(494c) Branch and Bound Method for Multiobjective Control Structure Design

Control structure design (CSD) deals with the selection of controlled and manipulated variables (CVs and MVs), and the pairings interconnecting these variables [1]. Simply stating, the objective of CSD is to decide upon: Given the process, where shall the controllers be placed? This choice is often not obvious for processes encountered in practice. This problem is further complicated by the increasing mass and energy integration among the different process units, which necessitates consideration of the whole plant together. CSD for complete chemical plants is also known as plantwide control.

With its practical implications, CSD has been extensively studied; see [2] for an overview of available techniques. In most of the available methods, the selection of variables and pairings is carried out sequentially, which often requires enumeration of every alternative. The rapid growth of alternatives with process dimensions makes CSD through an exhaustive search computationally forbidding for large-scale processes. Furthermore, the selection of CVs and MVs followed by pairing selection may result in sub-optimal control structures. For example, the selected CVs and MVs may lead to highly interacting control loops for all possible pairings rendering decentralized control difficult. This necessitates the consideration of different tasks of CSD together in a multiobjective optimization framework such that a set of promising solutions (Pareto-optimal set) can be found. Then, the practicing engineer can select the control structure from the Pareto-optimal set by trading-off different selection criteria.

Traditionally, multiobjective CSD problem has been solved by converting the multiobjective problem into an optimization problem with a single objective by weighing different objectives [3] or by converting all but one of the objectives to constraints [4]. In these approaches, the binary decision variables related to variable or pairing selection are relaxed as continuous variables. Subsequently, the Pareto-optimal set is obtained by repeatedly solving the mixed integer linear or nonlinear program (MILP or MINLP) with different weights or constraint limits. A drawback of these approaches is that the choice of weights and constraint limits is non-trivial. Furthermore, the Pareto-optimal set obtained using weighted objective function approach is not necessarily complete. Evolutionary algorithms can directly handle the multiobjective nature of CSD problem, but do not guarantee global optimality of the solution.

Recently, efficient branch and bound (BAB) methods have been developed for selection of CVs [5], [6], MVs [7], and pairings [8] by posing them as subset selection and permutation problems, respectively. These BAB methods guarantee globally optimal solution, while requiring several orders of magnitude lower computational times in comparison with exhaustive search. These methods, however, still need to be applied sequentially (CV and MV selection followed by pairing selection). Motivated by this drawback, we propose a BAB method to directly solve the multiobjective CSD problem in this paper.

The proposed BAB framework is general and can handle most of the available criteria for the selection of CVs, MVs and pairings. For illustration purposes, we consider a biobjective CSD problem in this work, where minimum singular value (MSV) rule [1] and mu-interaction measure (mu-IM) [9] are used for selection of CVs and pairings, respectively. The computational efficiency of the proposed method is demonstrated using randomly generated matrices and the large-scale case study of HDA process. These numerical tests show that the BAB method is able to reduce the solution time by several orders of magnitude in comparison with exhaustive search.

References

1. S. Skogestad and I. Postlethwaite, Multivariable Feedback Control: Analysis and Design, 1st ed. Chichester, UK: John Wiley & sons, 1996.

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5. Y. Cao and V. Kariwala, ?Bidirectional branch and bound for controlled variable selection Part I. Principles and minimum singular value criterion,? Comput. Chem. Engng., vol. 32, no. 10, pp. 2306?2319, 2008.

6. V. Kariwala and Y. Cao, ?Bidirectional branch and bound for controlled variable selection: Part II. Exact local method for self-optimizing control,? Comput. Chem. Eng., vol. In press, 2009.

7. Y. Cao and P. Saha, ?Improved branch and bound method for control structure screening,? Chem. Engg. Sci., vol. 60, no. 6, pp. 1555?1564, 2005.

8. V. Kariwala and Y. Cao, ?Efficient branch and bound methods for pairing selection,? in Proc. 17th IFAC World Congress, Seoul, Korea, 2008.

9. P. Grosdidier and M. Morari, ?Interaction measures for systems under decentralized control,? Automatica, vol. 22, no. 3, pp. 309?319, 1986.