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Unformatted text preview: ( Neural Networks a nd Fuzzy Systems Proceedings of the National Conference 23-25 July 1997 Editor K.M.Mchata School of Computer science and Engineering Anna University Chennai - 600 025. NEURAL NETWORKS FOR CG:.!TROL AND IDENTIFICATION: 2. There are needs today that cxnot be successfully addressed with the existing conventional control theory. They mainly pertain :o the area of uncertainty. Heurislic methods may be needed to tune the par;:rneters of an adaptwe control law. New COntrOi laws to perform novel control function, sbsz::: be designed v;hi!e the system is in OperatiOn. Learning from past experience ar,d planning control acticns may te necessary. Failure ?ons h a v e been pedormed in the past by detectlo” ;Ind identific;ltion i s needc4 Tnise f,!rcs h,,mwn r:,+wts.,~r. here we need an m;f:.:,>m! -sntrol re~:ac,nq !he h u m a n . The p r o b l e m o f identifG!::w conxtz o f set:,ng up ‘! zultabbj parameterized identification model and adjusting :“‘. 7 _I-,,. ,,;;:e rs ci the --o?ei. to optimize a performance function, based on the error between the plan! an d the identified model output. Structure of the identification model is chosen to be identical to that of the plant. By assumption , weight matrices of the neural networks in the identification model exists ,so that. for the same initial conditions, both plant and model have the same ou:qu: for any specified input. Hence identification procedure consists of adJust,i,, _’ F ; .:xwters. of the mxral network model using error correction learning.[3] 2.1. NONLINEAR SYSTEM IDENTIFICATION Multilayer Feed Forward Neural Network(MFNN) Universal function approximation property is exploited for the identification of nonlinear systems [I] - (21. A structure with single hidden neuron with sufficient neurons and a nonlinear activation function can approximate any continuous rwrZ:!-ir function. Consider thz following nonlinear almost discrete time Single Input Single Output(SIS0) system. y(k) = Jy(k -, I),y(k - 2), :. ,y(k - n), zr(k - I), u(k - 2), u(k - m)]. . . . . . . ...(l) Where u(k) and y(k) are measured inpur _ outpct, m and n ?re assumed known which determines the order of the system. Nonlinear idwitification problem can be stated as estimate the unknown nonlinear function f(.) using the measured input and output data (y(k).u(k)]. Let x(k) = b(k - l),y(k - 2),. .y(k - ,~),rr(k - I). rt(k - 2), .I/(k - m)]. ..(2) Than the neural network estimate of t(.j is ::(kj .j[r(k)] = I=, 2 w,(k#,[x(k)] (3) I and the logistic activation function is used .f(x) = 7 in the network for the neurons in the l i e hidden layer. The network architecture 1s shoijn in : 3 1 below . Fig .l MFNN Architecture for nonlinear system Identification. Conventional method of weight updation in >.*,TUN is the back-propagation algorithm [4] - [5]. But the convergence of the back-propagation algorithm is very slow because of its close relation with the Least Mean Square algorithm.ln this work for the identification of nonlinear systems, the MFNN is trained using an Extended Least square (Kalman) type filtering algorithm applied to back-propagation learning. The gso of RLS(Recursive Least square) algorithm offers the following advantages. 1) Fast speed of convergence. 2) Built-in learning rate parameter. The term extended is used here to account for the fact that the neurons of a multilayered perceptron are nonlinear, and therefore have to be linearized in order to accommodate the application of standard RI..? ‘iltering algorithm. [4] 3. EXTENDED KALMAN TYPE OF BACK-PROPAGATION LEARNING: Consider a MFNN charactenzed ‘?y a weight vector w representing the free parameters of ali the neurons in the network. The cost function to be minimized during training, is defined in terms of a total of a N Input-output examples as follows We simplify the application of ti:e ex:ended kalman filtering by partitioning the global problem into a number of subproblems. each one at a neuron level. Consider then neuron i which may be located anywhere in the network.During training the behaviour of the neuron i may ;: L -<jr,! s a nOnlinear dynamical system, which in the context of Kalman filtering theory may be described equation as follows w, (77 + 1) = lY;(/7). by the state and measurement (5) d;(,I) = (1(X,! i,r)w (,I!) 7 e;(/‘; ‘6) where X (n) is the input vector of neuron i. w (n) is the weight vector. Since the activation function is differentiable. applying Taylor series ti7 ei;:.?d (6) about, the current estimate w (n) and thereby linearize the actlvz!isn function as r$(X~(n)w(r,)) = (1; (n)w;(/I) -r [*(,u,r(,l)ri~(,,) - $‘(,l);u&f)]. .(7) where C/&I)‘= [w],,n,;,,n, using b(x) = 5 as the activation func!ion we get r,, (77) = j, (I))[! - ;, (,l)]X, (77). .@) The first term on the right of (7) is the desired linear term and the remaining terms representing the modeling error is neglected and (6) is obtained as di(77) = q’(II)IY,(/l) +e,(n) ,............. (9) Equations (5) and (9) describes the lineariztli dynamic behaviour of :he neuron i. The measurement error e (n) in eqn. (9) is a localized error the instantaneous estimate of which is given by c(u) = - -;>Y$ .(lO) Differentiation of eqn (10) corresponds to the back propagation of the global errc~r to the output of neuron i, just as done in the standard back-propagation From eqns. (5) and (9) the standar? ?..C; ,;gorithm is used t c :mke an csf:mate of the synaptic weight vector w (n) of neuron i. Ttw rcsulti;:a :.;!utior, IS dc:ined by the !ollowing system of recursive equations.(6] r,(u) = /:(,/)q,(/l) . ..(I I) kn(77) = /;(/t)[i -+I,’ (,~)c/,(rl)]“.... w;(,7+l)= ?:(,I + ~l~,(II)Te,(,l)ic;(“~ 1 ) (17) ( 1 3 = f,(n) - k,(7t)rr’(lzj ) .(14) Where n=l.Z,3,....N and N is the 1~17’ ?bmber of examples in the training set. P, (n) is the current estimate o: !he inverse of the co var~aixzs m?t:ix of q, (n). To start with P can be assumed to be a diagonal matrix with a large value as !he diagonal elements. k, (n) is the Kalman gain. The algorithm defined by eqns.(l 1) through (14) is called the Multiple extended Kalman algorithm (MEKA).Results 4. on the wnulation of nonlinear systems are as follows SIMULATION RESULTS The capability of neural networks to approximate almost all linear and nonlinear output maps makes it part;ctslarly suitabl- icr tne identification of nonlinear systems.ln this chapter some 4.1 exam;‘~2s * 3: nonlinear p’-ln:,der:ificatipn are :-zented. SINGLE INPUT SINGLE OUTPUT PLANT(SIS0): For the purpose of si~irla.i~on y(r) = (0.8 - 0.5c 1 O.lrr(/-I)rr(/-2) !hc following non!inear plant model was chosen [3]. ’ ‘“‘)y(i - I) - (iI 3 + 0.9c -““‘-“)y(l - 2) + I/(/ - I) + 0.2rr(t - 2) + .(l5) As explained before an MFNN architecture is chosen to model the nonlinear system From eqn.(l5). for estimating y(t) the samples required are y(t-l).y(t-2).u(t-1) and u(t-2). Thus the neural network model reqwr< 4 input nodes and 1 output node. A single hidden layer with 15 neurons is se!%:ed. Tn!s choice oi 15 is arbitrary. Training data : 500 training examples are generated using (15) with random inputs. W e i g h t U p d a t i o n : MFNN’s a r e trained u s i n g t h e E x t e n d e d K a l m a n t y p e o f b a c k propagation algorithm. Results: To speed up the iearning process the trairing data is shuffled at random from one epoch to the other, so that in effect the nclwork seems to perceive altogether a different training data from one epoch to another thereby making the search dovin the weight space stochastic. EPOCH 1850 ?ERFCl,i.$AIdCE INDEX O.CO18 Both step response and response !o a random input sequence is shown in figs 2 and 3. l7g3 Rnpuaeturslrpivp~I t+QRnpoacto~nNlominpl* 4.2 MULTI INPUT MULTI OUTPUT (MIMO) NONLINEAR SYSTEM: The system used for the purpose of simulation is an Antenna tracking system shown in fig 4..An ante?na is mounted on a rod which can rotate from 0’ to 180’ A motor capable of applying a rotational force is attached to the antenna arm. The torque is proportional to the input current u. Gravity pulls the antenna down and viscous friction resists motion. [7j Fig 4a. Antenna tracking system. The following equations will be treated as the real system that the neural network is required to identify. _....._....._... (16) B(I)=10sin(0(r))-2d(r)su(r).......(17) The objective is to train the neural ne:work !o oredict the :x?onse one time step ahead (time step is chosen as 0.05s). The system has 3 inputs and 2 outputs as shown in fig 4b. Fig 4b Nonlinear Plant model. Therefore the neural network model will also have 3 inputs and 2 outputs in addition to a threshold at the input node. An arbitrary choice of 10 neuron is made for the hidden layer. Trathtng data : We choose to make 700 measurements on the antenna tracking system. 500 of these measurements ‘i-re taken one time step after random initial angles, vetocities and CUW3ltS. 200 extra measu-emen!s v!ere t-ken in a similar way during steady state behaviour wherf input current had been held fixed long enough for velocity to go to aero, StoPPin Crfterla : The network is trained using the Extended Kalman type of hack ProPagation algorithm. till the sum of squared errors of output angle falls below 2’ . once the network iS trained to errcrs of this size. it can be a satisfactory model to the original system. ResuttS : QnCa the network has been trained it is tested with a step input for 2 different inktat ConditionsIt took 7990 epoch to train the network whereas the convantiohat bar-k . propagation algorithm would take 54.000 epochs.[Tj The Plot of output angle and rate of change of angle for initial condition of 100~ and 100 is shown in figs 5and 6 respectively. 0.a 5. CONCLUSIONS From the above studies on the simulat:on of SE0 and MIMO nonlinear systems the following can be concluded. l)The neural networks are ideally suited for the identification of nonlinear systems as many of the conventional methods for identification of nonlinear systems require much larger computational effort and there is no direct procedure that will guarantee the convergence of the estimated parameters 2) The design aspect of the neural network such as the number of hidden layers, the number of neurons etc..require extensive simulation to reach an optimum figure as there are no well defined rules for the choice of these. However- by training the neural network on larger number of samples it is possible to identify the system even when conventional nonlinear system identification methods may not yield satisfactory results in a reasonable amc’;nt cf cxcputational effort. 6. REFERENCES 1. SIMON HAYKIN. Neural Networks-A Comprehensive foundation - IEEE press1994. 2. MADAN M GUPTA and DANDINA H RAO: Neuro-Control Systems -Theory and Application: IEEE press-A selected Reprint Volume pp 247-256. I 3. S CHEN, .S A BILLINGS and GRANT: Nonlinear System using Neural Networks: Int. Journal Identification Of Control,Vol51 ,No.6, 1990, pp 209-214. - 4. KUMPATHI S NARENDRA and KANNAN PARTHASARATHY:ldentification Control of dynamic systems using Neural Neural.Networks.~‘ol 1 .No.l .March 1990.,~‘;- *’ and Networks:IEEE Transactions on ’ ’ 5. S JAGANNATHAN and F L LEWIS; Identification of Nonlinear Dynamical Systems Using Multilayered Neural Networks: Automatica Vol 32. No 12. Dee 1996, pp 1707-1712. : 6. A U LEVtN and K S NARENDRA: Recursive Identification UsingFeed forward Neural Networks: Int. Journa: Of Control, Vol 61, No 3. March 1995. pp 533-547. 7. Mark Beale and Howard Demuth. MATLAS Notes, Reprined. from Al Export. July 1992, Freeman Inc. ...
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