Stability Certification of Dynamical Systems: Lyapunov Logic Learning Machine

The stability of the dynamical system is associated with the concept of Region of Attraction (ROA), whose accurate estimation opens the door to multidisciplinary approaches involving control theory and machine learning. The Lyapunov theory provides sufficient conditions for stability and it can be applied to derive the ROA. However, finding the appropriate Lyapunov functions for accurate ROA estimation often is a major issue. The inherent region may be overly tight or, in the case of a multi-dimensional dynamical system, be difficult to understand in virtue of the inherent mathematical complexity (e.g., polynomials with high degree). The use of explainable machine learning overcomes this issue, by exploiting the model intelligibility to describe the ROA in terms of states. In this perspective, explainable machine learning and Lyapunov stability theory are jointly studied to let the ROA be intelligible and to simplify the optimization procedure for constructing positively invariant estimates of the ROA. Results on the Van der Pol oscillator show how this may lead to larger ROAs than via traditional methods.