Simon Michalowsky

Swinging up the Stephenson-Kapitza pendulum

We propose a novel vibrational control law for the Stephenson-Kapitza pendulum. The considered control law achieves practical asymptotic stabilization of the upper pendulum position and the region of attraction is large enough to swing up the pendulum. The control law includes a state dependent term that allows to approximate an LgV control law in a time averaged sense. The control law is still of vibrational type and, in a neighborhood of the upper pendulum position, it coincides with the open-loop vibrational input as it is used in the Stephenson-Kapitza pendulum.

Model-based extremum seeking for a class of nonlinear systems

In this paper we consider extremum seeking for plants that can be represented as a series connection of an integrator chain and an unknown nonlinear system. Using recent results on singularly perturbed Lie bracket approximations we develop an extremum seeking scheme that exploits the special structure of the plant. We illustrate our results by means of a source-seeking example for double integrator vehicles.

The multidimensional n-th order heavy ball method and its application to extremum seeking

In this paper the extension of the heavy ball method to n-th order integrator dynamics is considered. We propose a gradient based controller that achieves to find the extremum of a function depending on multiple variables and prove asymptotic stability for all functions from the set of strongly convex functions. Furthermore, we propose a gradient-free extremum seeking controller that approximates the proposed gradient-based controller and prove practical asymptotic stability of the extremum using Lie bracket averaging techniques. The result does not rely on singular perturbation methods and provides a new approach to extremum seeking for dynamic maps.

Global Output Regulation for the Rotational Dynamics of a Rigid Body / Globale Ausgangsregelung für die Drehbewegung eines Starrkörpersystems

Summary This paper considers a class of nonlinear output regulation problems for the rotational rigid body equations. The equations of motion for the rigid body are written in the natural SO(3) representation. The presented design consists of an observer and a certainty equivalence implementation of an independently designed state feedback law. The proposed solution is global and achieves almost global convergence in the closed loop Zusammenfassung Dieser Beitrag beschäftigt sich mit einer Klasse von nichtlinearen Ausgangsregelungsproblemen für die Drehbewegung von Starrkörpern. Die Bewegungsgleichungen der Starrkörper werden in der SO(3) Darstellung betrachtet. Der Regler besteht aus einem Beobachter und einer davon unabhängig entworfenen Zustandsrückführung, in welcher die Beobachterzustände verwendet werden. Der Entwurf resultiert in einem geschlossenen Kreis mit garantierter Konvergenz für fast alle Anfangsbedingungen

Extremum control of linear systems based on output feedback

In this paper we consider the problem of steering a linear system to an unknown setpoint that is defined as the minimizer of an optimization problem. We give a gradient-based solution as well as a gradient-free approximation that, in the spirit of extremum seeking control, only requires measurements of the objective function. We further show that in both cases dynamic output feedback can be used if full state measurements are not available.

Gradient approximation and extremum seeking via needle variations

We consider a gradient approximation scheme that is based on applying needle shaped inputs. By using ideas known from the classic proof of the Pontryagin Maximum Principle we derive an approximation that reveals that the considered system moves along a weighted averaged gradient. Moreover, based on the same ideas, we give similar results for arbitrary periodic inputs. We also present a new gradient-based optimization algorithm that is motivated by our calculations and that can be interpreted as a combination of the heavy ball method and Nesterovs method.