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(101a) Deterministic Global Dynamic Optimization of Hammerstein-Wiener Models

Authors: 
Kappatou, C. D. - Presenter, RWTH Aachen University
Sass, S., RWTH Aachen University
Najman, J., RWTH Aachen University
Bongartz, D., RWTH Aachen University
Mitsos, A., RWTH Aachen University
Hammerstein-Wiener (HW) models represent a significant class of data-driven models. They are block-structured models that consist of two static nonlinear blocks preceding and following a linear block with the system’s dynamics. Over the past years, numerous studies on identification of these systems have been reported in the literature, cf., e.g., [1-4]. HW models have been employed to model a wide spectrum of physical, chemical, and biological systems [5]. Especially in chemical engineering, they have been often used for process modeling and control, cf., e.g., [4,5].

The overall structure of HW models is inherently nonlinear, and thus optimization problems with these models embedded may exhibit suboptimal local minima. Local approaches are common practice in solving dynamic optimization problems, mainly due to their computational advantages. However, global optimization can be crucial for determining process validity (cf., e.g., parameter estimation studies [6, 7]) and can have a profound effect on process profitability [8]. Over the past few years, significant developments on global dynamic optimization have been accomplished [9]. Yet, research in this field lies still on an initial stage.

In this work, we present a novel computational algorithm to solve HW models to global optimality. More precisely, we expand recent theory for global optimization of systems with linear ODEs embedded [10] to HW models. In particular, we introduce additional optimization variables beside the degrees of freedom to enable the application of the theory of Singer and Barton [10]. For the numerical solution of the problem we apply a discretize-then-relax method to obtain a nonlinear program (NLP), which we formulate in a reduced-space fashion similar to [11]. The NLPs are solved with our open-source global optimization software MAiNGO [12]. MAiNGO is based on McCormick relaxations [13] and subgradient propagation as presented in [14]. We test our optimization strategy for different numerical case studies, both for off-line and on-line (nonlinear model predictive control) applications.

The results demonstrate the potential benefits of the presented approach over a wide field of applications. Computational effort of our approach scales favorably with refining state discretization, but is in general more sensitive to the number of control parameters. Therefore, our approach is particularly promising in the control field, where problems with short time horizons and few control elements are solved iteratively.

Acknowledgements:
The authors gratefully acknowledge the financial support of the Kopernikus project SynErgie by the Federal Ministry of Education and Research (BMBF) and the project supervision by the project management organization Projektträger Jülich. The authors would like to thank Adel Mhamdi for fruitful discussions and recommendations throughout the development of this work.

References:
[1] Bai EW. An optimal two-stage identification algorithm for Hammerstein–Wiener nonlinear systems. Automatica, 1998; 34(3):333–338.
[2] Zhu Y. Estimation of an N–L–N Hammerstein–Wiener model. Automatica, 2002; 38(9):1607–1614.
[3] Wills A, Schön TB, Ljung L and Ninness B. Identification of Hammerstein–Wiener models. Automatica, 2013; 49(1):70-81.
[4] Wang Z and Georgakis C. Identification of Hammerstein-Weiner models for nonlinear MPC from infrequent measurements in batch processes. Journal of Process Control, 2019; 82:58-69.
[5] Ławryńczuk M. Nonlinear predictive control for Hammerstein–Wiener systems. ISA transactions, 2015; 55:49-62.
[6] Papamichail I and Adjiman CS. Global optimization of dynamic systems. Computers & Chemical Engineering, 2004; 28(3):403-415.
[7] Singer AB, Taylor JW, Barton PI and Green WH. Global dynamic optimization for parameter estimation in chemical kinetics. The Journal of Physical Chemistry A, 2006; 110(3):971-976.
[8] Scott JK and Barton PI. Reachability analysis and deterministic global optimization of DAE models. In Surveys in Differential-Algebraic Equations III, 2015; pp. 61–116. Springer, Cham.
[9] Chachuat B, Singer AB and Barton PI. Global methods for dynamic optimization and mixed-integer dynamic optimization. Industrial & Engineering Chemistry Research, 2006; 45(25):8373–8392.
[10] Singer AB and Barton PI. Global solution of optimization problems with parameter-embedded linear dynamic systems. Journal of Optimization Theory and Applications, 2004; 121(3):613–646.
[11] Bongartz D and Mitsos A. Deterministic global optimization of process flowsheets in a reduced space using McCormick relaxations. Journal of Global Optimization, 2017; 69(4):761-796.
[12] Bongartz D, Najman J, Sass S and Mitsos A. MAiNGO: McCormick-based Algorithm for mixed-integer Nonlinear Global Optimization. Technical report, Process Systems Engineering (AVT.SVT), RWTH Aachen University, 2018.
[13] McCormick GP. Computability of global solutions to factorable nonconvex programs: Part I—convex underestimating problems. Mathematical programming, 1976; 10(1):147–175.
[14] Mitsos A, Chachuat B and Barton PI. McCormick-based relaxations of algorithms. SIAM Journal on Optimization, 2009; 20(2):573-601.

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