FEEDFORWARD NEURAL NETWORK COMBINED WITH NUMERICAL INTEGRATOR STRUCTURE FOR AN INVERTED PENDULUM MODELING AND SIMULATION.

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1 Anais d 12 O Encntr de Iniciaçã Científica e Pós-Graduaçã d ITA XII ENCITA / 26 Institut Tecnlógic de Aernáutica Sã Jsé ds Camps SP Brasil Outubr 16 a FEEDFORWARD NEURAL NETWORK COMBINED WITH NUMERICAL INTEGRATOR STRUCTURE FOR AN INVERTED PENDULUM MODELING AND SIMULATION. Daniel van Berghem Mtta Divisã de Ciência da Cmputaçã Institut Tecnlógic da Aernáutica - ITA Pç. Marechal Eduard Gmes 5 Vila das Acácias CEP Sã Jsé ds Camps SP Brasil dvbmtta@bl.cm.br Palma Maria Silva Rcha Departament de Engenharia Elétrica Universidade Estadual Paulista - UNESP Av. Dr. Aribert Pereira da Cunha 333 Pedregulh CEP Guaratinguetá SP Brasil. palma@feg.unesp.br Paul Marcel Tasinaff Divisã de Ciência da Cmputaçã Institut Tecnlógic da Aernáutica - ITA Pç. Marechal Eduard Gmes 5 Vila das Acácias CEP Sã Jsé ds Camps SP Brasil tasinaf@ita.br Abstract: This article aims t make the simulatin f nn-linearized inverted pendulum mdels using the set neural artificial netwrk cmbined with high accuracy numerical integratin. T achieve this gal we used a feedfward neural netwrk and the gradient and kalman algrithms t train this net. Using this algrithms we addressed the questin f cmputacinal csts t train and simulate these mdels. These csts includes CPU time t prcess the neural netwrk training and alcatin f memry t dispse data. As the gradient training methd in the neural netwrk gives the derivative functin f the system ( y`= f (yt) ) we used numeric integratrs t btain the utput f the system. S a furth rder Runge-Kutta and an Adams-Bashfrth integratr were matched with the utput f the neural netwrk ( y` ) s that we culd have y. Then it was pssible t analyze graphically the perfrmance f these integratin algrithms cmpared with the true utput f the physic dynamic system given by the slutin f the mdel`s analytic differential equatin. Key wrds: FeedFrward Neural Netwrk Inverted Pendulum Mdel Numerical Integratin nn-linear system Kalman Filter learning gradient methd. 1. Intrdutin Neural netwrks have been used widely fr identificatin f nn-linear dynamic systems (Carrara 1997; Chen et al 1992 and Hunt et al 1992). A feedfrward neural netwrk can be used in the structure f an rdinary differencial equatin numerical integratr algrithm t apprximate and t replace the derivative functin f the rdinary differential equatins dynamic system mathematical mdel a discrete mdel is btained which has quite advantageus characteristics (Ris Net 21 and Wang and Lin 1998). With this apprach the difficulty with t many inputs in the training f the neural netwrk is alleviated since it is nly necessary t learn an algebraic and static functin and the inputs are ccurrences f the state and cntrl variables in their envelpe f variatin. It is pssible fr example t train a feedfrward neural netwrk in instantaneus derivate methd with was prpsed by Wang and Lin in 1998 t represent nn-linear dynamic systems. In this paper we used a feedfward neural netwrk and instantaneus derivate methd t represent a nn-linear and a linear inverted pendulum mdel using Gradient and Kalman algrithms t train f the net. All this training used the sftware implemented by Tasinaff in 24. This sftware enabled us t vary several parameters f the net cnfiguratin such as integratin step and s we culd bserve their influence in the MEQ (Average Squared Errr) and hw they differently affect linear and nn-linear mdel training. Using this sftware we addressed the questin f cmputatinal csts t train and simulate these mdels. These csts includes CPU time t prcess the neural netwrk training and lcatin f memry t dispse data. Thugh this paper is fcused in testing the prcedure in an applicatin f interest it is rganized t guarantee a cmprehensive presentatin fr the benefit f thse nt familiar with the tested prcedure. In what fllws Sectin 2 presents the fundaments f feedfrward neural netwrks. Sectin 3 presents the cncepts f numerical integratr f discrete frward mdel and neural numerical integratr. In sectin 4 the simulatin and the results f the applicatin t the prblem f mdeling the inverted pendulum are presented. In the last sectin sme cnclusins and suggestins t future researches are drawn.

2 Anais d XII ENCITA 26 ITA Outubr Fundamentals: FeedFrward Neural Netwrks. It is nt easy t find ut a unique definitin abut what is an artificial neural netwrk (Zurada 22). In very general terms artificial neural netwrks are cmputatin mdels made up f artificial cmputatinally mdeled neurns that result frm explring analgies with bilgical neural netwrks f life rganisms. The feedfrward multiplayer perceptrn is a neural netwrk where Perceptrn type f artificial neurns (Braga et al 2) (Figure 1) are rganized in layers cnnected in a feedfrward architecture. The Perceptrn neurn is a classifier where inputs (xj s) are initially classified with respect t a hyperplane characterized by the weights (wij s) and a bias (bi) in series with a secnd nn linear classificatin dne by an activatin functin (fi(.)) which limits the classificatin t be inside asympttic nrmalized limits f (1) in the case f the Lgsig activatin r (-1+1) in the case f the Tansig activatin. nk-1 k k k k-1 k x i = f w ij x j b i p/ i = n k j= 1 (1) Figure 1 The Perceptrn Neurn. The Feedfrward neural netwrk is made up f clumns f Perceptrns (f s) and s the utput vectr x k f a layer k is illustrate and simplificatin as in figure 2. Figure 2 Input/Output f a General Hidden Layer k. 3. Numerical Integratr Discrete Frward Mdel and Neural Numerical Integratr Cnsider a dynamic system with a mathematical mdel given by a set f rdinary differential equatins: x = f(x u) (2) where x Є R n is the state vectr; u Є R m is the cntrl vectr; f( x u) is the derivative functin cming frm physical laws gverning the dynamic system. Cnsider nw an rdinary differential equatin (ODE) numerical integratr (Ster et al 198) t get a discrete apprximatin f the system f Eq.(3): x = x( t + t) f ( x( t) x( t t)... x( t n t); u( t)... u( t n t); t) (3) n

3 Anais d XII ENCITA 26 ITA Outubr f ( x( t) x( t t)... x( t n t); u( t)... u( t n t); t) is the functin that results frm using where n evaluatins f the derivative functin f Eq. (2) in the numerical integratr algrithm; n is related t the rder f the apprximatin; if its value is greater than zer ne has the situatin where a finite difference type f integratr is used (fr example an Adams-Bashfrth methd); if it is zer a single step type f integratr is used (fr example a Runge- Kutta methd); and t the step size is assumed sufficiently small t assure u(t) cnstant alng the discretizatin interval. The numerical integratr in Eq.(3) can be used recursively as an apprximate discrete predictive mdel f the dynamic system f Eq.(2) in internal mdel cntrl schemes. The errr in each step can be cntrlled by varying step size and r the rder f numerical integratr; and the resulting numerical algrithm can be prcessed in parallel fr each cmpnent f the state f the dynamic system. In the dynamic system f Eq. (2) the derivative functin in the ODE mathematical mdel is an algebraic functin that can be apprximated by a feedfrward neural netwrk: x = f ( x u) fˆ ( x u wˆ ) (4) f ( x u w ˆ ) is t represent the neural netwrk trained; and ŵ the neural netwrk learned weights. Cnsider nw this neural apprximatin f the derivative functin in the structure f an rdinary differential equatin (ODE) numerical integratr (Ster et al 198) t get a discrete apprximatin f the system f Eq.(2) : where ˆ x( t + t) fˆ ( x( t) x( t t)... x( t n t); u( t)... u( t n t); t; wˆ ) (5) n The neural numerical integratr in Eq.(5) is an apprximatin f the ODE numerical integratr f Eq. (3) and thus an apprximate discrete predictive mdel f the dynamic system f Eq.(2) which can be used as an internal mdel in cntrl schemes. 4. Simulatin and Results As shwn belw in figure 3 ur mdel cnsists f an inverted pendulum. We mdeled it using the appraches f a nn-linear mdel. The table 1 gives the nminal values used fr mdeling the bdy cnsiderate. Figure 3 Inverted pendulum. Table 1 - Inverted pendulum parameters. The nn-linear mdel can be described by the fllwing equatins where x and theta are as abve. M + m x + bx + ml θ θ ml θ θ = F 2 ( I + ml ) θ + mgl sinθ = mlx csθ (6) 2 ( ) cs sin

4 Anais d XII ENCITA 26 ITA Outubr Our bjective with this paper is divided in tw tasks: train a Neural Netwrk (NN) t learn the differential equatin in term f state variables that fits the nn-linear mdel f the inverted pendulum and after simulate the time respnse f the mdel t different initial cnditins. T train the NN we adjusted the number f training patterns given t the net and the learning rate s that we culd have a mdel gd enugh and minimize the csts in terms f prcessing. Besides we varied: x frm -5 t 5 meters; dx/dt frm -14 t 14 m/s; theta frm pi/12 t pi/12; d(theta)/dt: frm -1 t 1 rad/s F ( external excitatin f the system ) frm -1 t 1 Newtns We attempt t reach MEQ in the training f the NN abut 1^-7.Infrtunately the minimum MEQ we gt in nnlinear case was *1^-5. We used 35 patterns pints generated randmly within the state-space described abve. In additin we used tw algrithms t train the net : Kalman Filter and Gradient. The functining f this algrithms aren t in the scpe f ur wrk. After setting the main parameters f the simulatin as the step f integratin initial cnditin f state variables e time interval f simulatin we culd get sme results as shwn belw. Representaça Neural de Sistema Dinamic Na-Linear.2.1 c/ MEQ=1.642e-6 State Varible State Varible State Varible Original Furth Order.5Runge-Kutta 1 Neural Furth Nrmalized Order Runge-Kutta Time.6 Neural Furth Order Adams-Bashfrth State Varible Figure 4 Graphics shwing the evlutin in time interval [1] f state variables f the nn-linear mdel. As shwn in Figure 4 the time interval f simulatin was reduced frm ( as the initial time f simulatin ) until 1 ( end pint f simulatin ). It s easy t see that this simulatin was much mre successful than the first ne. It s due t the capability f apprximatin f the nn-linear mdel. Thus [1] is a reliable interval t this mdel. The prpagatin MEQ was drastically decreased with this change in simulatin interval. Representaça Neural de Sistema Dinamic Na-Linear.2 c/ MEQ=1.2812e-5 State Varible State Varible State Varible Original Furth Order Runge-Kutta Neural.4 Furth Order Runge-Kutta Neural Furth Order Adams-Bashfrth.2 State Varible Figure 5 Graphics shwing the evlutin in time interval [3] f state variables f the nn-linear mdel.

5 Anais d XII ENCITA 26 ITA Outubr In Figure 5 it s clear that we have reached a gd result with the nn-linear apprach with the new interval [ 3] secnd. This cnfirms the imprtance f the nn-linearity in the inverted pendulum mdel that itself can be applied besides the linear cnfinatin. It s very imprtant t ntice that the step f integratin cnsiderably affects the precisin f the neural netwrk estimatin. Nevertheless decreasing the step f integratin means an increase in the csts f cmputer prcessing time. 5. Cnclusins and suggestins t future researches This article reached the ur aims namely: the limitatins f a mathematical mdel has t represent a physical ne; the implementatins and simulatins f a nn-linear general mdel; the ptentialities and limitatins f Feedfrward neural netwrk with a single internal layer t represent and learn a nn-linear mdel and the influence f NN parameters and settings in the its perfrmance and thrughput. S we hpe we have cntributed t the understanding f FeedFrwad NN and its implementatin and as suggestin it wuld be very interesting t develp this prblem using a RBF Neural Netwrk architecture. In this case the RBF is a gd estimatr f linear parameters and pssibly the results f the linear mdel estimatin wuld be better. Afterwards it wuld be interesting as well as t implement a FeedFrward NN with mre than ne internal layer t see hw it wuld behave in terms f nn linear estimatin and cmputatinal csts. 6. References Braga A. P. Ludermir T. B. Carvalh A. C. P. L. F 2 Redes Neurais Artificiais Teria e Aplicações. LTC. Carrara V Redes Neurais Aplicadas a Cntrle de Atitude de Satélites cm Gemetria Variável Tese de Dutrad INPE Sã Jsé ds Camps. Chen S. Billings S. A Neural Netwrks fr Nnlinear Dynamic System Mdeling and Identificatin Int. J. Cntrl Vl. 56 N. 2 pp Hunt K. J. Sbarbar D. Zbikwski R. Gawthrp P. J Neural Netwrks fr Cntrl Systems A Survey Autmatica Vl. 28 N. 6 pp Ris Net A. 21 Dynamic Systems Numerical Integratrs in Neural Cntrl Schemes V Cngress Brasileir de Redes Neurais. Ster J. and Bulirsch R. 198 Intrductin t Numerical Analisys Springer Verlag N.Y.. Tassinaf M. 24 Estruturas de integraçã neural Feedfward testadas em prblemas de cntrle preditiv Tese de Dutrad INPE Sã Jsé ds Camps. Wang Y.-J. Lin C.-T Runge-Kutta Neural Netwrk fr Identificatin f Dynamical Systems in High Accuracy IEEE Transactins On Neural Netwrks Vl. 9 N. 2 pp Zurada J. M Intrductin t Artificial Neural System. West Pub. C.