This article presents an extremum‐seeking control (ESC) algorithm for unmodeled nonlinear systems with known steady‐state gain and generally non‐convex cost functions with bounded curvature. The main contribution of this article is a novel gradient estimator, which uses a polyhedral set that characterizes all gradient estimates consistent with the collected data. The gradient estimator is posed as a quadratic program, which selects the gradient estimate that provides the best worst‐case convergence of the closed‐loop Lyapunov function. We show that the polyhedral‐based gradient estimator ensures the stability of the closed‐loop system formed by the plant and optimization algorithm. Furthermore, the estimated gradient provably produces the optimal robust convergence. We demonstrate our ESC controller through three benchmark examples and one practical example, which shows our ESC has fast and robust convergence to the optimal equilibrium.
The nonlinear responses of species to environmental variability can play an important role in the maintenance of ecological diversity. Nonetheless, many models use parametric nonlinear terms which pre‐determine the ecological conclusions. Motivated by this concern, we study the estimate of the second derivative (curvature) of the link function in a functional single index model. Since the coefficient function and the link function are both unknown, the estimate is expressed as a nested optimization. We first estimate the coefficient function by minimizing squared error where the link function is estimated with a Nadaraya‐Watson estimator for each candidate coefficient function. The first and second derivatives of the link function are then estimated via local‐quadratic regression using the estimated coefficient function. In this paper, we derive a convergence rate for the curvature of the nonlinear response. In addition, we prove that the argument of the linear predictor can be estimated root‐
- NSF-PAR ID:
- 10453655
- Publisher / Repository:
- Wiley-Blackwell
- Date Published:
- Journal Name:
- Scandinavian Journal of Statistics
- Volume:
- 47
- Issue:
- 4
- ISSN:
- 0303-6898
- Page Range / eLocation ID:
- p. 1307-1338
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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