Modifier adaptation with guaranteed feasibility in the presence of gradient uncertainty

Highlights

• Modifier-adaptation scheme with guaranteed feasibility.

• Plant feasibility based on constraint upper bounds.

• Plant feasibility is guaranteed in the presence of gradient uncertainty.

• The approach is illustrated for the optimization of a chemical reactor.

Abstract

In the context of real-time optimization, modifier-adaptation schemes use estimates of the plant gradients to achieve plant optimality despite plant-model mismatch. Plant feasibility is guaranteed upon convergence, but not at the successive operating points computed by the algorithm prior to convergence. This paper presents a strategy for guaranteeing rigorous constraint satisfaction of all iterates in the presence of plant-model mismatch and uncertainty in the gradient estimates. The proposed strategy relies on constructing constraint upper-bounding functions that are robust to the gradient uncertainty that results when the gradients are estimated by finite differences from noisy measurements. The performance of the approach is illustrated for the optimization of a continuous stirred-tank reactor.

Introduction

The optimization of process operations is key to the economic success of continuous and batch industrial processes, for which the goal is to maximize profit or minimize cost subject to a number of operating constraints. An optimal operating point is typically found by solving a model-based optimization problem. Unfortunately, due to plant-model mismatch, the optimal solution obtained using the model does not, in general, correspond to the plant optimum, and may not even be feasible for the plant.

In order to deal with plant-model mismatch, real-time optimization (RTO) schemes use measurements to iteratively improve the quality of the model used for optimization. The most common RTO scheme consists in the two-step approach of parameter estimation followed by optimization using the updated model (Chen and Joseph, 1987, Darby et al., 2011). The performance of the two-step approach is highly dependent on the accuracy of the model, the choice of the adjustable parameters, and the choice of the measurements (Yip and Marlin, 2004, Quelhas et al., 2013). In the presence of structural plant-model mismatch, the model optimum will typically differ from the plant optimum. It is well known that a model that is adequate for optimization needs to be able to accurately approximate the plant necessary conditions of optimality (NCO), which involves the characterization of the set of active constraints and the gradients of the cost and constraint functions (Biegler et al., 1985, Marchetti et al., 2009). Hence, it is well understood in the RTO community that gradient estimates for the uncertain plant are key in achieving optimal plant operation in the presence of plant-model mismatch (Roberts, 1995, Marchetti et al., 2009, Gao and Engell, 2005).

For instance, modifier-adaptation (MA) schemes enforce plant optimality and feasibility upon convergence by making first-order corrections to the cost and constraint functions based on estimates of the plant cost and constraint gradients (Gao and Engell, 2005, Marchetti et al., 2009). Several gradient estimation methods can be considered for RTO applications (Mansour and Ellis, 2003, Srinivasan et al., 2011). Dual RTO schemes estimate the gradients by finite-difference approximation based on the past operating points (Brdyś and Tatjewski, 1994, Brdyś and Tatjewski, 2005, Marchetti et al., 2010, Marchetti, 2013). This requires accommodating two conflicting objectives: the “primal objective” aims at improving the cost, while the “dual objective” aims at estimating the gradients accurately. The dual objective is dealt with by introducing a duality constraint that considers the accuracy of the gradient estimates.

In addition to enforcing plant optimality upon convergence, it is desirable for an RTO algorithm to generate only feasible iterates, that is, the uncertain plant constraints are satisfied at all RTO iterates (Bunin et al., 2013). For continuous processes, it is possible to generate only feasible steady-state operating points by implementing the RTO results via a feedback control layer that controls the constrained quantities (Tatjewski et al., 2001, Marchetti et al., 2011, Navia et al., 2012). In many cases, however, the use of feedback controllers is not possible or inappropriate, for instance, when the constrained quantities cannot be measured with the required frequency. Another possibility to enforce feasibility of the points generated by the RTO algorithm is by using constraint functions that upper bound the constraints of the real system. The use of constraint upper bounds generates feasibility regions that are inner approximations of the true plant feasibility region. Bunin et al. (2013) proposed constraint upper bounds based on Lipschitz constants. These bounds match the value of the plant constraints at the current operating point and do not require the knowledge of the corresponding gradients. Nevertheless, since the approach may converge to a suboptimal point if the RTO iterates get too close to constraint activation, a projection is typically implemented to stay sufficiently deep inside the feasible region. This projection does require the knowledge of the gradients of the plant constraints. More recently, Singhal et al. (2016) considered the use of constraint upper bounds based on quadratic data-driven surrogate models. These constraint upper bounds match the values and the gradients of the plant constraints at the current operating point, and are valid under the assumption that the gradients of the plant constraints are perfectly known. However, in any real-world application, the plant gradients can only be approximated, and the feasibility guarantees provided by the constraint upper bounds are lost.

In this paper, we derive constraint upper bounds based on second-order corrections of a first-principles model for use in modifier-adaptation schemes, and we tackle the problem of rigorous feasibility guarantees for all RTO iterates in the presence of gradient uncertainty. In general, the level of gradient uncertainty depends on the approach used to estimate the gradients. Here, we consider finite-difference gradient estimation from the measurements obtained at the current and past operating points. For this case, an upper-bound on the gradient-error norm was obtained in Marchetti et al. (2010). In the present paper, we rely on this approach to construct bounds on the constraint gradient errors. We show how these bounds can be used to robustly guarantee plant feasibility of modifier-adaptation iterates in the presence of plant-model mismatch and additive gradient uncertainty.

The paper is organized as follows. Section 2 presents preliminary material that is used in this work, including the formulation of the static optimization problem, the introduction to modifier adaptation, the approach used to estimate the gradients from past operating points, the computation of upper bounds on the gradient error, and the introduction to dual modifier adaptation. The proposed approach for guaranteeing feasibility in RTO schemes by constructing robust constraint upper bounds is presented in Section 3. A dual modifier-adaptation algorithm is formulated that combines all the elements previously introduced. The performance of the RTO algorithm is illustrated via the case study of the Williams-Otto reactor in Section 4. Finally, conclusions are drawn in Section 5.