Lie Bracket Approximation Research Articles

Nonsmooth optimization by Lie bracket approximations into random directions

Nonsmooth optimization by Lie bracket approximations into random directions

ISS-like properties in Lie-bracket approximations and application to extremum seeking

In this paper, we consider a class of nonlinear time-varying systems that are subject to uncontrolled inputs. Those inputs may represent, for instance, disturbances, time-varying parameters, or model uncertainties. Exploiting the stability properties of an extended system, a so-called Lie-bracket system, we derive stability properties of the original system. Those stability results, in the spirit of input-to-state stability, are then applied to the case where the dynamical system is an extremum seeking system. Two scenarios are analyzed. We first consider that the uncontrolled input is a deterministic noise corrupting the cost measurement. We then examine the case where the uncontrolled input induces time-variations of the cost optimizer.

Extremum seeking control of nonlinear dynamic systems using Lie bracket approximations

SummaryIn this article, we consider extremum seeking problems for a general class of nonlinear dynamic control systems. The main result of the article is a broad family of control laws which optimize the steady‐state performance of the system. We prove practical asymptotic stability of the optimal steady‐state and, moreover, propose sufficient conditions for the asymptotic stability in the sense of Lyapunov. The results generalize and extend existing results which are based on Lie bracket approximations. In particular, our approach does not rely on singular perturbation theory, as commonly used in extremum seeking of nonlinear dynamic systems.

A Lie bracket approximation approach to distributed optimization over directed graphs

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.

Newton-based extremum seeking: A second-order Lie bracket approximation approach

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.

Underwater Acoustic Source Seeking Using Time-Difference-of-Arrival Measurements

The research presented in this paper is aimed at developing a control algorithm for an autonomous surface system carrying a two-sensor array consisting of two acoustic receivers, capable of measuring the time-difference-of-arrival (TDOA) of a quasiperiodic underwater acoustic signal and utilizing this value to steer the system toward the acoustic source in the horizontal plane. Stability properties of the proposed algorithm are analyzed using the Lie bracket approximation technique. Furthermore, simulation results are presented, where particular attention is given to the relationship between the time difference of arrival measurement noise and the sensor baseline-the distance between the two acoustic receivers. Also, the influence of a constant disturbance caused by sea currents is considered. Finally, experimental results in which the algorithm was deployed on two autonomous surface vehicles, each equipped with a single acoustic receiver, are presented. The algorithm successfully steers the vehicle formation toward the acoustic source, despite the measurement noise and intermittent measurements, thus showing the feasibility of the proposed algorithm in real-life conditions.

Extremum seeking control for an acceleration controlled unicycle

We study the problem of source seeking with an acceleration controlled unicycle without position measurements. It is assumed that only real-time measurements of the source signal are available. The proposed control law is based on an approximation of the symmetric product of suitably chosen vector fields. The approximation of the symmetric product can be traced back to known averaging results for mechanical systems under vibrational control. Compared to known extremum seeking control laws by means of Lie bracket approximations, the proposed approach has the advantage that the velocity remains bounded in the high-frequency limit. We prove semi-global practical asymptotic stability for the acceleration-driven unicycle system under the proposed control law.

Partial Stability Concept in Extremum Seeking Problems

The paper deals with the extremum seeking problem for a class of cost functions depending only on a part of state variables of a control system. This problem is related to the concept of partial asymptotic stability and analyzed by Lyapunov’s direct method and averaging schemes. Sufficient conditions for the practical partial stability of a system with oscillating inputs are derived with the use of Lie bracket approximation techniques. These conditions are exploited to describe a broad class of extremum-seeking controllers ensuring the partial stability of the set of minima of a cost function. The obtained theoretical results are illustrated by the Brockett integrator and rotating rigid body.

On extremum seeking controllers based on the Lie bracket approximation in domains with obstacles

AbstractIn this paper, we consider the problem of extremum seeking in the presence of obstacles. The analytical expression of the cost function as well as locations and shapes of the obstacles are assumed to be partially or completely unknown. We describe a broad family of control algorithms for a unicycle type system, which provides a solution of the above problem based on Lie bracket approximation ideas. The obtained controllers generalize some known results and allow to construct new control strategies. Furthermore, it is shown that gradient‐free control algorithms can be used for excluding the attraction of trajectories to undesirable equilibriuma in obstacle avoidance problems.

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.

Formation Shape Control Based on Distance Measurements Using Lie Bracket Approximations

We study the problem of distance-based formation control in autonomous multi-agent systems in which only distance measurements are available. This means that the target formations as well as the sensed variables are both determined by distances. We propose a fully distributed distance-only control law, which requires neither a time synchronization of the agents nor storage of measured data. The approach is applicable to point agents in the Euclidean space of arbitrary dimension. Under the assumption of infinitesimal rigidity of the target formations, we show that the proposed control law induces local uniform asymptotic stability. Our approach involves sinusoidal perturbations in order to extract information about the negative gradient direction of each agent's local potential function. An averaging analysis reveals that the gradient information originates from an approximation of Lie brackets of certain vector fields. The method is based on a recently introduced approach to the problem of extremum seeking control. We discuss the relation in the paper.

Extremum Seeking for Time-Varying Functions using Lie Bracket Approximations

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.

Extremum seeking for dynamic maps using Lie brackets and singular perturbations

We introduce a framework for the analysis of extremum seeking systems for dynamic maps based on Lie bracket approximations and singular perturbation theory. Using the introduced framework, we provide an extremum seeking regulator for a unicycle and prove its stability properties.

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.

Solving Potential Games With Dynamical Constraint.

We solve N -player potential games with dynamical constraint in this paper. Potential games with stable dynamics are first considered followed by one type of potential games without inherently stable dynamics. Different from most of the existing Nash seeking methods, we provide an extremum seeking-based method that does not require explicit information on the game dynamics or the payoff functions. Only measurements of the payoff functions are needed in the game strategy synthesis. Lie bracket approximation is used for the analysis of the proposed Nash seeking scheme. A singularly semi-globally practically uniformly asymptotically stable result is presented for potential games with stable dynamics and an ultimately bounded result is provided for potential games without inherently stable dynamics. For first-order perturbed integrator-type dynamics, we employ an extended-state observer to deal with the disturbance such that better convergence is achievable. Stability of the closed-loop system is proven and the ultimate bound is quantified. Numerical examples are presented to verify the effectiveness of the proposed methods.

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.

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.