Non-linear PI Control Inspired by 
Biological Control Systems 
Lyndon J. Brown Gregory E. Gonye James S. Schwaber * 
Experimental Station, E.I. DuPont deNemours & Co. Wilmington, DE 19880 
Abstract 
A non-linear modification to PI control is motivated by a model 
of a signal transduction pathway active in mammalian blood pres- 
sure regulation. This control algorithm, labeled PII (proportional 
with intermittent integral), is appropriate for plants requiring ex- 
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 II/AT1 signaling pathway are used to 
control the switching of the integral control. Improved performance 
over PI control is shown on a model of cyclopentenol production. 
A sign change in plant gain at the desirable operating point causes 
traditional PI control to result in an unstable system. Applica- 
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- 
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- 
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 
barorefiex, 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]. 
*lyndn'j'brwn@usa'dupnt'cm Address correspondence to this author 
Gregory. E.Gonye_PHD@usa.dupont.com James'S'Scwhaber@usa'dupnt'cm 
976 L. d. Brown, G. E. Gonye and d. S. Schwaber 
Neurons exhibit complex dynamic behavior that is not directly revealed by their 
electrical behavior, but is incorporated in biochemical signal transduction path- 
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- 
ways. It is logical that this richness and diversity play a 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 II/AT1 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/AT1 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 la 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- 
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 II/AT1 receptor Signal TYansduction 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. Portman 
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- 
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- 
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, G,, 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: 
1. 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; 
4. 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. d. Brown, G. E. Gonye and d. S. Schwaber 
h=hormone -hra 
r--receptor 
a=G alpha subunit 
b=G beta gamma subunits radb " h  rabd* 
d=GDP 
t=GTI, /  X'-._ 
l=PhospholipaseCBeta / \ \  .d 
i=P3 1 
p=PIP2 
k=Protein kinase C 1  { / 
Ca--Calcium 
Cytoplasm metaM}lit Diffusion 
Figure 1' Schematic and Surface Responses of Angiotensin II / AT1 Model 
smooth manner. This has been achieved by allowing the value of the integral gain, 
Ki, to decay exponentially. It is important that this is done in such a manner 
as not to affect the actual control signal. This can be achieved by adjusting the 
offset appropriately. Since u = Kpe + Kie/s and ti cr -Ki, then u can be made 
constant for constant e by adding offset Ko where /o cr Kie/s. The integral 
action is completely removed once Ki has decayed to the point where it is no longer 
significant. In order to make the effect of activation of the integrator correspond 
to the behavior of the angiotensin model, the integrated error is scaled by the time 
spent reaching the threshold when the integrator is turned on. This corresponds to 
point 3 above. 
If the error undergoes significant change when the integrator is already fully active 
the system will behave similarly to a system with a PI controller whose gains have 
been set too high. This may result in significant overshoot and possibly instability. 
There is a small chance that even with infrequent step changes, the residual error, or 
random disturbance could trigger the integrator immediately before a step change. 
In a biological control system, control does not rest in one neuron or necessarily in 
one signal transduction pathway but in multiple pathways. Furthermore, study of 
individual cells shows a great deal of variability in the details of their behavior. By 
implementing the intermittent integral control as a sum of many equivalent con- 
trollers, as in left side of Figure 2, with variability in their threshold parameters, 
a controller can be developed that is not subject to the chance of being fully acti- 
vated by random disturbance or residual error. During steady-state operation these 
integrators will quickly deactivate when noise or small disturbances trigger them, 
as the error will be less than the threshold. However, an actual step change in the 
error signal will result in all or most of the integrators activating, and remaining 
active until the error is compensated for. 
The block diagram on right side of Figure 2 and the time dependent definitions in 
Table I precisely define the control algorithm for the single integrator case. 
Non-Linear PI Control Inspired by Biological Control Systems 979 
Gain 
nltan 
13 Ki4 
14 Ki5 
Sum1 
kis=kil +ki2+ki3+ki4+ki5 
e 
xu=xul<xu2<xu3<xu4<xu5 
eu=eul<eu2<eu3<eu4<eu5 
Figure 2: Block Diagrams for Control Algorithm Implementations 
If 
t=to 
then 
Ki(t) - 0 and Ix(t)l > xu 
Igi(t)l > K and le(t)l < eu 
0 < IKi(t)l < K Ko(t +) = Ko(t) + Ki(t)x(t), 
Ki(t +)=0, x(t +):0, tt(t +) = t. 
Otherwise Ii(t) = O, Io(t) = O, 5: = e, fi(t): O. 
x(to) = 0, Ki(to)= O, Ko(to)= Kj, tt(to)= to. 
, x(t) 
Ki(t) = K i ,x(t +) - max(,:,(t-tt))' 
ti(t) = -KaecayKi(t), to(t) = KaecayKi(t)x(t). 
Table 1' Definition of Gains for PII Control 
3 Control of CSTR Reactor for Cyclopentenol Production 
The model of the CSTR reactor is taken from [9]. The basic process converts 
cyclopentadiene to cyclopentenol. Cyclopentenol can undergo a further undesirable 
reaction to form cyclopentadiol, and cyclopentadiene can undergo an alternative 
reaction to form dicyclopentadiene. The rates of the reactions are temperature 
dependent. Inputs to the model are flow rate, and the jacket temperature. The first 
input is the control input, and the jacket temperature is an unmeasured disturbance, 
with a root mean square deviation of 0.1 C about a nominal value of 130 C. The 
regulated output will be the cyclopentenol concentration in the outflow. 
The steady-state response of this process is shown in Figure 3. Operation in the 
region labeled II up to the peak of the curve labeled VIII has been considered. At 
the point labeled VIII, the steady-state gain of the plant goes to 0. Plants with 
steady-state gains which change sign can not be stably controlled with PI control. 
An additional complicating factor is that the plant has significant inverse response 
in this region. 
Criteria for this control design problem, in order of importance are 
operate between 45 and 60 1/hour with reasonable high frequency gain 
minimize the overshoot 
minimize rise time 
980 L. J. Brown, G. E. Gonye and J. S. Schwaber 
minimize the inverse response 
Satisfying the first and last criteria should ensure a robust controller. Precise nu- 
merical performance criteria for the rise time have not been specified as no fixed 
values are reasonable for the entire region. 
A PI controller, as well as a PII controller have been designed and the results are 
displayed in Figures 3. The controller parameters were Kp - 75, Ki = 7500 for 
the standard PI controller. The PII controller used 5 equally weighted parallel 
integrators with Kp = 125, total K - 10000 and Kdecau = 100. The threshold 
parameters were chosen as e - [4 3 2 I 1]  0.00025, x,, - [16 8 4 2 1]  0.00004, 
nncl T t. = T.*/. oo3 
20 40 60 80 100 10 140 160 
Fedra [l/hl 
0.02 
0.01 
~0 03 
Oulput from CSTR fo Cyclopernol 
1 2 3 4 5 6 7 8 9 10 11 
l,me (hour) 
-0 02 
r, me (hound) 
Figure 3: Steady-State Response of Cyclopentenol CSTR Reactor and Output Con- 
centration from CSTR Reactor 
The set-point was chosen to be a series of smoothed steps. Smoothing was per- 
formed with a first-order, low-pass filter with unity DC gain and a time constant of 
30 hours -. While operating in the region of design from 0 to 4.8 hours and 5.4 to 7 
hours, the PII controlled system, as compared to the PI controlled system, had re- 
duced inverse response, less worst-case overshoot, similar response times and greater 
disturbance attenuation. A closer examination of the PI controlled system, during 
the interval 4.8-5.4 hours, showed that at this extreme operating point, oscillations 
of a fixed period begin to appear. This indicates the existence of poorly damped 
poles. The PII controlled system did not show this degradation of performance. 
The set-point was raised to nearly the maximum achievable concentration. This 
allows examination of the behavior of the controller when operating near regions 
of uncertainty in the sign of the plant gain. This operating point achieves the 
maximum possible conversion to cyclopentenol and thus has significant economic 
advantages. In the region from 7.2s to 10s, there is a 10% reduction in the distur- 
bance response with the PII controller. At this operating point, the PI controlled 
system can be shown to be locally stable. However, the effects of integrated noise 
easily allow the system trajectory to escape the region of attraction. As expected, 
the PI controlled system went unstable. The PII controlled system remains well 
behaved. The simulation was run for a total simulated time of 43 hours at this op- 
erating point, and repeated many times without seeing any loss of stability with PII 
controller. With PI control, the system went unstable within 10 hours for each trial. 
Thus, PII control allows operation at set-points closer to maximums or minimums. 
4 Conclusion 
The mechanisms that biological control systems employ to successfully control non- 
linear, time varying, multivariable physiological systems under very demanding per- 
Non-Linear PI Control Inspired by Biological Control Systems 981 
formance requirements are likely to have application in process control problems. In 
addition to neural networks already incorporated in advanced controllers, cells pro- 
cess information through biochemical signal transduction networks that may also 
contain useful non-linear mechanisms. A model of one such pathway has been devel- 
oped, and features have been identified which can be used to develop an improved 
control system. 
The fundamental idea is to design two separate control laws, one intermittently 
used for cancelling infrequently changing but mostly predictable disturbances, and 
another for attenuating white disturbances. The first controller learns the simple 
characteristics of the predictable disturbance. When the predictable disturbance is 
learned, it can be canceled with an open loop controller, and no further learning 
takes place. However if it appears that the open loop controller is not cancelling the 
disturbance, further learning takes place until the disturbance is again successfully 
cancelled. The second controller is designed strictly for fast disturbance attenuation. 
Without the lag inherent in integration, the controller can be made more aggressive 
resulting in better performance. The two controllers can be integrated by applying 
the threshold and switching mechanisms identified in the signal transduction model. 
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