Sina Ober-Blöbaum

Optimal Reconfiguration of Formation Flying Satellites

We employ a recently developed new technique for the numerical treatment of optimal control problems for mechanical systems in order to compute optimal open loop control laws for the reconfiguration of a group of formation flying satellites. The method is based on a direct discretization of a variational formulation of the dynamical constraints. We extend the method by linearizing around a given nominal trajectory and incorporate certain collision avoidance strategies. We numerically illustrate the approach for a certain reconfiguration maneuver in the context of future formation flying missions and compare our method to a standard finite difference approach.

A Variational Approach to Multirate Integration for Constrained Systems

The simulation of systems with dynamics on strongly varying time scales is quite challenging and demanding with regard to possible numerical methods. A rather naive approach is to use the smallest necessary time step to guarantee a stable integration of the fast frequencies. However, this typically leads to unacceptable computational loads. Alternatively, multirate methods integrate the slow part of the system with a relatively large step size while the fast part is integrated with a small time step. In this work, a multirate integrator for constrained dynamical systems is derived in closed form via a discrete variational principle on a time grid consisting of macro and micro time nodes. Being based on a discrete version of Hamilton’s principle, the resulting variational multirate integrator is a symplectic and momentum preserving integration scheme and also exhibits good energy behaviour. Depending on the discrete approximations for the Lagrangian function, one obtains different integrators, e.g. purely implicit or purely explicit schemes, or methods that treat the fast and slow parts in different ways. The performance of the multirate integrator is demonstrated by means of several examples.

Set Oriented Methods for the Numerical Treatment of Multiobjective Optimization Problems

In many applications, it is required to optimize several conflicting objectives concurrently leading to a multobjective optimization problem (MOP). The solution set of a MOP, the Pareto set, typically forms a (k-1)-dimensional object, where k is the number of objectives involved in the optimization problem. The purpose of this chapter is to give an overview of recently developed set oriented techniques - subdivision and continuation methods - for the computation of Pareto sets \(\mathcal{P}\) of a givenMOP. All these methods have in common that they create sequences of box collections which aim for a tight covering of \(\mathcal{P}\). Further, we present a class of multiobjective optimal control problems which can be efficiently handled by the set oriented continuation methods using a transformation into high-dimensionalMOPs. We illustrate all the methods on both academic and real world examples.

Handling high-dimensional problems with multi-objective continuation methods via successive approximation of the tangent space

In many applications, several conflicting objectives have to be optimized concurrently leading to a multi-objective optimization problem. Since the set of solutions, the so-called Pareto set, typically forms a (k−1)-dimensional manifold, where k is the number of objectives considered in the model, continuation methods such as predictor–corrector (PC) methods are in certain cases very efficient tools for rapidly computing a finite size representation of the set of interest. However, their classical implementation leads to trouble when considering higher-dimensional models (i.e. for dimension n>1000 of the parameter space). In this work, it is proposed to perform a successive approximation of the tangent space which allows one to find promising predictor points with less effort in particular for high-dimensional models since no Hessians of the objectives have to be calculated. The applicability of the resulting PC variant is demonstrated on a benchmark model for up to n=100, 000 parameters.

Variational integrators for electric circuits

In this contribution, we develop a variational integrator for the simulation of (stochastic and multiscale) electric circuits. When considering the dynamics of an electric circuit, one is faced with three special situations: 1. The system involves external (control) forcing through external (controlled) voltage sources and resistors. 2. The system is constrained via the Kirchhoff current (KCL) and voltage laws (KVL). 3. The Lagrangian is degenerate. Based on a geometric setting, an appropriate variational formulation is presented to model the circuit from which the equations of motion are derived. A time-discrete variational formulation provides an iteration scheme for the simulation of the electric circuit. Dependent on the discretization, the intrinsic degeneracy of the system can be canceled for the discrete variational scheme. In this way, a variational integrator is constructed that gains several advantages compared to standard integration tools for circuits; in particular, a comparison to BDF methods (which are usually the method of choice for the simulation of electric circuits) shows that even for simple LCR circuits, a better energy behavior and frequency spectrum preservation can be observed using the developed variational integrator.

Construction and analysis of higher order Galerkin variational integrators

In this work we derive and analyze variational integrators of higher order for the structure-preserving simulation of mechanical systems. The construction is based on a space of polynomials together with Gauss and Lobatto quadrature rules to approximate the relevant integrals in the variational principle. The use of higher order schemes increases the accuracy of the discrete solution and thereby decrease the computational cost while the preservation properties of the scheme are still guaranteed. The order of convergence of the resulting variational integrators is investigated numerically and it is discussed which combination of space of polynomials and quadrature rules provide optimal convergence rates. For particular integrators the order can be increased compared to the Galerkin variational integrators previously introduced in Marsden and West (Acta Numerica 10:357–514 2001). Furthermore, linear stability properties, time reversibility, structure-preserving properties as well as efficiency for the constructed variational integrators are investigated and demonstrated by numerical examples.

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

Abstract In this paper we present a new approach to determine trajectories for changing the state of the double pendulum on a cart from one equilibrium to another and show the experimental realization on a test bench. The control of these transitions is accomplished by a two-degrees-of-freedom control scheme. For the design of the feedforward and feedback control of the system two models of the double pendulum on a cart are introduced. The feedforward control is achieved by the optimal control method Discrete Mechanics and Optimal Control (DMOC). The trajectories can be optimized with respect to energy consumption and transition time. Additionally, for the applicatory design, the implementation of a feedback control by means of gain-scheduling is explained. As experimental result the realization of a trajectory on the test bench is presented.

Solving multiobjective Optimal Control problems in space mission design using Discrete Mechanics and reference point techniques

Many different numerical methods have been developed to compute trajectories of optimal control problems on the one hand and to approximate Pareto sets of multiobjective optimization problems on the other hand. However, so far only few approaches exist for the numerical treatment of the combination of both problems leading to multiobjective optimal control problems. In this contribution we combine the optimal control method Discrete Mechanics and Optimal Control (DMOC) with reference point techniques to compute Pareto optimal control solutions. The presented approach is verified by determining a multiobjective optimal transfer for a space mission. Due to the approximation of the Pareto set, structurally different kind of mission trajectories can be detected which would not have been found by local single-objective optimal control methods and provide important information for a mission designer.

A variational approach to define robustness for parametric multiobjective optimization problems

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 external parameter