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Boosting the transient performance of reference tracking controllers with neural networks

Nicolas Kirsch, Leonardo Massai, Giancarlo Ferrari-Trecate

TL;DR

The paper addresses improving transient performance in nonlinear reference tracking while preserving $\mathcal{L}_p$ stability. It extends the Performance Boosting (PB) framework to reference tracking via an Internal Model Control (IMC) structure, introducing a neural-network operator $\boldsymbol{\mathcal{M}}$ that parameterizes all stability-preserving trackers and can be trained to optimize transient costs. It establishes robustness guarantees under model mismatch through incremental $L_p$-gain bounds and implements $\mathcal{M}$ as a composition of a contractive REN and an MLP bound by a sigmoid, enabling unconstrained optimization with finite parameters. The approach is validated in robotic simulations where the reference-tracking controller generalizes across multiple targets and improves transient behavior, highlighting practical potential for nonlinear control with learning-based components. Overall, the work provides a theoretically grounded, scalable method to fuse neural networks with stability guarantees for enhanced transient tracking in complex systems.

Abstract

Reference tracking is a key objective in many control systems, including those characterized by complex nonlinear dynamics. In these settings, traditional control approaches can effectively ensure steady-state accuracy but often struggle to explicitly optimize transient performance. Neural network controllers have gained popularity due to their adaptability to nonlinearities and disturbances; however, they often lack formal closed-loop stability and performance guarantees. To address these challenges, a recently proposed neural-network control framework known as Performance Boosting (PB) has demonstrated the ability to maintain $\mathcal{L}_p$ stability properties of nonlinear systems while optimizing generic transient costs. This paper extends the PB approach to reference tracking problems. First, we characterize the complete set of nonlinear controllers that preserve desired tracking properties for nonlinear systems equipped with base reference-tracking controllers. Then, we show how to optimize transient costs while searching within subsets of tracking controllers that incorporate expressive neural network models. Furthermore, we analyze the robustness of our method to uncertainties in the underlying system dynamics. Numerical simulations on a robotic system demonstrate the advantages of our approach over the standard PB framework.

Boosting the transient performance of reference tracking controllers with neural networks

TL;DR

The paper addresses improving transient performance in nonlinear reference tracking while preserving stability. It extends the Performance Boosting (PB) framework to reference tracking via an Internal Model Control (IMC) structure, introducing a neural-network operator that parameterizes all stability-preserving trackers and can be trained to optimize transient costs. It establishes robustness guarantees under model mismatch through incremental -gain bounds and implements as a composition of a contractive REN and an MLP bound by a sigmoid, enabling unconstrained optimization with finite parameters. The approach is validated in robotic simulations where the reference-tracking controller generalizes across multiple targets and improves transient behavior, highlighting practical potential for nonlinear control with learning-based components. Overall, the work provides a theoretically grounded, scalable method to fuse neural networks with stability guarantees for enhanced transient tracking in complex systems.

Abstract

Reference tracking is a key objective in many control systems, including those characterized by complex nonlinear dynamics. In these settings, traditional control approaches can effectively ensure steady-state accuracy but often struggle to explicitly optimize transient performance. Neural network controllers have gained popularity due to their adaptability to nonlinearities and disturbances; however, they often lack formal closed-loop stability and performance guarantees. To address these challenges, a recently proposed neural-network control framework known as Performance Boosting (PB) has demonstrated the ability to maintain stability properties of nonlinear systems while optimizing generic transient costs. This paper extends the PB approach to reference tracking problems. First, we characterize the complete set of nonlinear controllers that preserve desired tracking properties for nonlinear systems equipped with base reference-tracking controllers. Then, we show how to optimize transient costs while searching within subsets of tracking controllers that incorporate expressive neural network models. Furthermore, we analyze the robustness of our method to uncertainties in the underlying system dynamics. Numerical simulations on a robotic system demonstrate the advantages of our approach over the standard PB framework.

Paper Structure

This paper contains 11 sections, 2 theorems, 26 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Problem formulation
  3. Main results
  4. Model Mismatch
  5. Implementation of the operator $\mathcal{M}$
  6. Simulation examples
  7. Benchmark against original PB
  8. Mountain range example
  9. Conclusions
  10. Proof of Theorem 1
  11. Proof of Theorem 2

Key Result

Theorem 1

Assume that for any $\mathbf{x_{ref}} \in \mathbf{X_{ref}}$ the operator $\mathbfcal{F}$ is such that $\mathbf{e}$ defined in eq:error is an $\ell_p$ signal if $(\mathbf{w},\mathbf{u}) \in \ell_p$ and consider the evolution of eq:operator_form_state where $\mathbf{u}$ is chosen as for a causal operator $\boldsymbol{\mathcal{M}}:\ell^q\times \ell^n \rightarrow \ell^m$. Let $\mathbf{K}$ be the oper

Figures (4)

  • Figure 1: IMC architecture parametrizing all reference tracking controllers in terms of one operator $\boldsymbol{\mathcal{M}}$
  • Figure 2: Diagram of the rPB with reference governor architecture acting on the reference signal
  • Figure 3: Closed-loop trajectories for bPB and rPB controllers after training
  • Figure 4: Closed loop trajectories with and without the trained rPB controller. The full animated trajectories can be found at kirsch_gifs_rPB_2025

Theorems & Definitions (6)

  • Definition 1
  • Definition 2
  • Theorem 1
  • Remark 1
  • Definition 3
  • Theorem 2