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- Abstract
- Nowadays, we encounter on a daily basis systems of tremendous complexity. For instance, we rely ever more on cyber-physical systems, including self-driving cars, drones or smart assistive technologies. Deeply understanding the properties of these applications is essential to optimize their potential. However, frequently, due to their inherent complexity, there is no available closed-form model of these systems. On the other hand, as the development of new technologies grows massively, so does the amount of information-rich data we have access to. Hence, it is crucial to develop new methods meeting the challenge of analyzing complex systems without relying on model-based techniques, but rather on the unprecedented volume of data available. In this thesis, we propose new data-driven approaches providing probabilistic stability guarantees for black-box linear switched systems, a first class of complex systems whose model is unknown. We construct approximate Lyapunov functions decreasing along a finite number of sampled trajectories by solving a quasi-convex optimization problem, and deduce the stability of the underlying system with a user-defined confidence level. We also explicitly characterize the number of observations required to attain this confidence level. Recently, techniques relying on common quadratic Lyapunov functions (CQLFs) to provide probabilistic stability guarantees have been proposed. However, they require non trivial assumptions on the system analyzed, and suffer from conservativeness since many stable switched linear systems do in fact not admit a CQLF. Hence, in a first time, we overcome these issues and provide a broader class of probabilistic stability guarantees, by considering Sum of Squares (SOS) Lyapunov functions. Our contribution is twofold: first, we propose a novel approach for CQLFs, allowing to improve the existing guarantees and get rid of all assumptions. Then, we leverage our approach to the SOS setting, providing further improvement for the technique. In a second time, we propose an approach based on adaptive sampling to reduce the sample and computational complexity required to obtain stability guarantees holding with a given confidence level. We learn recursively, from small batches of trajectories, an optimal sampling strategy. This allows the data-driven approach to be tractable for high-dimensional systems, which is the case of the majority of real-world complex systems. We illustrate the performance of our approaches on several numerical experiments.
Citations
Rubbens, A. (2021). Data-driven stability analysis of black-box switched linear systems. https://dial-mem.test.bib.ucl.ac.be/handle/2078.2/23667