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Digitale Infotage für Schüler*innen, 7.-11. Februar, 15-19 Uhr, Anmeldungen unter Bildinformationen anzeigen

Digitale Infotage für Schüler*innen, 7.-11. Februar, 15-19 Uhr, Anmeldungen unter

Foto: Universität Paderborn

Prof. Dr. Sina Ober-Blöbaum


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Symplectic integration of learned Hamiltonian systems

C. Offen, S. Ober-Blöbaum, Chaos (2022), 32(1), pp. 013122

Hamiltonian systems are differential equations which describe systems in classical mechanics, plasma physics, and sampling problems. They exhibit many structural properties, such as a lack of attractors and the presence of conservation laws. To predict Hamiltonian dynamics based on discrete trajectory observations, incorporation of prior knowledge about Hamiltonian structure greatly improves predictions. This is typically done by learning the system's Hamiltonian and then integrating the Hamiltonian vector field with a symplectic integrator. For this, however, Hamiltonian data needs to be approximated based on the trajectory observations. Moreover, the numerical integrator introduces an additional discretisation error. In this paper, we show that an inverse modified Hamiltonian structure adapted to the geometric integrator can be learned directly from observations. A separate approximation step for the Hamiltonian data avoided. The inverse modified data compensates for the discretisation error such that the discretisation error is eliminated. The technique is developed for Gaussian Processes.


Multi-objective minimum time optimal control for low-thrust trajectory design

N. Vertovec, S. Ober-Blöbaum, K. Margellos, 2021

We propose a reachability approach for infinite and finite horizon multi-objective optimization problems for low-thrust spacecraft trajectory design. The main advantage of the proposed method is that the Pareto front can be efficiently constructed from the zero level set of the solution to a Hamilton-Jacobi-Bellman equation. We demonstrate the proposed method by applying it to a low-thrust spacecraft trajectory design problem. By deriving the analytic expression for the Hamiltonian and the optimal control policy, we are able to efficiently compute the backward reachable set and reconstruct the optimal trajectories. Furthermore, we show that any reconstructed trajectory will be guaranteed to be weakly Pareto optimal. The proposed method can be used as a benchmark for future research of applying reachability analysis to low-thrust spacecraft trajectory design.

    Efficient time stepping for numerical integration using reinforcement learning

    M. Dellnitz, E. Hüllermeier, M. Lücke, S. Ober-Blöbaum, C. Offen, S. Peitz, K. Pfannschmidt, in: arXiv:2104.03562, 2021

    Many problems in science and engineering require the efficient numerical approximation of integrals, a particularly important application being the numerical solution of initial value problems for differential equations. For complex systems, an equidistant discretization is often inadvisable, as it either results in prohibitively large errors or computational effort. To this end, adaptive schemes have been developed that rely on error estimators based on Taylor series expansions. While these estimators a) rely on strong smoothness assumptions and b) may still result in erroneous steps for complex systems (and thus require step rejection mechanisms), we here propose a data-driven time stepping scheme based on machine learning, and more specifically on reinforcement learning (RL) and meta-learning. First, one or several (in the case of non-smooth or hybrid systems) base learners are trained using RL. Then, a meta-learner is trained which (depending on the system state) selects the base learner that appears to be optimal for the current situation. Several examples including both smooth and non-smooth problems demonstrate the superior performance of our approach over state-of-the-art numerical schemes. The code is available under

    Bifurcation preserving discretisations of optimal control problems

    C. Offen, S. Ober-Blöbaum, 2021, pp. 334-339

    The first order optimality conditions of optimal control problems (OCPs) can be regarded as boundary value problems for Hamiltonian systems. Variational or symplectic discretisation methods are classically known for their excellent long term behaviour. As boundary value problems are posed on intervals of fixed, moderate length, it is not immediately clear whether methods can profit from structure preservation in this context. When parameters are present, solutions can undergo bifurcations, for instance, two solutions can merge and annihilate one another as parameters are varied. We will show that generic bifurcations of an OCP are preserved under discretisation when the OCP is either directly discretised to a discrete OCP (direct method) or translated into a Hamiltonian boundary value problem using first order necessary conditions of optimality which is then solved using a symplectic integrator (indirect method). Moreover, certain bifurcations break when a non-symplectic scheme is used. The general phenomenon is illustrated on the example of a cut locus of an ellipsoid.

    Explicit multiobjective model predictive control for nonlinear systems with symmetries

    S. Ober-Blöbaum, S. Peitz, International Journal of Robust and Nonlinear Control (2021), 31(2), pp. 380-403

    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.

    Variational integration of learned dynamical systems

    S. Ober-Blöbaum, C. Offen, 2021

    The principle of least action is one of the most fundamental physical principle. It says that among all possible motions connecting two points in a phase space, the system will exhibit those motions which extremise an action functional. Many qualitative features of dynamical systems, such as the presence of conservation laws and energy balance equations, are related to the existence of an action functional. Incorporating variational structure into learning algorithms for dynamical systems is, therefore, crucial in order to make sure that the learned model shares important features with the exact physical system. In this paper we show how to incorporate variational principles into trajectory predictions of learned dynamical systems. The novelty of this work is that (1) our technique relies only on discrete position data of observed trajectories. Velocities or conjugate momenta do not need to be observed or approximated and no prior knowledge about the form of the variational principle is assumed. Instead, they are recovered using backward error analysis. (2) Moreover, our technique compensates discretisation errors when trajectories are computed from the learned system. This is important when moderate to large step-sizes are used and high accuracy is required. For this, we introduce and rigorously analyse the concept of inverse modified Lagrangians by developing an inverse version of variational backward error analysis. (3) Finally, we introduce a method to perform system identification from position observations only, based on variational backward error analysis.


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

    C.I. Hernández Castellanos, S. Ober-Blöbaum, S. Peitz, International Journal of Robust and Nonlinear Control (2020), 30(17), pp. 7593-7618

    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.


      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.


        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.

          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.


            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.


              Self-optimizing Mechatronic Systems

              M. Dellnitz, K. Flaßkamp, P. Hartmann, M. Krüger, T. Meyer, C. Priesterjahn, S. Ober-Blöbaum, C. Rasche, W. Sextro, K. Stahl, A. Trächtler, in: Dependability of Self-optimizing Mechatronic Systems, Kapitel: 1.1, Springer-Verlag, 2014, pp. 3-12

              Development of the RailCab Vehicle

              K. Flaßkamp, S. Grösbrink, P. Hartmann, C. Heinzemann, B. Kleinjohann, L. Kleinjohann, M. Krüger, S. Ober-Blöbaum, C. Priesterjahn, C. Rasche, W. Schäfer, D. Steenken, A. Trächtler, H. Wehrheim, S. Ziegert, in: Dependability of Self-Optimizing Mechatronic Systems, Springer-Verlag , 2014, pp. 184-190

              Verification for Interacting Mechatronic Systems with Motion Profiles

              K. Flaßkamp, C. Heinzemann, M. Krüger, S. Ober-Blöbaum, W. Schäfer, D. Steenken, A. Trächtler, H. Wehrheim, in: Dependability of Self-optimizing Mechatronic Systems, Springer-Verlag, Heidelberg, Germany, 2014, pp. 119-128

              Control strategies on stable manifolds for energyefficient swing-ups of double pendula

              K. Flaßkamp, J. Timmermann, S. Ober-Blöbaum, A. Trächtler, International Journal of Control (2014), DOI: 10.1080/00207179.2014.893450, pp. 1-20

              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.


                Methods for the Design and Development

                H. Anacker, M. Dellnitz, K. Flaßkamp, S. Grösbrink, P. Hartmann, C. Heinzemann, C. Horenkamp, L. Kleinjohann, B. Kleinjohann, S. Korf, M. Krüger, W. Müller, S. Ober-Blöbaum, S. Oberthür, M. Porrmann, C. Priesterjahn, W. Radkowski, C. Rasche, J. Rieke, M. Ringkamp, K. Stahl, D. Steenken, J. Stöcklein, R. Timmermann, A. Trächtler, K. Witting, T. Xie, S. Ziegert, in: Design Methodology for Intelligent Technical Systems Systems – Develop Intelligent Technical Systems of the Future, Springer-Verlag, 2013, pp. 187-356

                Sichere Konvoibildung mit Hilfe optimaler Bremsprofile

                K. Flaßkamp, C. Heinzemann, M. Krüger, D. Steenken, S. Ober-Blöbaum, W. Schäfer, A. Trächtler, H. Wehrheim, in: 9. Paderborner Workshop Entwurf mechatronischer Systeme, Verlagsschriftenreihe des Heinz Nixdorf Instituts, Paderborn, 2013


                Optimal Control on Stable Manifolds for a Double Pendulum

                K. Flaßkamp, J. Timmermann, S. Ober-Blöbaum, M. Dellnitz, A. Trächtler, in: Proceedings in Applied Mathematics and Mechanics, 2012


                Discrete Mechanics and Optimal Control and its Application to a Double Pendulum on a Cart

                J. Timmermann, S. Khatab, S. Ober-Blöbaum, A. Trächtler, in: 18th IFAC World Congress 2011, 2011


                Optimal Control for a Pitcher's Motion Modeled as Constrained Mechanical System

                S. Ober-Blöbaum, J. Timmermann, in: ASME 2009 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference, ASME, 2009

                Optimal Control for a Pitcher's Motion Modeled as Constrained Mechanical System

                S. Ober-Blöbaum, J. Timmermann, in: ASME 2009 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference, ASME, 2009

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