Non-Linear PI Control Inspired by Biological Control Systems
Authors: Lyndon J. Brown, Gregory E. Gonye, James S. Schwaber
Published in: Advances in Neural Information Processing Systems 11  (NIPS 1998) (Main Conference Track), 1998

Abstract: A  non-linear  modification  to  PI  control  is  motivated  by  a  model  of a signal transduction pathway active in mammalian blood pres(cid:173) sure regulation.  This control algorithm,  labeled  PII  (proportional  with  intermittent integral),  is  appropriate for  plants requiring ex(cid:173) act set-point matching and disturbance attenuation in the presence  of infrequent  step  changes  in  load  disturbances or  set-point.  The  proportional  aspect  of the  controller is  independently designed  to  be  a  disturbance  attenuator  and  set-point  matching  is  achieved  by  intermittently invoking an integral controller.  The mechanisms  observed in the Angiotensin 11/ AT1  signaling pathway are used to  control the switching of the integral control.  Improved performance  over  PI control is  shown  on  a  model  of cyc1opentenol  production.  A sign change in  plant gain at the desirable operating point causes  traditional  PI  control  to  result  in  an  unstable  system.  Applica(cid:173) tion  of this  new  approach  to  this  problem  results  in  stable exact  set-point matching for  achievable set-points. 
Biological processes have evolved sophisticated mechanisms for solving difficult con(cid:173) trol  problems.  By analyzing and understanding these natural systems it is  possible  that principles can be derived which are applicable to general control systems.  This  approach has already been the basis for  the field of artificial neural networks, which  are loosely  based on a  model of the electrical signaling of neurons.  A suitable can(cid:173) didate system for  analysis is  blood pressure control.  Tight control of blood pressure  is  critical for  survival  of an  animal.  Chronically high  levels  can lead to  premature  death.  Low blood pressure can lead to oxygen and nutrient deprivation and sudden  load changes must be quickly responded to or loss of consciousness can result.  The  baroreflex,  reflexive  change  of heart  rate  in  response  to  blood  pressure  challenge,  has been previously studied in order to develop some insights into biological control  systems  [1,  2,  3]. 
·Jyndon.j .brown@usa.dupont.com 
Address correspondence to this author 
Gregory.E.Gonye-PHD@usa.dupont.com James.S.Scwhaber@usa.dupont.com 
976 
L. J.  Brown,  G. E. Gonye and J.  S.  Schwaber 
Neurons  exhibit  complex  dynamic  behavior  that  is  not  directly  revealed  by  their  electrical  behavior,  but  is  incorporated  in  biochemical  signal  transduction  path(cid:173) ways.  This is  an important basis for  plasticity of neural  networks.  The area of the  brain to which  the baroreceptor afferents  project is  the nucleus of tractus solitarus  (NTS).  The neurons in the NTS  are  rich  with  diverse receptors for  signaling path(cid:173) ways.  It  is  logical  that  this  richness  and  diversity  playa crucial  role  in  the signal  processing  that  occurs  here.  Hormonal  and  neurotransmitter signals  can  activate  signal  transduction  pathways  in  the  cell,  which  result  in  physical  modification  of  some components of a cell, or altered gene regulation.  Fuxe et al  [4]  have shown the  presence of the angiotensin 11/ AT!  receptor pathway in NTS  neurons, and Herbert  [5]  has demonstrated its ability to affect the baroreflex. 
To  develop understanding of the effects of biochemical pathways,  a detailed kinetic  model  of the  angiotensin/AT!  pathway  was  developed.  Certain  features  of  this  model and the baroreflex have interesting characteristics from  a control engineering  perspective.  These  features  have  been  used  to  develop  a  novel  control  strategy.  The resulting control algorithm utilizes a proportional controller that intermittently  invokes  integral  action  to  achieve  set-point  matching.  Thus  the  controller  will  be  labeled PII. 
The  use  of integral  control  is  popular  as  it  guarantees  cancellation  of offsets  and  ensures  exact  set-point  matching.  However,  the  use  of integral  control  does  have  drawbacks.  It  introduces  significant  lag  in  the  feedback  system,  which  limits  the  bandwidth of the system.  Increasing the integral gain, in order to improve response  time,  can  lead  to  systems  with  excessive  overshoot,  excessive  settling  times,  and  less  robustness  to  plant  changes  or  uncertainty.  Many  processes  in  the  chemical  industry  have  a  steady-state  response  curve  with  a  maximum  and  frequently,  the  optimal operating condition is  at this peak.  Unfortunately,  any controller with true  integral action will  be unstable at this operating point. 
In a crude sense, the integrator learns the constant control action required to achieve  set-point matching.  If the integral control is viewed as a simple learning device, than  a  logical  step  is  to remove it from  the feedback  loop  once  the  necessary  offset  has  been learned.  If the offset is  being successfully compensated for,  only noise remains  as a source for  learning.  It has been well established that learning based on nothing  but  noise  leads  to undesirable  results.  The maxim,  'garbage in,  garbage out'  will  apply.  Without integral control,  the  proportional controller  can be made more  ag(cid:173) gressive while maintaining stability margins and/or control actions at similar levels.  This control strategy will  be appropriate for  plants with infrequent step changes in  set-points or loads.  The challenge becomes deciding when,  and how  to perform this  switching so  that the resulting controller provides significant improvements. 
1  Angiotensin III ATI receptor  Signal Transduction Model 
Regulation of blood pressure is  a vital control problem in mammals.  Blood pressure  is  sensed by stretch sensitive  cells  in  the aortic  arch and carotid sinus.  These cells  transmit signals to neurons in the NTS which  are combined with other signals from  the  central  nervous  system  (CNS)  resulting  in  changes  to the  cardiac output  and  vascular tone [6].  This control is  implemented by two  parallel systems in  the CNS,  the  sympathetic  and  parasympathetic  nervous  systems.  The  sympathetic  system  primarily affects  the vascular tone  and the parasympathetic system affects  cardiac  output  [7].  Cardiac  control  can  have  a  larger  and  faster  effect,  but  long  term  application of this control is  injurious to the overall health of the animal.  Pottman  et  al  [2]  have  suggested  that  these  two  systems  separately  control  for  long  term  set-point control and fast  disturbance rejection. 
Non-Linear PI Control Inspired by Biological Control Systems 
977 
One  receptor  in  NTS  neuronal  cells  is  the  AT1  receptor  which  binds  Angiotensin  II.  The  NTS  is  located in  the brain stem where  much of the  processing of the au(cid:173) tonomic regulatory systems reside.  Angiotensin infusion in this region of the brain  has  been  shown  to  significantly  affect  blood  pressure  control.  In  order  to  under(cid:173) stand  this  aspect  of neuronal  behavior,  a  detailed  kinetic  model  of this  signaling  pathway  was  developed.  The  pathway is  presented  in  Figure  2.  The outputs  can  be  considered  to  be  the  concentrations  of Gq·GTP,  GO-y,  activated  protein  kinase  C,  and/or calmodulin dependent  protein kinase. 
Several reactions  in  the cascade are of interest.  The binding of phospholipase C  is  significantly  slower  than  the  other  steps  in  the  reaction.  This  can  be  modeled  as  a  first  order  transfer  function  with  a  long  time  constant  or  as  a  pure  integrator.  The  IP3 receptor  is  a  ligand  gated  channel  on  the  membrane  of the  endoplasmic  reticulum  (ER).  As  Figure  2  shows,  when  IP3  binds  to  this  receptor,  calcium  is  released  from  the  ER  into  the  cells  cytoplasm.  However  the  IP3  receptor  also  has  2  binding  sites  on  its  cytoplasmic  domain  for  binding  calcium.  The  first  has  relatively  fast  dynamics  and  causes  a  substantial  increase  in  the  channel  opening.  The second calcium  binding site has slower  dynamics  and inactivates  the  channel.  The effect of this first  binding site is to introduce positive feedback into the model.  In traditional control literature, positive feedback is  generally undesirable.  Thus it  is  very interesting to see  positive feedback in neuronal  control systems. 
A typical surface response for  the model,  comparing the time response of activated  calmodulin  versus  the  peak  concentration  of  a  pulse  of  angiotensin,  is  shown  in  Figure  1.  The  results  are  consistent  with  behavior  of  cells  measured  by  Li  and  Guyenet  [8].  The  output  level  is  seen  to  abruptly  rise  after  a  delay,  which  is  a  decreasing function of the magnitude of the input.  Unlike a linear system, both the  magnitude and speed of the response of the system are functions  of the magnitude  of the  input.  Further,  the  relaxing of the system to  its  equilibrium  is  a  very slow  response  as  compared  to  its  activation.  This  behavior  can  be  attributed  to  the  positive  feedback  response  inherent  to  the  IP3  receptor.  The  effect  of  the  slow  dynamics of the phospholipase C binding,  and the IP3 receptor dynamics results in  an activation behavior similar to a threshold detector on the integrated input signal.  However, removal of the input results in a slow recovery back to zero.  The activation  of the calcium calmodulin dependent protein kinase  can lead to phosphorilation of  channels  that result  in  synaptic conductance changes that are functionally  related  to  the  amount  of activated  kinase.  The  activation  of calcium  calmodulin  can  also  lead to changes in gene regulation that could potentially result in long term changes  in  the neurons  synaptic conductances. 
2  Proportional with Intermittent Integral  Control 
Key features  from  the model  that are incorporated in  the control  law  are: 

separate controllers for  set-point control and disturbance attenuation;  2.  activation of set-point controller when integrated error exceeds threshold;  3.  strength of integral  action  when  activated  will  be  a  function  of the  speed 

with  which  activation was  achieved; 

smooth removal of integral action,  without disruption of control action. 

The PII controller begins initially as a proportional controller with a nominal offset  added  to  its  output.  The  integrated  error  is  monitored.  The  integral  controller  is  turned  on  when  the  integrated  error  exceeds  a  threshold.  Once  the  integral  control action is  activated, it remains active as  long as the error is  excessive.  Once  the  error  is  not  significant,  then  the  integral  control  action  can  be  removed  in  a 
978 
L. J.  Brown,  G.  E.  Gonye and J.  S.  Schwaber