Learning to Boost the Performance of Stable Nonlinear Systems

TL;DR

This work develops a stability-by-design framework for boosting the transient performance of nonlinear discrete-time systems using data-driven controllers. By leveraging an Internal Model Control (IMC) structure, the authors characterize all stability-preserving controllers as those implementable via a freely chosen $\mathcal{L}_p$ operator $\mathbfcal{M}$, enabling unconstrained optimization over rich neural network classes while preserving $\ell_p$-stability even if optimization halts or the ground-truth model is uncertain. The paper further provides robustness results against model mismatch (with a small-gain like condition) and extends to distributed architectures, where local controllers respect network topology. Implementation is facilitated through Recurrent Equilibrium Networks (RENs) to parameterize $\mathbfcal{M}$, and the approach is validated through numerical experiments in cooperative robotics, including safety-invariant and temporal-logic guided tasks. The framework unifies stability guarantees with flexible, nonconvex cost shaping, offering a practical pathway to safe, high-performance learning-based control in complex, distributed settings.

Abstract

The growing scale and complexity of safety-critical control systems underscore the need to evolve current control architectures aiming for the unparalleled performances achievable through state-of-the-art optimization and machine learning algorithms. However, maintaining closed-loop stability while boosting the performance of nonlinear control systems using data-driven and deep-learning approaches stands as an important unsolved challenge. In this paper, we tackle the performance-boosting problem with closed-loop stability guarantees. Specifically, we establish a synergy between the Internal Model Control (IMC) principle for nonlinear systems and state-of-the-art unconstrained optimization approaches for learning stable dynamics. Our methods enable learning over arbitrarily deep neural network classes of performance-boosting controllers for stable nonlinear systems; crucially, we guarantee L_p closed-loop stability even if optimization is halted prematurely, and even when the ground-truth dynamics are unknown, with vanishing conservatism in the class of stabilizing policies as the model uncertainty is reduced to zero. We discuss the implementation details of the proposed control schemes, including distributed ones, along with the corresponding optimization procedures, demonstrating the potential of freely shaping the cost functions through several numerical experiments.

Figures (9)

• Figure 1: IMC architecture parametrizing of all stabilizing controllers in terms of one freely chosen operator $\mathbfcal{M} \in \mathcal{L}_p$.
• Figure 2: The closed-loop system when the nominal model ${\widehat{\mathbf{F}}}(\mathbf{x},\mathbf{u})$ used in the IMC controller and the real plant $\mathbf{F}(\mathbf{x},\mathbf{u})={\widehat{\mathbf{F}}}(\mathbf{x},\mathbf{u}) + {\mathbf{\Delta}}(\mathbf{x},\mathbf{u})$ differ by the perturbation ${\mathbf{\Delta}}\in\mathcal{L}_p$. Compared to Figure \ref{['fig:IMCscheme']} the blocks have been rearranged to highlight the subsystems used in the small-gain argument adopted in the proof of Theorem \ref{['th:result_robust']}.
• Figure 3: Example of networked dynamics \ref{['eq:sub-operator']} and decentralized IMC controller for agent $i=1$.
• Figure 4: Mountains --- Closed-loop trajectories before training (left) and after training (middle and right) over 100 randomly sampled initial conditions marked with $\circ$. Snapshots taken at time-instants $\tau$. Colored (gray) lines show the trajectories in $[0,\tau_i]$ ($[\tau_i,\infty)$). Colored balls (and their radius) represent the agents (and their size for collision avoidance).
• Figure 5: Mountains --- Closed-loop trajectories after 25%, 50% and 75% of the total training whose closed-loop trajectory is shown in Figure \ref{['fig:corridor']}. Even if the performance can be further optimized, stability is always guaranteed.

Theorems & Definitions (12)

• Definition 1
• Theorem 1
• proof
• Corollary 1
• proof
• Theorem 2: Nonlinear SLS parametrization ho2020system
• Theorem 3
• proof
• Remark 1: Robust stability of nonlinear SLS
• Proposition 1