On the Role of Constraints in Optimization under Uncertainty

Saurabh Anand Deshpande
EPFL, 2012
Thèse EPFL

Résumé

This thesis addresses the problem of industrial real-time process optimization that suffers from the presence of uncertainty. Since a process model is typically used to compute the optimal operating conditions, both plant-model mismatch and process disturbances can result in suboptimal or, worse, infeasible operation. Hence, for practical applications, methodologies that help avoid re-optimization during process operation, at the cost of an acceptable optimality loss, become important. The design and analysis of such approximate solution strategies in real-time optimization (RTO) demand a careful analysis of the components of the necessary conditions of optimality. This thesis analyzes the role of constraints in process optimality in the presence of uncertainty. This analysis is made in two steps. Firstly, a general analysis is developed to quantify the effect of input adaptation on process performance for static RTO problems. In the second part, the general features of input adaptation for dynamic RTO problems are analyzed with focus on the constraints. Accordingly, the thesis is organized in two parts: For static RTO, a joint analysis of the model optimal inputs, the plant optimal inputs and a class of adapted inputs, and For dynamic RTO, an analytical study of the effect of local adaptation of the model optimal inputs. The first part (Chapters 2 and 3) addresses the problem of adapting the inputs to optimize the plant. The investigation takes a constructive viewpoint, but it is limited to static RTO problems modeled as parametric nonlinear programming (pNLP) problems. In this approach, the inputs are not limited to being local adaptation of the model optimal inputs but, instead, they can change significantly to optimize the plant. Hence, one needs to consider the fact that the set of active constraints for the model and the plant can be different. It is proven that, for a wide class of systems, the detection of a change in the active set contributes only negligibly to optimality, as long as the adapted solution remains feasible. More precisely, if η denotes the magnitude of the parametric variations and if the linear independence constraint qualification (LICQ) and strong second-order sufficient condition (SSOSC) hold for the underlying pNLP, the optimality loss due to any feasible input that conserves only the strict nominal active set is of magnitude O(η2), irrespective of whether or not there is a change in the set of active constraints. The implication of this result for a static RTO algorithm is to prioritize the satisfaction of only a core set of constraints, as long as it is possible to meet the feasibility requirements. The second part (Chapters 4 and 5) of the thesis deals with a way of adapting the model optimal inputs in dynamic RTO problems. This adaptation is made along two sets of directions such that one type of adaptation does not affect the nominally active constraints, while the other does. These directions are termed the sensitivity-seeking (SS) and the constraint-seeking (CS) directions, respectively. The SS and CS directions are defined as elements of a fairly general function space of input variations. A mathematical criterion is derived to define SS directions for a general class of optimal control problems involving both path and terminal constraints. According to this criterion, the SS directions turn out to be solutions of linear integral equations that are completely defined by the model optimal solution. The CS directions are then chosen orthogonal to the subspace of SS directions, where orthogonality is defined with respect to a chosen inner product on the space of input variations. It follows that the corresponding subspaces are infinite-dimensional subspaces of the function space of input variations. It is proven that, when uncertainty is modeled in terms of small parametric variations, the aforementioned classification of input adaptation leads to clearly distinguishable cost variations. More precisely, if η denotes the magnitude of the parametric variations, adaptation of the model optimal inputs along SS directions causes a cost variation of magnitude O(η2). On the other hand, the cost variation due to input adaptation along CS directions is of magnitude O(η). Furthermore, a numerical procedure is proposed for computing the SS and CS components of a given input variation. These components are projections of the input variation on the infinite-dimensional subspaces of SS and CS directions. The numerical procedure consists of the following three steps: approximation of the optimal control problem by a pNLP problem, projection of the given direction on the finite-dimensional SS and CS subspaces of the pNLP and, finally, reconstruction of the SS and CS components of the original problem from those of the pNLP.

Official source

https://infoscience.epfl.ch/record/167920?ln=fr

À propos de ce résultat

Cette page est générée automatiquement et peut contenir des informations qui ne sont pas correctes, complètes, à jour ou pertinentes par rapport à votre recherche. Il en va de même pour toutes les autres pages de ce site. Veillez à vérifier les informations auprès des sources officielles de l'EPFL.

Concepts associés

Chargement

Publications associées

Chargement

Saurabh Anand Deshpande
EPFL, 2012
Thèse EPFL

Résumé

Official source

https://infoscience.epfl.ch/record/167920?ln=fr

À propos de ce résultat

Concepts associés (6)

Concepts associés

Chargement

Publications associées (3)

Publications associées

Chargement

Optimization is important in science and engineering as a way of finding ”optimal” situations, designs or operating conditions. Optimization is typically performed on the basis of a mathematical model of the process under investigation. In practice, optimization is complicated by the presence of uncertainty in the form of plant-model mismatch and unknown disturbances. Without uncertainty, one could use the model at hand, optimize it numerically off-line and implement the optimal inputs in an open-loop fashion. However, because of uncertainty, the inputs need to be re-computed or adjusted in real-time based on measurements. This is the field of real-time optimization, which is labeled RTO for static optimization problems [1]. A standard way of implementing RTO is via the so-called two-step approach of repeated parameter estimation and optimization. In this scheme, measurements are used to adapt the model parameters, and the updated model is used for optimization. The fixed point of this iterative scheme is an important issue. The quality of the model is often overlooked in real-time optimization. The model is typically inaccurate due to lack of time for detailed modeling or shear complexity. More surprising is the fact that a model can predict the plant outputs well but be inadequate to push the plant to its optimum, which means that the model does not adequately represent the optimality conditions of the plant. The problem of model selection in the two-step RTO approach has been discussed in [2]. If the model is structurally correct and the parameters are identifiable, convergence to the plant optimum can be achieved in a single iteration. However, in the presence of plant-model mismatch, whether the scheme converges, or to which point it does converge, becomes anyone’s guess. This is due to the fact that the objective for parameter adaptation might be unrelated to the cost and constraint values and gradients that drive optimality in the optimization problem. Hence, minimizing the mean-square error of the plant outputs may not help in our quest for feasibility and optimality. Convergence under plant-model mismatch has been addressed in [3] and [4], where it has been shown that optimal operation is reached if model adaptation leads to matched KKT conditions for the model and the plant. This contribution addresses the convergence of two-step RTO schemes in the presence of structural plant-model mismatch. We propose to investigate the parameter estimation and optimization problems in the light of the second-order sufficient conditions of optimality to show that, in general, there are too few degrees of freedom to be able to reach plant optimality. A possible solution to this problem consists in reconciling the objectives of the parameter estimation and optimization problems along the lines of “modeling for optimization” [5,6]. One could for example use as “measured outputs” in the parameter estimation problem estimates of the necessary conditions of optimality for the optimization problem. These issues will be investigated both theoretically and via the simulation of chemical processes at steady state. References [1] Marlin, T.E. and Hrymak, A.N. (1997). “Real-Time Operations Optimization of Continuous Processes”, In AIChE Symposium Series - CPC-V. Vol. 93. 156–164. [2] Forbes, J.F. and Marlin, T.E. (1996). “Design cost: A Systematic Approach to Technology Selection for Model-Based Real-Time Optimization Systems”, Comp. Chem. Eng. 20, 717–734. [3] Biegler, L.T., Grossmann, I.E. and Westerberg, A.W. (1985). “A Note on Approximation Techniques Used for Process Optimization”, Comp. Chem. Eng. 9, 201–206. [4] Forbes, J.F., Marlin, T. E. and MacGregor, J.F. (1994). “Model Adequacy Requirements for Optimizing Plant Operations”, Comp. Chem. Eng. 18(6), 497–510. [5] Srinivasan, B. and Bonvin, D. (2002). “Interplay Between Identification and Optimization in Run-to-Run Optimization Schemes”, ACC, Anchorage 2174–2179. [6] Bonvin, D. and Srinivasan, B. (2012). “On the Role of the Necessary Conditions of Optimality in Structuring Dynamic Real-Time Optimization Schemes”, submitted to Comp. Chem. Eng.

2012

Chargement

The process industries are characterized by a large number of continuously operating plants, for which optimal operation is of economic and ecological importance. Many industrial systems can be regarded as an arrangement of several subsystems, where outputs of certain subsystems are inputs to others. This gives rise to the notion of interconnected systems. Plant optimality is difficult to achieve when the model used in optimization is inaccurate or in the presence of process disturbances. However, in the presence of plant-model mismatch, optimal operation can be enforced via specific real-time optimization methods. Specifically, this thesis considers so-called Modifier-Adaptation schemes which achieve plant optimality by direct incorporation of process measurements in the form of first-order corrections. As a first contribution, this thesis proposes a novel problem formulation for modifier adaptation. Specifically, it is focused on plants consisting of multiple interconnected subsystems that allows problem decomposition and application of distributed optimization strategies. The underlying key idea is the use of measurements and global plant gradients in place of an interconnection model. As a second contribution, this thesis investigates modifier adaptation for interconnected systems relying on local gradients by using an interconnection model. We show that the use of local information in terms of model, gradients and measurements is sufficient to optimize the steady-state performance of the plant. Finally, we propose a distributed modifier-adaptation algorithm that, besides the interconnection model and local gradients, employs a coordinator. For this scheme, we prove feasible-side convergence to the plant optimum, where a coordinator ensures that the local optimal inputs computed for each subsystem are consistent with the interconnection model. The experimental effort necessary to estimate the plant gradients increases with the number of plant inputs and may become intractable and sometimes not feasible or reliable for large-scale interconnected systems. The proposed approaches that use the interconnection model and local gradients overcome this problem. As an application case study of industrial relevance, this thesis investigates the problem of optimal load-sharing for serial and parallel gas compressors. The aim of load-sharing optimization is operating compressor units in an energy-efficient way, while at the same time satisfying varying load demands. We show how the structure of both the parallel and serial compressor configurations can be exploited in the design of tailored modifier adaptation algorithms based on efficient estimation of local gradients. Our findings show that the complexity of this estimation is independent of the number of compressors. In addition, we discuss gradient estimation for the case where the compressors are operating close to the surge conditions, which induces discontinuities in the problem.

EPFL2018

Chargement

Robust control of constrained systems given an information structure

Luca Furieri

We study finite horizon optimal control where the controller is subject to sensor-information constraints, that is, each input has access to a fixed subset of states at all times. In particular, we consider linear systems affected by exogenous disturbances with state and input constraints. We establish the class of sensor-information structures that allows for the formulation of this optimization problem as a convex program. In the literature, Quadratic Invariance (QI) is a well-established result that is applicable to the infinite horizon unconstrained case. We show that, despite state and inputs constraints being enforced, QI results can be naturally adapted to our problem. To this end, we highlight and exploit the connection between Youla parametrization and disturbance-feedback policies. Additionally, we provide graph-theoretic visual insight which is consistent with Partially Nested (PN) interpretations.

IEEE2017

Chargement

2012

EPFL2018