(329f) Estimation of Experimental Gradients for Real-Time Optimization

Challenges in real-time process optimization arise
from the inability to build and adapt accurate models for complex
physico-chemical processes. Two main classes of optimization methods are
available for handling model uncertainty. In the absence of measurements, a
robust optimization approach is typically used, whereby conservatism is
introduced to guarantee feasibility for the entire range of expected variations
[1]. When measurements are available, adaptive optimization can help adjust to
process changes and disturbances, thereby reducing conservatism [2].

This presentation focuses on the latter class of
methods. The measurements can be used in different ways to compensate for model
uncertainty. A possible classification can be made as follows [3]: (i) model-adaptation
methods that
use the measurements to update the parameters of the process model before
repeating the optimization; (ii) modifier-adaptation methods that adapt constraint and
gradient modifiers; and (iii) input-adaptation methods that convert the optimization
problem into a feedback control problem. Here, we argue in favor of
modifier-adaptation methods, which use a parameterization and measurements that
are tailored to the tracking of the necessary conditions of optimality.

Perhaps the key issue in applying modifier-adaptation
methods is tied to the fact that the gradients of the plant outputs with
respect to the plant inputs, also called experimental gradients, must be available. An
accurate estimate of the experimental gradients is indeed necessary for the
iterates to yield a KKT point (relative to the plant) upon convergence. A novel
way of estimating the experimental gradients is described in this presentation.
It is based on the rather natural idea that the expected level of noise in the
gradient estimates, as induced by the measurement noise, can be kept
sufficiently small by ensuring a certain distance between the successive
operating points. In particular, a number of theoretical results are presented
regarding the choice of such a distance.

These developments are illustrated for an experimental
three-tank system. It consists of three cylinders interconnected in series by
two pipes; two pumps (driven by DC motors) supply the leftmost and rightmost
columns with liquid (water). The real-time optimization problem is formulated
so as to minimize the overall pumping energy needed to maintain the height of
liquid in the columns between given limits.

References

[1] Monnigmann M. and
Marquardt W., "Steady-state process optimization with guaranteed robust
stability and feasibility," AIChE J 49(12):3110-3126, 2003.

[2] Marlin T. E. and Hrymak A.
N., "Real-time operations optimization of continuous process," Proc 5^th
Int Conf on Chemical Process Control (CPC-5), Tahoe City NV, 1997.

[3] Chachuat B., Srinivasan B.
and Bonvin D., "Model parameterizaton tailored to real-time optimization," Proc.
18^th Eur Symp on Computer Aided Process Engineering (ESCAPE-18), Lyon, France, 2008.