Part of Advances in Neural Information Processing Systems 11 (NIPS 1998)
Lyndon Brown, Gregory Gonye, James Schwaber
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
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Gregory.E.Gonye-PHD@usa.dupont.com James.S.Scwhaber@usa.dupont.com
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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
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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:
with which activation was achieved;
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
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L. J. Brown, G. E. Gonye and J. S. Schwaber