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Persistently Exciting Online Feedback Optimization Controller with Minimal Perturbations

Tore Gude, Marta Anna Zagorowska, Lars Struen Imsland

TL;DR

This work tackles gradient estimation in Online Feedback Optimization when the input–output map $y = h(u)$ is unknown, by eliminating random perturbations in favor of a persistently exciting input design. A bilevel framework relaxes a nonconvex rank constraint with linear constraints and a penalty, yielding a convex lower-level problem solved via KKT conditions to produce minimal-disturbance perturbations that guarantee persistence. The approach is validated on a gas-lift optimization case for four wells, showing profits comparable to OFO with random perturbations and superior to OFO without perturbations, while avoiding excessive disturbances. The method offers practical benefits for real-time production optimization under limited resources and uncertain gradients, with potential extension to disturbed physical systems.

Abstract

This paper develops a persistently exciting input generating Online Feedback Optimization (OFO) controller that estimates the sensitivity of a process ensuring minimal deviations from the descent direction while converging. This eliminates the need for random perturbations in feedback loop. The proposed controller is formulated as a bilevel optimization program, where a nonconvex full rank constraint is relaxed using linear constraints and penalization. The validation of the method is performed in a simulated scenario where multiple systems share a limited, costly resource for production optimization, simulating an oil and gas resource allocation problem. The method allows for less input perturbations while accurately estimating gradients, allowing faster convergence when the gradients are unknown. In the case study, the proposed method achieved the same profit compared to an OFO controller with random input perturbations, and $1.4\%$ higher profit compared to an OFO controller without input perturbations.

Persistently Exciting Online Feedback Optimization Controller with Minimal Perturbations

TL;DR

This work tackles gradient estimation in Online Feedback Optimization when the input–output map is unknown, by eliminating random perturbations in favor of a persistently exciting input design. A bilevel framework relaxes a nonconvex rank constraint with linear constraints and a penalty, yielding a convex lower-level problem solved via KKT conditions to produce minimal-disturbance perturbations that guarantee persistence. The approach is validated on a gas-lift optimization case for four wells, showing profits comparable to OFO with random perturbations and superior to OFO without perturbations, while avoiding excessive disturbances. The method offers practical benefits for real-time production optimization under limited resources and uncertain gradients, with potential extension to disturbed physical systems.

Abstract

This paper develops a persistently exciting input generating Online Feedback Optimization (OFO) controller that estimates the sensitivity of a process ensuring minimal deviations from the descent direction while converging. This eliminates the need for random perturbations in feedback loop. The proposed controller is formulated as a bilevel optimization program, where a nonconvex full rank constraint is relaxed using linear constraints and penalization. The validation of the method is performed in a simulated scenario where multiple systems share a limited, costly resource for production optimization, simulating an oil and gas resource allocation problem. The method allows for less input perturbations while accurately estimating gradients, allowing faster convergence when the gradients are unknown. In the case study, the proposed method achieved the same profit compared to an OFO controller with random input perturbations, and higher profit compared to an OFO controller without input perturbations.

Paper Structure

This paper contains 13 sections, 19 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Online Feedback Optimization
  4. Gradient Estimation
  5. Persistently Exciting OFO Controller
  6. Random Perturbations
  7. Persistently Exciting Input Design
  8. Bilevel Formulation
  9. Case Study
  10. Gas Lift Optimization
  11. Setup
  12. Results
  13. Conclusion

Figures (4)

  • Figure 1: Schematic of the system of oil wells. The subsea hub located on the seabed distributes incoming gas lift to the oil wells and transports the fluid from the oil wells to the oil platform at sea level.
  • Figure 2: Illustration of the GLPR curves used in the case study. Red circle indicates the optimal operating point for the wells when the total available gas lift is at a certain capacity.
  • Figure 3: Objective function as a function of time step at the left y-axis, total gas lift availability at the right y-axis in red. Mean value of the OFO simulations is in blue with $\pm$ one standard deviations (lighter blue). The results from the PE OFO controller are in turquoise and the optimal solution is plotted with dotted gray lines. The results from the OFO controller without perturbations is plotted in dotted turquoise. The total gas lift availability is changed as a step every 100th time step.
  • Figure 4: Applied perturbation to each element in the input vector. The standard deviation of the Gaussian distribution is tuned so that the OFO controller reaches the time-varying optimum and divided with the stepsize for comparison.