A Stochastic-Optimization-Based Adaptive-Sampling Scheme for Data-Driven Stability Analysis of Switched Linear Systems

Abstract: We introduce a novel approach based on stochastic optimization to find the optimal sampling distribution for the data-driven stability analysis of switched linear systems. Our goal is to address limitations of existing approaches, in particular, the fact that these methods suffer from illconditioning of the optimal Lyapunov function, which was shown in recent work to be a direct consequence of the way the data is collected by sampling uniformly the state space. In this work, we formalize the notion of optimal sampling distribution, using the perspective of stochastic optimization. This allows us to leverage tools from stochastic optimization to estimate the optimal sampling distribution, and then use it to collect samples for the data-driven stability analysis of the system. We show in numerical experiments (on challenging systems of dimension up to five) that the overall procedure is highly favorable in terms of data usage compared to existing methods using fixed sampling distributions. Finally, we introduce a heuristic that combines data points from previous samples, and show empirically that this allows an additional substantial reduction in the number of samples required to achieve the same stability guarantees.