@inbook{16289, abstract = {{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.}}, author = {{Peitz, Sebastian and Klus, Stefan}}, booktitle = {{Lecture Notes in Control and Information Sciences}}, isbn = {{9783030357122}}, issn = {{0170-8643}}, pages = {{257--282}}, publisher = {{Springer}}, title = {{{Feedback Control of Nonlinear PDEs Using Data-Efficient Reduced Order Models Based on the Koopman Operator}}}, doi = {{10.1007/978-3-030-35713-9_10}}, volume = {{484}}, year = {{2020}}, } @article{16290, abstract = {{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.}}, author = {{Bieker, Katharina and Peitz, Sebastian and Brunton, Steven L. and Kutz, J. Nathan and Dellnitz, Michael}}, issn = {{0935-4964}}, journal = {{Theoretical and Computational Fluid Dynamics}}, pages = {{577–591}}, title = {{{Deep model predictive flow control with limited sensor data and online learning}}}, doi = {{10.1007/s00162-020-00520-4}}, volume = {{34}}, year = {{2020}}, } @article{16309, abstract = {{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.}}, author = {{Peitz, Sebastian and Otto, Samuel E. and Rowley, Clarence W.}}, journal = {{SIAM Journal on Applied Dynamical Systems}}, number = {{3}}, pages = {{2162--2193}}, title = {{{Data-Driven Model Predictive Control using Interpolated Koopman Generators}}}, doi = {{10.1137/20M1325678}}, volume = {{19}}, year = {{2020}}, } @article{16297, abstract = {{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.}}, author = {{Hernández Castellanos, Carlos Ignacio and Ober-Blöbaum, Sina and Peitz, Sebastian}}, journal = {{International Journal of Robust and Nonlinear Control}}, pages = {{7593--7618}}, title = {{{Explicit Multi-objective Model Predictive Control for Nonlinear Systems Under Uncertainty}}}, doi = {{10.1002/rnc.5197}}, volume = {{30(17)}}, year = {{2020}}, } @inbook{17994, abstract = {{In this work we review the novel framework for the computation of finite dimensional invariant sets of infinite dimensional dynamical systems developed in [6] and [36]. By utilizing results on embedding techniques for infinite dimensional systems we extend a classical subdivision scheme [8] as well as a continuation algorithm [7] for the computation of attractors and invariant manifolds of finite dimensional systems to the infinite dimensional case. We show how to implement this approach for the analysis of delay differential equations and partial differential equations and illustrate the feasibility of our implementation by computing the attractor of the Mackey-Glass equation and the unstable manifold of the one-dimensional Kuramoto-Sivashinsky equation.}}, author = {{Gerlach, Raphael and Ziessler, Adrian}}, booktitle = {{Advances in Dynamics, Optimization and Computation}}, editor = {{Junge, Oliver and Schütze, Oliver and Ober-Blöbaum, Sina and Padberg-Gehle, Kathrin}}, isbn = {{9783030512637}}, issn = {{2198-4182}}, pages = {{66--85}}, publisher = {{Springer International Publishing}}, title = {{{The Approximation of Invariant Sets in Infinite Dimensional Dynamical Systems}}}, doi = {{10.1007/978-3-030-51264-4_3}}, volume = {{304}}, year = {{2020}}, } @article{16712, abstract = {{We investigate self-adjoint matrices A∈Rn,n with respect to their equivariance properties. We show in particular that a matrix is self-adjoint if and only if it is equivariant with respect to the action of a group Γ2(A)⊂O(n) which is isomorphic to ⊗nk=1Z2. If the self-adjoint matrix possesses multiple eigenvalues – this may, for instance, be induced by symmetry properties of an underlying dynamical system – then A is even equivariant with respect to the action of a group Γ(A)≃∏ki=1O(mi) where m1,…,mk are the multiplicities of the eigenvalues λ1,…,λk of A. We discuss implications of this result for equivariant bifurcation problems, and we briefly address further applications for the Procrustes problem, graph symmetries and Taylor expansions.}}, author = {{Dellnitz, Michael and Gebken, Bennet and Gerlach, Raphael and Klus, Stefan}}, issn = {{1468-9367}}, journal = {{Dynamical Systems}}, number = {{2}}, pages = {{197--215}}, title = {{{On the equivariance properties of self-adjoint matrices}}}, doi = {{10.1080/14689367.2019.1661355}}, volume = {{35}}, year = {{2020}}, } @article{16710, abstract = {{In this work we present a set-oriented path following method for the computation of relative global attractors of parameter-dependent dynamical systems. We start with an initial approximation of the relative global attractor for a fixed parameter λ0 computed by a set-oriented subdivision method. By using previously obtained approximations of the parameter-dependent relative global attractor we can track it with respect to a one-dimensional parameter λ > λ0 without restarting the whole subdivision procedure. We illustrate the feasibility of the set-oriented path following method by exploring the dynamics in low-dimensional models for shear flows during the transition to turbulence and of large-scale atmospheric regime changes . }}, author = {{Gerlach, Raphael and Ziessler, Adrian and Eckhardt, Bruno and Dellnitz, Michael}}, issn = {{1536-0040}}, journal = {{SIAM Journal on Applied Dynamical Systems}}, pages = {{705--723}}, title = {{{A Set-Oriented Path Following Method for the Approximation of Parameter Dependent Attractors}}}, doi = {{10.1137/19m1247139}}, year = {{2020}}, } @article{21944, author = {{Nüske, Feliks and Boninsegna, Lorenzo and Clementi, Cecilia}}, issn = {{0021-9606}}, journal = {{The Journal of Chemical Physics}}, title = {{{Coarse-graining molecular systems by spectral matching}}}, doi = {{10.1063/1.5100131}}, year = {{2019}}, } @article{16709, author = {{Sahai, Tuhin and Ziessler, Adrian and Klus, Stefan and Dellnitz, Michael}}, issn = {{0924-090X}}, journal = {{Nonlinear Dynamics}}, title = {{{Continuous relaxations for the traveling salesman problem}}}, doi = {{10.1007/s11071-019-05092-5}}, year = {{2019}}, } @article{10593, abstract = {{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.}}, author = {{Peitz, Sebastian and Klus, Stefan}}, issn = {{0005-1098}}, journal = {{Automatica}}, pages = {{184--191}}, title = {{{Koopman operator-based model reduction for switched-system control of PDEs}}}, doi = {{10.1016/j.automatica.2019.05.016}}, volume = {{106}}, year = {{2019}}, } @article{10595, abstract = {{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.}}, author = {{Gebken, Bennet and Peitz, Sebastian and Dellnitz, Michael}}, issn = {{0925-5001}}, journal = {{Journal of Global Optimization}}, number = {{4}}, pages = {{891--913}}, title = {{{On the hierarchical structure of Pareto critical sets}}}, doi = {{10.1007/s10898-019-00737-6}}, volume = {{73}}, year = {{2019}}, } @inproceedings{10597, abstract = {{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.}}, author = {{Hanke, Soren and Peitz, Sebastian and Wallscheid, Oliver and Böcker, Joachim and Dellnitz, Michael}}, booktitle = {{2019 IEEE International Symposium on Predictive Control of Electrical Drives and Power Electronics (PRECEDE)}}, isbn = {{9781538694145}}, title = {{{Finite-Control-Set Model Predictive Control for a Permanent Magnet Synchronous Motor Application with Online Least Squares System Identification}}}, doi = {{10.1109/precede.2019.8753313}}, year = {{2019}}, } @article{16708, abstract = {{ In this work we extend the novel framework developed by Dellnitz, Hessel-von Molo, and Ziessler to the computation of finite dimensional unstable manifolds of infinite dimensional dynamical systems. To this end, we adapt a set-oriented continuation technique developed by Dellnitz and Hohmann for the computation of such objects of finite dimensional systems with the results obtained in the work of Dellnitz, Hessel-von Molo, and Ziessler. We show how to implement this approach for the analysis of partial differential equations and illustrate its feasibility by computing unstable manifolds of the one-dimensional Kuramoto--Sivashinsky equation as well as for the Mackey--Glass delay differential equation. }}, author = {{Ziessler, Adrian and Dellnitz, Michael and Gerlach, Raphael}}, issn = {{1536-0040}}, journal = {{SIAM Journal on Applied Dynamical Systems}}, number = {{3}}, pages = {{1265--1292}}, title = {{{The Numerical Computation of Unstable Manifolds for Infinite Dimensional Dynamical Systems by Embedding Techniques}}}, doi = {{10.1137/18m1204395}}, volume = {{18}}, year = {{2019}}, } @unpublished{16711, abstract = {{Embedding techniques allow the approximations of finite dimensional attractors and manifolds of infinite dimensional dynamical systems via subdivision and continuation methods. These approximations give a topological one-to-one image of the original set. In order to additionally reveal their geometry we use diffusion mapst o find intrinsic coordinates. We illustrate our results on the unstable manifold of the one-dimensional Kuramoto--Sivashinsky equation, as well as for the attractor of the Mackey-Glass delay differential equation.}}, author = {{Gerlach, Raphael and Koltai, Péter and Dellnitz, Michael}}, booktitle = {{arXiv:1902.08824}}, title = {{{Revealing the intrinsic geometry of finite dimensional invariant sets of infinite dimensional dynamical systems}}}, year = {{2019}}, } @unpublished{21634, abstract = {{Predictive control of power electronic systems always requires a suitable model of the plant. Using typical physics-based white box models, a trade-off between model complexity (i.e. accuracy) and computational burden has to be made. This is a challenging task with a lot of constraints, since the model order is directly linked to the number of system states. Even though white-box models show suitable performance in most cases, parasitic real-world effects often cannot be modeled satisfactorily with an expedient computational load. Hence, a Koopman operator-based model reduction technique is presented which directly links the control action to the system's outputs in a black-box fashion. The Koopman operator is a linear but infinite-dimensional operator describing the dynamics of observables of nonlinear autonomous dynamical systems which can be nicely applied to the switching principle of power electronic devices. Following this data-driven approach, the model order and the number of system states are decoupled which allows us to consider more complex systems. Extensive experimental tests with an automotive-type permanent magnet synchronous motor fed by an IGBT 2-level inverter prove the feasibility of the proposed modeling technique in a finite-set model predictive control application.}}, author = {{Hanke, Sören and Peitz, Sebastian and Wallscheid, Oliver and Klus, Stefan and Böcker, Joachim and Dellnitz, Michael}}, booktitle = {{arXiv:1804.00854}}, title = {{{Koopman Operator-Based Finite-Control-Set Model Predictive Control for Electrical Drives}}}, year = {{2018}}, } @article{21940, author = {{Litzinger, Florian and Boninsegna, Lorenzo and Wu, Hao and Nüske, Feliks and Patel, Raajen and Baraniuk, Richard and Noé, Frank and Clementi, Cecilia}}, issn = {{1549-9618}}, journal = {{Journal of Chemical Theory and Computation}}, pages = {{2771--2783}}, title = {{{Rapid Calculation of Molecular Kinetics Using Compressed Sensing}}}, doi = {{10.1021/acs.jctc.8b00089}}, year = {{2018}}, } @article{21941, author = {{Klus, Stefan and Nüske, Feliks and Koltai, Péter and Wu, Hao and Kevrekidis, Ioannis and Schütte, Christof and Noé, Frank}}, issn = {{0938-8974}}, journal = {{Journal of Nonlinear Science}}, pages = {{985--1010}}, title = {{{Data-Driven Model Reduction and Transfer Operator Approximation}}}, doi = {{10.1007/s00332-017-9437-7}}, year = {{2018}}, } @article{21942, author = {{Boninsegna, Lorenzo and Nüske, Feliks and Clementi, Cecilia}}, issn = {{0021-9606}}, journal = {{The Journal of Chemical Physics}}, title = {{{Sparse learning of stochastic dynamical equations}}}, doi = {{10.1063/1.5018409}}, year = {{2018}}, } @article{21943, author = {{Hruska, Eugen and Abella, Jayvee R. and Nüske, Feliks and Kavraki, Lydia E. and Clementi, Cecilia}}, issn = {{0021-9606}}, journal = {{The Journal of Chemical Physics}}, title = {{{Quantitative comparison of adaptive sampling methods for protein dynamics}}}, doi = {{10.1063/1.5053582}}, year = {{2018}}, } @inproceedings{8750, abstract = {{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.}}, author = {{Gebken, Bennet and Peitz, Sebastian and Dellnitz, Michael}}, booktitle = {{Numerical and Evolutionary Optimization – NEO 2017}}, isbn = {{9783319961033}}, issn = {{1860-949X}}, title = {{{A Descent Method for Equality and Inequality Constrained Multiobjective Optimization Problems}}}, doi = {{10.1007/978-3-319-96104-0_2}}, year = {{2018}}, }