Christian Ebenbauer

Dennis Gramlich, Patricia Pauli, Carsten W. Scherer et al.

This paper introduces a novel representation of convolutional Neural Networks (CNNs) in terms of 2-D dynamical systems. To this end, the usual description of convolutional layers with convolution kernels, i.e., the impulse responses of linear filters, is realized in state space as a linear time-invariant 2-D system. The overall convolutional Neural Network composed of convolutional layers and nonlinear activation functions is then viewed as a 2-D version of a Lur'e system, i.e., a linear dynamical system interconnected with static nonlinear components. One benefit of this 2-D Lur'e system perspective on CNNs is that we can use robust control theory much more efficiently for Lipschitz constant estimation than previously possible.

Christian Feller, Christian Ebenbauer

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.

Simon Michalowsky, Bahman Gharesifard, Christian Ebenbauer

We consider a group of computation units trying to cooperatively solve a distributed optimization problem with shared linear equality and inequality constraints. Assuming that the computation units are communicating over a network whose topology is described by a time-invariant directed graph, by combining saddle-point dynamics with Lie bracket approximation techniques we derive a methodology that allows to design distributed continuous-time optimization algorithms that solve this problem under minimal assumptions on the graph topology as well as on the structure of the constraints. We discuss several extensions as well as special cases in which the proposed procedure becomes particularly simple.

Simon Michalowsky, Bahman Gharesifard, Christian Ebenbauer

We consider the problem of solving a smooth convex optimization problem with equality and inequality constraints in a distributed fashion. Assuming that we have a group of agents available capable of communicating over a communication network described by a time-invariant directed graph, we derive distributed continuous-time agent dynamics that ensure convergence to a neighborhood of the optimal solution of the optimization problem. Following the ideas introduced in our previous work, we combine saddle-point dynamics with Lie bracket approximation techniques. While the methodology was previously limited to linear constraints and objective functions given by a sum of strictly convex separable functions, we extend these result here and show that it applies to a very general class of optimization problems under mild assumptions on the communication topology.

Simon Michalowsky, Christian Ebenbauer

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.

Ala Kolsi, Christian Ebenbauer, Bahman Gharesifard et al.

We study the problem of global exponential stabilization of a force- and torque-controlled unicycle model in discrete time. To this end, we extend a recently introduced approach to model predictive control (MPC) in which a flexible number of inputs is implemented in every iteration. We present the first flexible-step MPC protocol with state-dependent weights for average descent. Notably, the proposed method relies neither on a suitable design of running or terminal cost functions nor on a suitable choice of terminal constraints. Instead, stability is guaranteed through a generalized discrete-time control Lyapunov function. We establish a new theoretical framework for global exponential stabilization of general nonlinear discrete-time control systems by flexible-step MPC. The obtained results go beyond the unicycle example. However, given the importance of the unicycle dynamics, we make that a focal point of our work. For the particular case of the dynamic (second-order) unicycle model, we show that global exponential stability cannot be attained in the classical sense, but in a slightly weaker sense. The proposed flexible-step MPC method is shown to induce the best possible notion of global exponential stability for this model. We provide explicit rules for the choice of parameters, which guarantee feasibility and global exponential stability. Our numerical simulations show that the discrete MPC method also works very well in applications to a continuous-time torque-actuated unicycle.

Stefan Wildhagen, Simon Michalowsky, Jan Feiling et al.

We consider perturbation-based extremum seeking, which recovers an approximate gradient of an analytically unknown objective function through measurements. Using classical needle variation analysis, we are able to explicitly quantify the recovered gradient in the scalar case. We reveal that it corresponds to an averaged gradient of the objective function, even for very general extremum seeking systems. From this, we create a recursion which represents the learning dynamics along the recovered gradient. These results give rise to the interpretation that extremum seeking actually optimizes a function other than the original one. From this insight, a new perspective on global optimization of functions with local extrema emerges: because the gradient is averaged over a certain time period, local extrema might be evened out in the learning dynamics. Moreover, a multidimensional extension of the scalar results is given.