Internal Model Control design for systems learned by Control Affine Neural Nonlinear Autoregressive Exogenous Models

TL;DR

The study develops CA-NNARX models that replicate the control-affine structure of unknown plants and injects $\delta$ISS stability into training. It then derives an explicit model inverse to enable a streamlined Internal Model Control (IMC) design, with stability guarantees that extend to IOS in closed loop. The approach is validated on the Quadruple Tank benchmark, showing that CA-NNARX offers superior modeling accuracy over standard NNARX at similar sizes, and that IMC based on a $\delta$ISS CA-NNARX model achieves tracking performance comparable to nonlinear MPC but with orders of magnitude lower online computation. The results highlight the practical viability of stability-regularized learning for data-driven control, enabling robust, efficient model-based control in the presence of noise and plant-model mismatch. Future work will focus on tightening the $\delta$ISS conditions, robust bounds on modeling error, and scalability to larger systems.

Abstract

This paper explores the use of Control Affine Neural Nonlinear AutoRegressive eXogenous (CA-NNARX) models for nonlinear system identification and model-based control design. The idea behind this architecture is to match the known control-affine structure of the system to achieve improved performance. Coherently with recent literature of neural networks for data-driven control, we first analyze the stability properties of CA-NNARX models, devising sufficient conditions for their incremental Input-to-State Stability ($δ$ISS) that can be enforced at the model training stage. The model's stability property is then leveraged to design a stable Internal Model Control (IMC) architecture. The proposed control scheme is tested on a real Quadruple Tank benchmark system to address the output reference tracking problem. The results achieved show that (i) the modeling accuracy of CA-NNARX is superior to the one of a standard NNARX model for given weight size and training epochs, (ii) the proposed IMC law provides performance comparable to the ones of a standard Model Predictive Controller (MPC) at a significantly lower computational burden, and (iii) the $δ$ISS of the model is beneficial to the closed-loop performance.

Figures (10)

• Figure 1: Structure of the proposed CA-NNARX model.
• Figure 2: General scheme of Internal Model Control with reference tracking.
• Figure 3: Internal model control architecture with model reference and low-pass filter on the feedback.
• Figure 4: Quadruple Tank lab apparatus used for the closed-loop validation of the proposed approach.
• Figure 6: Open-loop prediction (red dashed line) vs ground truth (black solid line) on the test dataset of the four levels.

Theorems & Definitions (16)

• Definition 1: $\mathcal{K}_\infty$ function
• Definition 2: $\mathcal{KL}$ function
• Definition 3: $\delta$ISS bayer2013discrete
• Definition 4: IOS
• Proposition 1: $\delta$ISS implies IOS, bonassi2023reconciling
• Theorem 1
• proof
• Remark 1
• Remark 2
• Remark 3