arXiv subjects

Adrien Banse

Publications and source records attributed to Adrien Banse.

14 recordsLinked to original sources

arXiv2026

Robust Optimal Control of Arbitrarily Switched Systems: A Path-Complete Framework

This paper addresses the robust control of switched systems under arbitrary switching with performance guarantees. We propose a framework that jointly synthesizes a feedback policy and a certified upper bound on its corresponding infinite-horizon closed-loop value function. The proposed upper bound not only certifies the performance of the synthesized policy, but can also be optimized during controller synthesis. More precisely, our approach associates functions with the nodes of a path-complete graph and enforces graph-based Bellman inequalities along its edges. Exploiting a newly introduced notion of reachability graph, these functions are combined into both a feedback policy and a certified upper bound on its corresponding closed-loop value function, expressed as a pointwise min-max combination of the graph-indexed functions. For linear switched systems with quadratic stage costs, the proposed framework admits tractable computational formulations based on semidefinite programming and alternating optimization. Numerical experiments, including a building temperature regulation benchmark, demonstrate the practical usefulness of the proposed approach both for direct feedback control using the synthesized policy and for model predictive control using the certified upper bound as a terminal cost.

math.OC↗

arXiv2026

A Path-Complete Approach for Optimal Control of Switched Systems

We study the problem of estimating the value function of discrete-time switched systems under arbitrary switching. Unlike the switched LQR problem, where both inputs and mode sequences are optimized, we consider the case where switching is exogenous. For such systems, the number of possible mode sequences grows exponentially with time, making the exact computation of the value function intractable. This motivates the development of tractable bounds that approximate it. We propose a novel framework, based on path-complete graphs, for constructing computable upper bounds on the value function. In this framework, multiple quadratic functions are combined through a directed graph that encodes dynamic programming inequalities, yielding convex and sound formulations. For example, for switched linear systems with quadratic cost, we derive tractable LMI-based formulations and provide computational complexity bounds. We further establish approximation guarantees for the upper bounds and show asymptotic non-conservativeness using concepts from graph theory. Finally, we extend the approach to controller synthesis for systems with affine control inputs and demonstrate its effectiveness on numerical examples.

math.OC↗

arXiv2025

Comparing Labeled Markov Chains: A Cantor-Kantorovich Approach

Labeled Markov Chains (or LMCs for short) are useful mathematical objects to model complex probabilistic languages. A central challenge is to compare two LMCs, for example to assess the accuracy of an abstraction or to quantify the effect of model perturbations. In this work, we study the recently introduced Cantor-Kantorovich (or CK) distance. In particular we show that the latter can be framed as a discounted sum of finite-horizon Total Variation distances, making it an instance of discounted linear distance, but arising from the natural Cantor topology. Building on the latter observation, we analyze the properties of the CK distance along three dimensions: computational complexity, continuity properties and approximation. More precisely, we show that the exact computation of the CK distance is #P-hard. We also provide an upper bound on the CK distance as a function of the approximation relation between the two LMCs, and show that a bounded CK distance implies a bounded error between probabilities of finite-horizon traces. Finally, we provide a computable approximation scheme, and show that the latter is also #P-hard. Altogether, our results provide a rigorous theoretical foundation for the CK distance and clarify its relationship with existing distances.

cs.LO↗

arXiv2025

Agile Temporal Discretization for Symbolic Optimal Control

As control systems grow in complexity, abstraction-based methods have become essential for designing controllers with formal guarantees. However, a key limitation of these methods is their reliance on discrete-time models, typically obtained by discretizing continuous-time systems with a fixed timestep. This discretization leads to two major problems: when the timestep is small, the abstraction includes numerous stuttering and spurious trajectories, making controller synthesis suboptimal or even infeasible; conversely, a large time step may also render control design infeasible due to a lack of flexibility. In this work, drawing inspiration from Reinforcement Learning concepts, we introduce temporal abstractions, which allow for a flexible timestep. We provide a method for constructing such abstractions and formally establish their correctness in controller design. Furthermore we show how to apply these to optimal control under reachability specifications. Finally we showcase our methods on two numerical examples, highlighting that our approach leads to controllers that achieve a lower worst-case control cost.

eess.SY↗

arXiv2025

Memory-dependent abstractions of stochastic systems through the lens of transfer operators

With the increasing ubiquity of safety-critical autonomous systems operating in uncertain environments, there is a need for mathematical methods for formal verification of stochastic models. Towards formally verifying properties of stochastic systems, methods based on discrete, finite Markov approximations -- abstractions -- thereof have surged in recent years. These are found in contexts where: either a) one only has partial, discrete observations of the underlying continuous stochastic process, or b) the original system is too complex to analyze, so one partitions the continuous state-space of the original system to construct a handleable, finite-state model thereof. In both cases, the abstraction is an approximation of the discrete stochastic process that arises precisely from the discretization of the underlying continuous process. The fact that the abstraction is Markov and the discrete process is not (even though the original one is) leads to approximation errors. Towards accounting for non-Markovianity, we introduce memory-dependent abstractions for stochastic systems, capturing dynamics with memory effects. Our contribution is twofold. First, we provide a formalism for memory-dependent abstractions based on transfer operators. Second, we quantify the approximation error by upper bounding the total variation distance between the true continuous state distribution and its discrete approximation.

eess.SY↗

arXiv2024

A Cantor-Kantorovich Metric Between Markov Decision Processes with Application to Transfer Learning

We extend the notion of Cantor-Kantorovich distance between Markov chains introduced by (Banse et al., 2023) in the context of Markov Decision Processes (MDPs). The proposed metric is well-defined and can be efficiently approximated given a finite horizon. Then, we provide numerical evidences that the latter metric can lead to interesting applications in the field of reinforcement learning. In particular, we show that it could be used for forecasting the performance of transfer learning algorithms.

cs.LG↗

arXiv2024

Data-driven memory-dependent abstractions of dynamical systems via a Cantor-Kantorovich metric

Abstractions of dynamical systems enable their verification and the design of feedback controllers using simpler, usually discrete, models. In this paper, we propose a data-driven abstraction mechanism based on a novel metric between Markov models. Our approach is based purely on observing output labels of the underlying dynamics, thus opening the road for a fully data-driven approach to construct abstractions. Another feature of the proposed approach is the use of memory to better represent the dynamics in a given region of the state space. We show through numerical examples the usefulness of the proposed methodology.

eess.SY↗

arXiv2024

Dionysos.jl: a Modular Platform for Smart Symbolic Control

We introduce Dionysos.jl, a modular package for solving optimal control problems for complex dynamical systems using state-of-the-art and experimental techniques from symbolic control, optimization, and learning. More often than not with Cyber-Physical systems, the only sensible way of developing a controller is by discretizing the different variables, thus transforming the control task into a purely combinatorial problem on a finite-state mathematical object, called an abstraction of this system. Although this approach offers a safety-critical framework, the available techniques suffer important scalability issues. In order to render these techniques practical, it is necessary to construct smarter abstractions that differ from classical techniques by partitioning the state-space in a non trivial way.

eess.SY↗

arXiv2024

Federated Learning with Differential Privacy

Federated learning (FL), as a type of distributed machine learning, is capable of significantly preserving client's private data from being shared among different parties. Nevertheless, private information can still be divulged by analyzing uploaded parameter weights from clients. In this report, we showcase our empirical benchmark of the effect of the number of clients and the addition of differential privacy (DP) mechanisms on the performance of the model on different types of data. Our results show that non-i.i.d and small datasets have the highest decrease in performance in a distributed and differentially private setting.

cs.LG↗

arXiv2023

Data-driven abstractions via adaptive refinements and a Kantorovich metric [extended version]

We introduce an adaptive refinement procedure for smart, and scalable abstraction of dynamical systems. Our technique relies on partitioning the state space depending on the observation of future outputs. However, this knowledge is dynamically constructed in an adaptive, asymmetric way. In order to learn the optimal structure, we define a Kantorovich-inspired metric between Markov chains, and we use it as a loss function. Our technique is prone to data-driven frameworks, but not restricted to. We also study properties of the above mentioned metric between Markov chains, which we believe could be of application for wider purpose. We propose an algorithm to approximate it, and we show that our method yields a much better computational complexity than using classical linear programming techniques.

cs.LG↗

arXiv2023

Learning stability guarantees for constrained switching linear systems from noisy observations

We present a data-driven framework based on Lyapunov theory to provide stability guarantees for a family of hybrid systems. In particular, we are interested in the asymptotic stability of switching linear systems whose switching sequence is constrained by labeled graphs, namely constrained switching linear systems. In order to do so, we provide chance-constrained bounds on stability guarantees, that can be obtained from a finite number of noisy observations. We first present a method providing stability guarantees from sampled trajectories in the hybrid state-space of the system. We then study the harder situation where one only observes the continuous part of the hybrid states. We show that in this case, one may still obtain formal chance-constrained stability guarantees. For this latter result we provide a new upper bound of general interest, also for model-based stability analysis

eess.SY↗

arXiv2022

Data-driven memory-dependent abstractions of dynamical systems

We propose a sample-based, sequential method to abstract a (potentially black-box) dynamical system with a sequence of memory-dependent Markov chains of increasing size. We show that this approximation allows to alleviating a correlation bias that has been observed in sample-based abstractions. We further propose a methodology to detect on the fly the memory length resulting in an abstraction with sufficient accuracy. We prove that under reasonable assumptions, the method converges to a sound abstraction in some precise sense, and we showcase it on two case studies.

eess.SY↗

arXiv2022

Learning stability guarantees for data-driven constrained switching linear systems

We consider stability analysis of constrained switching linear systems in which the dynamics is unknown and whose switching signal is constrained by an automaton. We propose a data-driven Lyapunov framework for providing probabilistic stability guarantees based on data harvested from observations of the system. By generalizing previous results on arbitrary switching linear systems, we show that, by sampling a finite number of observations, we are able to construct an approximate Lyapunov function for the underlying system. Moreover, we show that the entropy of the language accepted by the automaton allows to bound the number of samples needed in order to reach some pre-specified accuracy.

eess.SY↗

arXiv2022

Black-box stability analysis of hybrid systems with sample-based multiple Lyapunov functions

We present a framework based on multiple Lyapunov functions to find probabilistic data-driven guarantees on the stability of unknown constrained switching linear systems (CSLS), which are switching linear systems whose switching signal is constrained by an automaton. The stability of a CSLS is characterized by its constrained joint spectral radius (CJSR). Inspired by the scenario approach and previous work on unconstrained switching systems, we characterize the number of observations needed to find sufficient conditions on the (in-)stability of a CSLS using the notion of CJSR. More precisely, our contribution is the following: we derive a probabilistic upper bound on the CJSR of an unknown CSLS from a finite number of observations. We also derive a deterministic lower bound on the CJSR. From this we obtain a probabilistic method to characterize the stability of an unknown CSLS.

eess.SY↗