Invariant set estimation for piecewise affine dynamical systems using piecewise affine barrier function

This paper introduces an algorithm for estimating the invariant set of closed-loop controlled dynamical systems identified using single-hidden layer Rectified linear units (ReLU) neural networks or piecewise affine ($\mathrm{PWA}$) functions, particularly addressing the challenge of providing safety guarantees for single-hidden layer ReLU networks commonly used in safety–critical applications. The invariant set of $\mathrm{PWA}$ dynamical system is estimated using single-hidden layer ReLU networks or its equivalent $\mathrm{PWA}$ function. This method entails formulating the barrier function as a $\mathrm{PWA}$ function and converting the search process into a linear optimization problem using vertices. We incorporate a domain refinement strategy to increase flexibility in case the optimization does not find a valid barrier function. Moreover, the objective of the optimization is to find a less conservative invariant set based on the current partition. Our experimental results demonstrate the effectiveness and efficiency of our approach, demonstrating its potential for ensuring the safety of $\mathrm{PWA}$ dynamical systems.