Sina Ober-Blöbaum

• University of Oxford

In this work, we propose and study a new approach to formulate the optimal control problem of second-order differential equations, with a particular interest in those derived from force-controlled Lagrangian systems. The formulation results in a new hyperregular control Langrangian and, thus, a new control Hamiltonian whose equations of motion prov...

In this work, we analyse the discretisation of a recently proposed new Lagrangian approach to optimal control problems of affine-controlled second-order differential equations with cost functions quadratic in the controls. We propose exact discrete and semi-discrete versions of the problem, providing new tools to develop numerical methods. Discrete...

We introduce a new classification of multimode states with a fixed number of photons. This classification is based on the factorizability of homogeneous multivariate polynomials and is invariant under unitary transformations. The classes physically correspond to field excitations in terms of single and multiple photons, each of which being in an ar...

Fractional dissipation is a powerful tool to study nonlocal physical phenomena such as damping models. The design of geometric, in particular, variational integrators for the numerical simulation of such systems relies on a variational formulation of the model. In Jiménez and Ober-Blöbaum (J Nonlinear Sci 31:46, 2021), a new approach is proposed to...

Motivated by mechanical systems with symmetries, we focus on optimal control problems possessing certain symmetries. Following recent works (Faulwasser in Math Control Signals Syst 34:759–788 2022; Trélat in Math Control Signals Syst 35:685–739 2023), which generalized the classical concept of static turnpike to manifold turnpike we extend the expo...

The depth of networks plays a crucial role in the effectiveness of deep learning. However, the memory requirement for backpropagation scales linearly with the number of layers, which leads to memory bottlenecks during training. Moreover, deep networks are often unable to handle time-series appearing at irregular intervals. These issues can be resol...

Differential equations posed on quadratic matrix Lie groups arise in the context of classical mechanics and quantum dynamical systems. Lie group numerical integrators preserve the constants of motions defining the Lie group. Thus, they respect important physical laws of the dynamical system, such as unitarity and energy conservation in the context...

Motivated by mechanical systems with symmetries, we focus on optimal control problems possessing symmetries. Following recent works, which generalized the classical concept of static turnpike to manifold turnpike, we extend the exponential turnpike property to the exponential trim turnpike for control systems with symmetries induced by abelian or n...

The simulation of systems that act on multiple time scales is challenging. A stable integration of the fast dynamics requires a highly accurate approximation whereas for the simulation of the slow part, a coarser approximation is accurate enough. With regard to the general goals of any numerical method, high accuracy and low computational costs, a...

Deep learning is widely used in tasks including image recognition and generation, in learning dynamical systems from data and many more. It is important to construct learning architectures with theoretical guarantees to permit safety in the applications. There has been considerable progress in this direction lately. In particular, symplectic networ...

A fundamental concern in progressing airborne wind energy (AWE) operations toward commercial success is guaranteeing that safety requirements placed on the systems are met. Due to the high-dimensional complexity of AWE systems, however, formal mathematical robustness guarantees become difficult to compute. We draw on research from Hamilton–Jacobi (...

We show how to learn discrete field theories from observational data of fields on a space–time lattice. For this, we train a neural network model of a discrete Lagrangian density such that the discrete Euler–Lagrange equations are consistent with the given training data. We, thus, obtain a structure-preserving machine learning architecture. Lagrang...

Direct methods for the simulation of optimal control problems apply a specific discretization to the dynamics of the problem, and the discrete adjoint method is suitable to calculate corresponding conditions to approximate an optimal solution. While the benefits of structure preserving or geometric methods have been known for decades, their explora...

We show how to learn discrete field theories from observational data of fields on a space-time lattice. For this, we train a neural network model of a discrete Lagrangian density such that the discrete Euler--Lagrange equations are consistent with the given training data. We, thus, obtain a structure-preserving machine learning architecture. Lagran...

The article shows how to learn models of dynamical systems from data which are governed by an unknown variational PDE. Rather than employing reduction techniques, we learn a discrete field theory governed by a discrete Lagrangian density \(L_d\) that is modelled as a neural network. Careful regularisation of the loss function for training \(L_d\) i...

Recently, Hamiltonian neural networks (HNNs) have been introduced to incorporate prior physical knowledge when learning the dynamical equations of Hamiltonian systems. Hereby, the symplectic system structure is preserved despite the data-driven modeling approach. However, preserving symmetries requires additional attention. In this research, we enh...

The article shows how to learn models of dynamical systems from data which are governed by an unknown variational PDE. Rather than employing reduction techniques, we learn a discrete field theory governed by a discrete Lagrangian density Ld that is modelled as a neural network. Careful regularisation of the loss function for training Ld is necessar...

The article shows how to learn models of dynamical systems from data which are governed by an unknown variational PDE. Rather than employing reduction techniques, we learn a discrete field theory governed by a discrete Lagrangian density $L_d$ that is modelled as a neural network. Careful regularisation of the loss function for training $L_d$ is ne...

Recently, Hamiltonian neural networks (HNN) have been introduced to incorporate prior physical knowledge when learning the dynamical equations of Hamiltonian systems. Hereby, the symplectic system structure is preserved despite the data-driven modeling approach. However, preserving symmetries requires additional attention. In this research, we enha...

One of the fundamental problems in spacecraft trajectory design is finding the optimal transfer trajectory that minimizes the propellant consumption and transfer time simultaneously. We formulate this as a multi-objective optimal control (MOC) problem that involves optimizing over the initial or final state, subject to state constraints. Drawing on...

A fundamental concern in progressing Airborne Wind Energy (AWE) operations towards commercial success, is guaranteeing that safety requirements placed on the systems are met. Due to the high dimensional complexity of AWE systems, however, formal mathematical robustness guarantees become difficult to compute. We draw on research from Hamilton-Jacobi...

When learning continuous dynamical systems with Gaussian Processes, computing trajectories requires repeatedly mapping the distributions of uncertain states through the distribution of learned nonlinear functions, which is generally intractable. Since sampling-based approaches are computationally expensive, we consider approximations of the output...

By one of the most fundamental principles in physics, a dynamical system will exhibit those motions which extremise an action functional. This leads to the formation of the Euler-Lagrange equations, which serve as a model of how the system will behave in time. If the dynamics exhibit additional symmetries, then the motion fulfils additional conserv...

One of the fundamental problems in spacecraft trajectory design is finding the optimal transfer trajectory that minimizes the propellant consumption and transfer time simultaneously. We formulate this as a multi-objective optimal control (MOC) problem that involves optimizing over the initial or final state, subject to state constraints. Drawing on...

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 equa...

Classical turnpikes correspond to optimal steady states which are attractors of infinite-horizon optimal control problems. In this paper, motivated by mechanical systems with symmetries, we generalize this concept to manifold turnpikes. Specifically, the necessary optimality conditions projected onto a symmetry-induced manifold coincide with those...

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 err...

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...

A nonlinear model predictive control (NMPC) approach is proposed based on a variational representation of the system model and the receding horizon optimal control problem. The proposed tube-based convex MPC approach provides improvements in model accuracy and computational efficiency, and allows for alternative means of computing linearization err...

To predict motions of dynamical systems with physical structure, incorporation of structure into learning greatly improves qualitative aspects of predictions. The poster shows that incorporating modified structures adapted to geometric integrators improves predictions even more.

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...

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 immediatel...

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 immediatel...

Turnpikes have recently gained significant research interest in optimal control, since they allow for pivotal insights into the structure of solutions to optimal control problems. So far, mainly steady state solutions which serve as optimal operation points, are studied. This is in contrast to time-varying turnpikes, which are in the focus of this...

We study the order of convergence of Galerkin variational integrators for ordinary differential equations. Galerkin variational integrators approximate a variational (Lagrangian) problem by restricting the space of curves to the set of polynomials of degree at most $s$ and approximating the action integral using a quadrature rule. We show that, if...

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 co...

Classical turnpikes correspond to optimal steady states which are attractors of optimal control problems. In this paper, motivated by mechanical systems with symmetries, we generalize this concept to manifold turnpikes. Specifically, the necessary optimality conditions on a symmetry-induced manifold coincide with those of a reduced-order problem un...

We deliver a novel approach towards the variational description of Lagrangian mechanical systems subject to fractional damping by establishing a restricted Hamilton’s principle. Fractional damping is a particular instance of non-local (in time) damping, which is ubiquitous in mechanical engineering applications. The restricted Hamilton’s principle...

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 prop...

We prove a Noether's theorem of the first kind for the so-called restricted fractional Euler-Lagrange equations and their discrete counterpart, introduced in [26,27], based in previous results [11,35]. Prior, we compare the restricted fractional calculus of variations to the asymmetric fractional calculus of variations, introduced in [14], and form...

We study the order of convergence of Galerkin variational integrators for ordinary differential equations. Galerkin variational integrators approximate a variational (Lagrangian) problem by restricting the space of curves to the set of polynomials of degree at most s and approximating the action integral using a quadrature rule. We show that, if th...

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 immediatel...

Recent advances in learning techniques have enabled the modelling of dynamical systems for scientific and engineering applications directly from data. However, in many contexts, explicit data collection is expensive and learning algorithms must be data-efficient to be feasible. This suggests using additional qualitative information about the system...

In this paper, we present a novel algorithm for the computation of lightly robust optimal solutions for multi-objective optimization problems. To this end, we adapt the generalized cell mapping, originally designed for the global analysis of dynamical systems, to the current context. This is the first time that a set-based method is developed for s...

Model predictive control is a prominent approach to construct a feedback control loop for dynamical systems. Due to real‐time constraints, solving model‐based optimal control problems in a very short amount of time can be challenging. For linear‐quadratic problems, Bemporad et al have proposed an explicit formulation where the underlying optimizati...

In real-world problems, uncertainties (eg, errors in the measurement, precision errors, among others) often lead to poor performance of numerical algorithms when not explicitly taken into account. This is also the case for control problems , where in the case of uncertainties, optimal solutions can degrade in quality or they can even become unfeasi...

We propose an optimal control method for simultaneous slewing and vibration control of flexible spacecraft. Considering dynamics on different time scales, the optimal control problem is discretized on micro and macro time grids using a multirate variational approach. The description of the system and the necessary optimality conditions are derived...

Many dynamical systems possess symmetries, e.g. rotational and translational invariances of mechanical systems. These can be beneficially exploited in the design of numerical optimal control methods. We present a model predictive control scheme which is based on a library of precomputed motion primitives. The primitives are equivalence classes w.r....

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...

Turnpikes have recently gained significant research interest in optimal control, since they allow for pivotal insights into the structure of solutions to optimal control problems. So far, mainly steady state solutions which serve as optimal operation points, are studied. This is in contrast to time-varying turnpikes, which are in the focus of this...

This paper uses physical arguments to derive variational integration schemes for dissipative mechanical systems. These integration algorithms find utility in the solution of the equations of motion and optimal control problems for these systems. Engineers usually represent dissipation effects using phenomenological devices such as “dampers”. In the...

In this paper, we present a novel evolutionary algorithm for the computation of approximate solutions for multi-objective optimization problems. These solutions are of particular interest to the decision-maker as backup solutions since they can provide solutions with similar quality but in different regions of the decision space. The novel algorith...

This book presents a collection of papers on recent advances in problems concerning dynamics, optimal control and optimization. In many chapters, computational techniques play a central role. Set-oriented techniques feature prominently throughout the book, yielding state-of-the-art algorithms for computing general invariant sets, constructing globa...

Symmetries, e.g. rotational and translational invariances for the class of mechanical systems, allow to characterize solution trajectories of nonlinear dynamical systems. Thus, the restriction to symmetry-induced dynamics, e.g. by using the concept of motion primitives, may be considered as a quantization of the system. Symmetry exploitation is wel...

The paper proposes first steps towards the formalization and characterization of time-varying turnpikes in optimal control of mechanical systems. We propose the concepts of velocity steady states, which can be considered as partial steady states, and hyperbolic velocity turnpike properties for analysis and control. We show for a specific example th...

We study the local truncation error of the so-called fractional variational integrators, recently developed in [1, 2] based on previous work by Riewe and Cresson [3, 4]. These integrators are obtained through two main elements: the enlarging of the usual mechanical Lagrangian state space by the introduction of the fractional derivatives of the dyna...

The paper proposes first steps towards the formalization and characterization of time-varying turnpikes in optimal control of mechanical systems. We propose the concepts of velocity steady states, which can be considered as partial steady states, and hyperbolic velocity turnpike properties for analysis and control. We show for a specific example th...

Symmetries, e.g. rotational and translational invariances for the class of mechanical systems, allow to characterize solution trajectories of nonlinear dynamical systems. Thus, the restriction to symmetry-induced dynamics, e.g. by using the concept of motion primitives, may be considered as a quantization of the system. Symmetry exploitation is wel...

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...

We deliver a novel approach towards the variational description of Lagrangian mechanical systems subject to fractional damping by establishing a restricted Hamilton's principle. Fractional damping is a particular instance of non-local (in time) damping, which is ubiquitous in mechanical engineering applications. The restricted Hamilton's principle...

The paper proposes first steps towards the formalization and characterization of time-varying turnpikes in optimal control of mechanical systems. We propose the concepts of velocity steady states, which can be considered as partial steady states, and hyperbolic velocity turnpike properties for analysis and control. We show for a specific example th...

Bei der Entwicklung eines technischen Systems möchte man dieses derart auslegen, dass es sich bzgl. der jeweiligen Anwendungssituation optimal verhält. Wird das System mit unterschiedlichen Anwendungssituationen im Betrieb konfrontiert, erfordert dies in der Regel unterschiedliche Zielsetzungen, welche sich im Allgemeinen widersprechen. Eine wichti...

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 underlyi...

We study the connection between Lagrangian and Hamiltonian descriptions of closed/open dynamics, for a collection of particles with quadratic interaction (closed system) and a sub-collection of particles with linear damping (open system). We consider both continuous and discrete versions of mechanics. We define the Damping Reduction as the mapping...

Employing a phase space which includes the (Riemann-Liouville) fractional derivative of curves evolving on real space, we develop a restricted variational principle for Lagrangian systems yielding the so-called restricted fractional Euler-Lagrange equations (both in the continuous and discrete settings), which, as we show, are invariant under linea...

Employing a phase space which includes the (Riemann-Liouville) fractional derivative of curves evolving on real space, we develop a restricted variational principle for Lagrangian systems yielding the so-called restricted fractional Euler-Lagrange equations (both in the continuous and discrete settings), which, as we show, are invariant under linea...

The considered systems contain dynamics on different time scales caused by different types and stiffnesses in potentials. For the slow motion, a coarse approximation suffices, but to resolve the fast motion, high accuracy is required. To avoid unnecessarily high computational costs, the idea here is to use polynomials of different degrees to approx...

In this work, variational integrators of higher order for dynamical systems with holonomic constraints are constructed and analyzed. The construction is based on approximating the configuration and the Lagrange multiplier via different polynomials. The splitting of the augmented Lagrangian in two parts enables the use of different quadrature formul...

In this contribution higher order variational integrators are developed and their relation to Runge-Kutta methods is investigated. While it is well known that particular higher order variational integrators are equivalent to symplectic partitioned Runge-Kutta methods ([1, 2]), this is not true for more generally constructed variational integrators....

The presented constrained Galerkin variational integrators base on the higher order variational integrators in [1], now applied to holonomically constrained systems and are an extension of the constrained Galerkin methods in [2]. Sufficient conditions are given to obtain a stiffly accurate integration scheme, its structure preserving properties are...

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...

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 t...

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...

For systems that contain slow and fast dynamics, variational multirate integration schemes are used. These schemes split the system into parts which are simulated using two time grids consisting of micro and macro nodes. This formulation can be extended for multi-body systems. The rigid multi-body system is described by the so called director formu...

The variational integrators presented in [5] are applied to systems with holonomic constraints, yielding constrained higher order variational integrators that are an extension of the constrained Galerkin methods in [4]. The construction of the integrators bases on a discrete version of Hamilton's principle. The inheritance of qualitative properties...

The solutions of mixed integer optimal control problems (MIOCPs) yield optimized trajectories for dynamical systems with instantly changing dynamical behavior. The instant change is caused by a changing value of the integer valued control function. For example, a changing integer value can cause a car to change the gear, or a mechanical system to c...

Switching time optimization arises in finite-horizon optimal control for switched systems where, given a sequence of continuous dynamics, one minimizes a cost function with respect to the switching times. We propose an efficient method for computing the optimal switching times for switched linear and nonlinear systems. A novel second-order optimiza...

Switching time optimization arises in finite-horizon optimal control for switched systems where, given a sequence of continuous dynamics, one minimizes a cost function with respect to the switching times. We propose an efficient method for computing the optimal switching times for switched linear and nonlinear systems. A novel second-order optimiza...

Nonlinear control systems with instantly changing dynamical behavior can be modeled by introducing an additional control function that is integer valued in contrast to a control function that is allowed to have continuous values. The discretization of a mixed integer optimal control problem (MIOCP) leads to a non differentiable optimization problem...

Mixed integer control systems are used to model dynamical behavior that can change instantly, for example a driving car with different gears. Changing a gear corresponds to an instant change of the differential equation what is achieved in the model by changing the value of the integer control function. The optimal control of a mixed integer contro...

In this contribution an efficient numerical method to solve multiobjective optimal control problems of fluid flow mixing governed by the advection equation is presented. To obtain well-mixed behavior for non-diffusive processes, the multi-scale mixing of measure introduced in [1] is used in the formulation of the multiobjective optimal control prob...

Symmetry properties such as invariances of mechanical systems can be beneficially exploited in solution methods for control problems. A recently developed approach is based on quantization by so called motion primitives. A library of these motion primitives forms an artificial hybrid system. In this contribution, we study the symmetry properties of...

In recent years, much effort in designing numerical methods for the simulation and optimization of mechanical systems has been put into schemes which are structure preserving. One particular class are variational integrators which are momentum preserving and symplectic. In this article, we develop two high order variational integrators which distin...

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...

The goal of this paper is to derive a structure preserving integrator for geometrically exact beam dynamics, by using a Lie group variational integrator. Both spatial and temporal discretization are implemented in a geometry preserving manner. The resulting scheme preserves both the discrete momentum maps and symplectic structures, and exhibits alm...

Planning and optimal control of mechanical systems are challenging tasks in robotics as well as in many other application areas, e.g. in automotive systems or in space mission design. This holds in particular for hybrid, i.e. mixed discrete and continuous dynamical models. In this contribution, we present an approach to solve control problems for h...

Optimal control problems for mechanical systems often arise in technical applications. To find solutions with minimal control effort, the system’s natural, uncontrolled dynamics can be used. Promising candidates to be considered for energy-efficient trajectories are highly dynamic, but uncontrolled motions on (un)stable manifolds of equilibria. In...

After the domain-spanning conceptual design, engineers from different domains work in parallel and apply their domain-specific methods and modeling languages to design the system. Vital for the successful design, are system optimization methods and the design of the reconfiguration behavior. The former methods enable the parametric adaption of the...

Machines are ubiquitous. They produce, they transport. Machines facilitate and assist with work. The increasing fusion of mechanical engineering with information technology has brought about considerable benefits. This situation is expressed by the term mechatronics, which means the close interaction of mechanics, electrics/electronics, control eng...

Consider a mechanical system that contains slow and fast dynamics. Let it be possible, to split the potential energy into a slow and a fast potential and the configuration vector into slow and fast variables. For such systems, multirate schemes split the system into different parts which can be simulated using different time steps with the goal to...

Optimal control of mechanical systems is an active area of research. However, so far, most contributions are taking only one single objective into account, whereas for many practical problems, one is interested in optimizing several conflicting objectives at the same time. Applying singleobjective optimization to each of them leads to several traje...

For the elastodynamic simulation of a spatially discretized beam, asynchronous variational integrators (AVI) offer the possibility to use different time steps for every element [1]. They are symplectic and conserve discrete momentum maps and since the presented integrator for geometrically exact beam dynamics [2] is derived in the Lie group setting...

Switching time optimization is a crucial topic in the optimal control of hybrid systems. Since it is rare that such problems can be solved analytically, the use of numerical discretization schemes for the integration of state and adjoint systems is indispensable. Thus, in this contribution, the switching time optimization problem is studied in a di...

In contrast to classical optimization problems, in multiobjective optimization several objective functions are considered at the same time. For these problems, the solution is not a single optimum but a set of optimal compromises, the so-called Pareto set. In this work, we consider multiobjective optimization problems that additionally depend on an...

Recently, a couple of approaches have been developed that combine multiobjective optimization with direct discretization methods to approximate trajectories of optimal control problems, resulting in restricted optimization problems of high dimension. The solution set of a multiobjective optimization problem is called the Pareto set which consists o...