Optimizing control based on output feedback

Abstract

In the framework of process optimization, the use of measurements to compensate the effect of uncertainty has re-emerged as an active area of research. One of the ideas therein is to adapt the inputs in order to track the active constraints and push certain sensitivities to zero. In perturbation-based optimization, the sensitivities are evaluated by perturbation of the inputs and measurement of the cost function, which can be experimentally time consuming. However, since more measurements (typically the outputs) than just the cost function are available, the idea developed in this paper is to incorporate the outputs in a measurement-based optimization framework. This is done using an extension to the neighboring-extremal scheme for the case of output measurements. If measurement noise can be neglected, the approach is shown to converge to the optimum in at most two input updates. The effect of measurement noise is * Current address: Department of Chemical Engineering, Ecole Polytechnique Montreal, Montreal, Canada also investigated. The strength of neighboring-extremal output feedback for optimization is illustrated on a continuous chemical reactor example.

Equivalence between Neighboring-Extremal Control and Self-Optimizing Control for the Steady-State Optimization of Dynamical Systems

“…It has been shown that a particular choice of CVs leads to an estimation of the cost gradient 13 , which is precisely the focus of NEC. Although the two approaches attempt to solve the same static optimization problem, they are still sometimes seen as being different 13 , which is probably due to the fact that they were introduced with different types of measurements, namely, at steady state for NEC 11 and in the transient for SOC 9 . By presenting the two approaches on the same footing, this paper shows that they are indeed very similar and, in fact, the two methods can be made strictly equivalent through appropriate tuning.…”

“…The first method, neighboring-extremal control (NEC), implements first-order optimality corrections using state feedback 10 . It has been shown that NEC is a first-order approximation to gradientbased optimization 11 . The second method is self-optimizing control (SOC) based on the null-space method, which proposes to determine CVs that ensure marginal optimality loss when maintained at their nominal optimal values 9,12 .…”

“…This equation is a first-order approximation to the gradient-based optimization scheme as was shown in 11 . Interestingly, equation (24) indicates that the NEC law has an integral term and is therefore able to force the estimated gradient to zero.…”

“…The technique, which has been revisited recently to handle parametric uncertainty and output feedback 11 , uses the first-order variations of the NCO to compute δ u in terms of the parametric disturbances δ θ as given in equation (18). More specifically, NEC relies on the implicit estimation of δ θ from δ y and δ u, which is described next.…”

Use of Convex Model Approximations for Real-Time Optimization via Modifier Adaptation

“…Again, the performance of such an approach will generally decrease when the number of inputs and plant constraints increases. Alternatively, gradient estimates can be obtained by using (i) a variational analysis of the modeled cost and constraint functions to obtain the gradients in terms of the inputs and outputs, and (ii) plant measurements to compute the gradient values at the current iteration 26 . This approach can lead to perfect gradient estimates provided the model uncertainty is of parametric nature and the identity of the uncertain parameters is known.…”