TY - JOUR
AB - 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.
AU - Klus, Stefan
AU - Nüske, Feliks
AU - Peitz, Sebastian
AU - Niemann, Jan-Hendrik
AU - Clementi, Cecilia
AU - Schütte, Christof
ID - 16288
JF - Physica D: Nonlinear Phenomena
SN - 0167-2789
TI - Data-driven approximation of the Koopman generator: Model reduction, system identification, and control
VL - 406
ER -
TY - JOUR
AB - 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.
AU - Bieker, Katharina
AU - Peitz, Sebastian
AU - Brunton, Steven L.
AU - Kutz, J. Nathan
AU - Dellnitz, Michael
ID - 16290
JF - Theoretical and Computational Fluid Dynamics
SN - 0935-4964
TI - Deep model predictive flow control with limited sensor data and online learning
ER -
TY - CHAP
AB - 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.
AU - Peitz, Sebastian
AU - Klus, Stefan
ID - 16289
SN - 0170-8643
T2 - Lecture Notes in Control and Information Sciences
TI - Feedback Control of Nonlinear PDEs Using Data-Efficient Reduced Order Models Based on the Koopman Operator
VL - 484
ER -
TY - GEN
AB - 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.
AU - Peitz, Sebastian
AU - Otto, Samuel E.
AU - Rowley, Clarence W.
ID - 16309
T2 - arXiv:2003.07094
TI - Data-Driven Model Predictive Control using Interpolated Koopman Generators
ER -
TY - JOUR
AB - 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.
AU - Schütze, Oliver
AU - Cuate, Oliver
AU - Martín, Adanay
AU - Peitz, Sebastian
AU - Dellnitz, Michael
ID - 10596
IS - 5
JF - Engineering Optimization
SN - 0305-215X
TI - Pareto Explorer: a global/local exploration tool for many-objective optimization problems
VL - 52
ER -
TY - GEN
AB - 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.
AU - Hernández Castellanos, Carlos Ignacio
AU - Ober-Blöbaum, Sina
AU - Peitz, Sebastian
ID - 16297
T2 - arXiv:2002.06006
TI - Explicit Multi-objective Model Predictive Control for Nonlinear Systems Under Uncertainty
ER -
TY - CONF
AB - In this paper, we first outline a Hypothetical Learning Trajectory (HLT), which aims at a formal understanding of the rules for manipulating integers. The HLT is based on task formats, which promote algebraic thinking in terms of generalizing rules from the analysis of patterns and should be familiar to students from their mathematics education experiences in elementary school. Second, we analyze two students' actual learning process based on Peircean semiotics. The analysis shows that the actual learning process diverges from the hypothesized learning process in that the students do not relate the different levels of the diagrams in a way that allows them to extrapolate the rule for the subtraction of negative numbers. Based on this finding, we point out consequences for the design of the tasks.
AU - Schumacher, Jan
AU - Rezat, Sebastian
ED - Jankvist, Uffe Thomas
ED - Van den Heuvel-Panhuizen, Marja
ED - Veldhuis, Michiel
ID - 13107
KW - diagrammatic reasoning
KW - hypothetical learning trajectory
KW - induction extrapolatory method
KW - integers
KW - negative numbers
KW - permanence principle
KW - semiotics
T2 - Proceedings of the Eleventh Congress of the European Society for Research in Mathematics Education (CERME11, February 6 – 10, 2019)
TI - A Hypothetical Learning Trajectory for the Learning of the Rules for Manipulating Integers
ER -
TY - GEN
AB - 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.
AU - Gebken, Bennet
AU - Peitz, Sebastian
ID - 16295
T2 - arXiv:1901.06141
TI - Inverse multiobjective optimization: Inferring decision criteria from data
ER -
TY - JOUR
AB - 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.
AU - Gebken, Bennet
AU - Peitz, Sebastian
AU - Dellnitz, Michael
ID - 10595
IS - 4
JF - Journal of Global Optimization
SN - 0925-5001
TI - On the hierarchical structure of Pareto critical sets
VL - 73
ER -
TY - CHAP
AB - Diagrammatisches Schlie{\ss}en wird im Zusammenhang mit dem Lernen von Mathmematik und ihrer Symbolsprache als wesentliche Theorie der Wissenskonstruktion diskutiert. Dabei wird h{\"{a}}ufig davon ausgegangen, dass die Wissenskonstruktion im Sinne diagrammatischen Schlie{\ss}ens erfolgt. Deskriptive Rekonstruktionen diagrammatischen Schlie{\ss}ens bei Lernenden stellen jedoch ein Desiderat der mathematikdidaktischen Forschung dar. Der vorliegende Beitrag befasst sich mit der Fragestellung, wie sich diagrammatisches Schlie{\ss}en bei Lernenden rekonstruieren l{\"{a}}sst. Als m{\"{o}}gliche Werkzeuge f{\"{u}}r eine solche Rekonstruktion werden Toulmins Argumentationsschema und Vergnauds Schema-Begriff exemplarisch angewandt, um das diagrammatische Schlie{\ss}en eines Sch{\"{u}}lerpaars beim Einstieg in die Subtraktion negativer Zahlen zu rekonstruieren. Abschlie{\ss}end wird die tats{\"{a}}chliche Eignung der beiden Ans{\"{a}}tze zur Rekonstruktion diagrammatischen Schlie{\ss}ens diskutiert.
AU - Schumacher, Jan
AU - Rezat, Sebastian
ED - Kadunz, Gert
ID - 13108
T2 - Semiotische Perspektiven auf das Lernen von Mathematik II
TI - Rekonstruktion diagrammatischen Schließens beim Erlernen der Subtraktion negativer Zahlen. Vergleich zweier methodischer Zugänge
ER -
TY - BOOK
ED - Rezat, Sebastian
ED - Fan, Lianghuo
ED - Hattermann, Mathias
ED - Schumacher, Jan
ED - Wuschke, Holger
ID - 13139
TI - Proceedings of the Third International Conference on Mathematics Textbook Research and Development: 16-19 September 2019 Paderborn, Germany
ER -
TY - GEN
AB - 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.
AU - Banholzer, Stefan
AU - Gebken, Bennet
AU - Dellnitz, Michael
AU - Peitz, Sebastian
AU - Volkwein, Stefan
ID - 16296
T2 - arXiv:1906.09075
TI - ROM-based multiobjective optimization of elliptic PDEs via numerical continuation
ER -
TY - CONF
AB - 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.
AU - Hanke, Soren
AU - Peitz, Sebastian
AU - Wallscheid, Oliver
AU - Böcker, Joachim
AU - Dellnitz, Michael
ID - 10597
SN - 9781538694145
T2 - 2019 IEEE International Symposium on Predictive Control of Electrical Drives and Power Electronics (PRECEDE)
TI - Finite-Control-Set Model Predictive Control for a Permanent Magnet Synchronous Motor Application with Online Least Squares System Identification
ER -
TY - JOUR
AB - 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.
AU - Peitz, Sebastian
AU - Ober-Blöbaum, Sina
AU - Dellnitz, Michael
ID - 8753
IS - 1
JF - Acta Applicandae Mathematicae
SN - 0167-8019
TI - Multiobjective Optimal Control Methods for the Navier-Stokes Equations Using Reduced Order Modeling
VL - 161
ER -
TY - JOUR
AB - 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.
AU - Peitz, Sebastian
AU - Klus, Stefan
ID - 10593
JF - Automatica
SN - 0005-1098
TI - Koopman operator-based model reduction for switched-system control of PDEs
VL - 106
ER -
TY - JOUR
AB - We address the general mathematical problem of computing the inverse p-th
root of a given matrix in an efficient way. A new method to construct iteration
functions that allow calculating arbitrary p-th roots and their inverses of
symmetric positive definite matrices is presented. We show that the order of
convergence is at least quadratic and that adaptively adjusting a parameter q
always leads to an even faster convergence. In this way, a better performance
than with previously known iteration schemes is achieved. The efficiency of the
iterative functions is demonstrated for various matrices with different
densities, condition numbers and spectral radii.
AU - Richters, Dorothee
AU - Lass, Michael
AU - Walther, Andrea
AU - Plessl, Christian
AU - Kühne, Thomas
ID - 21
IS - 2
JF - Communications in Computational Physics
TI - A General Algorithm to Calculate the Inverse Principal p-th Root of Symmetric Positive Definite Matrices
VL - 25
ER -
TY - GEN
AU - Jurgelucks, Benjamin
AU - Schulze, Veronika
AU - Feldmann, Nadine
AU - Claes, Leander
ID - 8482
T2 - GAMM 2019
TI - Arbitrary sensitivity for inverse problems in piezoelectricity
ER -
TY - CHAP
AU - Kuklinski, Christiane
AU - Leis, Elena
AU - Liebendörfer, Michael
AU - Hochmuth, Reinhard
ID - 8577
T2 - Beiträge zum Mathematikunterricht 2019
TI - Erklärung von Mathematikleistung im Ingenieursstudium
ER -
TY - CONF
AU - Schumacher, Jan
ID - 13106
T2 - Beiträge zum Mathematikunterricht 2019
TI - Rekonstruktion diagrammatischen Schließens am Beispiel der Subtraktion negativer Zahlen
ER -
TY - CHAP
AU - Liebendörfer, Michael
ED - Didaktik der Mathematik der Universität Paderborn, Fachgruppe
ID - 8573
T2 - Beiträge zum Mathematikunterricht 2018
TI - Psychologische Grundbedürfnisse im frühen Mathematikstudium
ER -