{"title": "Neural Control for Nonlinear Dynamic Systems", "book": "Advances in Neural Information Processing Systems", "page_first": 1010, "page_last": 1016, "abstract": null, "full_text": "Neural Control for Nonlinear Dynamic Systems \n\nSsu-Hsin Yu \n\nDepartment of Mechanical Engineering \nMassachusetts Institute of Technology \n\nCambridge, MA 02139 \nEmail:  hsin@mit.edu \n\nAnuradha M. Annaswamy \n\nDepartment of Mechanical Engineering \nMassachusetts Institute of Technology \n\nCambridge, MA 02139 \nEmail:  aanna@mit.edu \n\nAbstract \n\nA neural network based approach is presented for controlling two distinct \ntypes of nonlinear systems.  The first  corresponds to  nonlinear systems \nwith  parametric  uncertainties  where  the  parameters occur  nonlinearly. \nThe second corresponds to  systems  for  which  stabilizing control  struc(cid:173)\ntures cannot be determined.  The proposed neural controllers are shown \nto result in closed-loop system stability under certain conditions. \n\n1 \n\nINTRODUCTION \n\nThe  problem  that  we  address  here  is  the  control  of general  nonlinear dynamic  systems \nin  the  presence of uncertainties.  Suppose the  nonlinear dynamic  system  is  described  as \nx=  f(x , u , 0)  , y  =  h(x, u, 0)  where u  denotes an external input, y  is  the output, x  is  the \nstate, and 0 is the parameter which represents constant quantities in the system. The control \nobjectives  are  to  stabilize  the  system  in  the  presence  of disturbances  and  to  ensure  that \nreference trajectories can be tracked accurately, with  minimum delay.  While uncertainties \ncan  be  classified  in  many  different  ways,  we  focus  here  on  two  scenarios.  One  occurs \nbecause the changes in  the  environment and  operating conditions  introduce uncertainties \nin  the system parameter O.  As  a result, control objectives such as regulation and  tracking, \nwhich may be realizable using a continuous function u  =  J'(x, 0)  cannot be achieved since \no is  unknown.  Another class  of problems arises  due  to  the  complexity  of the  nonlinear \nfunction f. Even if 0,  f  and h can be precisely determined, the selection of an appropriate \nJ'  that  leads  to  stabilization  and  tracking  cannot  be  made  in  general.  In  this  paper,  we \npresent two  methods based  on  neural  networks which are shown to  be applicable to  both \nthe  above classes of problems.  In  both cases,  we  clearly  outline the  assumptions  made, \nthe requirements for adequate training of the neural  network, and the class of engineering \nproblems where the proposed methods are applicable.  The proposed approach significantly \nexpands  the  scope  of neural  controllers  in  relation  to  those  proposed  in  (Narendra and \nParthasarathy,  1990;  Levin  and  Narendra,  1993;  Sanner  and  Slotine,  1992;  Jordan  and \nRumelhart,  1992). \n\n\fNeural Control for  Nonlinear Dynamic  Systems \n\n1011 \n\nThe first  class  of problems we  shall  consider includes nonlinear systems  with  parametric \nuncertainties.  The field  of adaptive control  has  addressed  such  a  problem,  and  over  the \npast  thirty  years,  many  results  have  been derived  pertaining to  the  control  of both  linear \nand nonlinear dynamic systems (Narendra and Annaswamy, 1989).  A common assumption \nin  almost  all  of the  published  work  in  this  field  is  that  the  uncertain  parameters  occur \nlinearly.  In this paper, we consider the control of nonlinear dynamic systems with nonlinear \nparametrizations. We design a neural network based controller that adapts to the parameter \no and show that closed-loop system stability can be achieved under certain conditions.  Such \na  controller will  be referred  to as  a  O-adaptive  neural  controller.  Pertinent results  to  this \nclass are discussed in section 2. \n\nThe second class of problems includes nonlinear systems, which despite being completely \nknown, cannot be stabilized by conventional analytical techniques.  The obvious method for \nstabilizing nonlinear systems is to resort to linearization and use linear control design meth(cid:173)\nods.  This limits the scope of operation of the stabilizing controller.  Feedback linearization \nis another method by which nonlinear systems can be stably controlled (lsidori, 1989). This \nhowever requires fairly stringent set of conditions to be satisfied by the functions! and h. \nEven after these conditions are satisfied, one cannot always find  a closed-form solution to \nstabilize the system since  it  is  equivalent to  solving a  set of partial differential equations. \nWe consider in  this paper,  nonlinear systems,  where system models as  well  as  parameters \nare  known,  but controlIer structures  are  unknown.  A  neural  network  based  controller  is \nshown to exist and trained so that a stable closed-loop system  is  achieved.  We denote this \nclass of controllers as a stable neural controller. Pertinent results to this class are discussed \nin section 3. \n\n2  O-ADAPTIVE NEURAL CONTROLLER \n\nThe focus of the nonlinear adaptive controller to be developed in  this  paper is  on dynamic \nsystems that can be written in the d-step ahead predictor form as follows: \n\nYt+d  = !r(Wt,Ut,O) \n\n(I) \n. ,Yt-n+l, Ut-I, ' \", Ut-m-d+l],  n  ~ I, m  ~ 0,  d  ~ I, m  + d  =  n, \nwhere wi =  [Yt,\" \nYI, U I  C  ~ containing the origin and 8 1  C  ~k are open, ir : Y1 x U;n+d  x  8 1  - t  ~, Yt \nand Ut  are the output and the input of the system at time t respectively, and 0 is an unknown \nparameter and occurs nonlinearly in  ir.1  The goal  is to choose a control input 'It  such that \nthe system in (1) is  stabilized and the plant output is regulated around zero. \n\nLetxi  ~ [Yt+d-I , '\" \n,Yt+l , wil T , Am =  [e2,\"', en+d-I, 0 , en+d+I,\"', en +m+2d-2, \n0], Bm =  [el' en+d],  where e, is  an unit vector with the i-th term equal to  I.  The following \nassumptions are made regarding the system in Eq. (I ). \n\n(AI)  For every 0 E  8 1,  ir(O,O, O)  =  0. \nCA2)  There exist open and convex neighborhoods of the origin Y2  C  YI  and U2  C  U I, an \nopen and convex set 82  C  8 1,  and a function K  : 0.2 x  Y2  x  8 2 --->  U I  such that for \nevery Wt  E 0.2, Yt+d  E Y2  and 0 E  8 2,  Eq. (1) can be written as Ut  =  K(wt, Yt+d, 0), \nwhere 0.2  ~ Y2 X  u;,,+d-I. \n\n(A3)  K  is twice differentiable and has bounded first and second derivatives on EI  ~ 0.2  X \nY2  X  8 2 ,  while ir is differentiable and has a bounded derivative on 0.2  x  I{ (E I )  x  8 2 . \n\n(A4)  There exists bg  > \u00b0 such that for every YI  E  ir(o.2, K(o.2' 0 , 8 2), 8 2), W E  0.2 and \n\no BE  8 \n,2 ,  \n\nay \n\n11  - (8K(w,y ,O)  _  8K(w,y,9))1  _ \n\nay \n\nY - YI \n\n.  8f,(w ,u ,O) I -\n\nau \n\nU-UI \n\nI > b \n\ng' \n\n1 Here, as  well as in the following  sections,  An denotes the n-th product space of the set  A . \n\n\f1012 \n\nS.  YU, A. M. ANNASW AMY \n\n(A5)  There exist positive definite matrices P  and Q of dimensions (n + m + 2d - 2) such \nthat  x t  (AmPAm  - P)Xt+  J(  BrnPBmK + 2xt  ArnPBmK  ~ -Xt QXt,  where \n[( =  [0, K(wt, 0 , O)]T. \n\n-T  T \n\nT  T' \n\nT  T \n\nT \n\n-\n\n-\n\nSince the objective is  to control the system ~n (1) where 0 is unknown, in order to stabilize \nthe output Y at the origin with an estimate Of,  we choose the control input as \n\n(2) \n\n2.1  PARAMETER ESTIMATION SCHEME \n\nSuppose the estimation algorithm  for updating Ot  is  defined  recursively  as  /10t  ~  Ot(cid:173)\nOt-I  =  R(Yt,Wt-d,Ut-d,Ot-l) the  problem  is  to  determine  the  function  R  such  that Ot \nconverges to  0  asymptotical1y.  In general,  R  is  chosen  to  depend on Yt,  Wt-d,  1\u00a3t-d  and \nOt-l  since they  are measurable and contain information regarding O.  For example,  in  the \ncase of linear  systems  which can  be  cast in  the  input  predictor form,  1\u00a3t  =  <b[ 0,  a  wel\\-\nknown  linear  parameter estimation  method  is  to  adjust  /10  as  (Goodwin  and  Sin,  1984) \n/10t  =  1+4>'t~~t-'/ [1\u00a3t-d  - \u00a2LdOt-d\u00b7  In  other  words,  the  mechanism  for  carrying  out \nparameter estimation is  realized by  R.  In  the case of general  nonlinear systems, the task \nof determining such a  function  R  is  quite difficult,  especial\\y  when  the parameters occur \nnonlinearly.  Hence, we propose the use of a neural network parameter estimation algorithm \ndenoted O-adaptive neural network (TANN) (Annaswamy and Yu,  1996).  That is,  we adjust \nOt  as \n\nif /1Vd,  < - f  \notherwise \n\n(3) \n\nwhere the inputs of the neural network are Yt, Wt-d, 1\u00a3t-d  and Ot-I,  the output is  /10t,  and \nf  defines a dead-zone where parameter adaptation stops. \n\nThe  neural  network  is  to  be  trained  so  that  the  resulting  network  can  improve  the  pa(cid:173)\nrameter  estimation  over  time  for  any  possible  0  in  a  compact  set. \nIn  addition,  the \ntrained  network  must  ensure  that  the  overal1  system  in  Eqs.  (1),  (2)  and  (3)  is  stable. \nToward  this  end,  N  in  TANN algorithm  is  required to  satisfy  the  fol1owing  two  proper-\nties:  (PI) IN(Yt,wt - d,1\u00a3t - d,Ot-l)12  ~ a(llfJt;~~,~1~2)2uLd' and (P2) /1Vt  -/1Vd,  < fl, \nt- d' \nC(\u00a2t)  = (~~ (Wt,Yt+d,O)lo=oo)T,  Ut  = Ut  - K(Wt,Yt+d,Ot+d - I),  (fit  = [WT,Yt+djT, \na  E  (0, I) and 00  is the point where K  is  linearized and often chosen to be the mean  value \nof parameter variation. \n\n-a  2+ IC(<f;, _,,)1 1 \n)12)2 \n\n>  0  where  A T f \n\nIii  12  _  Iii \nUt \n\nUt - I, Ut  - Ut \n\nII  _  II  AV, \n\nu,  L.l.  d,  -\n\n, 4>,-./ \n\nL.l.Vt  -\n\n-\n\nIC( -\n\n(\n\n1+ \n\n12 \n\nii \n\n-\n\nfl \n\n, \n\n-\n\n1\u00a3-2 \n\n2.2  TRAINING OF TANN FOR CONTROL \n\nIn  the  previous  section,  we  proposed  an  algorithm  using  a  neural  network  for  adjusting \nthe control  parameters.  We  introduced  two  properties (PI) and  (P2) of the  identification \nalgorithm  that  the  neural  network  needs  to  possess  in  order  to  maintain  stability  of the \nclosed-loop system.  In this section, we discuss the training procedure by which the weights \nof the neural network are to be adjusted so that the network retains these properties, \n\nThe  training  set  is  constructed off-line and  should  compose of data  needed  in  the  train(cid:173)\ning  phase.  If we  wan..!.  the  algorithm  in  Eq.  (3)  to  be  valid  on  the  specified  sets  Y3  and \nU3  for  various  0  and  0  in  83,  the  training  set  should  cover those  variables  appearing  in \nEq.  (3)  in  their  respective  ranges.  Hence,  we  first  sample W in  the  set Y;- x  U;:+d-I, \n\n\fNeural Control for Nonlinear Dynamic Systems \n\n1013 \n\nth \n\necorrespon  mg \n\nd\u00b7  C(A-)  - BK  ( \n\nand  B,  8 in  the  set  83.  Their  values  are,  say,  WI,  BI  and  81  respectively.  For  the \nparticular  fh  and  BI  we  sample  8 again  in  the  set  {B  E  8 31 IB  - BII \n:s:  181 - BI I}, \nand  its  value  is,  say,  8t.  Once  WI,  BI ,  81  and  8t  are  sampled,  other  data  can  then \nbe  calculated,  such  as  UI  =  K(WI' 0, 8d  and  YI  =  fr(WI, UI, Bd.  We  can  also  ob-\n~)2  d \n. \ntam \nan \n_ \n\n_ \nLI  - a (1+IC(<I>I)I2)2  (UI  - UI)  ,where \u00a2I  -\nand  UI  - K(WI'  YI,( 1 )\u00b7  A  data \nelement can then be formed as  (Yl ,WI ,UI, 8t, BI ,  ~ Vd l , Ld.  Proceeding in the same man(cid:173)\nner,  by  choosing various ws ,  Bs ,  1f.  and  8~ in  their respective  ranges,  we form  a  typical \ntraining set Ttram  =  {  (Ys , W s, us,1f~ ,  Bs, ~ Vd d  Ls) 11  :s:  s  :s:  M}, where M  denotes the \ntotal  number of patterns in the training set.  If the quadratic penalty function method (Bert(cid:173)\nsekas,  1995) is  used,  properties (PI) and  (P2)  can  be satisfied  by  training the network on \nthe training set to minimize the following cost function: \n\n- ao  WI , YI,  0, i l   d l \nB)  All: \nT \n[WI  ,yJ} \n\n-a(I+IC(\u00a2I)i2)2  UI  -'ttl \n\n2+IC(\u00a2dI 2  ( \n\n'1'1 \n~  2 \n\nIC(\u00a2IW \n\n~  _ \n\n~d \n\n-\n\n-\n\n-\n\nT \n\nmJpl  ~ mJ,n~~{(max{0, ~VeJ)2+ ;2  (max{0, INi(W)12 - L t})2} \n\nM \n\n(4) \n\nTo  find  a  W  which  minimizes  the  above  unconstrained  cost  function  1,  we  can  apply \nalgorithms such as the gradient method and the Gauss-Newton method. \n\n2.3  STABILITY RESULT \n\nWith  the  plant  given  by  Eq.  (1),  the  controller  by  Eq.  (2),  and  the  TANN  parameter \nestimation algorithm by Eq. (3), it can be shown that the stability of the closed-loop system \nis  guaranteed. \n\nBased  on the  assumptions  of the  system  in  (1)  and  properties  (PI) and  (P2)  that TANN \nsatisfies,  the stability result of the closed-loop system can be concluded  in  the  following \ntheorem.  We refer the reader to (Yu  and Annaswamy, 1996) for further detail. \n\nTheorem 1  Given the compact sets Y;+ I  X U:;+d x 8 3  where the neural network in Eq. (3) \nis trained.  There  exist EI, E  >  0 such that for any interior point B of 8 3,  there  exist open \nsets Y4  C  Y3,  U4  C  U3  and a  neighborhood 8 4  of B such  that if Yo , ... , Yn+d-2  E  Y4, \nl ,  ... ,8n +d - 2  E  8 4,  then  all the  signals  in  the  closed-loop \nUo, .. .  , U n -2  E  U4 ,  and 8n -\nsystem remain bounded and Yt  converges to a neighborhood of the origin. \n\n2.4  SIMULATION RESULTS \n\n. \n\n. \n\nTh \n\ne system  IS  0 \n\nf  h  f \nt  e  orm Yt+1  =  I+e  U.USH\",  + Ut,  were \n\nIn  this  section,  we  present a  simulation example of the TANN controller proposed in  this \nb \nsectton. \nIS  t  e parameter to  e \ndetermined on-line.  Prior information regarding the system  is  that ()  E  [4,  10].  Based on \n' where Bt  denotes the parameter \nEq. (2), the controller was chosen to be Ut  =  -\nestimate at time t.  According to Eq.  (3), B was estimated using the TANN algorithm with \ninputs  YHI, Yt.  Ut  and~, and E  = 0.01.  N  is a Gaussian network with 700 centers.  The \ntraining set and the testing set were composed of 6,040 and 720 data elements respectively. \n\n8  ( I )  \n,y, 0 ,-;'Y' \nI+e - \u00b7  \"\" \n\nB\u00b7  h \n\nlIy, ( I-y,) \n\n~ \n\nh \n\nAfter the training  was  completed,  we  tested  the TANN controller on  the  system  with  six \ndifferent values  of B,  4.5, 5.5,  6.5,  7.~, 8.5  and 9.5,  while  the  initial  parameter estimate \nand  the initial  output were chosen as  BI  =  7  and Yo  =  -0.9 respectively.  The results  are \nplotted in Figure 1.  It can be seen that Yt  can be stabilized at the origin for all  these values \nof B.  For comparison, we also simulated the system under the same conditions but with 8 \n\n\f1014 \n\n-1 \n\n-2 \no \n\nS.  YU, A. M. ANNASWAMY \n\n~ 0 \n\n-1 \n\n-2 \no \n\n50 \n\n100  4 \n\n10 \n\n50 \n\n10 \n\n100  4 \n\nFigure 1:  Yt  (TANN Controller) \n\nFigure 2:  Yt  (Extended Kalman Filter) \n\nestimated using the extended Kalman  filter (Goodwin and Sin,  1984).  Figure 2 shows the \noutput responses.  It is  not surprising that for  some values of fJ,  especially when the initial \nestimation error is  large, the responses either diverge or exhibit steady state error. \n\n3  STABLE NEURAL CONTROLLER \n\n3.1  STATEMENT OF THE PROBLEM \n\nConsider the following nonlinear dynamical system \n\nX=  j(x,u), \n\nY =  h(x) \n\n(5) \n\nwhere  x  E  Rn  and u  E  RTn.  Our goal  is  to  construct a  stabilizing  neural  controller as \nu  =  N(y; W)  where N  is  a neural network with weights W, and establish the conditions \nunder which the closed-loop system is stable. \n\nThe nonlinear system in (5) is  expressed as a combination of a higher-order linear part and \na  nonlinear part as x=  A x  + Bu + RI (x, u)  and  y  =  Cx + R 2(x),  where  j(O, O)  =  0 \nand  h(O)  =  O.  We  make  the  following  assumptions:  (AI)  j, h  are  twice  continuously \ndifferentiable and are completely known.  (A2) There exists a  K  such that (A  - BKC) is \nasymptotically stable. \n\n3.2  TRAINING OF THE STABLE NEURAL CONTROLLER \n\nIn order for the neural controller in Section 3.1 to result in an asymptotically stable c1osed(cid:173)\nloop system, it is sufficient to establish that a continuous positive definite function of the state \nvariables decreases monotonically through output feedback.  In other words, if we can find a \nscalar definite function with a negative definite derivative of all points in  the state space, we \ncan guarantee stability  of the overall system.  Here,  we  limit our choices of the Lyapunov \nfunction  candidates to  the  quadratic form,  i.e.  V  =  x T Px,  where  P  is  positive definite, \nand the goal  is  to choose the controller so that V < 0 where V =  2xT P j(x, N(h(x), W)). \nBased on the above idea, we define a \"desired\" time-derivative V d  as V d=  -xTQx where \nQ  =  QT  > O.  We  choose P  and Q  matrices as follows.  First, according to (AI), we can \nfind  a  matrix  K  to  make  (A - BKC) asymptotically  stable.  We  can  then  find  a  (P, Q) \npair by choosing an arbitrary positive definite matrix Q and solving the Lyapunov equation, \n(A - BKC)T P  + P(A - BKC) =  -Q to obtain a positive definite P. \n\n\fNeural  Control  for  Nonlinear  Dynamic Systems \n\n1015 \n\nWith the contro\\1er of the form  in  Section 3.1, the goal is to find  W  in  the neural  network \nwhich yields V:::; V d  along the trajectories in  a neighborhood X  C  ~n of the origin in  the \nstate  space.  Let  Xi  denote the  value  of a  sample point where i  is  an  index  to  the sample \nvariable X  E  X  in  the state space.  To  establish V:::; V d,  it is  necessary that for every  Xl  in \na  neighborhood X  C  ~n of the origin, Vi:::;Vd\"  where Vi=  2x;Pf(x l ,N(h(:rl\n,  W)) \nand V d, =  -x; QXi .  That is,  the goal  is  to  find  a  W  such  that the inequality  constraints \ntlVe , \n- V d,  and  M  denotes the \ntotal  number of sample points in  X.  As  in the training of TANN controller, this can  also \nbe  posed  as  an  optimization  problem.  If the  same  quadratic  penalty  function  method  is \nused,  the  problem  is  to find  W  to  minimize the  fo\\1owing  cost function  over the training \nset,  which is  described as Ttrain  =  {(Xl' Yi, V d.)\\l  :::;  i  :::;  M}: \n\n:::;  0,  where i  =  1,\u00b7\u00b7\u00b7 , M, is  satisfied,  where tlVe ,  =V l \n\n)\n\nrwn J  6.  mJp 2 I: (max{O, tlVe ,})2 \n\n1M \n\ni= 1 \n\n(6) \n\n3.3  STABILITY OF THE CLOSED-LOOP SYSTEM \n\nAssum~tions (A 1)  and  (A2)  imply  that  a  stabilizing  controller u  =  - J( y  exists  so  that \nV  =  X  Px is  a candidate Lyapunov function .  More genera\\1y,  suppose a continuous but \nunknown function ,,((y) exists such that for V  = x T Px, a control  input 1t  = \"((y)  leads to \nV:::;  -xT Qx, then we can find a neural network N (y) which approximates \"((y) arbitrarily \nclosely in a compact set leading to closed-loop stability.  This is  summarized in Theorem 2 \n(Yu  and Annaswamy,  1995). \n\nTheorem 2  Let there be a  continuous function \"((h(x))  such that 2xT P f(x , \"((h(x)))  + \nxT Qx :::;  0 for every X E  X  where  X  is a  compact set containing the origin as an interior \npoint.  Then,  given a neighborhood 0  C  X  of the origin, there exists a neural controlierH = \nN(h(x); W)  and a  compact set Y  E  X  such that the solutions of x=  f(x , N(h(x); W)) \nconverge to 0, for every initial condition x(to)  E  y. \n\n3.4  SIMULATION RESULTS \n\nIn this section, we show simulation results  for  a discrete-time nonlinear systems using the \nproposed neural  network contro\\1er in  Section 3, and compare it  with a linear contro\\1er to \nillustrate the difference.  The system we considered is a second-order nonlinear system Xt  = \nf(xt-I , Ut-I), where f  = [II, 12]T, h  = Xl t _ 1 X  (1 +X2'_ 1 )+X2t-1 x (l-u t- I +uLI) and \n12  = XT'_I  + 2X2'_1  +Ut-I (1 + X2'_I)\u00b7  It was assumed that X is measurable, and we wished \nto stabilize the system  around the origin.  The controller is  of the form  Ht  =  N (x It, X2 t  ). \nThe neural  network N  used  is  a Gaussian  network with  120 centers.  The training set and \nthe testing set were composed of 441  and  121  data elements respectively. \n\nAfter the training  was done,  we plotted the actual  change of the Lyapunov function,  tl V, \nusing the linear controller U  =  - K x  and the  neural  network controller in  Figures 3 and 4 \nrespectively.  It can be observed from the two figures that if the neural network contro\\1er is \nused,  tl V  is  negative definite except in  a small  neighborhood of the origin, which assures \nthat the closed-loop system would converge to  vicinity of the origin; whereas, if the linear \ncontroller is used, tl V  becomes positive in some region away from the origin, which implies \nthat the  system can be unstable for  some  initial  conditions.  Simulation results confirmed \nour observation. \n\n\f1016 \n\nS. YU, A.  M.  ANNASW AMY \n\n-0 01 \n\n- 0  J \n\n-0  J \n\n- 0  I \n\n-()2 \n\n-O J \n\nFigure 3:  ~V(u = -Kx ) \n\nFigure 4:  ~V(u =  N(x)) \n\nAcknowledgments \n\nThis  work  is  supported in  part by  Electrical  Power Research  Institute under contract No. \n8060-13 and in part by National Science Foundation under grant No. ECS-9296070. \n\nReferences \n\n[1]  A.  M.  Annaswamy  and  S.  Yu.  O-adaptive  neural  networks:  A  new  approach  to \n\nparameter estimation. IEEE Transactions on Neural Networks,  (to appear)  1996. \n\n[2]  D.  P. Bertsekas.  Nonlinear Programming.  Athena Scientific, Belmont, MA,  1995. \n[3]  G.  C.  Goodwin and K.  S.  Sin.  Adaptive Filtering  Prediction and Control.  Prentice(cid:173)\n\nHall, Inc., 1984. \n\n[4]  A. Isidori.  Nonlinear Control Systems.  Springer-Verlag, New York, NY,  1989. \n[5]  M. L Jordan and D. E. Rumelhart. Forward models:  Supervised learning with a distal \n\nteacher.  Cognitive Science, 16:307-354, 1992. \n\n[6]  A. U. Levin and K. S.  Narendra. Control of nonlinear dynamical systems using neural \nnetworks:  Controllability and stabilization.  IEEE Transactions on Neural Networks, \n4(2): 192-206, March  1993. \n\n[7]  K. S.  Narendra and A . M. Annaswamy. Stable Adaptive Systems.  Prentice-Hall, Inc., \n\n1989. \n\n[8]  K.  S.  Narendra and K.  Parthasarathy.  Identification and control of dynamical systems \nusing  neural  networks.  IEEE Transactions  on  Neural Networks,  1 (I ):4-26, March \n1990. \n\n[9]  R. M.  Sanner and J.-J. E.  Slotine. Gaussian networks for direct adaptive control.  IEEE \n\nTransactions on Neural Networks,  3(6):837-863, November 1992. \n\n[10]  S.  Yu  and A. M.  Annaswamy.  Adaptive control of nonlinear dynamic systems using \nO-adaptive  neural  networks.  Technical  Report 9601 ,  Adaptive Control  Laboratory, \nDepartment of Mechanical Engineering, M.LT.,  1996. \n\n[11]  S.-H.  Yu  and  A.  M.  Annaswamy.  Control  of nonlinear dynamic  systems  using  a \nstability based neural network approach.  In  Technical report 9501, Adaptive Control \nLaboratory, MIT,  Submitted to Proceedings of the 34th IEEE Conference on Decision \nand Control, New Orleans, LA,  1995. \n\n\f", "award": [], "sourceid": 1043, "authors": [{"given_name": "Ssu-Hsin", "family_name": "Yu", "institution": null}, {"given_name": "Anuradha", "family_name": "Annaswamy", "institution": null}]}