system agrees well wth the ones reported n lterature and seems to be accurate from the aerodynamcs/physcs of the system.

14 VII. CONCLUSIONS Network parameters # I/p nodes # O/p nodes # hdden nodes Error NARX ABC - NARX AIS - NARX PSO - NARX Table 3. Comparson of NARX and PSO/ABC/AIS-NARX model The models proposed n ths paper have successfully crcumvented the maor drawback of the NARX model of havng to know system nformaton before hand retanng ts other superor characterstcs. The use of nature nspred technques for tranng the network effectvely overcomes the problems assocated wth back-propagaton. Nature nspred technques wth ther stochastc means are less lkely to get stuck n local mnma, makng them very robust and flexble. From the results t can be seen the correlaton between the predcted response and actual response s satsfactorly accurate n case of ABC-NARX and PSO-NARX. Further n comparson wth NARX, PSO-NARX and AIS-NARX, ABC-NARX evolved model s more effcent and also the tranng error s sgnfcantly reduced. The proposed model s computatonally qute resource ntensve; ths s because of the hgh degree of complexty nvolved wth the two ter archtecture, wth havng to execute the entre second ter for every obectve evaluaton of the frst ter algorthm. Further work has to be drected n adaptng ths model for a parallel envronment and makng t mplementaton more effcent. Also, n the current work we have restrcted the use of ths model to dentfy only the longtudnally uncoupled dynamcs of a helcopter. In future work, ths model can be used to dentfy the helcopter dynamcs more comprehensvely by consderng the laterally uncoupled dynamcs, and coupled dynamcs also. Ths would provde more nformaton on the capabltes and lmtatons of the model. Further these models can be used n vared applcatons such as flght smulators, modelng, control etc... VIII. REFERENCES Mane, R. E., & Ilff, K. W. (1985). Identfcaton of Dynamc Systems. AGARD AG-300, Vol. 2. Anon (1991). Rotorcraft System Identfcaton. AGARD-AR-280. Mller, W. T., Sutton, R. S., & Werbos, P.J. (1990). Neural Networks for Control. Cambrdge, MA. MIT Press. Narendra K. S. & Parthasarathy, K. (1989). Adaptve dentfcaton and control of dynamc systems, usng neural networks, n Proc. 28th CDC. Narendra. K. S. & Parthasarathy, K. (1990). Identfcaton and control of dynamcal systems usng neural networks. IEEE Transactons on Neural Networks, vol. 1, no. 1.

15 Narendra. K. S. & Parthasarathy, K. (1991). Gradent methods for optmzaton of dynamcal systems contanng neural networks. IEEE Transactons on Neural Networks. vol. 2, no. 2. Ichkawa. Y and Sawa. T, (1992). Neural network applcaton for drect feedback controllers. IEEE Transactons on Neural Networks, vol. 3,no. 2. Sastry, P. S., Santharam G. & Unnkrshnan K. P., (1994). Memory Neuron Networks for Identfcaton and Control of Dynamcal Systems. IEEE Transactons on Neural Networks, vol. 5, no. 2. Chen, S., Bllngs, S. A., & Grant, P. M., (1990). Non-lnear system dentfcaton usng neural networks. Internatonal Journal of Controls, vol. 51, no. 6. Hoskns, D. A., Hwang, J. N., & Vagners J., (1992). Iteratve nverson of neural networks and ts applcaton to adaptve control. IEEE Transactons on Neural Networks, vol. 3, no. 2. Tschler, M. B., (1996). System Identfcaton Methods for Arcraft Flght Control Development and Valdaton. Advances n Arcraft Flght Control, London, pp Vaya Kumar, M., Omkar, S. N., Ranan Gangul, Prasad Sampath, & Suresh S., (2003). Identfcaton of Helcopter Dynamcs usng Recurrent Neural Networks and Flght Data. In proc of 59th Annual Forum of the Amercan Helcopter Socety, Phoenx, Arzona, USA. Mudgere, D., Omkar, S. N., Vay Kumar, M., (2008). Identfcaton of helcopter dynamcs usng a PSO based approach. In proc of 64 th Annual Forum and Technology Dsplay, Amercan Helcopter Socety, Montreal, Canada. Eberchart R, & Kennedy J, (1995). A new optmzer usng partcle swarm theory. In Proc. Int. Sym. Mcro Machne and Human Scence, Japan. Eberchart, R., & Kennedy, J., (1995). Partcle swarm optmzaton. In Proc.of IEEE Int. Conf. Neural Networks, Perth, Australa. Karaboga, D., & Basturk, B., (2008). On the performance of artfcal bee colony (ABC) algorthm. Appled Soft Computng 8 (1), pp Karaboga, D., Akay, B., & Ozturk, C., (2007). Artfcal Bee Colony (ABC) Optmzaton Algorthm for Tranng Feed-Forward Neural Networks. Vol pp De Castro, L. N., & Von Zuben, F. J., (2000). The clonal selecton algorthm wth engneerng applcatons, In: Workshop on Artfcal Immune Systems and ther Applcatons. GECCO, pp Dasgupta D., (1999). Artfcal Immune Systems and ther Applcaton.Sprnger, Berln (Educaton seres).

16 Bäck, T., & Schwefel, H. P., (1993). An overvew of evolutonary algorthms for parameter optmzaton. Evolutonary Computng. vol. 1, no. 1. Haykn, S., (1994). Neural Networks A Comprehensve Foundaton. Second Edton. Macmllan College, New York. Goldberg, D. E., (1989). Genetc Algorthms n Search Optmzaton and Machne Learnng. Readng. MA: Addson-Wesley. De Castro, L.N., Tmms, J., (2002). An artfcal mmune network for multmodal functon optmzaton. In: Proceedngs of CEC 2002: IEEE Congress on Evolutonary Computaton, pp Forrest, S., & Perelson, A.S., (1991). Genetc Algorthms and the Immune System Parallel Problem Solvng from Nature. Lecture Notes n Computer Scence, vol Sprnger, Berln, pp McCall, J., (2005). Genetc algorthms for modellng and optmzaton. Journal of Computatonal and Appled Mathematcs. 184 (1), pp Coello Coello, C.A., Cruz Cortes, N., (2002). An Approach to Solve Multobectve Optmzaton Problems Based on an Artfcal Immune System. ICARIS 2002, pp Astrom, K. J., & Eykhoff, P. E., (1987). System Identfcaton- A Survey. Automatca, Vol. 7, 1987, pp Kalaba, R., & Spngarn, K., (1982). Control, Identfcaton and Input Optmzaton. Plenum Press. Kuzta, B., (1983). Modelng and Identfcaton of Dynamc Systems. Van Nostrand Renhold Company. Tschler, M. B., & Cauffman, M. G., (1992). Frequency-Response Method for Rotorcraft System Identfcaton: Flght Applcatons to BO 105 Coupled Rotor/Fuselage Dynamcs. Journal of the Amercan Helcopter Socety, Vol. 37, (3), pp Fu, K.H., & Marchand, M., (1983). Helcopter System Identfcaton n the Frequency Doman. 9th European Rotorcraft Forum, Stresa, Italy. Fu, K. H., & Kaletka, J., (1990). Frequency-Doman Identfcaton of BO 105: Dervatves Models wth Rotor Degrees of Freedom. 16th European Rotorcraft Forum, Glasgow, Unted Kngdom. Engelbrecht, P., & Ismal, A., (1999). Tranng product unt neural networks. Stablty Control: Theory Appl., vol. 2, no. 1 2.