Michael Dellnitz

Multiobjective Optimal Control Methods for the Development of an Intelligent Cruise Control

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.

Transition manifolds of complex metastable systems: Theory and data-driven computation of effective dynamics

We consider complex dynamical systems showing metastable behavior, but no local separation of fast and slow time scales. The article raises the question of whether such systems exhibit a low-dimensional manifold supporting its effective dynamics. For answering this question, we aim at finding nonlinear coordinates, called reaction coordinates, such that the projection of the dynamics onto these coordinates preserves the dominant time scales of the dynamics. We show that, based on a specific reducibility property, the existence of good low-dimensional reaction coordinates preserving the dominant time scales is guaranteed. Based on this theoretical framework, we develop and test a novel numerical approach for computing good reaction coordinates. The proposed algorithmic approach is fully local and thus not prone to the curse of dimension with respect to the state space of the dynamics. Hence, it is a promising method for data-based model reduction of complex dynamical systems such as molecular dynamics.

Set-Oriented Multiobjective Optimal Control of PDEs using Proper Orthogonal Decomposition

In this chapter, we combine a global, derivative-free subdivision algorithm for multiobjective optimization problems with a posteriori error estimates for reduced-order models based on Proper Orthogonal Decomposition in order to efficiently solve multiobjective optimization problems governed by partial differential equations. An error bound for a semilinear heat equation is developed in such a way that the errors in the conflicting objectives can be estimated individually. The resulting algorithm constructs a library of locally valid reduced-order models online using a Greedy (worst-first) search. Using this approach, the number of evaluations of the full-order model can be reduced by a factor of more than 1000.

On the computation of attractors for delay differential equations

In this work we present a novel framework for the computation of nfinite dimensional invariant sets of infinite dimensional dynamical nsystems. It extends a classical subdivision technique [7] for the computation of nsuch objects of finite dimensional systems to the infinite ndimensional case by utilizing results on embedding techniques for ninfinite dimensional systems. We show how to implement this approach for nthe analysis of delay differential equations and illustrate the nfeasibility of our implementation by computing invariant sets for nthree different delay differential equations.

Gradient-Based Multiobjective Optimization with Uncertainties

In this article we develop a gradient-based algorithm for the solution of multiobjective optimization problems with uncertainties. To this end, an additional condition is derived for the descent direction in order to account for inaccuracies in the gradients and then incorporated into a subdivision algorithm for the computation of global solutions to multiobjective optimization problems. Convergence to a superset of the Pareto set is proved and an upper bound for the maximal distance to the set of substationary points is given. Besides the applicability to problems with uncertainties, the algorithm is developed with the intention to use it in combination with model order reduction techniques in order to efficiently solve PDE-constrained multiobjective optimization problems.

A Set-Oriented Numerical Approach for Dynamical Systems with Parameter Uncertainty

In this article, we develop a set-oriented numerical methodology which allows us to perform uncertainty quantification (UQ) for dynamical systems from a global point of view. That is, for systems with uncertain parameters we approximate the corresponding global attractors and invariant measures in the related stochastic setting. Our methods do not rely on generalized polynomial chaos techniques. Rather, we extend classical set-oriented methods designed for deterministic dynamical systems [M. Dellnitz and A. Hohmann, Numer. Math., 75 (1997), pp. 293--317; M. Dellnitz and O. Junge, SIAM J. Numer. Anal., 36 (1999), pp. 491--515] to the UQ-context, and this allows us to analyze the long-term uncertainty propagation. The algorithms have been integrated into the software package GAIO [M. Dellnitz, G. Froyland, and O. Junge, Ergodic Theory, Analysis, and Efficient Simulation of Dynamical Systems, Springer, Berlin, 2001, pp. 145--174], and we illustrate the use and efficiency of these techniques with a couple of numerical examples.

Age- and Expertise-Related Differences of Sensorimotor Network Dynamics during Force Control

Age-related deterioration of force control is evident on behavioral and neural levels. Extensive and deliberate practice can decrease these changes. This study focused on detecting electrophysiological correlates of age- and expertise-related differences in force control. We examined young (20-27u202fyears) and late middle-aged (57-67u202fyears) novices as well as late middle-aged experts in the field of fine motor control. Therefore, EEG data were recorded while participants performed a force maintenance task. Variability and complexity of force data were analyzed. To detect electrophysiological correlates, dynamic mode decomposition (DMD) was applied to EEG data. DMD allows assessing brain network dynamics by extracting electrode interrelations and their dynamics. Defining clusters of electrodes, we focused on sensorimotor and attentional networks. We confirmed that force control in late middle-aged novices was more variable and less complex than in other groups. Analysis of task-related overall network characteristics, showed a decrease within the α band and increase within low β, high β, and u202fθ u202fband. Compared to the other groups young novices presented a decreased α magnitude. High β magnitude was lower in late middle-aged novices than for other groups. Comparing left and right hands performance, young novices showed higher low β magnitude for the left hand. Late middle-aged novices showed high values for both hands while late middle-aged experts showed higher values for the right than for their left hand. Activation of attentional networks was lower in late middle-aged experts compared to novices. These results may relate to different control strategies of the three groups.

A Descent Method for Equality and Inequality Constrained Multiobjective Optimization Problems

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.

Sensing and control in symmetric networks

ABSTRACT In engineering applications, one of the major challenges today is to develop reliable and robust control algorithms for complex networked systems. Controllability and observability of such systems play a crucial role in the design process. The underlying network structure may contain symmetries – caused, for example, by the coupling of identical building blocks – and these symmetries lead to repeated eigenvalues in a generic way. This complicates the design of controllers since repeated eigenvalues decrease the controllability of the system. In this paper, we will analyze the relationship between the controllability and observability of complex networked systems and symmetries using results from group representation theory. Furthermore, we will propose an algorithm to compute sparse input and output matrices based on projections onto corresponding isotypic components. We will illustrate our results with the aid of two guiding examples, a network with D4 symmetry and the Petersen graph.