Abstract
This paper proposes a gradient extremum seeking method to address locally quadratic static maps in the presence of time-varying delays. Accommodating nonconstant delays has a strong impact in the predictor construction in terms of the associated transport partial differential equation (PDE) with variable convection speeds beyond the restrictions imposed on the delay regarding its arbitrary duration but bounded variation. A novel predictor design using perturbation-based estimates of the unknown Gradient and Hessian of the map must be introduced to handle this variable nature of the delays, which can arise both in the input and output channels of the nonlinear map to be optimized. Local exponential stability and convergence to a small neighborhood of the unknown extremum point are guaranteed. This technical result is assured by using backstepping transformation and averaging theory in infinite dimensions. Implementation aspects of the presented predictor for variable delays as well as the extension to state-dependent delays are also discussed. At last, we introduce the first results in the topic of extremum seeking control for cascades of transport PDEs that are interconnected through boundary conditions. Such a PDE-PDE cascade is useful to represent simultaneous time- and state-dependent delays. A simulation example illustrates the effectiveness of the proposed predictor-based extremum seeking approach for time-delay compensation.
Highlights
Extremum Seeking (ES) is a sort of adaptive control methodology which does not follow directly the standard paradigm of reference model, where the control objective of stabilizing the error between the reference trajectory and the output signal of the plant with known parametrization is studied [1]–[3]
The main goal of this paper is to generalize our previous results on prediction-feedback for ES in order to include a more general class of variable delays, time-varying delays but delays depending on the system states and time, simultaneously
AVERAGING FOR NONCONSTANT DELAYS In Theorem 1, we proved that the average closed-loop system (53)–(55) with transport partial differential equation (PDE) for delay representation is exponentially stable
Summary
Extremum Seeking (ES) is a sort of adaptive control methodology which does not follow directly the standard paradigm of reference model, where the control objective of stabilizing the error between the reference trajectory and the output signal of the plant with known parametrization is studied [1]–[3]. By employing periodic dithers (sinusoidal perturbations), the estimation in the average sense of the Gradient and the Hessian of the nonlinear map is attained so that the local stability proof for the overall closed-loop system and the convergence to a small neighborhood of the extremum can be assured. Depending on the specific application, such kind of delays may be: known/unknown, small/large, deterministic/stochastic or constant/variable [16], [17]. The latter type of classification is the main topic of interest in this paper
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