Adaptive Tuning of Online Feedback Optimization for Process Control Applications

Abstract: Online Feedback Optimization leverages properties of optimization algorithms to develop controllers for systems with limited model availability, which is often the case in process control. The interplay between the parameters of the chosen optimization algorithm, as well as lack of direct connection to the characteristics of the underlying process make their tuning challenging. We propose a method for adaptive tuning of Online Feedback Optimization controllers based on scaled projected gradient descent by using sensitivity of the desired objective to the parameters of the algorithm. The proposed adaptive tuning method limits the operator-tunable parameters to scalar values that represent how much the control inputs and the objective can change between iterations without requiring either additional information about the controlled system or repeated experiments. Numerical studies on a gas lift and a continuously-stirred tank reactor processes confirm that our adaptive scheme improves closed-loop performance of Online Feedback optimization compared to standard manual tuning methods.

• The paper presents a sensitivity-driven scheme for adapting both step size and scaling matrix in online feedback optimization, enhancing convergence.
• The paper demonstrates significant improvements in convergence speed and tracking accuracy across benchmark examples and real-world process control cases.
• The paper establishes a low-dimensional, robust adaptation mechanism that minimizes manual hyperparameter tuning in complex, nonlinear systems.

Adaptive Tuning in Online Feedback Optimization for Process Control

Introduction

This paper addresses a significant challenge in applying Online Feedback Optimization (OFO) to process control systems: the tuning of controller hyperparameters, specifically the step size and the scaling matrix in projected gradient-based feedback optimization schemes. Unlike most forms of Model Predictive Control (MPC) or classical optimization-based control, OFO is designed for environments with limited model availability and relies directly on real-time system measurements. However, the performance of OFO depends critically on parameter tuning, which, if done manually, requires extensive system experiments—an impractical and inefficient process for industrial applications.

Problem Statement and Motivation

The tuning landscape for OFO is problematic because the algorithmic hyperparameters have no clear connection to process dynamics and disturbance characteristics. Existing adaptive methods either handle only specific classes of systems (e.g., linear-quadratic or steady-state with simple constraints) or depend on information (e.g., Hessians, full input-output mappings) that is often unobservable in practice. The main contribution here is a generic adaptive scheme that is both model-light and data-efficient, avoiding repeated open-loop experiments.

Methodology

Sensitivity-Based Adaptive Tuning

The authors propose a sensitivity-driven scheme for simultaneously adapting both the step size and the scaling matrix in a scaled projected gradient descent (SPGD) form of OFO. The approach leverages known input-output sensitivities—typically available in process control via measurement or estimation—and computes the effect of parameter updates directly on the objective.

• For the scaling matrix adaptation, a semi-definite programming relaxation is constructed to enforce monotonic decrease in the objective, relying only on available sensitivities and output measurements rather than full model knowledge. The update requires solving an auxiliary optimization problem at each iteration to ensure positive definiteness and limit norm-based variations for stability.
• In the diagonal scaling special case, a rule resembling a two-way backtracking update adjusts each matrix entry based on the sign and magnitude of the objective's sensitivity, permitting practical and computationally efficient tuning for large-scale and constrained systems.
• The step size adaptation uses a quadratic approximation of the objective landscape based on recent iterates, and the optimal step size at each iteration is selected by minimizing this surrogate function over an admissible range.

A single set of scalar metaparameters (allowable input and objective changes per iteration) controls the adaptation, greatly reducing the dimensionality and complexity of the tuning problem compared to manual methods.

Numerical Evaluation and Results

Numerical case studies demonstrate the efficacy of the adaptive tuning scheme across multiple canonical and industrially relevant scenarios:

1. Accelerated Convergence in Benchmark Examples: Compared to fixed-parameter SPGD, adaptive tuning dramatically reduces the number of iterations required to reach the optimum (e.g., from 82 to 5 iterations in a constrained nonconvex quadratic optimization benchmark), while avoiding infeasibility.
2. Decoupled Influence of Step Size and Scaling Matrix: Using the Rosenbrock function, the experiments elucidate that both parameter families (scaling matrix and step size) have non-redundant impact on performance. Adaptation of both yields fast, oscillation-free convergence, while adapting either alone produces suboptimal or oscillatory trajectories.
3. Gas Lift Optimization: In a high-dimensional, coupled-input process model of gas-lifted oil wells, diagonal scaling matrix adaptation enables rapid constraint-aware optimization, outperforming both fixed-parameter and heuristic adaptation approaches in terms of rate of convergence and optimality under active constraint scenarios.
4. Continuous Stirred Tank Reactor (CSTR) Case Study: In the CSTR control task, adaptively tuned OFO achieves up to 25% better setpoint tracking accuracy on the tuning trajectory and 20% improvement on a validation trajectory relative to state-of-the-art manual tuning schemes. Notably, this is obtained without any repeated or time-consuming open-loop experiments.

These results substantiate the claim that the proposed adaptation method consistently improves closed-loop OFO performance and significantly curtails the time and domain expertise burden associated with manual hyperparameter selection.

Implications and Future Directions

Theoretically, the adaptive tuning framework brings OFO closer to the domain of self-configuring optimization-based controllers, which is especially valuable given the data-driven and safety-critical nature of industrial process control. On the practical side, the ability to tune only a handful of scalar values and robustly adapt to system changes holds substantial promise for deploying OFO in settings with variable process characteristics, operational constraints, and non-stationary disturbances.

Two important future research directions are identified:

• Convergence and Constraint Satisfaction Guarantees: While the adaptation algorithms are validated empirically, rigorous analysis of stability and constraint satisfaction under noise, approximation, and system time-variance will further solidify their practical deployability.
• Robustification to Measurement Noise: Since sensitivity-based adaptation is potentially data-noise-sensitive, augmenting the method with robust estimation or filtering mechanisms is critical for real-world deployment, particularly in low signal-to-noise environments typical for many process industries.

Moreover, the generality of the adaptation procedure suggests its applicability beyond process control, to other domains employing OFO or similar online optimization-in-the-loop paradigms.

Conclusion

The proposed sensitivity-based adaptive tuning methodology for online feedback optimization eliminates the need for time and resource-intensive manual hyperparameter tuning in process control. By embedding a low-dimensional, robust adaptation mechanism within the OFO controller's iterative loop, the approach ensures accelerated convergence, improved tracking accuracy, and consistent constraint satisfaction across a variety of nonlinear and high-dimensional systems. The work lays foundational groundwork for more practical, scalable, and autonomous deployment of optimization-based controllers in industrial and other complex cyber-physical systems.