Comparison of Path-Complete Stability Criteria via Quantifier Elimination

Comparison of Path-Complete Stability Criteria via Quantifier Eliminatio

On Path-Complete Lyapunov Functions : comparison between a graph and its expansion

We study the stability of switching dynamical systems with the following dynamics

Template-Dependent Lifts for Path-Complete Stability Criteria and Application to Positive Switching Systems

In the framework of discrete-time switching systems, we analyze and compare various stability certificates relying on graph constructions. To this aim, we define several abstract expansions of graphs (so-called lifts), which depend on the chosen family of candidate Lyapunov functions (the template). We show that the validity of a given lift is linked with the analytical properties of the template. This allows us to generate new lifts, and as a byproduct, to obtain comparison criteria that go beyond the concept of simulation recently introduced in the literature. We apply our constructions to the case of copositive linear norms for positive switching systems, leading to novel stability criteria that outperform the state of the art. We provide further results relying on convex duality and we demonstrate via numerical examples how the comparison among different stability criteria is affected by the properties of the copositive norms template

Characterization of the ordering of path-complete stability certificates with addition-closed templates

As part of the development of Lyapunov techniques for cyberphysical systems, we study and compare graph-based stability certificates with respect to their conservatism. Previous work have highlighted the dependence of this ordering with respect to the properties of the chosen template of candidate Lyapunov functions. We extend here previous results from the literature to the case of templates closed under addition, as for instance the set of quadratic functions. In this context, we provide a characterization of the ordering, using an approach based on abstract operations on graphs, called lifts, which encode in a combinatorial way the algebraic properties of the chosen template. We finally provide a numerical method to algorithmically check the ordering relation

Comparison of Path-Complete Lyapunov Functions via Template-Dependent Lifts

This paper investigates, in the context of discrete-time switching systems, the problem of comparison for path-complete stability certificates. We introduce and study abstract operations on path-complete graphs, called lifts, which allow us to recover previous results in a general framework. Moreover, this approach highlights the existing relations between the analytical properties of the chosen set of candidate Lyapunov functions (the template) and the admissibility of certain lifts. This provides a new methodology for the characterization of the order relation of path-complete Lyapunov functions criteria, when a particular template is chosen. We apply our results to specific templates, notably the sets of primal and dual copositive norms, providing new stability certificates for positive switching systems. These tools are finally illustrated with the aim of numerical examples.Comment: 23 pages, 7 figures, submissio

Necessary and Sufficient Conditions for Template-Dependent Ordering of Path-Complete Lyapunov Methods

In the context of discrete-time switched systems, we study the comparison of stability certificates based on path-complete Lyapunov methods. A characterization of this general ordering has been provided recently, but we show here that this characterization is too strong when a particular template is considered, as it is the case in practice. In the present work we provide a characterization for templates that are closed under pointwise minimum/maximum, which covers several templates that are often used in practice. We use an approach based on abstract operations on graphs, called lifts, to highlight the dependence of the ordering with respect to the analytical properties of the template. We finally provide more preliminary results on another family of templates: those that are closed under addition, as for instance the set of quadratic functions

Formal Synthesis of Lyapunov Stability Certificates for Linear Switched Systems using ReLU Neural Networks

This paper presents a neural network-based algorithm with soundness guarantees to study the stability of discrete-time linear switched systems. This algorithm is cast as a counterexample guided inductive synthesis (CEGIS) architecture: an iterative structure which alternates between the learner, which provides a candidate Lyapunov function, and the verifier which checks its validity over the whole domain. We choose a ReLU neural network as learner to take advantage of its expressivity power and flexibility, and a satisfiability module theories (SMT) solver as verifier. In addition, we introduce a post processing step to leverage a valid Lyapunov function from the neural network in case of failure of the CEGIS loop. We provide several examples to illustrate the entire algorithm

Stability Analysis of Switched Linear Systems with Neural Lyapunov Functions

Neural-based, data-driven analysis and control of dynamical systems have been recently investigated and have shown great promise, e.g. for safety verification or stability analysis. Indeed, not only do neural networks allow for an entirely model-free, data-driven approach, but also for handling arbitrary complex functions via their power of representation (as opposed to, e.g. algebraic optimization techniques that are restricted to polynomial functions). Whilst classical Lyapunov techniques allow to provide a formal and robust guarantee of stability of a switched dynamical system, very little is yet known about correctness guarantees for Neural Lyapunov functions, nor about their performance (amount of data needed for a certain accuracy). We thus formally introduce neural Lyapunov functions for the stability analysis of switched linear systems: we benchmark them on this paradigmatic problem, which is notoriously difficult (and in general Turing-undecidable), but which admits recently-developed technologies and theoretical results. Inspired by switched systems theory, we provide theoretical guarantees on the representative power of neural networks, leveraging recent results from the ML community. We additionally experimentally display how neural Lyapunov functions compete with state-of-the-art results and techniques, while admitting a wide range of improvement, both in theory and in practice. This study intends to improve our understanding of the opportunities and current limitations of neural-based data-driven analysis and control of complex dynamical systems