{"title": "Non-Linear PI Control Inspired by Biological Control Systems", "book": "Advances in Neural Information Processing Systems", "page_first": 975, "page_last": 981, "abstract": null, "full_text": "Non-linear  PI  Control Inspired  by \n\nBiological  Control Systems \n\nLyndon J.  Brown  Gregory E.  Gonye \n\nJames S.  Schwaber  * \n\nExperimental Station,  E.!.  DuPont deNemours &  Co.  Wilmington,  DE 19880 \n\nAbstract \n\nA  non-linear  modification  to  PI  control  is  motivated  by  a  model \nof a signal transduction pathway active in mammalian blood pres(cid:173)\nsure regulation.  This control algorithm,  labeled  PII  (proportional \nwith  intermittent integral),  is  appropriate for  plants requiring ex(cid:173)\nact set-point matching and disturbance attenuation in the presence \nof infrequent  step  changes  in  load  disturbances or  set-point.  The \nproportional  aspect  of the  controller is  independently designed  to \nbe  a  disturbance  attenuator  and  set-point  matching  is  achieved \nby  intermittently invoking an integral controller.  The mechanisms \nobserved in the Angiotensin 11/ AT1  signaling pathway are used to \ncontrol the switching of the integral control.  Improved performance \nover  PI control is  shown  on  a  model  of cyc1opentenol  production. \nA sign change in  plant gain at the desirable operating point causes \ntraditional  PI  control  to  result  in  an  unstable  system.  Applica(cid:173)\ntion  of this  new  approach  to  this  problem  results  in  stable exact \nset-point matching for  achievable set-points. \n\nBiological processes have evolved sophisticated mechanisms for solving difficult con(cid:173)\ntrol  problems.  By analyzing and understanding these natural systems it is  possible \nthat principles can be derived which are applicable to general control systems.  This \napproach has already been the basis for  the field of artificial neural networks, which \nare loosely  based on a  model of the electrical signaling of neurons.  A suitable can(cid:173)\ndidate system for  analysis is  blood pressure control.  Tight control of blood pressure \nis  critical for  survival  of an  animal.  Chronically high  levels  can lead to  premature \ndeath.  Low blood pressure can lead to oxygen and nutrient deprivation and sudden \nload changes must be quickly responded to or loss of consciousness can result.  The \nbaroreflex,  reflexive  change  of heart  rate  in  response  to  blood  pressure  challenge, \nhas been previously studied in order to develop some insights into biological control \nsystems  [1,  2,  3]. \n\n\u00b7Jyndon.j .brown@usa.dupont.com \n\nAddress correspondence to this author \n\nGregory.E.Gonye-PHD@usa.dupont.com James.S.Scwhaber@usa.dupont.com \n\n\f976 \n\nL. J.  Brown,  G. E. Gonye and J.  S.  Schwaber \n\nNeurons  exhibit  complex  dynamic  behavior  that  is  not  directly  revealed  by  their \nelectrical  behavior,  but  is  incorporated  in  biochemical  signal  transduction  path(cid:173)\nways.  This is  an important basis for  plasticity of neural  networks.  The area of the \nbrain to which  the baroreceptor afferents  project is  the nucleus of tractus solitarus \n(NTS).  The neurons in the NTS  are  rich  with  diverse receptors for  signaling path(cid:173)\nways.  It  is  logical  that  this  richness  and  diversity  playa crucial  role  in  the signal \nprocessing  that  occurs  here.  Hormonal  and  neurotransmitter signals  can  activate \nsignal  transduction  pathways  in  the  cell,  which  result  in  physical  modification  of \nsome components of a cell, or altered gene regulation.  Fuxe et al  [4]  have shown the \npresence of the angiotensin 11/ AT!  receptor pathway in NTS  neurons, and Herbert \n[5]  has demonstrated its ability to affect the baroreflex. \n\nTo  develop understanding of the effects of biochemical pathways,  a detailed kinetic \nmodel  of the  angiotensin/AT!  pathway  was  developed.  Certain  features  of  this \nmodel and the baroreflex have interesting characteristics from  a control engineering \nperspective.  These  features  have  been  used  to  develop  a  novel  control  strategy. \nThe resulting control algorithm utilizes a proportional controller that intermittently \ninvokes  integral  action  to  achieve  set-point  matching.  Thus  the  controller  will  be \nlabeled PII. \n\nThe  use  of integral  control  is  popular  as  it  guarantees  cancellation  of offsets  and \nensures  exact  set-point  matching.  However,  the  use  of integral  control  does  have \ndrawbacks.  It  introduces  significant  lag  in  the  feedback  system,  which  limits  the \nbandwidth of the system.  Increasing the integral gain, in order to improve response \ntime,  can  lead  to  systems  with  excessive  overshoot,  excessive  settling  times,  and \nless  robustness  to  plant  changes  or  uncertainty.  Many  processes  in  the  chemical \nindustry  have  a  steady-state  response  curve  with  a  maximum  and  frequently,  the \noptimal operating condition is  at this peak.  Unfortunately,  any controller with true \nintegral action will  be unstable at this operating point. \n\nIn a crude sense, the integrator learns the constant control action required to achieve \nset-point matching.  If the integral control is viewed as a simple learning device, than \na  logical  step  is  to remove it from  the feedback  loop  once  the  necessary  offset  has \nbeen learned.  If the offset is  being successfully compensated for,  only noise remains \nas a source for  learning.  It has been well established that learning based on nothing \nbut  noise  leads  to undesirable  results.  The maxim,  'garbage in,  garbage out'  will \napply.  Without integral control,  the  proportional controller  can be made more  ag(cid:173)\ngressive while maintaining stability margins and/or control actions at similar levels. \nThis control strategy will  be appropriate for  plants with infrequent step changes in \nset-points or loads.  The challenge becomes deciding when,  and how  to perform this \nswitching so  that the resulting controller provides significant improvements. \n\n1  Angiotensin III ATI receptor  Signal Transduction Model \n\nRegulation of blood pressure is  a vital control problem in mammals.  Blood pressure \nis  sensed by stretch sensitive  cells  in  the aortic  arch and carotid sinus.  These cells \ntransmit signals to neurons in the NTS which  are combined with other signals from \nthe  central  nervous  system  (CNS)  resulting  in  changes  to the  cardiac output  and \nvascular tone [6].  This control is  implemented by two  parallel systems in  the CNS, \nthe  sympathetic  and  parasympathetic  nervous  systems.  The  sympathetic  system \nprimarily affects  the vascular tone  and the parasympathetic system affects  cardiac \noutput  [7].  Cardiac  control  can  have  a  larger  and  faster  effect,  but  long  term \napplication of this control is  injurious to the overall health of the animal.  Pottman \net  al  [2]  have  suggested  that  these  two  systems  separately  control  for  long  term \nset-point control and fast  disturbance rejection. \n\n\fNon-Linear PI Control Inspired by Biological Control Systems \n\n977 \n\nOne  receptor  in  NTS  neuronal  cells  is  the  AT1  receptor  which  binds  Angiotensin \nII.  The  NTS  is  located in  the brain stem where  much of the  processing of the au(cid:173)\ntonomic regulatory systems reside.  Angiotensin infusion in this region of the brain \nhas  been  shown  to  significantly  affect  blood  pressure  control.  In  order  to  under(cid:173)\nstand  this  aspect  of neuronal  behavior,  a  detailed  kinetic  model  of this  signaling \npathway  was  developed.  The  pathway is  presented  in  Figure  2.  The outputs  can \nbe  considered  to  be  the  concentrations  of Gq\u00b7GTP,  GO-y,  activated  protein  kinase \nC,  and/or calmodulin dependent  protein kinase. \n\nSeveral reactions  in  the cascade are of interest.  The binding of phospholipase C  is \nsignificantly  slower  than  the  other  steps  in  the  reaction.  This  can  be  modeled  as \na  first  order  transfer  function  with  a  long  time  constant  or  as  a  pure  integrator. \nThe  IP3 receptor  is  a  ligand  gated  channel  on  the  membrane  of the  endoplasmic \nreticulum  (ER).  As  Figure  2  shows,  when  IP3  binds  to  this  receptor,  calcium  is \nreleased  from  the  ER  into  the  cells  cytoplasm.  However  the  IP3  receptor  also \nhas  2  binding  sites  on  its  cytoplasmic  domain  for  binding  calcium.  The  first  has \nrelatively  fast  dynamics  and  causes  a  substantial  increase  in  the  channel  opening. \nThe second calcium  binding site has slower  dynamics  and inactivates  the  channel. \nThe effect of this first  binding site is to introduce positive feedback into the model. \nIn traditional control literature, positive feedback is  generally undesirable.  Thus it \nis  very interesting to see  positive feedback in neuronal  control systems. \n\nA typical surface response for  the model,  comparing the time response of activated \ncalmodulin  versus  the  peak  concentration  of  a  pulse  of  angiotensin,  is  shown  in \nFigure  1.  The  results  are  consistent  with  behavior  of  cells  measured  by  Li  and \nGuyenet  [8].  The  output  level  is  seen  to  abruptly  rise  after  a  delay,  which  is  a \ndecreasing function of the magnitude of the input.  Unlike a linear system, both the \nmagnitude and speed of the response of the system are functions  of the magnitude \nof the  input.  Further,  the  relaxing of the system to  its  equilibrium  is  a  very slow \nresponse  as  compared  to  its  activation.  This  behavior  can  be  attributed  to  the \npositive  feedback  response  inherent  to  the  IP3  receptor.  The  effect  of  the  slow \ndynamics of the phospholipase C binding,  and the IP3 receptor dynamics results in \nan activation behavior similar to a threshold detector on the integrated input signal. \nHowever, removal of the input results in a slow recovery back to zero.  The activation \nof the calcium calmodulin dependent protein kinase  can lead to phosphorilation of \nchannels  that result  in  synaptic conductance changes that are functionally  related \nto  the  amount  of activated  kinase.  The  activation  of calcium  calmodulin  can  also \nlead to changes in gene regulation that could potentially result in long term changes \nin  the neurons  synaptic conductances. \n\n2  Proportional with Intermittent Integral  Control \n\nKey features  from  the model  that are incorporated in  the control  law  are: \n\n1.  separate controllers for  set-point control and disturbance attenuation; \n2.  activation of set-point controller when integrated error exceeds threshold; \n3.  strength of integral  action  when  activated  will  be  a  function  of the  speed \n\nwith  which  activation was  achieved; \n\n4.  smooth removal of integral action,  without disruption of control action. \n\nThe PII controller begins initially as a proportional controller with a nominal offset \nadded  to  its  output.  The  integrated  error  is  monitored.  The  integral  controller \nis  turned  on  when  the  integrated  error  exceeds  a  threshold.  Once  the  integral \ncontrol action is  activated, it remains active as  long as the error is  excessive.  Once \nthe  error  is  not  significant,  then  the  integral  control  action  can  be  removed  in  a \n\n\f978 \n\nL. J.  Brown,  G.  E.  Gonye and J.  S.  Schwaber \n\n~hra~ \n\np=PIP2 \n\nr:ti*rsuoorut \n\nf;~f,:(~I:~:: :::~ ((adb \n. h  ra~  . ld \n~~~:L~  !-' \n~ ~  \\ -\n-<'Jf:-.  '~=' \n\n~S;7' \n\n~- \u2022\u2022 = \n\nCell Membrane  d a r   atlp ) \n\nCaM \n\nCytoplasm \n\nt\u00b7~ metabollt \n\n_________________ .  M, \nDi:uslon \n;ER  I~ Receptor  ~ -- - ---- ---- -~ C~ CaM \n: \n,-- ------ --------_. \n\nCa r!umr-----------\n\nFigure  1:  Schematic and Surface Responses  of Angiotensin II I ATI  Model \n\nsmooth manner.  This has been  achieved by allowing the value of the integral gain, \nKi,  to  decay  exponentially.  It is  important  that  this  is  done  in  such  a  manner \nas  not  to  affect  the  actual  control  signal.  This  can  be  achieved  by  adjusting  the \noffset  appropriately.  Since  u  =  Kpe + Kiels  and  Ki  ex:  -Ki' then  u  can be  made \nconstant  for  constant  e  by  adding  offset  Ko  where  Ko  ex:  Kiel s.  The  integral \naction is  completely removed once Ki has decayed to the point where it is  no longer \nsignificant.  In  order to  make  the  effect  of activation  of the  integrator  correspond \nto the behavior of the angiotensin model,  the integrated error is  scaled by  the time \nspent reaching the threshold when the integrator is turned on.  This corresponds to \npoint  3 above. \nIf the error undergoes significant change when  the integrator is  already fully  active \nthe system will  behave similarly to a  system with  a  PI controller whose gains have \nbeen set too high.  This may result in  significant overshoot and possibly instability. \nThere is  a small chance that even with infrequent step changes, the residual error, or \nrandom disturbance could trigger the integrator immediately before  a step change. \nIn  a  biological control system, control does  not rest in  one neuron or necessarily in \none signal  transduction pathway but in  multiple pathways.  Furthermore,  study of \nindividual cells  shows a  great deal of variability in the details of their behavior.  By \nimplementing  the  intermittent  integral  control  as  a  sum  of many  equivalent  con(cid:173)\ntrollers,  as  in  left  side  of Figure  2,  with  variability  in  their  threshold  parameters, \na  controller can be developed  that is  not  subject  to the chance  of being fully  acti(cid:173)\nvated by random disturbance or residual error.  During steady-state operation these \nintegrators  will  quickly  deactivate  when  noise  or small  disturbances  trigger  them, \nas the error will  be less  than the threshold.  However,  an actual step change in  the \nerror  signal  will  result  in  all  or most  of the  integrators  activating,  and  remaining \nactive until the error is  compensated for. \n\nThe block  diagram on right side of Figure 2 and the time dependent definitions  in \nTable 1 precisely define  the control algorithm for  the single  integrator case. \n\n\fNon-Linear PI Control Inspired by Biological Control Systems \n\n11 \n\n\"'2 \n\n,..----+1+ \n\n....--+1+ \n\n+ \n5uml \n\ne \n\n979 \n\n+  u \n+ \n\nkis=ki 1 +ki2+ki3+ki4+ki5 \n\nxu=xu1 <xu2<xu3<xu4<xu5 \n\neu=eu1<eu2<eu3<eu4<eu5 \n\nFigure 2:  Block Diagrams for  Control Algorithm Implementations \n\nIf \nt = to \nKi(t) = 0 and  Ix(t)1  > Xu \nIKi(t)1  > Ki  and le(t)1  < eu  Ki(t)  = -KdecayKi(t),  Ko(t)  = KdecayKi(t)X(t). \n0<  IKi(t)1  < Ki \n\nX(tO)  = 0,  Ki(tO) = 0,  Ko(to)  = K;,  tt(tO)  = to. \n\nKo(t+)  = Ko(t) + Ki(t)X(t), \n\n- max(l,K.(t-tz)). \n\nthen \n\nKi(t+) = 0,  x(t+) = 0,  tt(t+) = t. \n\nK  ( ) - K* \n\nit  -\n\ni'X t \n\n(+) _ \n\nx(t) \n\nOtherwise \n\nKi(t) = 0,  Ko(t)  = 0,  x = e, \n\nit(t) = o. \n\nTable  1:  Definition of Gains for  PII Control \n\n3  Control of CSTR Reactor for  Cyclopentenol Production \n\nThe  model  of  the  CSTR  reactor  is  taken  from  [9].  The  basic  process  converts \ncyclopentadiene to cyclopentenol.  Cyclopentenol can undergo a further undesirable \nreaction  to  form  cyclopentadiol,  and  cyclopentadiene  can  undergo  an  alternative \nreaction  to  form  dicyclopentadiene.  The  rates  of the  reactions  are  temperature \ndependent.  Inputs to the model are flow  rate, and the jacket temperature.  The first \ninput is the control input, and the jacket temperature is an unmeasured disturbance, \nwith  a  root  mean  square deviation  of 0.1  C  about  a  nominal  value  of 130  C.  The \nregulated output will  be the cyclopentenol concentration in  the outflow. \n\nThe  steady-state  response  of this  process  is  shown  in  Figure  3.  Operation in  the \nregion labeled II up to the peak of the curve labeled  VIII has been considered.  At \nthe  point  labeled  VIII,  the  steady-state gain  of the  plant  goes  to  o.  Plants  with \nsteady-state gains which  change sign can not be stably controlled with  PI control. \nAn  additional complicating factor is  that the plant has significant inverse response \nin  this region. \nCriteria for  this control design problem, in  order of importance are \n\n\u2022  operate between 45  and 60  ljhour with reasonable high frequency  gain \n\u2022  minimize the overshoot \n\u2022  minimize rise time \n\n\f980 \n\nL. J.  Brown, G.  E.  Gonye and J.  S.  Schwaber \n\n\u2022  minimize the inverse response \n\nSatisfying  the  first  and  last  criteria should  ensure  a  robust  controller.  Precise nu(cid:173)\nmerical  performance  criteria for  the  rise  time  have  not  been  specified  as  no  fixed \nvalues  are reasonable for  the entire region. \n\nA PI controller,  as  well  as  a  PH controller have been  designed  and the results  are \ndisplayed  in  Figures  3.  The  controller  parameters  were  Kp  = 75,  Ki  = 7500  for \nthe  standard  PI  controller.  The  PH  controller  used  5  equally  weighted  parallel \nintegrators  with  Kp  = 125,  total  K;  = 10000  and  Kdecay  = 100.  The  threshold \nparameters  were  chosen as  eu  = [4  3  2 1  11  * 0.00025,  x., = r16  8  4  2  11  * 0.00004, \n\n:n:~ K'  = K::~'= \n\n. \n\n! i~~':'\"'''''' 'I \n\n.., \n\n1 \u00b7\" \n\n~ \n\n0 \n\n1 \n\n2 \n\nJ \n\n.. \n\n5 \n\n6 \n\n7 \n\nI \n\n9 \n\n10 \n\n1l \n\n......... \n\n.. ~ . .  .... ... ..  ..... \n\n0.8 o \n\n20 \n\n40 \n\n60 \n\nM \n\n100 \n\n120 \n\n140 \n\n160 \n\nFoedtalo (lib) \n\n\"\n\n\" \n\n: \n\n- 0 01 \n\n-0015 \n\n~~ \n-0 025 \n\n.' \n... ~.:  , \n\n.; .\n\n.... \n\n{:, \n\n-0 03 \"~-\"'':-2 --:' .. -:--~ \u2022 \u2022 :--..:''':--~S--:':5 2:---:':\":---::5 .--::,,---'. \n\nTme(t'lOUnl) \n\nr'. \n\nPI! \n\nR .. I'M'ICe., \n\nFigure 3:  Steady-State Response of Cyclopentenol CSTR Reactor and Output Con(cid:173)\ncentration from  CSTR Reactor \nThe  set-point  was  chosen  to  be  a  series  of smoothed  steps.  Smoothing  was  per(cid:173)\nformed with a first-order , low-pass filter  with unity DC gain and a time constant of \n30 hours-i .  While operating in the region of design from 0 to 4.8 hours and 5.4 to 7 \nhours, the PH controlled system, as  compared to the PI controlled system, had re(cid:173)\nduced inverse response, less worst-case overshoot, similar response times and greater \ndisturbance attenuation.  A closer examination of the PI controlled system,  during \nthe interval 4.8-5.4 hours, showed that at this extreme operating point, oscillations \nof a  fixed  period  begin  to appear.  This  indicates  the  existence  of poorly damped \npoles.  The PH controlled system did not show this degradation of performance. \n\nThe  set-point  was  raised  to  nearly  the  maximum  achievable  concentration.  This \nallows  examination  of the  behavior  of the  controller  when  operating  near  regions \nof  uncertainty  in  the  sign  of  the  plant  gain.  This  operating  point  achieves  the \nmaximum  possible  conversion  to  cyclopentenol  and  thus  has  significant  economic \nadvantages.  In the region from  7.2s  to  lOs, there is  a  10%  reduction in the distur(cid:173)\nbance response with  the PH controller.  At  this operating point,  the PI controlled \nsystem can be shown  to be locally  st able.  However,  the effects  of integrated noise \neasily allow  the system trajectory to escape the region of attraction.  As  expected, \nthe  PI controlled  system  went  unstable.  The  PH controlled  system  remains  well \nbehaved.  The simulation was  run for  a  total simulated time of 43  hours at this op(cid:173)\nerating point, and repeated many times without seeing any loss of stability with PH \ncontroller.  With PI control, the system went unst able within 10 hours for each trial. \nThus, PH control allows  operation at set-points closer to maximums or minimums. \n\n4  Conclusion \n\nThe mechanisms that biological control systems employ to successfully control non(cid:173)\nlinear, time varying, multivariable physiological systems under very demanding per-\n\n\fNon-Linear PI Control Inspired by Biological Control Systems \n\n981 \n\nformance requirements are likely to have application in process control problems.  In \naddition to neural networks already incorporated in advanced controllers, cells pro(cid:173)\ncess  information  through  biochemical  signal  transduction  networks  that  may  also \ncontain useful non-linear mechanisms.  A model of one such pathway has been devel(cid:173)\noped,  and features  have  been identified  which  can be used  to develop  an improved \ncontrol system. \n\nThe  fundamental  idea  is  to  design  two  separate  control  laws,  one  intermittently \nused  for  cancelling infrequently  changing but mostly predictable disturbances,  and \nanother for  attenuating white  disturbances.  The first  controller  learns  the simple \ncharacteristics of the predictable disturbance.  When the predictable disturbance is \nlearned,  it  can  be  canceled  with  an  open  loop  controller,  and  no  further  learning \ntakes place.  However if it appears that the open loop controller is not cancelling the \ndisturbance, further  learning takes place until the disturbance is  again successfully \ncancelled.  The second controller is designed strictly for fast disturbance attenuation. \nWithout the lag inherent in integration, the controller can be made more aggressive \nresulting in better performance.  The two  controllers can be integrated by  applying \nthe threshold and switching mechanisms identified in the signal transduction model. \n\nReferences \n\n[1]  M.  A.  Henson,  B.  A.  Ogunnaike,  J.  S.  Schwaber,  and  F.  J.  Doyle  III,  \"The \nbaroreceptor reflex:  A  biological  control  system  with  applications  in  chemical \nprocess control,\"  I&EC Research,  vol.  33,  pp.  2453- 2465,  1994. \n\n[2]  M.  Pottman, M.  A.  Henson, B.  A. Ogunnaike, and J. S.  Schwaber,  \"A  parrallel \ncontrol strategy abstracted from the baroreceptor reflex,\"  Chemical Engineering \nScience,  vol.  51,  pp.  931-945,  1996. \n\n[3]  H.  S.  Kwatra,  F.  J.  Doyle  III,  and  J.  S.  Schwaber,  \"Dynamic  gain  scheduled \n\nprocess control,\"  Chemical  Engineering  Science,  1997. \n\n[4]  K.  Fuxe  and  B.  B.  et  aI,  \"Pre- and  post-synaptic  features  of  the  central  an(cid:173)\n\ngiotensin  systems:  Indications  for  a  role  of  angiotensin  peptides  in  volume \ntransmission  and  for  interactions  with  central  monamine  neurons,\"  Clin  Exp \nHypertens  [Theory  Practj,  vol.  Ala, pp.  143- 168,  1988. \n\n[5]  J.  Herbert,  \"Studying  the  central  actions  of angiotensin  using  the  expression \nof immediate-early genes:  Expectations  and  limitations,\"  Regulatory  Peptides, \nvol.  66,  pp.  13-18, 1996. \n\n[6]  K. M.  Spyer,  \"The central nervous organization of reflex circulatory control,\"  in \nClin Exp Hypertens [Theory  PractjCentral Regulation of Automanomic Fuctions \n(A.  D.  Loewy  and  K.  M.  Spyer,  eds.),  p.  168,  New  York:  Oxford  University \nPress,  1990. \n\n[7]  M.  N.  Kumada,  N.  Terui,  and  T.  Kuwaki,  \"Arterial  baroreceptor  reflex:  Its \ncentral and peripheral neural mechanisms,\"  Progr.  Neurophysiol.,  vol. 35, p. 331, \n1988. \n\n[8]  Y.  Li and P.  G.  Guyenet,  \"Angiotensin II decreases a resting K+ conductance in \nrat bulbospinal  neurons of the c1  area,\"  Circulatiob  Research,  vol.  78,  pp.  274-\n282,1996. \n\n[9]  B.  Ogunnaike  and  W.  H.  Ray,  Process  dynamics,  Modeling  and  Control.  New \n\nYork:  Oxford University Press,  1995. \n\n\f", "award": [], "sourceid": 1600, "authors": [{"given_name": "Lyndon", "family_name": "Brown", "institution": null}, {"given_name": "Gregory", "family_name": "Gonye", "institution": null}, {"given_name": "James", "family_name": "Schwaber", "institution": null}]}