Kathrin Flaßkamp

Solving Optimal Control Problems by Exploiting Inherent Dynamical Systems Structures

Computing globally efficient solutions is a major challenge in optimal control of nonlinear dynamical systems. This work proposes a method combining local optimization and motion planning techniques based on exploiting inherent dynamical systems structures, such as symmetries and invariant manifolds. Prior to the optimal control, the dynamical system is analyzed for structural properties that can be used to compute pieces of trajectories that are stored in a motion planning library. In the context of mechanical systems, these motion planning candidates, termed primitives, are given by relative equilibria induced by symmetries and motions on stable or unstable manifolds of e.g. fixed points in the natural dynamics. The existence of controlled relative equilibria is studied through Lagrangian mechanics and symmetry reduction techniques. The proposed framework can be used to solve boundary value problems by performing a search in the space of sequences of motion primitives connected using optimized maneuvers. The optimal sequence can be used as an admissible initial guess for a post-optimization. The approach is illustrated by two numerical examples, the single and the double spherical pendula, which demonstrates its benefit compared to standard local optimization techniques.

Sequential Action Control for Tracking of Free Invariant Manifolds

Abstract This paper presents a hybrid control method that controls to unstable equilibria of nonlinear systems by taking advantage of systems’ stable manifold of free dynamics. Resulting nonlinear controllers are closed-loop and can be computed in real-time. Thus, we present a computationally efficient approach to optimization-based switching control design using a manifold tracking objective. Our method is validated for the cart-pendulum and the pendubot inversion problems. Results show the proposed approach conserves control effort compared to tracking the desired equilibrium directly. Moreover, the method avoids parameter tuning and reduces sensitivity to initial conditions. Finally, when compared to existing energy based swing-up strategies, our approach does not rely on pre-derived, system-specific switching controllers. We use hybrid optimization to automate switching control synthesis on-line for nonlinear systems.

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

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 this contribution, we propose a control strategy for mechanical systems which sequences uncontrolled trajectories on (un)stable manifolds with short control manoeuvres to design a feedforward control. In particular, we present optimal swing-up solutions for a double pendulum which are based on trajectories on the stable manifold of the pendulum’s up–up equilibrium. To demonstrate the advantages of our approach compared to a black-box optimisation, we perform a post-optimisation with the optimal control sequence as an initial guess. The numerical results are evaluated in a simulation environment for the double pendulum on a cart and applied to a real test rig.

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.

Methods of Improving the Dependability of Self-optimizing Systems

Various methods have been developed in the Collaborative Research Center 614 which can be used to improve the dependability of self-optimizing systems. These methods are presented in this chapter. They are sorted into two categories with regard to the development process of self-optimizing systems. On one hand, there are methods which can be applied during the Conceptual Design Phase. On the other hand, there are methods that are applicable during Design and Development.

Energy efficient control for mechanical systems based on inherent dynamical structures

Many optimal control problems for mechanical systems include the computation of control policies that are optimal with respect to the energetic effort. In this contribution, we will demonstrate how to exploit inherent dynamical properties of the dynamical system to build up a motion planning framework that can in particular be used to compute good initial guesses for energy efficient solutions. This method is based on a motion planning library consisting of three types of trajectories: (1.) motions induced by the systems symmetry, (2.) trajectories on (un)stable manifolds of the natural dynamics, and (3.) controlled maneuvers to sequence the so called motion primitives. We demonstrate our proposed optimal control method on the example of a double spherical pendulum.

Variational Integrators for Structure-Preserving Filtering

Estimation and filtering are important tasks in most modern control systems. These methods rely on accurate discretetime approximations of the system dynamics. We present filtering algorithms that are based on discrete mechanics techniques (variational integrators), which are known to preserve system structures (momentum, symplecticity, constraints, for instance) and have stable long-term energy behavior. These filtering methods show increased performance in simulations and experiments on a real digital control system. The particle filter as well as the extended Kalman filter benefit from the statistics-preserving properties of a variational integrator discretization, especially in low bandwidth applications. Moreover, it is shown how the optimality of the Kalman filter can be preserved through discretization by means of modified discrete-time Riccati equations for the covariance updates. This leads to further improvement in filter accuracy, even in a simple test example.

The Paradigm of Self-optimization

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 engineering and software engineering to improve the behavior of a technical system. The integration of cognitive functions into mechatronic systems enables systems to have inherent partial intelligence. The behavior of these future systems is formed by the communication and cooperation of the intelligent system elements. From an information processing point of view, we consider these distributed systems to be multi-agent-systems. These capabilities open up fascinating prospects regarding the design of future technical systems. The term self-optimization characterizes this perspective: the endogenous adaptation of the system’s objectives due to changing operational conditions. This resuls in an autonomous adjustment of system parameters or system structure and consequently of the system’s behavior. In this chapter self-optimizing systems are described in detail. The long term aim of the Collaborative Research Centre 614 ”Self-Optimizing Concepts and Structures in Mechanical Engineering” is to open up the active paradigm of self-optimization for mechanical engineering and to enable others to develop these systems. For this, developers have to face a number of challenges, e.g. the multidisciplinarity and the complexity of the system. This book povides a design methodology that helps to master these challenges and to enable third parties to develop self-optimizing systems by themselves.

Introduction to Self-optimization and Dependability

This chapter gives an introduction to self-optimizing mechatronic systems and the risks and possibilities that arise with these. Self-optimizing mechatronic systems have capabilities that go far beyond those of traditional mechatronic systems. They are able to autonomously adapt their behavior and so react to outer influences, which can originate e.g. from the environment, changed user requirements or the current system status. The basic process of self-optimization, the procedures employed within and the main components of a self-optimizing system are explained here.

Motion Planning for Mechanical Systems with Hybrid Dynamics

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 hybrid dynamical systems by motion planning with motion primitives. These canonical motions either origin from inherent symmetry properties of the systems or they are controlled maneuvers that allow sequencing of several primitives. The motion primitives are collected in a motion planning library. A solution to a specific optimal control problem can then be found by searching for the optimal sequence of concatenated primitives. Energy efficiency often forms an important objective in control applications. We therefore extend the motion planning framework by primitives that are motions along invariant manifolds of the uncontrolled dynamics, e.g. trajectories on (un)stable manifolds of equilibria. The approach is illustrated by an academic example motivated by an operating scenario of an open-chain jointed robot.