arXiv subjects

Christoph Kawan

Publications and source records attributed to Christoph Kawan.

At least 19 recordsLinked to original sources

arXiv2025

The Small-Gain Condition for Infinite Networks Modeled on $\ell^{\infty}$-Spaces

In recent years, attempts have been made to extend nonlinear small-gain theorems for input-to-state stability (ISS) from finite networks to countably infinite networks with finite indegrees. Under specific assumptions about the interconnection gains and the ISS formulation, corresponding infinite-dimensional small-gain results have been proven. However, concerning these assumptions, the results are still too narrow to be considered a full extension of the state-of-the-art for finite networks. We take a step to closing this gap by developing a general technical framework within which the small-gain condition for both finite and infinite networks can be analyzed. This includes a thorough investigation of various monotone operators associated with a network and a specific ISS formulation. Our results extend and generalize the existing theory for finite networks and yield complete characterizations of the small-gain condition for specific ISS formulations. They also provide much deeper insight into the different characterizations of the small-gain condition, including the no-joint-increase condition $\Gamma(s) \not\geq s$.

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arXiv2023

A convex optimization approach to the Lyapunov exponents

The aim of this paper is to shed more light on some recent ideas about Lyapunov exponents and clarify the formal structures behind these ideas. In particular, we show that the vector of averaged Lyapunov exponents of a smooth measure-preserving dynamical system can be regarded as the solution to a vector-valued optimization problem on a space $\mathcal{M}$ of Riemannian metrics. Similar results were first proved by Jairo Bochi and Andr\'es Navas in the language of linear cocycles and their conjugacies. We go one step further and prove that the optimization problem is geodesically convex with respect to the $L^2$-metric on $\mathcal{M}$. Moreover, we derive some consequences of this fact.

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arXiv2022

Review on contraction analysis and computation of contraction metrics

Contraction analysis considers the distance between two adjacent trajectories. If this distance is contracting, then trajectories have the same long-term behavior. The main advantage of this analysis is that it is independent of the solutions under consideration. Using an appropriate metric, with respect to which the distance is contracting, one can show convergence to a unique equilibrium or, if attraction only occurs in certain directions, to a periodic orbit. Contraction analysis was originally considered for ordinary differential equations, but has been extended to discrete-time systems, control systems, delay equations and many other types of systems. Moreover, similar techniques can be applied for the estimation of the dimension of attractors and for the estimation of different notions of entropy (including topological entropy). This review attempts to link the references in both the mathematical and the engineering literature and, furthermore, point out the recent developments and algorithms in the computation of contraction metrics.

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arXiv2022

A Projected Subgradient Method for the Computation of Adapted Metrics for Dynamical Systems

In this paper, we extend a recently established subgradient method for the computation of Riemannian metrics that optimizes certain singular value functions associated with dynamical systems. This extension is threefold. First, we introduce a projected subgradient method which results in Riemannian metrics whose parameters are confined to a compact convex set and we can thus prove that a minimizer exists; second, we allow inexact subgradients and study the effect of the errors on the computed metrics; and third, we analyze the subgradient algorithm for three different choices of step sizes: constant, exogenous and Polyak. The new methods are illustrated by application to dimension and entropy estimation of the Hénon map.

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arXiv2022

A Lyapunov-based ISS small-gain theorem for infinite networks of nonlinear systems

In this paper, we show that an infinite network of input-to-state stable (ISS) subsystems, admitting ISS Lyapunov functions, itself admits an ISS Lyapunov function, provided that the couplings between the subsystems are sufficiently weak. The strength of the couplings is described in terms of the properties of an infinite-dimensional nonlinear positive operator, built from the interconnection gains. If this operator induces a uniformly globally asymptotically stable (UGAS) system, a Lyapunov function for the infinite network can be constructed. We analyze necessary and sufficient conditions for UGAS and relate them to small-gain conditions used in the stability analysis of finite networks.

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arXiv2021

A small-gain theory for infinite networks via infinite-dimensional gain operators

In this paper, we develop a new approach to study gain operators built from the interconnection gains of infinite networks of dynamical systems. Our focus is on the construction of paths of strict decay which are used for building Lyapunov functions for the network and thus proving various stability properties, including input-to-state stability. Our approach is based on the study of an augmented gain operator whose fixed points are precisely the points of decay for the original gain operator. We show that plenty of such fixed points exist under a uniform version of the no-joint-increase condition. Using these fixed points to construct a path of strict decay, in general, requires specific dynamical properties of associated monotone operators. For particular types of gain operators such as max-type operators and subadditive operators, these properties follow from uniform global asymptotic stability of the system induced by the original gain operator. This is consistent with former results in the literature which can readily be recovered from our theory.

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arXiv2021

Numerical over-approximation of invariance entropy via finite abstractions

For a closed-loop control system with a digital channel between the sensor and the controller, the notion of invariance entropy quantifies the smallest average rate of information above which a given compact subset of the state space can be made invariant. There exist different versions of this quantity for deterministic and uncertain systems, which are equivalent in the deterministic case. In this work, we present algorithms for the numerical computation of these two quantities. In particular, given a subset $Q$ of the state set, we first partition it. Then a controller, in the form of a lookup table that assigns a set of control values to each cell of the partition, is computed to enforce invariance of $Q$. After determinizing the controller, a weighted directed graph is constructed. For deterministic systems, the logarithm of the spectral radius of a transition matrix obtained from the graph gives an upper bound of the entropy. For uncertain systems, the maximum mean cycle weight of the graph upper bounds the entropy. With three deterministic examples, for which the exact value of the invariance entropy is known or can be estimated by other means, we demonstrate that the upper bound obtained by our algorithm is of the same order of magnitude as the actual value. Additionally, our algorithm provides a static coder-controller scheme corresponding to the obtained data-rate bound. Finally, we present the computed upper bounds of invariance entropy for an uncertain linear control system as well.

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arXiv2021

Ergodicity of Controlled Stochastic Nonlinear Systems under Information Constraints: Refined Bounds via Splitting

This paper considers the problem of stabilizing a discrete-time non-linear stochastic system over a finite capacity noiseless channel. Our focus is on systems which decompose into a stable and unstable component, and the stability notion considered is asymptotic ergodicity of the $\mathbb{R}^N$-valued state process. We establish a necessary lower bound on channel capacity for the existence of a coding and control policy which renders the closed-loop system stochastically stable. In the literature, it has been established that under technical assumptions, the channel capacity must not be smaller than the logarithm of the determinant of the system linearization, averaged over the noise and ergodic state measures. In this paper, we establish that for systems with a stable component, it suffices to consider only the unstable dimensions, providing a refinement on the general channel capacity bound for a large class of systems. The result is established using the notion of stabilization entropy, a notion adapted from invariance entropy, used in the study of noise-free systems under information constraints.

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arXiv2021

Nonlinear small-gain theorems for input-to-state stability of infinite interconnections

We consider infinite heterogeneous networks, consisting of input-to-state stable subsystems of possibly infinite dimension. We show that the network is input-to-state stable, provided that the gain operator satisfies a certain small-gain condition. We show that for finite networks of nonlinear systems this condition is equivalent to the so-called strong small-gain condition of the gain operator (and thus our results extend available results for finite networks), and for infinite networks with a linear gain operator they correspond to the condition that the spectral radius of the gain operator is less than one. We provide efficient criteria for input-to-state stability of infinite networks with linear gains, governed by linear and homogeneous gain operators, respectively.

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arXiv2021

A subgradient algorithm for data-rate optimization in the remote state estimation problem

In the remote state estimation problem, an observer tries to reconstruct the state of a dynamical system at a remote location, where no direct sensor measurements are available. The observer only has access to information sent through a digital communication channel with a finite capacity. The recently introduced notion of restoration entropy provides a way to determine the smallest channel capacity above which an observer can be designed that observes the system without a degradation of the initial observation quality. In this paper, we propose a subgradient algorithm to estimate the restoration entropy via the computation of an appropriate Riemannian metric on the state space, which allows to determine the approximate value of the entropy from the time-one map (in the discrete-time case) or the generating vector field (for ODE systems), respectively.

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arXiv2021

Control of chaos with minimal information transfer

This paper studies set-invariance and stabilization of hyperbolic sets over rate-limited channels for discrete-time control systems. We first investigate structural and control-theoretic properties of hyperbolic sets, in particular such that arise by adding small control terms to uncontrolled systems admitting (classical) hyperbolic sets. Then we derive a lower bound on the invariance entropy of a hyperbolic set in terms of the difference between the unstable volume growth rate and the measure-theoretic fiber entropy of associated random dynamical systems. We also prove that our lower bound is tight in two extreme cases. Furthermore, we apply our techniques to the problem of local uniform stabilization to a hyperbolic set. Finally, we discuss an example built on the Hénon horseshoe.

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arXiv2021

ISS small-gain criteria for infinite networks with linear gain functions

This paper provides a Lyapunov-based small-gain theorem for input-to-state stability (ISS) of networks composed of infinitely many finite-dimensional systems. We model these networks on infinite-dimensional $\ell_{\infty}$-type spaces. A crucial assumption in our results is that the internal Lyapunov gains, modeling the influence of the subsystems on each other, are linear functions. Moreover, the gain operator built from the internal gains is assumed to be subadditive and homogeneous, which covers both max-type and sum-type formulations for the ISS Lyapunov functions of the subsystems. As a consequence, the small-gain condition can be formulated in terms of a generalized spectral radius of the gain operator. Through an example, we show that the small-gain condition can easily be checked if the interconnection topology of the network has some kind of symmetry. While our main result provides an ISS Lyapunov function in implication form for the overall network, an ISS Lyapunov function in a dissipative form is constructed under mild extra assumptions.

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arXiv2021

Ergodicity Conditions For Controlled Stochastic Non-Linear Systems Under Information Constraints

Consider a stochastic nonlinear system controlled over a possibly noisy communication channel. An important problem is to characterize the largest class of channels for which there exist coding and control policies so that the closed-loop system is stochastically stable. In this paper, we consider the stability notion of (asymptotic) ergodicity. We prove lower bounds on the channel capacity necessary to achieve the stability criterion. Under mild technical assumptions, we obtain that the necessary channel capacity is lower bounded by the log-determinant of the linearization, double-averaged over the state and noise space. We prove this bound by introducing a modified version of invariance entropy and utilizing the almost sure convergence of sample paths guaranteed by the pointwise ergodic theorem. The fundamental bounds obtained generalize well-known formulas for linear systems, and are in some cases more refined than those obtained for nonlinear systems via information-theoretic methods.

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arXiv2020

A Lyapunov-based small-gain theorem for infinite networks

This paper presents a small-gain theorem for networks composed of a countably infinite number of finite-dimensional subsystems. Assuming that each subsystem is exponentially input-to-state stable, we show that if the gain operator, collecting all the information about the internal Lyapunov gains, has a spectral radius less than one, the overall infinite network is exponentially input-to-state stable. The effectiveness of our result is illustrated through several examples including nonlinear spatially invariant systems with sector nonlinearities and a road traffic network.

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arXiv2020

Invariance Properties of Controlled Stochastic Nonlinear Systems under Information Constraints

Given a stochastic nonlinear system controlled over a possibly noisy communication channel, the paper studies the largest class of channels for which there exist coding and control policies so that the closed-loop system is stochastically stable. The stability criterion considered is asymptotic mean stationarity (AMS). We develop a general method based on ergodic theory and probability to derive fundamental bounds on information transmission requirements leading to stabilization. Through this method we develop a new notion of entropy which is tailored to derive lower bounds for asymptotic mean stationarity for both noise-free and noisy channels. The bounds obtained through probabilistic and ergodic-theoretic analysis are more refined in comparison with the bounds obtained earlier via information-theoretic methods. Moreover, our approach is more versatile in view of the models considered and allows for finer lower bounds when the AMS measure is known to admit further properties such as moment bounds.

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arXiv2020

Numerical Estimation of Invariance Entropy for Nonlinear Control Systems

For a closed-loop control system with a digital channel between the sensor and the controller, the notion of invariance entropy quantifies the smallest average rate of information transmission above which a given compact subset of the state space can be made invariant. In this work, we present for the first time an algorithm to numerically compute upper bounds of invariance entropy. With three examples, for which the exact value of the invariance entropy is known to us or can be estimated by other means, we demonstrate that the upper bound obtained by our algorithm is of the same order of magnitude as the actual value. Additionally, our algorithm provides a static coder-controller scheme corresponding to the obtained data-rate bound.

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arXiv2020

Small-gain theorem for stability, cooperative control and distributed observation of infinite networks

Motivated by a paradigm shift towards a hyper-connected world, we develop a computationally tractable small-gain theorem for a network of infinitely many systems, termed as infinite networks. The proposed small-gain theorem addresses exponential input-to-state stability with respect to closed sets, which enables us to analyze diverse stability problems in a unified manner. The small-gain condition, expressed in terms of the spectral radius of a gain operator collecting all the information about the internal Lyapunov gains, can be numerically computed for a large class of systems in an efficient way. To demonstrate broad applicability of our small-gain theorem, we apply it to the stability analysis of infinite time-varying networks, to consensus in infinite-agent systems, as well as to the design of distributed observers for infinite networks.

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arXiv2018

On the relation between topological entropy and restoration entropy

In the context of state estimation under communication constraints, several notions of dynamical entropy play a fundamental role, among them: topological entropy and restoration entropy. In this paper, we present a theorem which demonstrates that for most dynamical systems restoration entropy strictly exceeds topological entropy. This implies that robust estimation policies in general require a higher rate of data transmission than non-robust ones. The proof of our theorem is quite short, but uses sophisticated tools from the theory of smooth dynamical systems.

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