Robust Stability of Neural Network Control Systems with Interval Matrix Uncertainties

TL;DR

This work tackles robust stability certification for neural-network control systems with interval matrix uncertainties by combining quadratic-constraint based neural abstractions with linear matrix inequality certificates. It introduces a vertex-based certificate and three relaxed LMIs that reduce computational burden while preserving feasibility, enabling stable operation and inner regions of attraction around the origin. The paper connects the relaxations to classic results in robust stability for linear interval systems and demonstrates the approach on inverted pendulum and mass-spring-damper examples, highlighting both accuracy and scalability. The methodology provides a formal, scalable route to stability guarantees for NNCS in the presence of model uncertainties, with potential extensions to broader neural architectures.

Abstract

Neural networks have become increasingly popular in controller design due to their versatility and efficiency. However, their integration into feedback systems can pose stability challenges, particularly in the presence of uncertainties. This work addresses the problem of certifying robust stability in neural network control systems with interval matrix uncertainties. Leveraging classical robust stability techniques and the quadratic constraint-based method to characterize the input-output behavior of neural networks, we derive novel robust stability certificates formulated as linear matrix inequalities. To reduce computational complexity, we introduce three relaxed sufficient conditions and establish their equivalence in terms of feasibility. Additionally, we explore their connections to existing robust stability results. The effectiveness of the proposed approach is demonstrated through inverted pendulum and mass-spring-damper examples.

Figures (6)

• Figure 1: Feedback system that contains a plant with interval matrix uncertainties and NN controller $\pi$.
• Figure 2: (LMI-I), (LMI-II), and (LMI-III) all serve as certificates for the robust stability of the NNCS in \ref{['close-sys']} with interval uncertainties in both the state and input matrices. These LMIs extend the corresponding robust stability results for linear systems with interval state matrix uncertainties in alamo2008newben2002tractablemao2003quadratic. The feasibility equivalence of the three proposed LMIs also implies the previously unestablished feasibility equivalence of the corresponding LMIs in alamo2008newben2002tractablemao2003quadratic.
• Figure 3: Inner-approximations of the robust ROA for the inverted pendulum model with uncertainty level $\delta = 0.01$. The three inner-approximations corresponding to the three LMIs are congruent. Trajectories with randomly selected initial states on the boundary are plotted in green.
• Figure 4: Inner-approximations of the robust ROA for the inverted pendulum model with varying uncertainty levels.
• Figure 5: Mass-spring-damper system with $n_c$ carts.