Hans-Bernd Dürr

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.

A smooth vector field for saddle point problems

In this paper we propose a novel smooth vector field whose trajectories globally converge to the saddle point of the Lagrangian associated with a convex and constrained optimization problem. Under suitable assumptions, we prove global convergence of the trajectories for the class of strictly convex problems and we propose a vector field for linear programs.

A Lie Bracket Approximation for Extremum Seeking Vehicles

Abstract In this paper we propose a novel methodology for the analysis of autonomous vehicles seeking the extremum of an arbitrary smooth nonlinear map in the plane. By interpreting the extremum seeking schemes as input-affine systems with periodic excitations and by using the methodology of Lie brackets, we calculate a simplified system which approximates the qualitative behavior of the original one better than existing methods. By examining this approximate Lie bracket system, we are able to directly derive properties of the original one. Thus, by showing that the Lie bracket direction is directly related to the unknown gradient of the objective function we prove global uniform practical asymptotic stability of the extremum point for vehicles modeled as single integrators and non-holonomic unicycles. We illustrate the proposed method through simulations.

Saddle Point Seeking for Convex Optimization Problems

Abstract In this paper, we consider convex optimization problems with constraints. By combining the idea of a Lie bracket approximation for extremum seeking systems and saddle point algorithms, we propose a feedback which steers a single-integrator system to the set of saddle points of the Lagrangian associated to the convex optimization problem. We prove practical uniform asymptotic stability of the set of saddle points for the extremum seeking system for strictly convex as well as linear programs. Using a numerical example we illustrate how the approach can be used in distributed optimization problems.

Feedback design for multi-agent systems: A saddle point approach

In this paper we propose a dynamic state feedback controller for an input affine nonlinear system that asymptotically stabilizes a point in the output space that is implicitly given as the solution to a convex optimization problem. The construction of the feedback law is based on saddle point flows for convex optimization problems and a backstepping technique. An explicit solution of the optimization problem is not needed for the controller design. We show how the design approach can be applied to multi-agent systems, yielding a decentralized controller. For a particular example, we extend the controller to the output feedback case.

Extremum seeking on submanifolds in the Euclidian space

Extremum seeking is a powerful control method to steer a dynamical system to an extremum of a partially unknown function. In this paper, we introduce extremum seeking systems on submanifolds in the Euclidian space. Using a trajectory approximation technique based on Lie brackets, we prove that uniform asymptotic stability of the so-called Lie bracket system on the manifold implies practical uniform asymptotic stability of the corresponding extremum seeking system on the manifold. We illustrate the approach with an example of extremum seeking on a torus.

A smooth vector field for quadratic programming

In this paper we consider the class of convex optimization problems with affine inequality constraints and focus hereby on the class of quadratic programs. We propose a smooth vector field that is constructed such that its trajectories converge to the saddle point of the Lagrangian function associated to the convex optimization problem. We establish global asymptotic stability as well as exponential stability under mild assumptions for different variants of the vector field and propose a continuous-time Nesterov method.

Singularly Perturbed Lie Bracket Approximation

We consider the interconnection of two dynamical systems where one has an input-affine vector field. By employing a singular perturbation and a Lie bracket analysis technique, we show how the trajectories can be approximated by two decoupled systems. From this trajectory approximation result and the stability properties of the decoupled systems, we derive stability properties of the overall system.

Examples of distance-based synchronization: An extremum seeking approach

In this paper we consider two examples of synchronization problems, i.e., a network of oscillators and a network of rigid bodies. We propose a controller that requires only the knowledge of the relative distances among the neighboring systems in the network. The controller is based on an extremum seeking controller, that steers the overall system to the minimum of an optimization problem on a manifold. Using a Lie bracket approximation for extremum seeking systems, we show that the controller leads to a synchronization of the overall network in both examples.

On a Class of Smooth Optimization Algorithms with Applications in Control

Abstract In this paper, some recent results and ideas on a class of smooth optimization algorithms for convex optimization problems are presented. These algorithms are formulated as ordinary differential equations whose solutions converge to saddle points of the Lagrangian function associated to the convex optimization problem. Specifically in this paper, the global stability behavior for general convex programs as well as linear and quadratic programs are discussed. Furthermore, a continuous Nesterov-like fast gradient variant as well as an interior-point variant of these continuous-time saddle point algorithms are proposed.