@inproceedings{30733, abstract = {{Hamilton-Jacobi reachability methods for safety-critical control have been well studied, but the safety guarantees derived rely on the accuracy of the numerical computation. Thus, it is crucial to understand and account for any inaccuracies that occur due to uncertainty in the underlying dynamics and environment as well as the induced numerical errors. To this end, we propose a framework for modeling the error of the value function inherent in Hamilton-Jacobi reachability using a Gaussian process. The derived safety controller can be used in conjuncture with arbitrary controllers to provide a safe hybrid control law. The marginal likelihood of the Gaussian process then provides a confidence metric used to determine switches between a least restrictive controller and a safety controller. We test both the prediction as well as the correction capabilities of the presented method in a classical pursuit-evasion example.}}, author = {{Vertovec, Nikolaus and Ober-Blöbaum, Sina and Margellos, Kostas}}, location = {{London}}, pages = {{1870--1875}}, title = {{{Verification of safety critical control policies using kernel methods}}}, year = {{2022}}, } @article{44624, author = {{Faulwasser, Timm and Flaßkamp, Kathrin and Ober-Blöbaum, Sina and Schaller, Manuel and Worthmann, Karl}}, journal = {{Mathematics of Control, Signals, and Systems}}, pages = {{759--788}}, publisher = {{Springer}}, title = {{{Manifold turnpikes, trims, and symmetries}}}, volume = {{34}}, year = {{2022}}, } @article{23382, abstract = {{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.}}, author = {{Offen, Christian and Ober-Blöbaum, Sina}}, journal = {{Chaos: An Interdisciplinary Journal of Nonlinear Science}}, publisher = {{AIP}}, title = {{{Symplectic integration of learned Hamiltonian systems}}}, doi = {{10.1063/5.0065913}}, volume = {{32(1)}}, year = {{2022}}, } @inproceedings{29421, author = {{Ober-Blöbaum, Sina and Vermeeren, M.}}, booktitle = {{7th IIFAC Workshop on Lagrangian and Hamiltonian Methods for Nonlinear Control LHMNC}}, pages = {{327--333}}, title = {{{Superconvergence of galerkin variational integrators}}}, volume = {{54(19)}}, year = {{2021}}, } @article{16294, abstract = {{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.}}, author = {{Ober-Blöbaum, Sina and Peitz, Sebastian}}, journal = {{International Journal of Robust and Nonlinear Control}}, pages = {{380--403}}, title = {{{Explicit multiobjective model predictive control for nonlinear systems with symmetries}}}, doi = {{10.1002/rnc.5281}}, volume = {{31(2)}}, year = {{2021}}, } @inproceedings{22894, abstract = {{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.}}, author = {{Offen, Christian and Ober-Blöbaum, Sina}}, issn = {{2405-8963}}, keywords = {{optimal control, catastrophe theory, bifurcations, variational methods, symplectic integrators}}, location = {{Berlin, Germany}}, pages = {{334--339}}, title = {{{Bifurcation preserving discretisations of optimal control problems}}}, doi = {{https://doi.org/10.1016/j.ifacol.2021.11.099}}, volume = {{54(19)}}, year = {{2021}}, } @inproceedings{21572, author = {{Ridderbusch, Steffen and Offen, Christian and Ober-Blöbaum, Sina and Goulart, Paul}}, booktitle = {{2021 60th IEEE Conference on Decision and Control (CDC)}}, location = {{Austin, TX, USA}}, pages = {{2896}}, publisher = {{IEEE}}, title = {{{Learning ODE Models with Qualitative Structure Using Gaussian Processes }}}, doi = {{10.1109/CDC45484.2021.9683426}}, year = {{2021}}, } @inproceedings{21592, abstract = {{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.}}, author = {{Vertovec, Nikolaus and Ober-Blöbaum, Sina and Margellos, Kostas}}, location = {{Rotterdam, the Netherlands}}, pages = {{1975--1980}}, title = {{{Multi-objective minimum time optimal control for low-thrust trajectory design}}}, year = {{2021}}, } @inproceedings{29868, author = {{Jiménez, F. and Ober-Blöbaum, Sina}}, booktitle = {{Nichtlineare Sci 31}}, title = {{{Fractional Damping Through Restricted Calculus of Variations}}}, volume = {{46}}, year = {{2021}}, } @inproceedings{47146, author = {{Lishkova, Y. and Cannon, M. and Ober-Blöbaum, Sina}}, publisher = {{European Control Conference (ECC), IEEE}}, title = {{{A multirate variational approach to Nonlinear MPC}}}, year = {{2021}}, } @article{29399, author = {{Limebeer, D. J. N. and Ober-Blöbaum, Sina and Farshi, F. H.}}, journal = {{IEEE Transactions on Automatic Control}}, pages = {{1381--1396}}, title = {{{Variational integrators for dissipative systems}}}, volume = {{65(4)}}, 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}}, } @article{29398, author = {{Hernández Castellanos, C. I. O. and Schütze, G. and Sun, J.-Q. and Ober-Blöbaum, Sina and Morales-Luna, G.}}, journal = {{Mathematics}}, title = {{{Numerical computation of lightly multi-objective robust optimal solutions by means of generalized cell mapping}}}, volume = {{8(11):1959}}, year = {{2020}}, } @inproceedings{29422, author = {{Lishkova, Y. and Ober-Blöbaum, Sina and Cannon, M. and Leyendecker, S.}}, booktitle = {{Accepted for publication in Proceedings of 2020 AAS/AIAA Astrodynamics Specialist Conference - Lake Tahoe}}, title = {{{A multirate variational approach to simulation and optimal control for flexible spacecraft}}}, year = {{2020}}, } @inproceedings{29423, author = {{Faulwasser, T. and Flaßkamp, K. and Ober-Blöbaum, Sina and Worthmann, K. }}, booktitle = {{24th International Symposium on Mathematical Theory of Networks and Systems}}, title = {{{A dissipativity characterization of velocity turnpikes in optimal control problems for mechanical systems}}}, year = {{2020}}, } @inproceedings{29424, author = {{Cresson, J. and Jiménez, F. and Ober-Blöbaum, Sina}}, booktitle = {{24th International Symposium on Mathematical Theory of Networks and Systems}}, title = {{{Modelling of the convection-diffusion equation through fractional restricted calculus of variations}}}, year = {{2020}}, } @inbook{29413, author = {{Flaßkamp, K. and Ober-Blöbaum, Sina and Peitz, S. }}, booktitle = {{Advances in Dynamics, Optimization and Computation}}, editor = {{Junge, Oliver and Schütze, Oliver and Froyland, Gary and Ober-Blöbaum, Sina and Padberg-Gehle, Kathrin}}, pages = {{209--237}}, publisher = {{Springer International Publishing}}, title = {{{Symmetry in optimal control: A multiobjective model predictive control approach}}}, year = {{2020}}, } @article{19993, author = {{Flaßkamp, Kathrin and Ober-Blöbaum, Sina and Worthmann, Karl}}, journal = {{MCSS}}, pages = {{455--485}}, title = {{{Symmetry and motion primitives in model predictive control}}}, volume = {{31}}, year = {{2019}}, } @inproceedings{29867, author = {{Faulwasser, Tim and Flaßkamp, K. and Ober-Blöbaum, Sina and Worthmann, Karl}}, pages = {{490--495}}, title = {{{Towards velocity turnpikes in optimal control of mechanical systems}}}, volume = {{52(16)}}, year = {{2019}}, } @article{8753, abstract = {{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.}}, author = {{Peitz, Sebastian and Ober-Blöbaum, Sina and Dellnitz, Michael}}, issn = {{0167-8019}}, journal = {{Acta Applicandae Mathematicae}}, number = {{1}}, pages = {{171–199}}, title = {{{Multiobjective Optimal Control Methods for the Navier-Stokes Equations Using Reduced Order Modeling}}}, doi = {{10.1007/s10440-018-0209-7}}, volume = {{161}}, year = {{2018}}, }