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Constrained Performance Boosting Control for Nonlinear Systems via ADMM

Gianluca Giacomelli, Danilo Saccani, Siep Weiland, Giancarlo Ferrari-Trecate, Valentina Breschi

TL;DR

This work tackles boosting nonlinear system performance under hard input and state constraints while preserving stability. It extends the prior Performance Boosting (PB) approach by embedding it in an ADMM framework (ADMM-PB) to handle constraints without altering the original neural controller architecture, leveraging an IMC-based stability guarantee. The method introduces copy variables, an augmented Lagrangian, and alternating optimization over a learnable operator $\mathbfcal{M}$ parameterized by $\theta$, together with projection steps onto convex constraint sets. Numerical results on a constrained point-mass robot benchmark show that ADMM-PB yields smoother training and better constraint satisfaction than a barrier-based baseline, with a trade-off toward more conservative trajectories at higher constraint emphasis. This approach provides a principled, scalable route to performance enhancement for nonlinear systems where constraint satisfaction is critical in safety-critical applications.

Abstract

We present the Alternating Direction Method of Multipliers for Performance Boosting (ADMM-PB), an approach to design performance boosting controllers for stable or pre-stabilized nonlinear systems, while explicitly seeking input and state constraint satisfaction. Rooted on a recently proposed approach for designing neural-network controllers that guarantees closed-loop stability by design while minimizing generic cost functions, our strategy integrates it within an alternating direction method of multipliers routine to seek constraint handling without modifying the controller structure of the aforementioned seminal strategy. Our numerical results showcase the advantages of the proposed approach over a baseline penalizing constraint violation through barrier-like terms in the cost, indicating that ADMM-PB can lead to considerably lower constraint violations at the price of inducing slightly more cautious closed-loop behaviors.

Constrained Performance Boosting Control for Nonlinear Systems via ADMM

TL;DR

This work tackles boosting nonlinear system performance under hard input and state constraints while preserving stability. It extends the prior Performance Boosting (PB) approach by embedding it in an ADMM framework (ADMM-PB) to handle constraints without altering the original neural controller architecture, leveraging an IMC-based stability guarantee. The method introduces copy variables, an augmented Lagrangian, and alternating optimization over a learnable operator parameterized by , together with projection steps onto convex constraint sets. Numerical results on a constrained point-mass robot benchmark show that ADMM-PB yields smoother training and better constraint satisfaction than a barrier-based baseline, with a trade-off toward more conservative trajectories at higher constraint emphasis. This approach provides a principled, scalable route to performance enhancement for nonlinear systems where constraint satisfaction is critical in safety-critical applications.

Abstract

We present the Alternating Direction Method of Multipliers for Performance Boosting (ADMM-PB), an approach to design performance boosting controllers for stable or pre-stabilized nonlinear systems, while explicitly seeking input and state constraint satisfaction. Rooted on a recently proposed approach for designing neural-network controllers that guarantees closed-loop stability by design while minimizing generic cost functions, our strategy integrates it within an alternating direction method of multipliers routine to seek constraint handling without modifying the controller structure of the aforementioned seminal strategy. Our numerical results showcase the advantages of the proposed approach over a baseline penalizing constraint violation through barrier-like terms in the cost, indicating that ADMM-PB can lead to considerably lower constraint violations at the price of inducing slightly more cautious closed-loop behaviors.

Paper Structure

This paper contains 12 sections, 1 theorem, 49 equations, 4 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Setting and goals
  3. ADMM-PB: ADMM for Constrained Performance Boosting
  4. ADMM reformulation of performance boosting
  5. Augmented performance boosting
  6. Projection into the constraint sets
  7. Practical aspects of ADMM-PB
  8. Termination criteria
  9. Tuning the step sizes in ADMM-PB
  10. Benchmark example: constrained control of a point-mass robot
  11. Comparison with Control Barrier Functions (CBFs)
  12. Conclusions

Key Result

Theorem 1

Let Assumption assumption:L_p_map hold and consider eq:operator_form with the input sequence chosen as for a causal operator $\mathbfcal{M}: \ell^{n} \rightarrow \ell^{m}$. Let $\mathbf{K}$ be the operator such that $\mathbf{u}=\mathbf{K}(\mathbf{x})$ is equivalent to eq:input_sequence_thm. Then, $(i)$ if $\mathbfcal{M} \in \pazocal{L}_{p}$, then the closed-loop system is $\ell_p$-stable, $(ii)$

Figures (4)

  • Figure 1: Scheme of the adopted IMC architecture (see furieri2024learning).
  • Figure 2: ADMM-PB vs CBF-based baseline: robot position over time (as indicated in the colorbar) with red $\times$ denoting the initial positions. Each trajectory refers to one of the $5$ testing scenarios, showing that overall the baseline with $\omega=10^3$ leads to a faster convergence to the origin, yet trajectories are slightly less consistent across scenarios.
  • Figure 3: Evolution of ADMM's step size consequent to the update rules in \ref{['eq:update_rho']} and \ref{['eq:step_size_choice']}.
  • Figure 4: ADMM-PB vs CBF-based baseline: speed of the robots over time for both Euclidean coordinates $x$ and $y$. The dashed lines indicate the speed constraints in \ref{['eq:speed_constraints']}, while the bold colored lines are the minimum and maximum speed achieved by the robot at each time across the $5$ testing scenarios. The red vertical lines are the instants at which constraint violations occur.

Theorems & Definitions (6)

  • Definition 1
  • Theorem 1: furieri2024learning
  • Remark 1: Practical parameterization of $\mathbfcal{M}$
  • Remark 2: Ordering of ADMM-PB's steps
  • Remark 3: Stability & early stopping
  • Remark 4: Copy variables & their shortcomings