Estimating Directional Inflection Points of Multivariable Static Maps using Second-Order Newton-Based Extremum Seeking

Core Concepts

A second-order Newton-based extremum seeking (SONES) algorithm is proposed to estimate the directional inflection points of multivariable static maps without requiring information about the curvature of the map and its gradient.

Abstract

The paper presents a second-order Newton-based extremum seeking (SONES) algorithm to estimate the directional inflection points of multivariable static maps. The key highlights are:

The conventional extremum seeking (ES) algorithms aim to drive the system towards the extremum point of the map. In contrast, the proposed SONES algorithm targets the directional inflection points, where the curvature of the map changes direction along a specific axis.

The SONES algorithm requires accurate estimates of the second-order derivative (Hessian matrix) and the inverse of the third-order derivative of the map. The paper provides perturbation matrices to generate these estimates using a carefully chosen set of probing frequencies.

A differential Riccati filter is used to calculate the inverse of the third-order derivative, which is necessary for the SONES algorithm.

The local stability of the SONES algorithm is proven for general multivariable static maps using averaging analysis. The algorithm ensures uniform convergence towards the directional inflection point without requiring information about the curvature of the map and its gradient.

Simulation results demonstrate the effectiveness of the proposed SONES algorithm in estimating the directional inflection point of a multivariable static map.

Stats

The map has an inflection point at θ∗ = [1 2]⊤ along the θ1-axis near a local minimum and a saddle point located at [-0.07 2.22]⊤ and [1.26 2.62]⊤, respectively.

Quotes

None.

Key Insights Distilled From

Second-Order Newton-Based Extremum Seeking for Multivariable Static Maps

by Azad Ghaffar... at arxiv.org 04-02-2024

https://arxiv.org/pdf/2404.01103.pdf

Deeper Inquiries

How can the SONES algorithm be extended to handle dynamic maps with time-varying parameters

To extend the SONES algorithm to handle dynamic maps with time-varying parameters, one can introduce a time-varying parameter estimation scheme. This would involve updating the estimates of the second- and third-order derivatives in real-time as the parameters of the map change. By incorporating adaptive mechanisms that track the variations in the parameters, the algorithm can adjust its probing frequencies and perturbation matrices accordingly. Additionally, the use of dynamic filters or observers can help in estimating the time-varying derivatives accurately, enabling the SONES algorithm to effectively optimize dynamic maps with changing parameters.

What are the potential applications of the SONES algorithm in real-world engineering systems beyond the static map optimization problem

The SONES algorithm has a wide range of potential applications in real-world engineering systems beyond static map optimization. Some of these applications include:

Robotics: SONES can be utilized in robotic systems for trajectory optimization, motion planning, and obstacle avoidance by dynamically seeking extremum points in the environment.
Power Systems: In power systems, SONES can optimize energy generation and distribution by adjusting parameters to maximize efficiency or minimize losses.
Aerospace: Extremum seeking with SONES can be applied in aerospace systems for optimal control, flight path planning, and aerodynamic parameter optimization.
Biomedical Engineering: SONES can optimize parameters in medical devices, such as drug delivery systems or prosthetics, to enhance performance and patient outcomes.
Environmental Monitoring: SONES can be used in environmental monitoring systems to optimize sensor networks, data collection strategies, and resource allocation for efficient data analysis.

Can the SONES algorithm be adapted to handle stochastic or uncertain multivariable maps, where the gradient and curvature information is not precisely known

Adapting the SONES algorithm to handle stochastic or uncertain multivariable maps is feasible by incorporating robust control techniques and stochastic optimization methods. To address the uncertainty in the gradient and curvature information, one can introduce probabilistic models or Bayesian estimation approaches to update the derivatives based on noisy measurements. Additionally, the algorithm can be augmented with adaptive mechanisms that adjust the probing frequencies and perturbation matrices to account for uncertainties in the map. By integrating robust optimization strategies and stochastic modeling, the SONES algorithm can effectively handle stochastic or uncertain multivariable maps, ensuring reliable performance in real-world applications.