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Christian Ebenbauer - One of the best experts on this subject based on the ideXlab platform.
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a Lie Bracket approximation approach to distributed optimization over directed graphs
Automatica, 2020Co-Authors: Simon Michalowsky, Bahman Gharesifard, Christian EbenbauerAbstract:Abstract 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.
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newton based extremum seeking a second order Lie Bracket approximation approach
Automatica, 2019Co-Authors: Christophe Labar, Emanuele Garone, Michel Kinnaert, Christian EbenbauerAbstract:In this paper, we present novel multi-variable Newton-based extremum seeking systems, based on Lie Bracket approximation methods. More precisely, we consider cost functions with an unknown mathematical description, but whose value can be measured on-line. We propose extremum seeking systems that approximate the Newton-based optimization law, by combining the on-line measurement of the cost with time-periodic excitation signals. The inversion of the Hessian matrix is avoided by introducing a first order dynamical system, whose output approximates the Newton step. This provides practical robustness with respect to ill-conditioned Hessian matrices. Semi-global stability properties of the proposed schemes are demonstrated both for static cost functions and for cost functions associated with a general non-linear dynamical system. The effectiveness of the approach is shown in simulations.
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constrained extremum seeking a modified barrier function approach
IFAC-PapersOnLine, 2019Co-Authors: Christophe Labar, Emanuele Garone, Michel Kinnaert, Christian EbenbauerAbstract:Abstract In this paper, we address the problem of steering the input of a convex function to a value that minimizes the function under a convex constraint. We consider the case where the constraint cannot be violated of more than a user-defined value during the whole transient phase. The mathematical expression of both the cost function and the constraint are assumed to be unknown. The only information available are the on-line values of the cost and the constraint. To tackle this problem, an optimization law, based on a modified-barrier function, and involving the gradient of both the cost function and the constraint, is firstly designed. The Lie Bracket formalism is then exploited to approximate this law, by combining time-periodic signals with the on-line measurements of both the cost and the constraint. The stability property of the resulting constrained extremum seeking system is proved, and its effectiveness is shown in simulation.
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on the Lie Bracket approximation approach to distributed optimization extensions and limitations
European Control Conference, 2018Co-Authors: Simon Michalowsky, Bahman Gharesifard, Christian EbenbauerAbstract: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.
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gradient based extremum seeking performance tuning via Lie Bracket approximations
European Control Conference, 2018Co-Authors: Christophe Labar, Jan Feiling, Christian EbenbauerAbstract:In this paper, we propose model-free extremum seeking systems approximating a filtered-gradient descent law, instead of a simple gradient descent law. Namely, we consider that the gradient is low pass filtered before being fed in the gradient descent law. Exploiting the Lie Bracket formalism, we derive general classes of systems that approximate the filtered- gradient descent law, and we focus on four particular schemes. The first ensures the boundedness of the update rates. The last three adapt the dither amplitude to enhance the steady state accuracy. The performances of those schemes are analyzed in simulation and compared with the performances of extremum seeking systems approximating a simple gradient descent law.
Hans-bernd Dürr - One of the best experts on this subject based on the ideXlab platform.
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Extremum Seeking for Time-Varying Functions using Lie Bracket Approximations
IFAC-PapersOnLine, 2017Co-Authors: Victoria Grushkovskaya, Christian Ebenbauer, Hans-bernd Dürr, Alexander ZuyevAbstract:Abstract The paper presents a control algorithm that steers a system to an extremum point of a time-varying function. The proposed extremum seeking law depends on values of the cost function only and can be implemented without knowing analytical expression of this function. By extending the Lie Brackets approximation method, we prove the local and semi-global practical uniform asymptotic stability for time-varying extremum seeking problems. For this purpose, we consider an auxiliary non-autonomous system of differential equations and propose asymptotic stability conditions for a family of invariant sets. The obtained control algorithm ensures the motion of a system in a neighborhood of the curve where the cost function takes its minimal values. The dependence of the radius of this neighborhood on the bounds of the derivative of a time-varying function is shown.
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iterative learning extremum seeking tracking via Lie Bracket approximation
arXiv: Optimization and Control, 2015Co-Authors: Zhixing Cao, Christian Ebenbauer, Hans-bernd Dürr, Frank Allgower, Furong GaoAbstract:In this paper, we develop an extremum seeking control method integrated with iterative learning control to track a time-varying optimizer within finite time. The behavior of the extremum seeking system is analyzed via an approximating system - the modified Lie Bracket system. The modified Lie Bracket system is essentially an online integral-type iterative learning control law. The paper contributes to two fields, namely, iterative learning control and extremum seeking. First, an online integral type iterative learning control with a forgetting factor is proposed. Its convergence is analyzed via $k$-dependent (iteration- dependent) contraction mapping in a Banach space equipped with $\lambda$-norm. Second, the iterative learning extremum seeking system can be regarded as an iterative learning control with "control input disturbance." The tracking error of its modified Lie Bracket system can be shown uniformly bounded in terms of iterations by selecting a sufficiently large frequency. Furthermore, it is shown that the tracking error will finally converge to a set, which is a $\lambda$-norm ball. Its center is the same with the limit solution of its corresponding "disturbance-free" system (the iterative learning control law); and its radius can be controlled by the frequency.
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singularly perturbed Lie Bracket approximation
IEEE Transactions on Automatic Control, 2015Co-Authors: Hans-bernd Dürr, Miroslav Krstic, Alexander Scheinker, Christian EbenbauerAbstract: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.
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extremum seeking on submanifolds in the euclidian space
Automatica, 2014Co-Authors: Hans-bernd Dürr, Milos S. Stankovic, Karl Henrik Johansson, Christian EbenbauerAbstract: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.
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saddle point seeking for convex optimization problems
IFAC Proceedings Volumes, 2013Co-Authors: Hans-bernd Dürr, Chen Zeng, Christian EbenbauerAbstract: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.
Christophe Labar - One of the best experts on this subject based on the ideXlab platform.
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newton based extremum seeking a second order Lie Bracket approximation approach
Automatica, 2019Co-Authors: Christophe Labar, Emanuele Garone, Michel Kinnaert, Christian EbenbauerAbstract:In this paper, we present novel multi-variable Newton-based extremum seeking systems, based on Lie Bracket approximation methods. More precisely, we consider cost functions with an unknown mathematical description, but whose value can be measured on-line. We propose extremum seeking systems that approximate the Newton-based optimization law, by combining the on-line measurement of the cost with time-periodic excitation signals. The inversion of the Hessian matrix is avoided by introducing a first order dynamical system, whose output approximates the Newton step. This provides practical robustness with respect to ill-conditioned Hessian matrices. Semi-global stability properties of the proposed schemes are demonstrated both for static cost functions and for cost functions associated with a general non-linear dynamical system. The effectiveness of the approach is shown in simulations.
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constrained extremum seeking a modified barrier function approach
IFAC-PapersOnLine, 2019Co-Authors: Christophe Labar, Emanuele Garone, Michel Kinnaert, Christian EbenbauerAbstract:Abstract In this paper, we address the problem of steering the input of a convex function to a value that minimizes the function under a convex constraint. We consider the case where the constraint cannot be violated of more than a user-defined value during the whole transient phase. The mathematical expression of both the cost function and the constraint are assumed to be unknown. The only information available are the on-line values of the cost and the constraint. To tackle this problem, an optimization law, based on a modified-barrier function, and involving the gradient of both the cost function and the constraint, is firstly designed. The Lie Bracket formalism is then exploited to approximate this law, by combining time-periodic signals with the on-line measurements of both the cost and the constraint. The stability property of the resulting constrained extremum seeking system is proved, and its effectiveness is shown in simulation.
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gradient based extremum seeking performance tuning via Lie Bracket approximations
European Control Conference, 2018Co-Authors: Christophe Labar, Jan Feiling, Christian EbenbauerAbstract:In this paper, we propose model-free extremum seeking systems approximating a filtered-gradient descent law, instead of a simple gradient descent law. Namely, we consider that the gradient is low pass filtered before being fed in the gradient descent law. Exploiting the Lie Bracket formalism, we derive general classes of systems that approximate the filtered- gradient descent law, and we focus on four particular schemes. The first ensures the boundedness of the update rates. The last three adapt the dither amplitude to enhance the steady state accuracy. The performances of those schemes are analyzed in simulation and compared with the performances of extremum seeking systems approximating a simple gradient descent law.
Simon Michalowsky - One of the best experts on this subject based on the ideXlab platform.
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a Lie Bracket approximation approach to distributed optimization over directed graphs
Automatica, 2020Co-Authors: Simon Michalowsky, Bahman Gharesifard, Christian EbenbauerAbstract:Abstract 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.
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on the Lie Bracket approximation approach to distributed optimization extensions and limitations
European Control Conference, 2018Co-Authors: Simon Michalowsky, Bahman Gharesifard, Christian EbenbauerAbstract: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.
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a family of extremum seeking laws for a unicycle model with a moving target theoretical and experimental studies
European Control Conference, 2018Co-Authors: Victoria Grushkovskaya, Alexander Zuyev, Simon Michalowsky, Max May, Christian EbenbauerAbstract:In this paper, we propose and practically evaluate a class of gradient-free control functions ensuring the motion of a unicycle-type system towards the extremum point of a time- varying cost function. We prove that the unicycle is able to track the extremum point, and illustrate our results by numerical simulations and experiments that show that the proposed control functions exhibit an improved tracking performance in comparison to standard extremum seeking laws based on Lie Bracket approximations.
Miroslav Krstic - One of the best experts on this subject based on the ideXlab platform.
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singularly perturbed Lie Bracket approximation
IEEE Transactions on Automatic Control, 2015Co-Authors: Hans-bernd Dürr, Miroslav Krstic, Alexander Scheinker, Christian EbenbauerAbstract: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.
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non c2 Lie Bracket averaging for nonsmooth extremum seekers
Journal of Dynamic Systems Measurement and Control-transactions of The Asme, 2014Co-Authors: Alexander Scheinker, Miroslav KrsticAbstract:A drawback of extremum seeking-based control is the introduction of a high frequency oscillation into a system’s dynamics, which prevents even stable systems from settling at their equilibrium points. In this paper, we develop extremum seeking-based controllers whose control efforts, unlike that of traditional extremum seeking-based schemes, vanish as the system approaches equilibrium. Because the controllers that we develop are not differentiable at the origin, in proving a form of stability of our control scheme we start with a more general problem and extend the semiglobal practical stability result of Moreau and Aeyels to develop a relationship between systems and their averages even for systems which are nondifferentiable at a point. More specifically, in order to apply the practical stability results to our control scheme, we extend the Lie Bracket averaging result of Kurzweil, Jarnik, Sussmann, Liu, Gurvits, and Li to non-C functions. We then improve on our previous results on model-independent semiglobal exponential practical stabilization for linear time-varying single-input systems under the assumption that the time-varying input vector, which is otherwise unknown, satisfies a persistency of excitation condition over a sufficiently short window. [DOI: 10.1115/1.4025457]
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minimum seeking for clfs universal semiglobally stabilizing feedback under unknown control directions
IEEE Transactions on Automatic Control, 2013Co-Authors: Alexander Scheinker, Miroslav KrsticAbstract:Employing extremum seeking (ES) for seeking minima of control Lyapunov function (CLF) candidates, we develop 1) the first systematic design of ES controllers for unstable plants, 2) a simple non-model based universal feedback law that emulates, in an average sense, the “ $L_{g}V$ controllers” for stabilization with inverse optimality, and 3) a new strategy for stabilization of systems with unknown control directions, as an alternative to Nussbaum gain controllers that lack exponential stability, lack transient performance guarantees, and lack robustness to changes in the control direction. The stability analysis that underLies our designs is inspired by an analysis approach synthesized in a recent work by Durr, Stankovic, and Johansson, which combines a Lie Bracket averaging result of Gurvits and Li with a semiglobal practical stability result under small parametric perturbations by Moreau and Aeyels.
