arXiv subjects

Raphaël M. Jungers

Publications and source records attributed to Raphaël M. Jungers.

At least 37 records · Page 2Linked to original sources

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.

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arXiv2024

Smart abstraction based on iterative cover and non-uniform cells

We propose a multi-scale approach for computing abstractions of dynamical systems, that incorporates both local and global optimal control to construct a goal-specific abstraction. For a local optimal control problem, we not only design the controller ensuring the transition between every two subsets (cells) of the state space but also incorporate the volume and shape of these cells into the optimization process. This integrated approach enables the design of non-uniform cells, effectively reducing the complexity of the abstraction. These local optimal controllers are then combined into a digraph, which is globally optimized to obtain the entire trajectory. The global optimizer attempts to lazily build the abstraction along the optimal trajectory, which is less affected by an increase in the number of dimensions. Since the optimal trajectory is generally unknown in practice, we propose a methodology based on the RRT* algorithm to determine it incrementally. Finally, we provide a tractable implementation of this algorithm for the optimal control of L-smooth nonlinear dynamical systems.

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arXiv2024

Memoryless concretization relation

We introduce the concept of memoryless concretization relation (MCR) to describe abstraction within the context of controller synthesis. This relation is a specific instance of alternating simulation relation (ASR), where it is possible to simplify the controller architecture. In the case of ASR, the concretized controller needs to simulate the concurrent evolution of two systems, the original and abstract systems, while for MCR, the designed controllers only need knowledge of the current concrete state. We demonstrate that the distinction between ASR and MCR becomes significant only when a non-deterministic quantizer is involved, such as in cases where the state space discretization consists of overlapping cells. We also show that any abstraction of a system that alternatingly simulates a system can be completed to satisfy MCR at the expense of increasing the non-determinism in the abstraction. We clarify the difference between the MCR and the feedback refinement relation (FRR), showing in particular that the former allows for non-constant controllers within cells. This provides greater flexibility in constructing a practical abstraction, for instance, by reducing non-determinism in the abstraction. Finally, we prove that this relation is not only sufficient, but also necessary, for ensuring the above properties.

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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.

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arXiv2023

Graph-Based Conditions for Feedback Stabilization of Switched and LPV Systems

This paper presents novel stabilizability conditions for switched linear systems with arbitrary and uncontrollable underlying switching signals. We distinguish and study two particular settings: i) the \emph{robust} case, in which the active mode is completely unknown and unobservable, and ii) the \emph{mode-dependent} case, in which the controller depends on the current active switching mode. The technical developments are based on graph-theory tools, relying in particular on the path-complete Lyapunov functions framework. The main idea is to use directed and labeled graphs to encode Lyapunov inequalities to design robust and mode-dependent piecewise linear state-feedback controllers. This results in novel and flexible conditions, with the particular feature of being in the form of linear matrix inequalities (LMIs). Our technique thus provides a first controller-design strategy allowing piecewise linear feedback maps and piecewise quadratic (control) Lyapunov functions by means of semidefinite programming. Numerical examples illustrate the application of the proposed techniques, the relations between the graph order, the robustness, and the performance of the closed loop.

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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

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arXiv2023

Learning stability of partially observed switched linear systems

This paper deals with learning stability of partially observed switched linear systems under arbitrary switching. Such systems are widely used to describe cyber-physical systems which arise by combining physical systems with digital components. In many real-world applications, the internal states cannot be observed directly. It is thus more realistic to conduct system analysis using the outputs of the system. Stability is one of the most frequent requirement for safety and robustness of cyber-physical systems. Existing methods for analyzing stability of switched linear systems often require the knowledge of the parameters and/or all the states of the underlying system. In this paper, we propose an algorithm for deciding stability of switched linear systems under arbitrary switching based purely on observed output data. The proposed algorithm essentially relies on an output-based Lyapunov stability framework and returns an estimate of the joint spectral radius (JSR). We also prove a probably approximately correct error bound on the quality of the estimate of the JSR from the perspective of statistical learning theory.

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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.

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arXiv2022

State-feedback Abstractions for Optimal Control of Piecewise-affine Systems

In this manuscript, we investigate symbolic abstractions that capture the behavior of piecewise-affine systems under input constraints and bounded external noise. This is accomplished by considering local affine feedback controllers that are jointly designed with the symbolic model, which ensures that an alternating simulation relation between the system and the abstraction holds. The resulting symbolic system is called a state-feedback abstraction and we show that it can be deterministic even when the original piecewise-affine system is unstable and non-deterministic. One benefit of this approach is the fact that the input space need not be discretized and the symbolic-input space is reduced to a finite set of controllers. When ellipsoidal cells and affine controllers are considered, we present necessary and sufficient conditions written as a semi-definite program for the existence of a transition and a robust upper bound on the transition cost. Two examples illustrate particular aspects of the theory and its applicability.

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arXiv2022

An Efficient Method to Verify the Inclusion of Ellipsoids

We present a novel method for deciding whether a given n-dimensional ellipsoid contains another one (possibly with a different center). This method consists in constructing a particular concave function and deciding whether it has any value greater than -1 in a compact interval that is a subset of [0,1]. This can be done efficiently by a bisection algorithm method that is guaranteed to stop in a finite number of iterations. The initialization of the method requires O(n^3) floating-point operations and evaluating this function and its derivatives requires O(n). This can be also generalized to compute the smallest level set of a convex quadratic function containing a finite number of n-ellipsoids. In our benchmark with randomly generated ellipsoids, when compared with a classic method based on semidefinite programming, our algorithm performs 27 times faster for ellipsoids of dimension n=3 and 2294 times faster for dimension n=100. We illustrate the usefulness of this method with a problem in the control theory field.

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arXiv2022

Interpretability of Path-Complete Techniques and Memory-based Lyapunov functions

We study path-complete Lyapunov functions, which are stability criteria for switched systems, described by a combinatorial component (namely, an automaton), and a functional component (a set of candidate Lyapunov functions, called the template). We introduce a class of criteria based on what we call memory-based Lyapunov functions, which generalize several techniques in the literature. Our main result is an equivalence result: any path-complete Lyapunov function is equivalent to a memory-based Lyapunov function, however defined on another template. We show the usefulness of our result in terms of numerical efficiency via an academic example.

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arXiv2022

Data-driven invariant subspace identification for black-box switched linear systems

We present an algorithmic framework for the identification of candidate invariant subspaces for switched linear systems. Namely, the framework allows to compute an orthonormal basis in which the matrices of the system are close to block-triangular matrices, based on a finite set of observed one-step trajectories and with a priori confidence level. The link between the existence of an invariant subspace and a common block-triangularization of the system matrices is well known. Under some assumptions on the system, one can also infer the existence of an invariant subspace when the matrices are close to be block-triangular. Our approach relies on quadratic Lyapunov analysis and recent tools in scenario optimization. We present two applications of our results for problems of consensus and opinion dynamics; the first one allows to identify the disconnected components in a switching hidden network, while the second one identifies the stationary opinion vector of a switching gossip process with antagonistic interactions.

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arXiv2022

Almost sure Stability of Stochastic Switched Systems: Graph lifts-based Approach

In this paper, we develop tools to establish almost sure stability of stochastic switched systems whose switching signal is constrained by an automaton. After having provided the necessary generalizations of existing results in the setting of stochastic graphs, we provide a characterization of almost sure stability in terms of multiple Lyapunov functions. We introduce the concept of lifts, providing formal expansions of stochastic graphs, together with the guarantee of conserving the underlying probability framework. We show how these techniques, firstly introduced in the deterministic setting, provide hierarchical methods in order to compute tight upper bounds for the almost sure decay rate. The theoretical developments are finally illustrated via a numerical example.

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arXiv2022

Computation of invariant sets via immersion for discrete-time nonlinear systems

In this paper, we propose an approach for computing invariant sets of discrete-time nonlinear systems by lifting the nonlinear dynamics into a higher dimensional linear model. In particular, we focus on the \emph{maximal admissible invariant set} contained in some given constraint set. For special types of nonlinear systems, which can be exactly immersed into higher dimensional linear systems with state transformations, invariant sets of the original nonlinear system can be characterized using the higher dimensional linear representation. For general nonlinear systems without the immersibility property, \emph{approximate immersions} are defined in a local region within some tolerance and linear approximations are computed by leveraging the fixed-point iteration technique for invariant sets. Given the bound on the mismatch between the linear approximation and the original system, we provide an invariant inner approximation of the \emph{maximal admissible invariant set} by a tightening procedure.

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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.

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arXiv2022

Data-driven control of switched linear systems with probabilistic stability guarantees

This paper tackles state feedback control of switched linear systems under arbitrary switching. We propose a data-driven control framework that allows to compute a stabilizing state feedback using only a finite set of observations of trajectories with quadratic and sum of squares (SOS) Lyapunov functions. We do not require any knowledge on the dynamics or the switching signal, and as a consequence, we aim at solving \emph{uniform} stabilization problems in which the feedback is stabilizing for all possible switching sequences. In order to generalize the solution obtained from trajectories to the actual system, probabilistic guarantees on the obtained quadratic or SOS Lyapunov function are derived in the spirit of scenario optimization. For the quadratic Lyapunov technique, the generalization relies on a geometric analysis argument, while, for the SOS Lyapunov technique, we follow a sensitivity analysis argument. In order to deal with high-dimensional systems, we also develop parallelized schemes for both techniques. We show that, with some modifications, the data-driven quadratic Lyapunov technique can be extended to LQR control design. Finally, the proposed data-driven control framework is demonstrated on several numerical examples.

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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.

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arXiv2022

Probabilistic guarantees on the objective value for the scenario approach via sensitivity analysis

This paper is concerned with objective value performance of the scenario approach for robust convex optimization. A novel method is proposed to derive probabilistic bounds for the objective value from scenario programs with a finite number of samples. This method relies on a max-min reformulation and the concept of complexity of robust optimization problems. With additional continuity and regularity conditions, via sensitivity analysis, we also provide explicit bounds which outperform an existing result in the literature. To illustrate the improvements of our results, we also provide a numerical example.

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