arXiv subjects

Christian Ebenbauer

Publications and source records attributed to Christian Ebenbauer.

43 records · Page 3Linked to original sources

arXiv2017

On a class of generating vector fields for the extremum seeking problem: Lie bracket approximation and stability properties

In this paper, we describe a broad class of control functions for extremum seeking problems. We show that it unifies and generalizes existing extremum seeking strategies which are based on Lie bracket approximations, and allows to design new controls with favorable properties in extremum seeking and vibrational stabilization tasks. The second result of this paper is a novel approach for studying the asymptotic behavior of extremum seeking systems. It provides a constructive procedure for defining frequencies of control functions to ensure the practical asymptotic and exponential stability. In contrast to many known results, we also prove asymptotic and exponential stability in the sense of Lyapunov for the proposed class of extremum seeking systems under appropriate assumptions on the vector fields.

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arXiv2016

A stabilizing iteration scheme for model predictive control based on relaxed barrier functions

We propose and analyze a stabilizing iteration scheme for the algorithmic implementation of model predictive control for linear discrete-time systems. Polytopic input and state constraints are considered and handled by means of so-called relaxed logarithmic barrier functions. The required on-line optimization is based on warm starting and performs only a limited, possibly small, number of optimization algorithm iterations between two consecutive sampling instants. The optimization algorithm dynamics as well as the resulting suboptimality of the applied control input are taken into account explicitly in the stability analysis, and the origin of the resulting overall closed-loop system, consisting of state and optimization algorithm dynamics, is proven to be asymptotically stable. The corresponding constraint satisfaction properties are also analyzed. The theoretical results and a presented numerical example illustrate the fact that asymptotic stability as well as a satisfactory closed-loop performance can be achieved by performing only a single optimization algorithm iteration at each sampling step.

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arXiv2016

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 Nesterov's method.

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arXiv2015

Ensemble Observability of Linear Systems

We address the observability problem for ensembles that are described by probability distributions. The problem is to reconstruct a probability distribution of the initial state from the time-evolution of the probability distribution of the output under a classical finite-dimensional linear system. We present two solutions to this problem, one based on formulating the problem as an inverse problem and the other one based on reconstructing all the moments of the distribution. The first approach leads us to a connection between the reconstruction problem and mathematical tomography problems. In the second approach we use the framework of tensor systems to describe the dynamics of the moments which leads to a more systems theoretic treatment of the reconstruction problem. Furthermore we show that both frameworks are inherently related. The appeal of having two dual view points, the first being more geometric and the second one being more systems theoretic, is illuminated in several examples of theoretical or practical importance.

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arXiv2015

Relaxed Logarithmic Barrier Function Based Model Predictive Control of Linear Systems

In this paper, we investigate the use of relaxed logarithmic barrier functions in the context of linear model predictive control. We present results that allow to guarantee asymptotic stability of the corresponding closed-loop system, and discuss further properties like performance and constraint satisfaction in dependence of the underlying relaxation. The proposed stabilizing MPC schemes are not necessarily based on an explicit terminal set or state constraint and allow to characterize the stabilizing control input sequence as the minimizer of a globally defined, continuously differentiable, and strongly convex function. The results are illustrated by means of a numerical example.

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arXiv2013

On the differential equation $\dotΘ=(Θ^{T}-Θ)Θ$ with $Θ\in{SO(n)}$

In this note we consider the global convergence properties of the differential equation $\dotΘ=(Θ^{T}-Θ)Θ$ with $Θ\in{SO(n)}$, which is a gradient flow of the function $f:SO(n)\rightarrow\mathbb{R},Θ\mapsto{2n-2\trΘ}$. Many of the presented results are not new, but scattered throughout literature. The motivation of this note is to summarize and extend the convergence results known from literature. Rather than giving an exhaustive list of references, the results are presented in a self-contained fashion.

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arXiv2012

Lie Bracket Approximation of Extremum Seeking Systems

Extremum seeking feedback is a powerful method to steer a dynamical system to an extremum of a partially or completely unknown map. It often requires advanced system-theoretic tools to understand the qualitative behavior of extremum seeking systems. In this paper, a novel interpretation of extremum seeking is introduced. We show that the trajectories of an extremum seeking system can be approximated by the trajectories of a system which involves certain Lie brackets of the vector fields of the extremum seeking system. It turns out that the Lie bracket system directly reveals the optimizing behavior of the extremum seeking system. Furthermore, we establish a theoretical foundation and prove that uniform asymptotic stability of the Lie bracket system implies practical uniform asymptotic stability of the corresponding extremum seeking system. We use the established results in order to prove local and semi-global practical uniform asymptotic stability of the extrema of a certain map for multi-agent extremum seeking systems.

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