Ensuring Both Positivity and Stability Using Sector-Bounded Nonlinearity for Systems with Neural Network Controllers
This work addresses stability challenges in dynamic systems with neural network controllers, which is incremental as it builds on existing theories like positive Lur'e systems and the positive Aizerman conjecture.
The paper tackles the problem of ensuring stability in positive feedback systems controlled by fully connected feedforward neural networks, by establishing sector bounds for these networks and proving global exponential stability for linear systems under such control.
This paper introduces a novel method for the stability analysis of positive feedback systems with a class of fully connected feedforward neural networks (FFNN) controllers. By establishing sector bounds for fully connected FFNNs without biases, we present a stability theorem that demonstrates the global exponential stability of linear systems under fully connected FFNN control. Utilizing principles from positive Lur'e systems and the positive Aizerman conjecture, our approach effectively addresses the challenge of ensuring stability in highly nonlinear systems. The crux of our method lies in maintaining sector bounds that preserve the positivity and Hurwitz property of the overall Lur'e system. We showcase the practical applicability of our methodology through its implementation in a linear system managed by a FFNN trained on output feedback controller data, highlighting its potential for enhancing stability in dynamic systems.