{"title": "An Integrated Architecture of Adaptive Neural Network Control for Dynamic Systems", "book": "Advances in Neural Information Processing Systems", "page_first": 1031, "page_last": 1038, "abstract": null, "full_text": "An Integrated Architecture of Adaptive Neural Network \n\nControl for Dynamic Systems \n\nRobert L.  Tokar2 \n\nBrian D.McVey2 \n\n'Center for Nonlinear Studies,  2Applied Theoretical Physics Division \n\nLos Alamos National Laboratory,  Los Alamos, NM, 87545 \n\nAbstract \n\nIn  this  study,  an  integrated  neural  network  control  architecture  for  nonlinear  dynamic  systems  is \npresented.  Most of the recent emphasis in  the neural network control field has no error feedback as the \ncontrol  input,  which  rises  the  lack  of adaptation  problem.  The  integrated  architecture  in  this  paper \ncombines  feed  forward  control  and error  feedback  adaptive  control  using  neural  networks.  The  paper \nreveals the  different  internal  functionality  of these  two  kinds of neural  network  controllers  for  certain \ninput  styles,  e.g.,  state  feedback  and error feedback.  With  error  feedback,  neural  network  controllers \nlearn  the  slopes  or  the  gains  with  respect  to  the  error  feedback,  producing  an  error  driven  adaptive \ncontrol  systems.  The  results  demonstrate  that  the  two  kinds  of control  scheme  can  be  combined  to \nrealize  their individual  advantages.  Testing  with disturbances added to  the  plant  shows  good  tracking \nand adaptation with the integrated neural control architecture. \n\n1  INTRODUCTION \n\nNeural  networks  are  used  for  control  systems  because  of their  capability  to  approximate  nonlinear \nsystem  dynamics.  Most  neural  network  control  architectures  originate  from  work  presented  by \nNarendra[I),  Psaltis[2)  and  Lightbody[3) .  In  these  architectures,  an  identification  neural  network  is \ntrained to function as a model for  the  plant.  Based on the neural network identification model,  a neural \nnetwork  controller  is  trained  by  backpropagating  the  error  through  the  identification  network.  After \ntraining,  the  identification  network  is  replaced  by  the  real  plant.  As  is  illustrated  in  Figure  1,  the \ncontroller  receives  external  inputs  as  well  as  plant  state  feedback  inputs.  Training  procedures  are \nemployed  such  that  the  networks  approximate  feed  forward  control  surfaces  that  are  functions  of \nexternal inputs and state feedbacks of the plant (or the identification network during training). \nIt is  worth noting that in  this architecture,  the  error between  the plant output and the  desired output of \nthe reference model  is not fed  back to the controller, after the training phase.  In other words, this error \ninformation is  ignored when  the neural network applies  its  control.  It is  well  known  in  control  theory \nthat  the  error  feedback  plays a  significant role  in  adaptation.  Therefore,  when  model  uncertainty  or \nnoise/disturbances are  present,  a  feed  forward  neural  network  controller  with  only  state  feedback  will \nnot adaptively update the control  signal.  On line training for the neural controller has been proposed to \nobtaip  adaptive  ability[I)[3).  However,  the  stability  for  the  on  line  training  of the  neural  network \ncontroller is unresolved[1][4]. \nIn  this  study,  an  additional  nonlinear  recurrent  network  is  combined  with  the  feed  forward  neural \nnetwork  controller  to  form  an  adaptive  controller.  This  added  neural  network  uses  feedback  error \nbetween the reference model output and the plant output as an  input  In  addition,  the  system's external \n\n\f1032 \n\nLiu  Ke,  Robert L.  Tokar,  Brian D. McVey \n\ninputs and the plant states are also input to the feedback network.  This architecture is used in the control \ncommunity,  but not  with  neural  network  components.  The  approach  differs  from  a  conventional  error \nfeedback  controller,  such as a gain  scheduled  PID  controller,  in  that the  neural  network  error  feedback \ncontroller implements a continuous nonlinear gain  scheduled hypersurface,  and  after  training,  adaptive \nmodel  reference  control  for  nonlinear  dynamic  systems  is  achieved  without  further  parameter \ncomputation.  The  approach is tested on  well-known  nonlinear control  problems  in  the  neural  network \nliterature, and good results are obtained. \n\n2  NEURAL NETWORK CONTROL \n\nIn  this section, several different neural network control  architectures are  presented.  In  these  structures, \nidentification neural networks, viewed as accurate models for real plants, are used. \n\n2.1  NEURAL NETWORK FEED FORWARD CONTROL \n\nThe  neural  network  controllers  are  trained  by  backpropagation  of errors  through  a  well  trained  neural \nidentification network.  In this architecture, the state variable yet) of the system is  sent back to the neural \nnetwork, and the external input x(t) also is input to  the network.  With  these  inputs,  the neural network \nestabJishes a feed  forward  mapping  from  the  external  input x(t) to  the  control signal  u(t).  This control \nmapping is expressed as a function  of the external input x(t) and the plant state yet): \n\nu(t)==j(x(t), yet\u00bb~ \n\n(1) \n\nwhere x(t)=[x(t), x(t-l), .. J, andy(t)=[y(t), y(t-l), . .Y. \nThis  neural  network  control  architecture  is  denoted  in  this  study  as  feed  forward  neural  control  even \nthough  it  includes  state  feedback.  Neural  control  with  error  feedback  is  denoted  as  feedback  neural \ncontrol. \n\nx(t) \n-..:...r-----~Ref. Modelf-------, \n\nx(t) \n-~----~Ref. Modell-----.., \n\ne(t+ 1) \n\nControl NN \n\ny(t+ 1) \n\nu(t)  ,-------, \n\nf----+-.. \n\ny(t+ 1) \n\nFigure  I Neural Network Control Architecture. \nID  NN represents the identification network. \nRef. Model means reference model, and NN \nmeans neural network. \n\nFigure 2 Neural Network Feedback Control \nArchitecture \n\nDuring  the  training  phases,  based  on  the  assumption  that  the  neural  identification  network  provides  a \nmodel for  the plant,  the gradient information needed for error backpropagation is obtained by calculating \nthe  Jacobian of the  identification network.  The following  equation describes this process for the  control \narchitecture shown in Figure  I.  If the cost function is defined as E,  then the gradient of the cost function \nwith respect to weight w of the neural controller is \n\n\fAn Integrated Architecture of Adaptive Neural  Network Control for Dynamic Systems \n\n1033 \n\na E  J a Yt-l \na E  a E a u \na: = a; a w  +  a u a Yt-l + a Yt-l  --a;-\n\n(a E  a u \n\n(2) \n\nwhere u is  tbe control signal and YI-1  is tbe plant feedback state. \nAfter tbe  training  stage,  tbe neural network supplies a control  law.  Because neural networks have  the \nability  to  approximate  any  arbitrary nonlinear functions[5],  a feed forward neural network  can  build a \nnonlinear controller, which is crucial to tbe use of tbe neural network in control engineering.  Also, since \nall  tbe  parameters  of the  neural  network  identification  model  and  tbe  neural  network  controller  are \nobtained  from  learning  through  samples,  matbematically  untraceable  features  of  tbe  plant  can  be \nextracted from tbe samples and imbedded into tbe control system. \nHowever, because tbe  feed forward controller has no error feedback,  tbe  controller can not adapt to  tbe \ndisturbances occurring in  tbe plant or tbe reference model.  This problem is of substantial importance in \ntbe  context of adaptive  control.  In  tbe  next  subsection,  error feedback  between  tbe  reference  models \nand tbe plant outputs is introduced into neural network controllers for adaptation. \n\n2.2  NEURAL ADAPTIVE CONTROL WITH ERROR FEEDBACK \n\nIt  is  known  that  feedback  errors  from  the  system  are  important  for  adaptation.  Due  to  the  flexibility  of the \nneural  network  architecture,  the  error  between  the  reference  model  and  the  plant  can  be  sent  back  to  the \ncontroller  as  an  extra  input. \nIn  such  an  architecture,  neural  networks  become  nonlinear  gain  scheduled \ncontrollers with  smooth continuous gains.  Figure 2 shows the architecture for the feedback neural control. \nWith tbis architecture,  tbe  neural network control  surface is not tbe  fixed mapping from  tbe x(t) to u(t) \nfor  each  state y(t),  but instead  it  learns  tbe  slope  or  tbe  gain  referring  to  tbe  feedback  error  e(t)  for \ncontrol.  This gain  is  a continuous nonlinear  function  of tbe  external  input x(t) and  tbe  state  feedback \nyet).  Figure  3  shows  tbe  recurrent network architecture  of tbe  feedback  neural  controller.  The  output \nnode  needs  to  be  recurrent  because  tbe  output witbout  tbe  recurrent  link  from  tbe  neural  controller  is \nonly a correction to  tbe old control  signal, and  tbe new control  signal should be tbe  combination of old \ncontrol signal and tbe  correction.  The otber nodes of tbe  network can be feed forward  or recurrent.  If \nwe denote tbe weight for tbe output node's recurrent link as w.,  tben  tbe output from  tbe recurrent link is \nw.u(t-l).  The following equation describes the feedback network. \n\nu(t) = wbu(t-I )+j(X(t), y(t), e(t\u00bb \n\n(3) \n\nwhere j(.) is  a  nonlinear function  established by  tbe  network for  which  tbe  recurrent  link  output is  not \nincluded and e(t)=[e(t), e(t-I), ... f \nTo  compare  tbe  control  gain  expression  with  conventional  control  theory,  consider  tbe  Taylor  series \nexpansion of tbe network forward mappingj(.), equation (3) becomes \n\nu(t) = w.u(t-l) + !'(x(t). yet\u00bb~ e(t)+ j\"(x(t), yet\u00bb~ e2(t)+... \n\n(4) \n\nwhere f'(x(t),  y(t\u00bb=[  i1j(x(t),  y(t),  e(t\u00bb/ae(t),  aJ!:x(t),  y(t),  e(t\u00bbIi1e(t-I),  ... ]. \nignored and gO representsf'O, we get \n\nIf  high  order  terms  are \n\nu(t) = wbu(t-I)+ g(x(t), yet\u00bb~ e(t) \n\n(5) \n\n\f1034 \n\nLiu  Ke,  Robert L. Tokar,  Brian D.  McVey \n\nwhich is a gain scheduled controller and the gain is the function of external input x(/) and the plant state \nIt is clear that when w.=l.O,  g(.)  is a  constant vector and e(/)=[e(t),  e(t-l), e(t-2)]T,  the  feedback \ny(/). \nneural  network  controller  degenerates  to  a  discrete  PID  controUer.  Because  the  neural  network  can \napproximate arbitrary nonlinear functions through learning,  the neural network  feedback  controller can \ngenerate a nonlinear continuous gain hypersurface. \n\nRef. Model \n\nFigure 3 Feedback Neural Network Controller \n\nFigure 4 Integrated NN Control Architeture. \n\nIn the training process,  error backpropagating through the identification network is used.  The process is \nsimilar to the training of a feed forward neural controller, but the resulting control surface is completely \ndifferent due  to the different inputs.  After training,  the  neural network is  able  to  provide a  nonlinear \ncontrol  law, \nthat  is,  the  desired  model  following  response  can  be  obtained  with  fixed  controller \nparameters for  nonlinear dynamic systems.  Traditionally,  the  control of the  nonlinear plant is derived \nfrom continuous computing of the controller gains. \nThis  feedback  controller  is  error  driven.  As  long  as  an  error  exists, \naccording to the error and the gain.  This kind of neural controller is an adaptive controller in principle. \n\nthe  control  signal  is  updated \n\n2.3  INTEGRATED NEURAL NETWORK CONTROLLER \n\nThe  characteristics  of feed  forward  and error  feedback  neural  control  networks  are  described  in  the \nprevious subsections.  In this section. the two controllers are combined.  Figure 4 shows the architecture. \n\nIn this architecture,  we include both feed forward and feedback neural network controllers.  The control \nsignal is the combination from  these  two networks'  outputs.  In  the  training  stage,  it is our experience \nthat  the  feed  forward  network  should  be  trained  first.  The  feedback  network  is  not  included  while \ntraining the feed forward network.  After training the feed forward controller, the error feedback network \nis  trained  with  the  feed  forward  network,  but  the  feed  forward  networks'  weights  are  unchanged. \nBackpropagating  the  error  through  the  identification  network  is  applied  for  the  training  of  both \nnetworks. \n\nWhen training the feedback control network,  the feed forward calculation is \n\nu(t) = ujt)+u/b(t). \n\ny(t+ 1) = P(x(t), y(t), u(t\u00bb, \n\n(6) \n\n(7) \n\nwhere  uj/)  is  the  output  from  the  feed  forward  controller  network  and  u,..(t)  is  the  output  from  the \nfeedback controller network, P(.) is the identification mapping. \n\n\fAn Integrated Architecture of Adaptive Neural Network Control for Dynamic Systems \n\n1035 \n\n3  CONTROL ON EXAMPLE PROBLEMS \n\nIn  this  section,  the  control  architecture  described  above  is  applied  to  a  well-known  problem  from  the \nliterature[I).  The  plants and the  reference  model  of the  sample  problems are  described  by  difference \nequations \n\nplant: \n\ny(t + 1)  = \n\nyet) \n2 \n\n1.0+ Y  (t) \n\n+ (u(t) -1.O)u(t)(u(t) + 1.0) \n\nreference model: \n\ny(t + 1) = 0.6y(t) + u(t) \n\nThis is a nonlinear time varying dynamic system with no analytical inverse. \n\n3.1  FEED FORWARD CONTROL \n\n(II) \n\n(12) \n\nA feed forward neural network is trained to control the system to follow  the reference model.  The plant \nstate yet) and external inputx(t) are fed  to the controller.  During the training, the x(t) is randomly \ngenerated.  After training, the controller generates a control signal u(t) such that the plant can follow the \nreference model output.  Figure 5 shows the testing result of the reference model output and the \ncontrolled plant output. The input function is x(t)=sin(21ttf25)+sin(21tt/1O).  The controller network \narchitecture is (2, 20,  1). \n\n4 \n\n2 \n\nOJ \n0 c \n\nQ) \n10-\nOJ \n\n~  0 \n'\" 1J \n\nc \n0 \n'\" \n\n-2 \n\n20 \n\n40 \n\n60 \n\n80 \n\n100 \n\n2 \n\n::J \n0 \n~ \nc \n0 \nu \n\n-1 \n\n? \n(j \n~ \n\n0 \n() \n\n2 \n\n0 \n\n..... 1 \n\n..... 2 \n\nFigure 5 Tracking Result From the Feed Forward NN. \nOutput of reference (solid line) and plant (dash line). \n\nFigure 6 Feed Forward Control Surface \n\nThe output surface of  the  controller network is shown  in Figure 6.  By examining the controller output \nsurface, we can see that the neural network builds a  feed  forward  mapping  from  x(t)  to u(t).  This feed \nforward  mapping  is  also  a  function  of  the  plant  state  yet).  Under  each  state,  the  neural  network \ncontroller accepts input x(t) to produce control signal u(t) such that the plant follows the reference model \nreasonably  well.  In  Figure 6,  the x  axis is the  external  input x(t) and the  y axis  is  the  plant feedback \noutput yet).  The z axis represents the control surface. \nThe  feed  forward  controller  laCks  the  ability  to  adapt  to  plant  uncertainty,  noise  or  changes  in  the \nreference model.  As an example, we apply this feed forward controller to the disturbed plant with a bias \n0.5 added to the original plant.  The tracking result is shown in Figure 7.  With this slight bias, the plant \ndoes not follow the reference model.  Clearly, the  feed forward controller has no adaptive  ability to this \nmodel bias. \n\n\f1036 \n\nLiu Ke,  Robert L.  Tokar,  Brian D.  McVey \n\n3.2  FEEDBACK CONTROL \n\nFtrSt,  we  compare  the  neural  network  feedback  controller  with  fixed  gain  PID  controllers.  For many \nnonlinear systems,  the  fixed gain  PID controllers  will  give poor tracking  and continuous adaptation of \nthe controller parameters is needed.  The neural network approach offers an alternative control approach \nfor  nonlinear  systems.  Through  the  training,  control  gains,  imbedded  in  the  neural  network,  are \nestablished as a continuous function of system external inputs x(t) and plant states yet). \n\nThe sample problem in the above section is now employed to describe how the neural network creates a \nnonlinear control gain surface with error feedback and additional inputs.  First, we show one simple case \nof neural adaptive feedback controller.  This controller can only adapt to the system nonlinearity with a \nfixed  linear input pattern.  The reason  to  show  this simple adaptation  case  first  is that its control  gain \nsurface can be illustrated graphically. \nFigure 8 illustrates, for the system in equations (11) and (12)  that a fixed gain PI controller fails to track \nthe reference model,  for even one fixed linear input pattern x(t)=0.2t-2.5, because the plant nonlinearity. \nFigure 9 illustrates the result from  a recurrent neural network with feedback error e(t) and x(t) as  inputs. \nThe neural network is trained by backpropagation error through the identification network.  Compared to \nthe flXed gain PI controller,  the neural network improves the tracking ability significantly. \n\n., \n.. ., \n0 c ., \n'!! \n\n>. \n-.::I \nC \n0 \n>. \n\n4 \n\n2 \n\n0 \n\n-2 \n\n6 \n\n3 \n\n0 \n\n- 3 \n\nOJ \n<J \nC \n~ \n.2 \n~ \n>. \n'0 \nc \n0 \n>. \n\n, \n\nI \nI \n\n20 \n\n40 \n\nt \n\n60 \n\n80 \n\n100 \n\no \n\n5 \n\n10  15  20  25  30  35 \n\nt \n\nFigure 7 Tracking Result for Shifted Plant, plant \noutput (dash line) and reference output (solid line). \n\nFigure 8 Reference Model Output (solid line) \nand PID Controlled Plant Output (dashed line) \n\nThe control surface of the updating output fl.) is shown in Figure  10, which is the output from  the neural \nnetwork controller without recurrent link (see equation (3\u00bb.  We plot the  surface of the updating output \nfrom  the controller with respect to input x(t) and error feed back input e(t).  The gain of the controller is \nequivalent to the updating output from  the network when error=l.O.  As shown in the figure,  the gain in \nthe neighborhood about x(t)=O changes largely according to the  direction of changes in  the  plant in  the \ncorresponding  region.  The  updating  surface  for  a  PID  controller  is  a  plane.  The  neural  network \nimplements a nonlinear continuous control gain surface. \n\nFor a more complicated case,  we addx(t-I) as another input to the neural network as well as e(t-l),  and \ntrain by error backpropagation through the identification network.  These two inputs, x(t) and x(t-I) add \ndifference  information  to the  network.  The network  can adapt to  not only  different operating regions \nindicated  by  x(t),  but  also  different  input  patterns.  Figure  11  shows  the  tracking  results  with  two \ndifferent  input  patterns.  In  Figure  II  (a),  input  pattern  is  x(t)=4.0sin(tI4.0). \nIn  Figure  11  (b)  input \npattern is x(t)=sin(21t1!25)+sin(21t111O). \n\n\fAn Integrated Architecture of Adaptive Neural Network  Control for Dynamic  Systems \n\n1037 \n\n6 \n\n3 \n\nQ) \nu \nc \n~ \n2 \n, \n~ \n, \n>. \n-0.  -3  I \nc \n0 \n>.  - 6 \n\n0 \n\nI \n\no \n\n5 \n\n1 0 \n\n1 5  20  25  30  35 \n\nt \n\nFigure 9 Reference Model Output (solid line) and \nNeural Network Controled Output (dashed line) \n\nFigure  10 Feedback Neural Controller Updating Surface \n\nOJ \nu \nc \n~ \n~ \n~ \n>. \n\n0 \n- 2 \n\" \nc: \n- 4 \n0 \n>.  -6 \n- 8 \n-10 \n0 \n\n20 \n\n40 \n(a) \n\n60 \n\n80 \n\n10C \n\n., \n\n5 \n4 \n3 \nu \nc \n2 \n~ \n1 \nOJ \nl' \n0 \n>.  -1 \n\"  - 2 \nc \n- 3 \n0 \n>.  - 4 \n-5 \n-6 \n0 \n\n20 \n\n40 \n\n60 \n\n80 \n\n100 \n\n(b) \n\nFigure 11  Output of the Reference Model (solid line) and the Plant (dash line) \n\n3.3  INTEGRA TED NEURAL CONTROLLER \n\nAs shown in the above section,  when  only error feedback neural controller is  used,  the  control result is \nnot very  accurate.  Now  we  combine  feed  forward  and  feedback  control  to  realize  good  tracking  and \nadaptation.  Figure  12 shows  the  control  result from  the  integrated controller  when  the  plant is  shifted \nO.S.  Compared  to  only  feed  forward  control (Figure  7),  the  integrated  controller  has  much  better \nadaptation to the shifted plant. \nWhen the plant changes,  adding an extra feed back controller can avoid on-line training of feed forward \nnetwork  which  may  induce  potential instability,  and  the  adaptation  is  achieved.  The  output from  the \nfeedback network controller is driven by the error between the reference model and the plant. \n\n4  DISCUSSIONS \n\nWe have emphasized in  the above sections that a  feed  forward controller with only  state  feedback  does \nnot  adapt  when  model  uncertainties  or  noise/disturbance  are  present.  The  presence  of a  feed  back \ncontroller  can  make  the  on  line  training  of  the  feed  forward  network  unnecessary,  thus  avoiding \npotential  instability.  The  main  reason  for  the  instability  of on-line  training  is  the  incompleteness  of \nsample sets, which is referred to as a lack of persistent excitation in  control  theory[6].  First, it leads to \nan  inaccurate  identification  network.  Training  with  this  network  can  result  in  an  unstable  controller. \nSecond,  it  makes  the  training  of controller  away  from  global  representation.  With  an  error  feedback \nadaptive  network,  the  output from  the  feedback  network  controller  is  driven  by  the  error  between  the \nreference model  and  the  plant.  In  the simplest case  when  all  the activity  functions  are  linear and only \nthe feedback errors are  inputs,  this kind of neural network is equivalent to a PID controller.  However, \n\n\f1038 \n\nLiu Ke,  Robert L  Tokar,  Brian D.  McVey \n\nbeyond the  scope of PID  controllers,  the  neural networks are  capable  to  approximating nonlinear  time \nvariant control gain surfaces corresponding to different operating regions.  Also, unlike a PID controller, \nthe coefficients for the neural adaptive controller are obtained through a training procedure. \n\n4 \n\nQ) \n\nu c \n1:' \n0; ... \n\nQ) \n\n>, \n\"0 \nC \n0 \n>, \n\no \n\n20 \n\n40 \n\n60 \n\n80 \n\n100 \n\nFigure 12 Integrated Network Controller Tracking Result for Shifted Plant. \n\nPlant Output (dash line) and Reference Output (solid line). \n\nIt has  rise  time,  overshoot \nThe  error  feedback  network  behaves  as  a  gain  scheduling  controller. \nconsideration  and  delay  problem.  Feed  forward  control  can  compensate  for  these  problems  to  some \ndegree.  For example,  the  feed  forward  network can  perform  a  nonlinear  mapping  with  designed  time \ndelay.  Therefore with the feed forward network, the delay problem maybe overcame significantly.  Also \nthe feed forward controller can help to reduce rise time compare to use only feedback controller. \n\nWith the feed forward network, the feedback network controller can have much smaller gains compared \nto using  a  feedback  network  alone.  This  increases  the  noise  rejection  ability.  Also  this  reduces  the \novershoot as well as settle time. \n\nThe neural  network  control  architecture offers an alternative  to  the  conventional  approach.  It gives a \ngeneric model for the broadest class of systems considered in control theory .  However this model needs \nto  be  configured  depending  on  the  details of the  control  problem.  With  different  inputs,  the  neural \nnetwork  controllers  establish  different  internal  hyperstates.  When  plant  states  are  fed  back  to  the \nnetwork,  a  feed forward mapping  is established as a  function  of the plant states by  the neural network \ncontroller.  When  the errors between  the  reference  model  and the  plant are  used as  the  error feedback \ninputs  to  a  dynamic  neural  network  controller, \nthe  network  functions  as  an  associative  memory \nnonlinear gain  scheduled  controller.  The  above  two  kinds  of neural  controller can  be combined  and \ncomplemented to achieve accurate tracking and adaptation. \n\nReferences \n\n[1] Kumpati S. Narendra and Kannan Parthasarathy. \"Gradient Methods for the Optimization of DynamiCal \n\nSystems Containing Neural Networks,\"  IEEE Trans. Neural Networks. vol. 2. pp252-262 Mar.  1991 \n\n[2] Psaltis. D .\u2022 Sideris. A.  and Yamamura. A., \"Neural controllers.\"  Proc. of 1st International Conference on \n\nNeural Networks.  Vol. 4. pp551-558. San Diego. CA.  1987 \n\n[3) G.  lightbOdy. Q. H. Wu and G. W. Irwin. \"Control applications for feed forward networks.\"  Chapter 4. \n\nNeural Networks  for Control and Systems.  Edited by K.warwich, G. W. Irwin and  K. J. Hunt 1992 \n\n[4) R. Abikowski and P. 1. Gawthrop. \"A survey of neural networks for control\" Chapter 3. NeUral Networks \nfor Control and  Systems. ISBN 0-86341-279-3.  Edited by K.warwich. G. W. Irwin and K. 1.  Hunt 1992 \n[5] John Hertz. Anders Krogh and Richard G.  Palmer. \"Introduction to the Theory of Neural Computation.\" \n[6J Thomas Miller. RiChard S. Sutton and Paul 1. Werbos. \"Neural Networks for Control\" \n\n\f", "award": [], "sourceid": 967, "authors": [{"given_name": "Ke", "family_name": "Liu", "institution": null}, {"given_name": "Robert", "family_name": "Tokar", "institution": null}, {"given_name": "Brain", "family_name": "McVey", "institution": null}]}