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Learning stabilising policies for constrained nonlinear systems

Daniele Ravasio, Danilo Saccani, Marcello Farina, Giancarlo Ferrari-Trecate

TL;DR

The paper tackles constrained nonlinear control for systems modeled by recurrent neural networks under additive disturbances, where both input and output constraints must be satisfied. It introduces a two-layer architecture: a stabilising base controller guarantees $\delta$ISS and constraint satisfaction within a robustly positive invariant set, and a performance-boosting internal-model-control layer, implemented as a stable neural operator, optimises closed-loop performance via unconstrained learning while preserving stability and safety. The stabilization is achieved through an LMI-based synthesis that yields a gain $K$ and an RPI set, ensuring safety bounds; the boosting layer uses a projected IMC with a learnable operator $\mathbfcal M(\theta)$ parameterised by a Recurrent Equilibrium Network, trained over disturbance samples and projected onto a safe boosting set. A key theoretical result shows that all feasible, $\mathcal{L}_p$-stable closed-loop maps can be realized within the proposed framework, and stability is preserved even when the boosting optimization is stopped early. The approach is validated on a pH-neutralisation benchmark, demonstrating effective tracking, constraint adherence, and notable improvements in performance metrics, illustrating practical applicability to safety-critical constrained nonlinear control tasks.

Abstract

This work proposes a two-layered control scheme for constrained nonlinear systems represented by a class of recurrent neural networks and affected by additive disturbances. In particular, a base controller ensures global or regional closed-loop l_p-stability of the error in tracking a desired equilibrium and the satisfaction of input and output constraints within a robustly positive invariant set. An additional control contribution, derived by combining the internal model control principle with a stable operator, is introduced to improve system performance. This operator, implemented as a stable neural network, can be trained via unconstrained optimisation on a chosen performance metric, without compromising closed-loop equilibrium tracking or constraint satisfaction, even if the optimisation is stopped prematurely. In addition, we characterise the class of closed-loop stable behaviours that can be achieved with the proposed architecture. Simulation results on a pH-neutralisation benchmark demonstrate the effectiveness of the proposed approach.

Learning stabilising policies for constrained nonlinear systems

TL;DR

The paper tackles constrained nonlinear control for systems modeled by recurrent neural networks under additive disturbances, where both input and output constraints must be satisfied. It introduces a two-layer architecture: a stabilising base controller guarantees ISS and constraint satisfaction within a robustly positive invariant set, and a performance-boosting internal-model-control layer, implemented as a stable neural operator, optimises closed-loop performance via unconstrained learning while preserving stability and safety. The stabilization is achieved through an LMI-based synthesis that yields a gain and an RPI set, ensuring safety bounds; the boosting layer uses a projected IMC with a learnable operator parameterised by a Recurrent Equilibrium Network, trained over disturbance samples and projected onto a safe boosting set. A key theoretical result shows that all feasible, -stable closed-loop maps can be realized within the proposed framework, and stability is preserved even when the boosting optimization is stopped early. The approach is validated on a pH-neutralisation benchmark, demonstrating effective tracking, constraint adherence, and notable improvements in performance metrics, illustrating practical applicability to safety-critical constrained nonlinear control tasks.

Abstract

This work proposes a two-layered control scheme for constrained nonlinear systems represented by a class of recurrent neural networks and affected by additive disturbances. In particular, a base controller ensures global or regional closed-loop l_p-stability of the error in tracking a desired equilibrium and the satisfaction of input and output constraints within a robustly positive invariant set. An additional control contribution, derived by combining the internal model control principle with a stable operator, is introduced to improve system performance. This operator, implemented as a stable neural network, can be trained via unconstrained optimisation on a chosen performance metric, without compromising closed-loop equilibrium tracking or constraint satisfaction, even if the optimisation is stopped prematurely. In addition, we characterise the class of closed-loop stable behaviours that can be achieved with the proposed architecture. Simulation results on a pH-neutralisation benchmark demonstrate the effectiveness of the proposed approach.

Paper Structure

This paper contains 11 sections, 4 theorems, 44 equations, 4 figures.

Table of Contents

  1. INTRODUCTION
  2. PROBLEM STATEMENT
  3. The plant model
  4. The control problem
  5. Proposed control scheme
  6. STABILISING CONTROLLER DESIGN
  7. PERFORMANCE-BOOSTING DESIGN
  8. The projected internal model control
  9. Main results
  10. Learning the performance map
  11. CASE STUDY

Key Result

Corollary 1

Let Assumptions ass:sigmoid_function, ass:disturbance, and ass:compact_sets hold. Also, assume that $u_{\mathrm{b}}(k)\in\mathbb U_{\mathrm{b}}$ for $k\geq0$, and define $Q_{w_\mathrm{s}}^0=\mathop{\mathrm{diag}}\nolimits{(Q_w^0/2,(G_\mathrm{b}^\top G_\mathrm{b})/(2m))}$, $D_{\mathrm s}=[I_n,\,B]$, where, for all $i\in\mathcal{I}(H_{\mathrm s})$, then, setting $K=ZQ_\mathrm{s}^{-1}$ and $P_\math

Figures (4)

  • Figure 1: Control architecture. The red dashed block represents the pre-stabilised system.
  • Figure 2: Closed-loop output performance: system trajectories starting from different initial conditions (blue lines) are compared with the reference (red dash-dotted line).
  • Figure 3: Evolution of the control input. The black dashed line represents the upper saturation limit.
  • Figure 4: Evolution of the performance-boosting input contribution (expressed in normalised scale). Black dashed lines represent the associated constraints.

Theorems & Definitions (7)

  • Corollary 1
  • Proposition 2
  • Theorem 3
  • Remark IV.1
  • Remark IV.2
  • Remark IV.3
  • Lemma 4