Physics-guided neural networks for feedforward control with input-to-state stability guarantees

TL;DR

The paper tackles the challenge of achieving high-precision feedforward control that remains safe outside training data by proposing physics-guided neural networks (PGNNs) that merge a physics-based layer with a neural layer. It introduces a regularized identification approach to maintain physics interpretability, and derives ISS guarantees for PGNN feedforward controllers using refined Lipschitz bounds. The approach includes optimized parameter initialization and extrapolation-enhancing strategies to improve robustness in unseen operating conditions. Experimental validation on a real coreless linear motor and a nonminimum-phase rotating-translating mass demonstrates that PGNNs can double the accuracy of physics-based feedforward and outperform pure neural or PINN controllers, with better extrapolation and stability properties.

Abstract

The increasing demand on precision and throughput within high-precision mechatronics industries requires a new generation of feedforward controllers with higher accuracy than existing, physics-based feedforward controllers. As neural networks are universal approximators, they can in principle yield feedforward controllers with a higher accuracy, but suffer from bad extrapolation outside the training data set, which makes them unsafe for implementation in industry. Motivated by this, we develop a novel physics-guided neural network (PGNN) architecture that structurally merges a physics-based layer and a black-box neural layer in a single model. The parameters of the two layers are simultaneously identified, while a novel regularization cost function is used to prevent competition among layers and to preserve consistency of the physics-based parameters. Moreover, in order to ensure stability of PGNN feedforward controllers, we develop sufficient conditions for analyzing or imposing (during training) input-to-state stability of PGNNs, based on novel, less conservative Lipschitz bounds for neural networks. The developed PGNN feedforward control framework is validated on a real-life, high-precision industrial linear motor used in lithography machines, where it reaches a factor 2 improvement with respect to physics-based mass-friction feedforward and it significantly outperforms alternative neural network based feedforward controllers.

Figures (15)

• Figure 1: Feedback--feedforward control architecture.
• Figure 2: Identified friction $F_{\textup{fric}}$ for different model parametrizations obtained by computing the feedforward signal for varying reference positions $r(k)$ and velocities $\dot{r}(k)$ with zero acceleration $\ddot{r}(k) = 0$.
• Figure 3: Physics--guided neural network architecture with physics and NN layers.
• Figure 4: Reference signal (top window) and feedforward signal generated by the PGNN \ref{['eq:PGNNGeneral']} trained according to identification criterion \ref{['eq:IdentificationCriterion']} with cost function \ref{['eq:CostFunctionMSE']} (middle window), and cost function \ref{['eq:CostFunctionPGNN']} with $\Lambda_{\textup{phy}} = \textup{diag} (\theta_{\textup{phy}}^*)^{-1}$ and $\Lambda_{\textup{NN}} = 0$ (bottom window).
• Figure 5: Training process of the NN \ref{['eq:NNParametrization']} and PGNN \ref{['eq:PGNNGeneral']} for $5$ independent trainings with random weight initialization and optimized initialization \ref{['eq:OptimizedInitialization']}. For comparison, $\frac{1}{N} \sum_{i=0}^{N-1} u_i^2 = 1477$$N^2$.

Theorems & Definitions (27)

• Remark 1.1
• Remark 2.1
• Remark 2.2
• Definition 2.1
• Remark 2.3
• Definition 3.1
• Definition 3.2
• Definition 3.3
• Proposition 3.1: PGNN consistency
• proof