Efficient multiple objective optimal control of dynamic systems with integer controls
Introduction
Many industrial processes can be accurately modelled by differential equations. Optimising their design and control, hence, gives rises to dynamic optimisation or optimal control problems, which have been studied extensively over the last 60 years (see, e.g., [1] for a historical review). However, much less results have been reported for specific subclasses, e.g., (i) mixed-integer optimal control problems (MIOCPs) with time-dependent control variables which can only take values from a finite set, or (ii) multiple objective optimal control problems (MOOCPs) with multiple and conflicting objective functions. This lack of generic results is not surprising since both classes are computationally challenging in nature. MIOCPs are closely related to the mixed-integer nonlinear programming problem (MINLP) class, which has been shown to be NP-hard [2]. MOOCPs typically give rise to a set of equally valid optimal solutions (i.e., the so-called Pareto set) instead of one single optimum [3]. Nevertheless, both classes are important for practical applications. MIOCPs are found, e.g., in car driving due to gear shifts [4], [5] or in process industry due to on-off valve switchings [6], while MOOCPs are encountered whenever trade-offs, e.g., between production and energy consumption, have to be accounted for (see, e.g., [7] for a review).
However, a lot of progress has been made over the last decade. For both problem classes generic and systematic solution approaches based on efficient deterministic procedures have been developed, resulting in a tremendous decrease in computation time. Driven by advances in scalar multiple objective optimisation (e.g., the development of generic deterministic approaches like normal boundary intersection (NBI) [8] and normalised normal constraint (NNC) [9]), procedures to quickly and efficiently generate the Pareto set for MOOCPs have lately been proposed by Logist et al. [10], [11]. For the mixed-integer optimal control cases, efficient convexification based direct multiple shooting approaches have recently been reported by Sager et al. [12]. An overview of (recent) advances in MIOCPs and references to the literature are provided in [13].
The aim of this paper is to design a generic, accurate and fast solution strategy for generating the Pareto set in MO-MIOCPs. The rationale behind the proposed strategy is a synergy between deterministic techniques from the fields of multiple objective optimisation (MOO) and mixed-integer optimal control (MIOC). Here, the exploitation of deterministic convexification techniques for MIOCPs does not only allow to quickly approximate the exact MIOCP solution to any desired accuracy, but also enables a synergistic coupling with accurate deterministic reformulation approaches from continuous scalar MOO. In particular, scalarisation methods as NBI and NNC (which have been shown to mitigate the intrinsic drawbacks of the classic weighted sum (WS)) are used to convert the MO-MIOCP into a series of parametric single objective MIOCPs, while each of these MIOCPs is efficiently solved by a direct multiple shooting [14] approach, which exploits convex relaxations of the integer requirements. In summary, a synergistic effect is obtained between scalarisation methods which accurately yield the Pareto set for continuous scalar multiple objective optimisation problems (MOOPs), and convexification techniques which efficiently provide an accurate solution to the MIOCPs, resulting in highly competitive computation times. As a result, the Pareto optimal set can be computed up to any desired accuracy, without the need for solving integer problems. This is very important—while for stochastic approaches it is an advantage if the search space is limited to discrete values, for derivative-based deterministic approaches a discrete nature typically leads to an exponential increase in runtime. Hence, with the current approach the limitations of stochastic approaches can be overcome for an important problem class.
In Section 2 the mathematical formulation of a general MO-MIOCP is first introduced, then typical aspects and methods for MOO and MIOC are reviewed. Afterwards, the proposed approach for MO-MIOCPs is described. Section 3 introduces the case studies: (i) a testdrive case study which involves a detailed car model with gear shifts and exhibits conflicting minimum time–minimal fuel consumption objectives and (ii) a jacket tubular reactor for which only cooling fluid at certain temperatures is available in view of conflicting conversion, heat transfer and installation costs. The results are discussed in Section 4. Section 5 summarises the main conclusions.
Section snippets
Multiple-objective mixed-integer optimal control
In this section, first the general formulation of an MO-MIOCP is specified. Afterwards, specific concepts and existing approaches for scalar MOOPs and MIOCPs are reviewed. Finally, an alternative MO-MIOCP approach is presented. The rationale behind the proposed approach is that the convex relaxation strategies exploited to quickly solve the MIOCPs at any desired accuracy (without the need for solving integer problems), enable also the coupling to accurate deterministic MOO procedures that
Case studies
To test the proposed MO-MIOCP approach, two cases are studied: (i) a testdrive of car and (ii) the design of a jacketed tubular reactor, which exhibit two and three conflicting objectives, respectively.
Results and discussion
In this part the results obtained with the scalarisation based approaches for the double-lane change manoeuvre and the jacketed tubular reactor are presented in Sections 4.1 Testdrive: Pareto set and optimal trajectories, 4.2 Tubular reactor: Pareto set and optimal trajectories, respectively. Here, it will be shown that Algorithm 2.4 allows to compute the Pareto set to any desired accuracy without the need of actually solving integer problems. In Section 4.3 – whenever possible – a comparison
Conclusions
In the current paper a generic approach to efficiently solve MO-MIOCPs has been proposed. The rationale is a synergy between complementary approaches from the fields of continuous scalar MOO and MIOC. To tackle the multiple objective aspect, deterministic scalarisation methods as NBI and NNC have been exploited, which transform the original MO-MIOCP into a series of parametric single objective MIOCPs. These MIOCPs are then solved efficiently by deterministic direct multiple shooting techniques,
Acknowledgements
Work supported in part by Projects OT/03/30-OT/09/025/TBA, OPTEC (Center-of-Excellence Optimization in Engineering) EF/05/006 and SCORES4CHEM KP/09/005 of the Katholieke Universiteit Leuven, and by the Belgian Program on Interuniversity Poles of Attraction, initiated by the Belgian Federal Science Policy Office, and by the Heidelberg Graduate School Mathematical and Computational Methods for the Sciences. J.F. Van Impe holds the chair Safety Engineering sponsored by the Belgian chemistry and
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