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Der Campus im Frühling. Bildinformationen anzeigen

Der Campus im Frühling.

Foto: Universität Paderborn, Kamil Glabica.

Dr. Sebastian Peitz

Kontakt
Profil
Vita
Publikationen
Dr. Sebastian Peitz

Lehrstuhl für Angewandte Mathematik

Mitglied - Akademischer Rat a. Z.

Institut für Industriemathematik

Geschäftsführer - Akademischer Rat a. Z.

Telefon:
+49 5251 60-5021
Fax:
+49 5251 60-4216
Büro:
TP21.1.25
Besucher:
Technologiepark 21
33098 Paderborn
1. Forschungsthemen
  • Mehrzieloptimierung und -optimalsteuerung,
    • Beschleunigung von Algorithmen durch das Ausnutzen von Strukturen
    • Echtzeit-Anwendbarkeit in Kombination Model Predictive Control
    • Lösen von PDE-restringierten Problemen mittels Modellreduktion
       
  • Modellreduktion für komplexe Systeme, insbesondere Partielle Differentialgleichungen, mittels
    • Proper Orthogonal Decomposition (POD)
    • Koopman Operator-Ansätzen
    • Maschineller Lernverfahren
       
  • Kontrolle komplexer dynamischer Systeme, insbesondere von Fluidströmungen
2. Forschungsprojekte
  • Titel: Multiobjective Optimization of Non-Smooth PDE-Constrained Problems - Switches, State Constraints and Model Order Reduction
    Beschreibung: Gemeinsames Projekt mit der Universität Konstanz (Prof. Dr. Stefan Volkwein), in dem wir Mehrzieloptimierungs- und Optimalsteuerungsprobelme mit nicht-glatten PDE-Nebenbedingungen lösen. Zur Erhöhung der numerischen Effizienz werden Modellreduktionsverfahren wie die Proper Orthogonal Decomposition (POD) eingesetzt, und wir entwickelnt Fehlerschranken, um konvergente Algorithmen zu konstruieren.
    Förderung: DFG Priority Programme 1962 "Non-Smooth and Complementarity-Based Distributed Parameter Systems: Simulation and Hierarchical Optimization"
     
  • Titel: Simultaneous Development and Testing of Cyber Physical Systems (CPS) using the example of autonomous electric vehicles (SET CPS)
    Beschreibung: Gemeinsames Projekt mit dSPACE und e.GO, in dem wir den Entwicklungs- und Testprozess von autonomen elektrischen Fahrzeugen mit Hilfe von Mehrzieloptimierungsalgorithmen miteinander kombinieren, um die Entwicklungszeit neuer Autos signifikant zu reduzieren (Weitere Informationen)
    Förderung: EFRE.NRW
3. Vorträge und Präsentationen

Workshop on Optimal Control
02.12. - 03.12.2019 - University of Konstanz, Germany
"The Koopman Operator in PDE-Constrained Optimal Control"

103rd Meeting of the GOR Working Group "Real World Optimization" on "Multicriteria Optimization and Decision Support in Industry"
21.11. - 22.11.2019 - Fraunhofer ITWM, Kaiserslautern, Germany
"Autonomous driving with multiobjective model predictive control"

6th International Conference on Continuous Optimization
05.08. - 08.08.2019 - Berlin, Germany
"Feedback control of nonlinear PDEs using Koopman-Operator-based switched systems"

SIAM Conference on Applications of Dynamical Systems
19.05. - 23.05.2019 - Snowbird, Utah, USA
"Data Driven Feedback Control of Nonlinear PDEs Using the Koopman Operator"

SIAM Conference on Computational Science and Engineering
25.02. - 01.03.2019 - Spokane, WA, USA
"Feedback Control of Nonlinear PDEs using Data-efficient Reduced Order Models Based on the Koopman Operator"

Symposium On Machine Learning for Dynamical Systems
11.02. - 13.02.2019 - London, UK
"Low-dimensional data-based surrogate models for analysis, simulation and control of high-dimensional dynamical systems"

Forschungsseminar der Fachgruppe CSC, Max-Planck-Institut für Dynamik Komplexer Technischer Systeme
20.11.2018 - Magedburg, Germany
"Data driven feedback control of nonlinear PDEs using the Koopman operator"

5th European Conference on Computational Optimization (EUCCO) 2018
10.09. - 12.09.2018 - Trier, Germany
"Data driven feedback control of nonlinear PDEs using the Koopman operator"

NUMDIFF-15
03.09. - 07.09.2018 - Halle (Saale), Germany
"Data driven feedback control of nonlinear PDEs using the Koopman operator"

Oxford Universitxy - Control Engineering Group Seminar
14.05.2018 - Oxford, UK
"Data driven feedback control of nonlinear PDEs using the Koopman operator"

89th Annual Meeting of GAMM
19.03. - 23.03.2018 - Munich, Germany
"Set-Oriented Multiobjective Optimal Control of PDEs using Certified ROMs"

4th CRITICS Workshop and Winter School on “Critical Transitions in Complex Systems: Mathematical theory and applications”
12.03. - 16.03.2018, Wöltingerode, Germany
"Controlling the Navier-Stokes equations using low-dimensional bilinear approximations obtained from data"

Center for Industrial Mathematics - University of Bremen
14.12.2017, Bremen, Germany
"Exploiting structure in multiobjective optimal control for real-time applicability and PDE constraints"

it's OWL Strategietagung
06.12.2017, Paderborn, Germany
"Mathematik in der Industrie - von der Mehrzieloptimierung zu intelligenten Systemen"

18th French - German - Italian Conference on Optimization
25.09. - 28.09.2017 - Paderborn, Germany
"Multiobjective optimization with uncertainties and the application to reduced order models for PDEs"

8th International Workshop on Set Oriented Numerics (SON)
12.09. - 15.09.2017 - Santa Barbara, California, USA
"A new framework for Koopman operator based open and closed loop control"

IFAC 2017 World Congress
19.07. - 24.07.2017 - Toulouse, France
"A Multiobjective MPC Approach for Autonomously Driven Electric Vehicle"

SIAM Conference on Applications of Dynamical Systems
21.05. - 25.05.2017 - Snowbird, Utah, USA
"A Multiobjective MPC Approach for Autonomously Driven Electric Vehicles"

88th Annual Meeting of GAMM
06.03. - 10.03.2017 - Weimar, Germany
"POD-based Set Oriented Multiobjective Optimal Control"

Workshop on Optimal and Feedback Control of Differential Equations
21.11. - 23.11.2016 - Konstanz, Germany
"Multiobjective Optimal Control of the Navier-Stokes Equations Using Reduced Order Modeling"

KoMSO Challenge Workshop "Reduced-Order Modeling for Simulation and Optimization"
07.11. - 08.11.2016 - Robert BOSCH Campus, Renningen, Germany
"ROM-based Multiobjective Optimal Control of the Navier-Stokes Equations"

4th European Conference on Computational Optimization
12.09. - 14.09.2016 - Leuven, Belgium
"Reduced Order Model Based Multiobjective Optimal Control of Nonlinear PDEs"

24th International Congress of Theoretical and Applied Mechanics
21.08. - 26.08.2016 - Montreal, Canada
"Reduced Order Model based Multiobjective Optimal Control of Fluids"

11th AIMS Conference on Dynamical Systems, Differential Equations and Applications
01.07. - 05.07.2016 - Orlando, Florida, USA
"Multiobjective Optimal Control of Coherent Structures Using Reduced Order Modeling"

Paderborner Wissenschaftstage
27.06.2016, Paderborn, Germany
"Energieeffizientes autonomes Fahren mit Hilfe mathematischer Optimierungsverfahren"

SON2015 - 6th International Workshop on Set-Oriented Numerics
28.09. - 01.10.2015, London, England
"Multiobjective Optimal Control Methods for Fluid Flow Using Reduced Order Modeling"

SIAM Conference on Control & it's Applications
08.07. - 10.07.2015, Paris, France
"Multiobjective Optimal Control Techniques in Energy Management" (within the Minisymposium "Controls, Optimization and Estimation for Building Energy Managment")

GAMM 86th Annual Meeting
23.03. - 27.03.2015, Lecce, Italy
"Multiobjective Optimization of the Flow Around a Cylinder Using Model Order Reduction"

Oberwolfach Seminar: Projection Based Model Reduction
23.09. - 29.09.2014, Oberwolfach, Germany

SON2014 - 5th International Workshop on Set-Oriented Numerics
01.09. - 05.09.2014, Christchurch, New Zealand
"Multiobjective Optimal Control Methods for the Development of an Intelligent Cruise Control"

SysInt2014 - Conference on System-Integrated Intelligence
02.07. - 04.07.2014, Bremen, Germany
"Development of an Intelligent Cruise Control using Optimal Control Methods"

ECMI2014 - European Consortium for Mathematics in Industry
09.06. - 13.06.2014, Taormina, Italy
"Multiobjective Optimal Control Methods for the Development of an Intelligent Cruise Control"

6th Elgersburg School on Hybrid Dynamical Systems and Mathematical Biology
16.03. - 22.03.2014, Elgersbrug, Germany

Dr. Sebastian Peitz
01.10.2017 - heute

Geschäftsführer

Institut für Industriemathematik, Universität Paderborn

01.10.2013 - 30.09.2017

Wissenschaftlicher Mitarbeiter

Lehrstuhl für Angewandte Mathematik und Institut für Industriemathematik (Prof. Dr. Michael Dellnitz, Universität Paderborn)

08.08.2017

Promotion

Titel: "Exploiting Structure in Multiobjective Optimization and Optimal Control"

01.10.2007 - 18.07.2013

Studium

Maschinenbau (RWTH Aachen)
Abschluss: 18.07.2013 (M. Sc.)

16.06.2006

Abitur

Hans-Ehrenberg-Gymnasium (Sennestadt)


Liste im Research Information System öffnen

2020

Data-driven approximation of the Koopman generator: Model reduction, system identification, and control

S. Klus, F. Nüske, S. Peitz, J. Niemann, C. Clementi, C. Schütte, Physica D: Nonlinear Phenomena (2020), 406

We derive a data-driven method for the approximation of the Koopman generator called gEDMD, which can be regarded as a straightforward extension of EDMD (extended dynamic mode decomposition). This approach is applicable to deterministic and stochastic dynamical systems. It can be used for computing eigenvalues, eigenfunctions, and modes of the generator and for system identification. In addition to learning the governing equations of deterministic systems, which then reduces to SINDy (sparse identification of nonlinear dynamics), it is possible to identify the drift and diffusion terms of stochastic differential equations from data. Moreover, we apply gEDMD to derive coarse-grained models of high-dimensional systems, and also to determine efficient model predictive control strategies. We highlight relationships with other methods and demonstrate the efficacy of the proposed methods using several guiding examples and prototypical molecular dynamics problems.


Deep model predictive flow control with limited sensor data and online learning

K. Bieker, S. Peitz, S.L. Brunton, J.N. Kutz, M. Dellnitz, Theoretical and Computational Fluid Dynamics (2020)

The control of complex systems is of critical importance in many branches of science, engineering, and industry, many of which are governed by nonlinear partial differential equations. Controlling an unsteady fluid flow is particularly important, as flow control is a key enabler for technologies in energy (e.g., wind, tidal, and combustion), transportation (e.g., planes, trains, and automobiles), security (e.g., tracking airborne contamination), and health (e.g., artificial hearts and artificial respiration). However, the high-dimensional, nonlinear, and multi-scale dynamics make real-time feedback control infeasible. Fortunately, these high- dimensional systems exhibit dominant, low-dimensional patterns of activity that can be exploited for effective control in the sense that knowledge of the entire state of a system is not required. Advances in machine learning have the potential to revolutionize flow control given its ability to extract principled, low-rank feature spaces characterizing such complex systems.We present a novel deep learning modelpredictive control framework that exploits low-rank features of the flow in order to achieve considerable improvements to control performance. Instead of predicting the entire fluid state, we use a recurrent neural network (RNN) to accurately predict the control relevant quantities of the system, which are then embedded into an MPC framework to construct a feedback loop. In order to lower the data requirements and to improve the prediction accuracy and thus the control performance, incoming sensor data are used to update the RNN online. The results are validated using varying fluid flow examples of increasing complexity.


Pareto Explorer: a global/local exploration tool for many-objective optimization problems

O. Schütze, O. Cuate, A. Martín, S. Peitz, M. Dellnitz, Engineering Optimization (2020), 52(5), pp. 832-855

Multi-objective optimization is an active field of research that has many applications. Owing to its success and because decision-making processes are becoming more and more complex, there is a recent trend for incorporating many objectives into such problems. The challenge with such problems, however, is that the dimensions of the solution sets—the so-called Pareto sets and fronts—grow with the number of objectives. It is thus no longer possible to compute or to approximate the entire solution set of a given problem that contains many (e.g. more than three) objectives. On the other hand, the computation of single solutions (e.g. via scalarization methods) leads to unsatisfying results in many cases, even if user preferences are incorporated. In this article, the Pareto Explorer tool is presented—a global/local exploration tool for the treatment of many-objective optimization problems (MaOPs). In the first step, a solution of the problem is computed via a global search algorithm that ideally already includes user preferences. In the second step, a local search along the Pareto set/front of the given MaOP is performed in user specified directions. For this, several continuation-like procedures are proposed that can incorporate preferences defined in decision, objective, or in weight space. The applicability and usefulness of Pareto Explorer is demonstrated on benchmark problems as well as on an application from industrial laundry design.


Feedback Control of Nonlinear PDEs Using Data-Efficient Reduced Order Models Based on the Koopman Operator

S. Peitz, S. Klus, in: Lecture Notes in Control and Information Sciences, Springer, 2020, pp. 257-282

In the development of model predictive controllers for PDE-constrained problems, the use of reduced order models is essential to enable real-time applicability. Besides local linearization approaches, proper orthogonal decomposition (POD) has been most widely used in the past in order to derive such models. Due to the huge advances concerning both theory as well as the numerical approximation, a very promising alternative based on the Koopman operator has recently emerged. In this chapter, we present two control strategies for model predictive control of nonlinear PDEs using data-efficient approximations of the Koopman operator. In the first one, the dynamic control system is replaced by a small number of autonomous systems with different yet constant inputs. The control problem is consequently transformed into a switching problem. In the second approach, a bilinear surrogate model is obtained via a convex combination of these autonomous systems. Using a recent convergence result for extended dynamic mode decomposition (EDMD), convergence of the reduced objective function can be shown. We study the properties of these two strategies with respect to solution quality, data requirements, and complexity of the resulting optimization problem using the 1-dimensional Burgers equation and the 2-dimensional Navier–Stokes equations as examples. Finally, an extension for online adaptivity is presented.


Data-Driven Model Predictive Control using Interpolated Koopman Generators

S. Peitz, S.E. Otto, C.W. Rowley, in: arXiv:2003.07094, 2020

In recent years, the success of the Koopman operator in dynamical systems analysis has also fueled the development of Koopman operator-based control frameworks. In order to preserve the relatively low data requirements for an approximation via Dynamic Mode Decomposition, a quantization approach was recently proposed in [Peitz & Klus, Automatica 106, 2019]. This way, control of nonlinear dynamical systems can be realized by means of switched systems techniques, using only a finite set of autonomous Koopman operator-based reduced models. These individual systems can be approximated very efficiently from data. The main idea is to transform a control system into a set of autonomous systems for which the optimal switching sequence has to be computed. In this article, we extend these results to continuous control inputs using relaxation. This way, we combine the advantages of the data efficiency of approximating a finite set of autonomous systems with continuous controls. We show that when using the Koopman generator, this relaxation --- realized by linear interpolation between two operators --- does not introduce any error for control affine systems. This allows us to control high-dimensional nonlinear systems using bilinear, low-dimensional surrogate models. The efficiency of the proposed approach is demonstrated using several examples with increasing complexity, from the Duffing oscillator to the chaotic fluidic pinball.


Explicit Multi-objective Model Predictive Control for Nonlinear Systems Under Uncertainty

C.I. Hernández Castellanos, S. Ober-Blöbaum, S. Peitz, in: arXiv:2002.06006, 2020

In real-world problems, uncertainties (e.g., errors in the measurement, precision errors) often lead to poor performance of numerical algorithms when not explicitly taken into account. This is also the case for control problems, where optimal solutions can degrade in quality or even become infeasible. Thus, there is the need to design methods that can handle uncertainty. In this work, we consider nonlinear multi-objective optimal control problems with uncertainty on the initial conditions, and in particular their incorporation into a feedback loop via model predictive control (MPC). In multi-objective optimal control, an optimal compromise between multiple conflicting criteria has to be found. For such problems, not much has been reported in terms of uncertainties. To address this problem class, we design an offline/online framework to compute an approximation of efficient control strategies. This approach is closely related to explicit MPC for nonlinear systems, where the potentially expensive optimization problem is solved in an offline phase in order to enable fast solutions in the online phase. In order to reduce the numerical cost of the offline phase, we exploit symmetries in the control problems. Furthermore, in order to ensure optimality of the solutions, we include an additional online optimization step, which is considerably cheaper than the original multi-objective optimization problem. We test our framework on a car maneuvering problem where safety and speed are the objectives. The multi-objective framework allows for online adaptations of the desired objective. Alternatively, an automatic scalarizing procedure yields very efficient feedback controls. Our results show that the method is capable of designing driving strategies that deal better with uncertainties in the initial conditions, which translates into potentially safer and faster driving strategies.


An efficient descent method for locally Lipschitz multiobjective optimization problems

B. Gebken, S. Peitz, in: arXiv:2004.11578, 2020

In this article, we present an efficient descent method for locally Lipschitz continuous multiobjective optimization problems (MOPs). The method is realized by combining a theoretical result regarding the computation of descent directions for nonsmooth MOPs with a practical method to approximate the subdifferentials of the objective functions. We show convergence to points which satisfy a necessary condition for Pareto optimality. Using a set of test problems, we compare our method to the multiobjective proximal bundle method by M\"akel\"a. The results indicate that our method is competitive while being easier to implement. While the number of objective function evaluations is larger, the overall number of subgradient evaluations is lower. Finally, we show that our method can be combined with a subdivision algorithm to compute entire Pareto sets of nonsmooth MOPs.


2019

On the hierarchical structure of Pareto critical sets

B. Gebken, S. Peitz, M. Dellnitz, Journal of Global Optimization (2019), 73(4), pp. 891-913

In this article we show that the boundary of the Pareto critical set of an unconstrained multiobjective optimization problem (MOP) consists of Pareto critical points of subproblems where only a subset of the set of objective functions is taken into account. If the Pareto critical set is completely described by its boundary (e.g., if we have more objective functions than dimensions in decision space), then this can be used to efficiently solve the MOP by solving a number of MOPs with fewer objective functions. If this is not the case, the results can still give insight into the structure of the Pareto critical set.


Inverse multiobjective optimization: Inferring decision criteria from data

B. Gebken, S. Peitz, in: arXiv:1901.06141, 2019

It is a very challenging task to identify the objectives on which a certain decision was based, in particular if several, potentially conflicting criteria are equally important and a continuous set of optimal compromise decisions exists. This task can be understood as the inverse problem of multiobjective optimization, where the goal is to find the objective vector of a given Pareto set. To this end, we present a method to construct the objective vector of a multiobjective optimization problem (MOP) such that the Pareto critical set contains a given set of data points or decision vectors. The key idea is to consider the objective vector in the multiobjective KKT conditions as variable and then search for the objectives that minimize the Euclidean norm of the resulting system of equations. By expressing the objectives in a finite-dimensional basis, we transform this problem into a homogeneous, linear system of equations that can be solved efficiently. There are many important potential applications of this approach. Besides the identification of objectives (both from clean and noisy data), the method can be used for the construction of surrogate models for expensive MOPs, which yields significant speed-ups. Both applications are illustrated using several examples.


ROM-based multiobjective optimization of elliptic PDEs via numerical continuation

S. Banholzer, B. Gebken, M. Dellnitz, S. Peitz, S. Volkwein, in: arXiv:1906.09075, 2019

Multiobjective optimization plays an increasingly important role in modern applications, where several objectives are often of equal importance. The task in multiobjective optimization and multiobjective optimal control is therefore to compute the set of optimal compromises (the Pareto set) between the conflicting objectives. Since the Pareto set generally consists of an infinite number of solutions, the computational effort can quickly become challenging which is particularly problematic when the objectives are costly to evaluate as is the case for models governed by partial differential equations (PDEs). To decrease the numerical effort to an affordable amount, surrogate models can be used to replace the expensive PDE evaluations. Existing multiobjective optimization methods using model reduction are limited either to low parameter dimensions or to few (ideally two) objectives. In this article, we present a combination of the reduced basis model reduction method with a continuation approach using inexact gradients. The resulting approach can handle an arbitrary number of objectives while yielding a significant reduction in computing time.


Finite-Control-Set Model Predictive Control for a Permanent Magnet Synchronous Motor Application with Online Least Squares System Identification

S. Hanke, S. Peitz, O. Wallscheid, J. Böcker, M. Dellnitz, in: 2019 IEEE International Symposium on Predictive Control of Electrical Drives and Power Electronics (PRECEDE), 2019

In comparison to classical control approaches in the field of electrical drives like the field-oriented control (FOC), model predictive control (MPC) approaches are able to provide a higher control performance. This refers to shorter settling times, lower overshoots, and a better decoupling of control variables in case of multi-variable controls. However, this can only be achieved if the used prediction model covers the actual behavior of the plant sufficiently well. In case of model deviations, the performance utilizing MPC remains below its potential. This results in effects like increased current ripple or steady state setpoint deviations. In order to achieve a high control performance, it is therefore necessary to adapt the model to the real plant behavior. When using an online system identification, a less accurate model is sufficient for commissioning of the drive system. In this paper, the combination of a finite-control-set MPC (FCS-MPC) with a system identification is proposed. The method does not require high-frequency signal injection, but uses the measured values already required for the FCS-MPC. An evaluation of the least squares-based identification on a laboratory test bench showed that the model accuracy and thus the control performance could be improved by an online update of the prediction models.


Multiobjective Optimal Control Methods for the Navier-Stokes Equations Using Reduced Order Modeling

S. Peitz, S. Ober-Blöbaum, M. Dellnitz, Acta Applicandae Mathematicae (2019), 161(1), pp. 171–199

In a wide range of applications it is desirable to optimally control a dynamical system with respect to concurrent, potentially competing goals. This gives rise to a multiobjective optimal control problem where, instead of computing a single optimal solution, the set of optimal compromises, the so-called Pareto set, has to be approximated. When the problem under consideration is described by a partial differential equation (PDE), as is the case for fluid flow, the computational cost rapidly increases and makes its direct treatment infeasible. Reduced order modeling is a very popular method to reduce the computational cost, in particular in a multi query context such as uncertainty quantification, parameter estimation or optimization. In this article, we show how to combine reduced order modeling and multiobjective optimal control techniques in order to efficiently solve multiobjective optimal control problems constrained by PDEs. We consider a global, derivative free optimization method as well as a local, gradient-based approach for which the optimality system is derived in two different ways. The methods are compared with regard to the solution quality as well as the computational effort and they are illustrated using the example of the flow around a cylinder and a backward-facing-step channel flow.


Koopman operator-based model reduction for switched-system control of PDEs

S. Peitz, S. Klus, Automatica (2019), 106, pp. 184-191

We present a new framework for optimal and feedback control of PDEs using Koopman operator-based reduced order models (K-ROMs). The Koopman operator is a linear but infinite-dimensional operator which describes the dynamics of observables. A numerical approximation of the Koopman operator therefore yields a linear system for the observation of an autonomous dynamical system. In our approach, by introducing a finite number of constant controls, the dynamic control system is transformed into a set of autonomous systems and the corresponding optimal control problem into a switching time optimization problem. This allows us to replace each of these systems by a K-ROM which can be solved orders of magnitude faster. By this approach, a nonlinear infinite-dimensional control problem is transformed into a low-dimensional linear problem. Using a recent convergence result for the numerical approximation via Extended Dynamic Mode Decomposition (EDMD), we show that the value of the K-ROM based objective function converges in measure to the value of the full objective function. To illustrate the results, we consider the 1D Burgers equation and the 2D Navier–Stokes equations. The numerical experiments show remarkable performance concerning both solution times and accuracy.


2018

A Survey of Recent Trends in Multiobjective Optimal Control—Surrogate Models, Feedback Control and Objective Reduction

S. Peitz, M. Dellnitz, Mathematical and Computational Applications (2018), 23(2)

Multiobjective optimization plays an increasingly important role in modern applications, where several criteria are often of equal importance. The task in multiobjective optimization and multiobjective optimal control is therefore to compute the set of optimal compromises (the Pareto set) between the conflicting objectives. The advances in algorithms and the increasing interest in Pareto-optimal solutions have led to a wide range of new applications related to optimal and feedback control, which results in new challenges such as expensive models or real-time applicability. Since the Pareto set generally consists of an infinite number of solutions, the computational effort can quickly become challenging, which is particularly problematic when the objectives are costly to evaluate or when a solution has to be presented very quickly. This article gives an overview of recent developments in accelerating multiobjective optimal control for complex problems where either PDE constraints are present or where a feedback behavior has to be achieved. In the first case, surrogate models yield significant speed-ups. Besides classical meta-modeling techniques for multiobjective optimization, a promising alternative for control problems is to introduce a surrogate model for the system dynamics. In the case of real-time requirements, various promising model predictive control approaches have been proposed, using either fast online solvers or offline-online decomposition. We also briefly comment on dimension reduction in many-objective optimization problems as another technique for reducing the numerical effort.


POD-based multiobjective optimal control of PDEs with non-smooth objectives

D. Beermann, M. Dellnitz, S. Peitz, S. Volkwein, in: PAMM, 2018, pp. 51-54

A framework for set‐oriented multiobjective optimal control of partial differential equations using reduced order modeling has recently been developed [1]. Following concepts from localized reduced bases methods, error estimators for the reduced cost functionals are utilized to construct a library of locally valid reduced order models. This way, a superset of the Pareto set can efficiently be computed while maintaining a prescribed error bound. In this article, this algorithm is applied to a problem with non‐smooth objective functionals. Using an academic example, we show that the extension to non‐smooth problems can be realized in a straightforward manner. We then discuss the implications on the numerical results.


Controlling nonlinear PDEs using low-dimensional bilinear approximations obtained from data

S. Peitz, in: arXiv:1801.06419, 2018

In a recent article, we presented a framework to control nonlinear partial differential equations (PDEs) by means of Koopman operator based reduced models and concepts from switched systems. The main idea was to transform a control system into a set of autonomous systems for which the optimal switching sequence has to be computed. These individual systems can be approximated very efficiently by reduced order models obtained from data, and one can guarantee equality of the full and the reduced objective function under certain assumptions. In this article, we extend these results to continuous control inputs using convex combinations of multiple Koopman operators corresponding to constant controls, which results in a bilinear control system. Although equality of the objectives can be carried over when the PDE depends linearly on the control, we show that this approach is also valid in other scenarios using several flow control examples of varying complexity.


Analyzing high-dimensional time-series data using kernel transfer operator eigenfunctions

S. Klus, S. Peitz, I. Schuster, in: arXiv:1805.10118, 2018

Kernel transfer operators, which can be regarded as approximations of transfer operators such as the Perron-Frobenius or Koopman operator in reproducing kernel Hilbert spaces, are defined in terms of covariance and cross-covariance operators and have been shown to be closely related to the conditional mean embedding framework developed by the machine learning community. The goal of this paper is to show how the dominant eigenfunctions of these operators in combination with gradient-based optimization techniques can be used to detect long-lived coherent patterns in high-dimensional time-series data. The results will be illustrated using video data and a fluid flow example.


Set-Oriented Multiobjective Optimal Control of PDEs Using Proper Orthogonal Decomposition

D. Beermann, M. Dellnitz, S. Peitz, S. Volkwein, in: Reduced-Order Modeling (ROM) for Simulation and Optimization, 2018, pp. 47-72

In this chapter, we combine a global, derivative-free subdivision algorithm for multiobjective optimization problems with a posteriori error estimates for reduced-order models based on Proper Orthogonal Decomposition in order to efficiently solve multiobjective optimization problems governed by partial differential equations. An error bound for a semilinear heat equation is developed in such a way that the errors in the conflicting objectives can be estimated individually. The resulting algorithm constructs a library of locally valid reduced-order models online using a Greedy (worst-first) search. Using this approach, the number of evaluations of the full-order model can be reduced by a factor of more than 1000.


Explicit multiobjective model predictive control for nonlinear systems with symmetries

S. Ober-Blöbaum, S. Peitz, in: arXiv:1809.06238, 2018

Model predictive control is a prominent approach to construct a feedback control loop for dynamical systems. Due to real-time constraints, the major challenge in MPC is to solve model-based optimal control problems in a very short amount of time. For linear-quadratic problems, Bemporad et al.~have proposed an explicit formulation where the underlying optimization problems are solved a priori in an offline phase. In this article, we present an extension of this concept in two significant ways. We consider nonlinear problems and -- more importantly -- problems with multiple conflicting objective functions. In the offline phase, we build a library of Pareto optimal solutions from which we then obtain a valid compromise solution in the online phase according to a decision maker's preference. Since the standard multi-parametric programming approach is no longer valid in this situation, we instead use interpolation between different entries of the library. To reduce the number of problems that have to be solved in the offline phase, we exploit symmetries in the dynamical system and the corresponding multiobjective optimal control problem. The results are verified using two different examples from autonomous driving.


A Descent Method for Equality and Inequality Constrained Multiobjective Optimization Problems

B. Gebken, S. Peitz, M. Dellnitz, in: Numerical and Evolutionary Optimization – NEO 2017, 2018

In this article we propose a descent method for equality and inequality constrained multiobjective optimization problems (MOPs) which generalizes the steepest descent method for unconstrained MOPs by Fliege and Svaiter to constrained problems by using two active set strategies. Under some regularity assumptions on the problem, we show that accumulation points of our descent method satisfy a necessary condition for local Pareto optimality. Finally, we show the typical behavior of our method in a numerical example.


Tensor-based dynamic mode decomposition

S. Klus, P. Gelß, S. Peitz, C. Schütte, Nonlinearity (2018), 31(7), pp. 3359-3380

Dynamic mode decomposition (DMD) is a recently developed tool for the analysis of the behavior of complex dynamical systems. In this paper, we will propose an extension of DMD that exploits low-rank tensor decompositions of potentially high-dimensional data sets to compute the corresponding DMD modes and eigenvalues. The goal is to reduce the computational complexity and also the amount of memory required to store the data in order to mitigate the curse of dimensionality. The efficiency of these tensor-based methods will be illustrated with the aid of several different fluid dynamics problems such as the von Kármán vortex street and the simulation of two merging vortices.


2017

A Multiobjective MPC Approach for Autonomously Driven Electric Vehicles

S. Peitz, K. Schäfer, S. Ober-Blöbaum, J. Eckstein, U. Köhler, M. Dellnitz, IFAC-PapersOnLine (2017), 50(1), pp. 8674-8679

We present a new algorithm for model predictive control of non-linear systems with respect to multiple, conflicting objectives. The idea is to provide a possibility to change the objective in real-time, e.g. as a reaction to changes in the environment or the system state itself. The algorithm utilises elements from various well-established concepts, namely multiobjective optimal control, economic as well as explicit model predictive control and motion planning with motion primitives. In order to realise real-time applicability, we split the computation into an online and an offline phase and we utilise symmetries in the open-loop optimal control problem to reduce the number of multiobjective optimal control problems that need to be solved in the offline phase. The results are illustrated using the example of an electric vehicle where the longitudinal dynamics are controlled with respect to the concurrent objectives arrival time and energy consumption.


Gradient-Based Multiobjective Optimization with Uncertainties

S. Peitz, M. Dellnitz, in: NEO 2016, 2017, pp. 159-182

In this article we develop a gradient-based algorithm for the solution of multiobjective optimization problems with uncertainties. To this end, an additional condition is derived for the descent direction in order to account for inaccuracies in the gradients and then incorporated into a subdivision algorithm for the computation of global solutions to multiobjective optimization problems. Convergence to a superset of the Pareto set is proved and an upper bound for the maximal distance to the set of substationary points is given. Besides the applicability to problems with uncertainties, the algorithm is developed with the intention to use it in combination with model order reduction techniques in order to efficiently solve PDE-constrained multiobjective optimization problems.


Multiobjective Optimal Control Methods for the Development of an Intelligent Cruise Control

M. Dellnitz, J. Eckstein, K. Flaßkamp, P. Friedel, C. Horenkamp, U. Köhler, S. Ober-Blöbaum, S. Peitz, S. Tiemeyer, in: Progress in Industrial Mathematics at ECMI 2014 , Springer International Publishing, 2017, pp. 633-641

During the last years, alternative drive technologies, for example electrically powered vehicles (EV), have gained more and more attention, mainly caused by an increasing awareness of the impact of CO2 emissions on climate change and by the limitation of fossil fuels. However, these technologies currently come with new challenges due to limited lithium ion battery storage density and high battery costs which lead to a considerably reduced range in comparison to conventional internal combustion engine powered vehicles. For this reason, it is desirable to increase the vehicle range without enlarging the battery. When the route and the road slope are known in advance, it is possible to vary the vehicles velocity within certain limits in order to reduce the overall drivetrain energy consumption. This may either result in an increased range or, alternatively, in larger energy reserves for comfort functions such as air conditioning. In this presentation, we formulate the challenge of range extension as a multiobjective optimal control problem. We then apply different numerical methods to calculate the so-called Pareto set of optimal compromises for the drivetrain power profile with respect to the two concurrent objectives battery state of charge and mean velocity. In order to numerically solve the optimal control problem by means of a direct method, a time discretization of the drivetrain power profile is necessary. In combination with a vehicle dynamics simulation model, the optimal control problem is transformed into a high dimensional nonlinear optimization problem. For the approximation of the Pareto set, two different optimization algorithms implemented in the software package GAIO are used. The first one yields a global optimal solution by applying a set-oriented subdivision technique to parameter space. By construction, this technique is limited to coarse discretizations of the drivetrain power profile. In contrast, the second technique, which is based on an image space continuation method, is more suitable when the number of parameters is large while the number of objectives is less than five. We compare the solutions of the two algorithms and study the influence of different discretizations on the quality of the solutions. A MATLAB/Simulink model is used to describe the dynamics of an EV. It is based on a drivetrain efficiency map and considers vehicle properties such as rolling friction and air drag, as well as environmental conditions like slope and ambient temperature. The vehicle model takes into account the traction battery too, enabling an exact prediction of the batterys response to power requests of drivetrain and auxiliary loads, including state of charge.


Exploiting structure in multiobjective optimization and optimal control

S. Peitz, 2017

Multiobjective optimization plays an increasingly important role in modern applications, where several criteria are often of equal importance. The task in multiobjective optimization and multiobjective optimal control is therefore to compute the set of optimal compromises (the Pareto set) between the conflicting objectives. Since – in contrast to the solution of a single objective optimization problem – the Pareto set generally consists of an infinite number of solutions, the computational effort can quickly become challenging. This is even more the case when many problems have to be solved, when the number of objectives is high, or when the objectives are costly to evaluate. Consequently, this thesis is devoted to the identification and exploitation of structure both in the Pareto set and the dynamics of the underlying model as well as to the development of efficient algorithms for solving problems with additional parameters, with a high number of objectives or with PDE-constraints. These three challenges are addressed in three respective parts. In the first part, predictor-corrector methods are extended to entire Pareto sets. When certain smoothness assumptions are satisfied, then the set of parameter dependent Pareto sets possesses additional structure, i.e. it is a manifold. The tangent space can be approximated numerically which yields a direction for the predictor step. In the corrector step, the predicted set converges to the Pareto set at a new parameter value. The resulting algorithm is applied to an example from autonomous driving. In the second part, the hierarchical structure of Pareto sets is investigated. When considering a subset of the objectives, the resulting solution is a subset of the Pareto set of the original problem. Under additional smoothness assumptions, the respective subsets are located on the boundary of the Pareto set of the full problem. This way, the “skeleton” of a Pareto set can be computed and due to the exponential increase in computing time with the number of objectives, the computations of these subsets are significantly faster which is demonstrated using an example from industrial laundries. In the third part, PDE-constrained multiobjective optimal control problems are addressed by reduced order modeling methods. Reduced order models exploit the structure in the system dynamics, for example by describing the dynamics of only the most energetic modes. The model reduction introduces an error in both the function values and their gradients, which has to be taken into account in the development of algorithms. Both scalarization and set-oriented approaches are coupled with reduced order modeling. Convergence results are presented and the numerical benefit is investigated. The algorithms are applied to semi-linear heat flow problems as well as to the Navier-Stokes equations.


2016

A Comparison of two Predictive Approaches to Control the Longitudinal Dynamics of Electric Vehicles

J. Eckstein, S. Peitz, K. Schäfer, P. Friedel, U. Köhler, M.H. Molo, S. Ober-Blöbaum, M. Dellnitz, in: Procedia Technology, 2016, pp. 465-472

In this contribution we compare two different approaches to the implementation of a Model Predictive Controller in an electric vehicle with respect to the quality of the solution and real-time applicability. The goal is to develop an intelligent cruise control in order to extend the vehicle range, i.e. to minimize energy consumption, by computing the optimal torque profile for a given track. On the one hand, a path-based linear model with strong simplifications regarding the vehicle dynamics is used. On the other hand, a nonlinear model is employed in which the dynamics of the mechanical and electrical subsystem are modeled.


Multiobjective Model Predictive Control of an Industrial Laundry

S. Peitz, M. Gräler, C. Henke, M.H. Molo, M. Dellnitz, A. Trächtler, in: Procedia Technology, 2016, pp. 483-490

In a wide range of applications, it is desirable to optimally control a system with respect to concurrent, potentially competing goals. This gives rise to a multiobjective optimal control problem where, instead of computing a single optimal solution, the set of optimal compromises, the so-called Pareto set, has to be approximated. When it is not possible to compute the entire control trajectory in advance, for instance due to uncertainties or unforeseeable events, model predictive control methods can be applied to control the system during operation in real time. In this article, we present an algorithm for the solution of multiobjective model predictive control problems. In an offline scenario, it can be used to compute the entire set of optimal compromises whereas in a real time scenario, one optimal compromise is computed according to an operator's preference. The results are illustrated using the example of an industrial laundry. A logistics model of the laundry is developed and then utilized in the optimization routine. Results are presented for an offline as well as an online scenario.


2015

Multiobjective Optimization of the Flow Around a Cylinder Using Model Order Reduction

S. Peitz, M. Dellnitz, in: PAMM, 2015, pp. 613-614

n this article an efficient numerical method to solve multiobjective optimization problems for fluid flow governed by the Navier Stokes equations is presented. In order to decrease the computational effort, a reduced order model is introduced using Proper Orthogonal Decomposition and a corresponding Galerkin Projection. A global, derivative free multiobjective optimization algorithm is applied to compute the Pareto set (i.e. the set of optimal compromises) for the concurrent objectives minimization of flow field fluctuations and control cost. The method is illustrated for a 2D flow around a cylinder at Re = 100.


2014

Development of an Intelligent Cruise Control Using Optimal Control Methods

M. Dellnitz, J. Eckstein, K. Flaßkamp, P. Friedel, C. Horenkamp, U. Köhler, S. Ober-Blöbaum, S. Peitz, S. Tiemeyer, in: Procedia Technology, 2014, pp. 285-294

In this contribution, the range extension problem of electric vehicles is addressed. To this aim, an intelligent cruise control is developed based on the formulation of an optimal control problem. Solutions of this optimal control problem are energy efficient accelerator pedal position profiles. They can be computed numerically by a direct optimal control method using sequential quadratic programming. The approach is applied to two different driving scenarios. The results show that the energy efficiency is increased by using optimal control for both an artificial and a realistic scenario.


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