skip to main content
US FlagAn official website of the United States government
dot gov icon
Official websites use .gov
A .gov website belongs to an official government organization in the United States.
https lock icon
Secure .gov websites use HTTPS
A lock ( lock ) or https:// means you've safely connected to the .gov website. Share sensitive information only on official, secure websites.
Attention:The NSF Public Access Repository (NSF-PAR) system and access will be unavailable from 7:00 AM ET to 7:30 AM ET on Friday, April 24 due to maintenance. We apologize for the inconvenience.

Explore Research Products in the PAR
It may take a few hours for recently added research products to appear in PAR search results.


Title: Higher-order Lie bracket approximation and averaging of control-affine systems with application to extremum seeking
This paper provides a rigorous derivation for what is known in the literature as the Lie bracket approximation of control-affine systems in a more general and sequential framework for higher-orders. In fact, by using chronological calculus, we show that said Lie bracket approximations can be derived, and considered, as higher-order averaging terms. Hence, the theory provided in this paper unifies both averaging and approximation theories of control-affine systems. In particular, the Lie bracket approximation of order (n) turns out to be a higher-order averaging of order (n + 1). The derivation and formulation provided in this paper can be directly reduced to the first and second order. Lie bracket approximations available in the literature. However, we do not need to make many of the assumptions that were needed/provided in the literature and show that they are in fact natural corollaries from our work. Moreover, we use our results to show that important and useful information about control-affine extremum seeking systems can be obtained and used for significant performance improvement, including a faster convergence rate influenced by higher-order derivatives. We provide multiple numerical simulations to demonstrate both the conceptual elements of this work as well as the significance of our results on extremum seeking with comparison against the literature.  more » « less
Award ID(s):
2318772
PAR ID:
10676269
Author(s) / Creator(s):
;
Publisher / Repository:
Elsevier
Date Published:
Journal Name:
Automatica
Volume:
188
Issue:
C
ISSN:
0005-1098
Page Range / eLocation ID:
112950
Subject(s) / Keyword(s):
Control-affine systems, Higher-order averaging, Lie bracket, Chronological calculus, Geometric control, Extremum seeking
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
More Like this
  1. ; (, Journal of Dynamic Systems, Measurement, and Control)
    null (Ed.)
    Abstract This paper addresses the compensation of wave actuator dynamics in scalar extremum seeking (ES) for static maps. Infinite-dimensional systems described by partial differential equations (PDEs) of wave type have not been considered so far in the literature of ES. A distributed-parameter-based control law using back-stepping approach and Neumann actuation is initially proposed. Local exponential stability as well as practical convergence to an arbitrarily small neighborhood of the unknown extremum point is guaranteed by employing Lyapunov–Krasovskii functionals and averaging theory in infinite dimensions. Thereafter, the extension for wave equations with Dirichlet actuation, antistable wave PDEs as well as the design for the delay-wave PDE cascade are also discussed. Numerical simulations illustrate the theoretical results. 
    more » « less
  2. ; ; ; (, IFAC-PapersOnLine)
    The problem of hovering in flapping insects/hummingbirds, and potential bio-mimicry by micro aerial vehicles (MAVs), have been studied for decades by scientists and engineers. Said communities often study hovering in flapping systems as either an open-loop or closed-loop system to analyze stability and/or propose control designs. Recently, a fundamentally novel result has been achieved in the literature of the hovering problem. That is, hovering in flapping insects/hummingbirds can be characterized/mimicked as a stable, model-free, real-time extremum seeking control (ESC) feedback system. In this paper we aim at two contributions: (i) provide a novel open-loop, optimal control characterization of hovering; and (ii) compare the performance of closed-loop, real-time ESC in hovering vs. the provided open-loop, non-real-time optimal control in hovering. 
    more » « less
  3. ; ; ; ; (, Forum of Mathematics, Sigma)
    Abstract We consider the Vlasov equation in any spatial dimension, which has long been known [ZI76, Mor80, Gib81, MW82] to be an infinite-dimensional Hamiltonian system whose bracket structure is ofLie–Poisson type. In parallel, it is classical that the Vlasov equation is amean-field limitfor a pairwise interacting Newtonian system. Motivated by this knowledge, we provide a rigorous derivation of the Hamiltonian structure of the Vlasov equation, both the Hamiltonian functional and Poisson bracket, directly from the many-body problem. One may view this work as a classical counterpart to [MNP+20], which provided a rigorous derivation of the Hamiltonian structure of the cubic nonlinear Schrödinger equation from the many-body problem for interacting bosons in a certain infinite particle number limit, the first result of its kind. In particular, our work settles a question of Marsden, Morrison and Weinstein [MMW84] on providing a ‘statistical basis’ for the bracket structure of the Vlasov equation. 
    more » « less
  4. ; (, Proceedings of the 30th Mediterranean Conference on Control and Automation (MED 2022))
    Throughout the past decades, many different versions of the widely used first-order Cell-Transmission Model (CTM) have been proposed for optimal traffic control. Highway traffic management techniques such as Ramp Metering (RM) are typically designed based on an optimization problem with nonlinear constraints originating in the flow-density relation of the Fundamental Diagram (FD). Most of the extended CTM versions are based on the trapezoidal approximation of the flow-density relation of the Fundamental Diagram (FD) in an attempt to simplify the optimization problem. However, this relation is naturally nonlinear, and crude approximations can greatly impact the efficiency of the optimization solution. In this study, we propose a class of extended CTMs that are based on piecewise affine approximations of the flow-density relation such that (a) the integrated squared error with respect to the true relation is greatly reduced in comparison to the trapezoidal approximation, and (b) the optimization problem remains tractable for real-time application of ramp metering optimal controllers. A two-step identification method is used to approximate the FD with piecewise affine functions resulting in what we refer to as PWA-CTMs. The proposed models are evaluated by the performance of the optimal ramp metering controllers, e.g. using the widely used PI-ALINEA approach, in complex highway traffic networks. Simulation results show that the optimization problems based on the PWA-CTMs require less computation time compared to other CTM extensions while achieving higher accuracy of the flow and density evolution. Hence, the proposed PWA-CTMs constitute one of the best approximation approaches for first-order traffic flow models that can be used in more general and challenging modeling and control applications. 
    more » « less
  5. ; ; ; (, IEEE Transactions on Automatic Control)
    This paper presents a time-invariant extremum seeking controller (ESC) for nonlinear autonomous systems with limit cycles. For this time-invariant ESC, we propose a method to prove the closed loop system has an asymptotically stable limit cycle. The method is based on a perturbation theorem for maps, and, unlike existing techniques that use averaging and singular perturbation tools, it is not limited to weakly nonlinear systems. We use a typical example system to show that our method does indeed establish asymptotic stability of the limit cycle with minimal amplitude. Utilizing the example, we provide a general guide for analytic computations that are required to apply our method. The corresponding Mathematica code is available as supplementary material. 
    more » « less
  • Citation Formats
  • MLA
  • APA
  • Chicago
  • BibTeX
  • Text Encoded Conversion
  • XML
  • Save / Share this Record
  • Send to Email