Feedback data rates for nonlinear systems

Nonlinearity, 2011

The paper deals with the problem of estimation of the topological entropy for non-autonomous systems of differential equations via the second (direct) Lyapunov method. The main result of the paper is illustrated by examples concerning the Lorenz system and Duffing oscillator.

IEEE Transactions on Control of Network Systems, 2015

We consider the problem of making a set of states invariant for a network of controlled systems. We assume that the subsystems, initially uncoupled, must be interconnected through controllers to be designed with a constraint on the data rate obtained by every subsystem from all the other subsystems. We introduce the notion of subsystem invariance entropy, which is a measure for the smallest data rate arriving at a fixed subsystem, above which the overall system is able to achieve the control goal. Moreover, we associate to a network of n subsystems a closed convex set of R n encompassing all possible combinations of data rates within the network that guarantee the existence of corresponding feedback strategies for making a given set invariant. The extremal points of this convex set can be regarded as Pareto-optimal data rates for the control problem, expressing a trade-off between the data rates required by different systems. We characterize these quantities for linear systems, and for synchronization of chaos.

Automatica

A critical notion in the ÿeld of communication-limited control is the smallest data rate above which there exists a stabilising coding and control law for a given plant. This quantity measures the lowest rate at which information can circulate in a stable feedback loop and provides a practical guideline for the allocation of communication resources. In this paper, the exponential stabilisability of ÿnite-dimensional LTI plants with limited feedback data rates is investigated. By placing a probability density on the initial state and casting the objective in terms of state moments, the problem is shown to be similar to one in asymptotic quantisation. Quantisation theory is then applied to obtain the inÿmum stabilising data rate over all causal coding and control laws, under mild requirements on the initial state density. ?

IFAC-PapersOnLine, 2019

In this paper, on the example of the Rössler systems, the application of the Pyragas time-delay feedback control technique for verification of Eden's conjecture on the maximum of local Lyapunov dimension, and for the estimation of the topological entropy is demonstrated. To this end, numerical experiments on computation of finite-time local Lyapunov dimensions and finite-time topological entropy on a Rössler attractor and embedded unstable periodic orbits are performed. The problem of reliable numerical computation of the mentioned dimension-like characteristics along the trajectories over large time intervals is discussed.

Journal of Mathematical Physics, 2010

Invariant manifolds provide geometric structures for understanding dynamical behavior of nonlinear systems. However, these nonlinear systems are often subject to random fluctuations or noises. It is thus desirable to quantify the impact of noises on the invariant manifolds. When the noise intensity is small, this impact is estimated via asymptotic analysis in the context of Liapunov–Perron formulation. Namely, the random invariant manifold is represented as a perturbation of the deterministic invariant manifold, with a well-defined bound for the deviation.

2012

For deterministic control systems with digital communication constraints, an approach is explained which, in particular, permits to determine minimal bit rates and entropy for exponential stabilization. In the case of linear systems, a formula for the entropy associated with exponential stabilization is provided. Key words. digital communication, topological entropy, stabilization AMS subject classi cations. 93D15, 94A17, 37B40 1. Introduction. The problem to determine minimal data rates for performing control tasks has been considered for quite a while, see the survey Nair, Fagnani, Zampieri and Evans [27]. Early landmarks are the papers by Delchamps [14], who considered quantized information for stabilization and proposed to use statistical methods from ergodic theory, and Wong and Brockett [30] who discussed stabilization of linear systems via coding. From the wealth of literature on this topic we also cite Tatikonda and Mitter [28], Delvenne [15], Fagnani and Zampieri [17], Bull...

Physical Review Research

Uncertainty in the initial conditions of dynamical systems can cause exponentially fast divergence of trajectories, a signature of deterministic chaos, or be suppressed by the dissipation of energy. Here, we derive a classical uncertainty relation that sets a speed limit on the rates of local observables underlying these behaviors. For systems with a time-invariant stability matrix, the speed limit we derive simplifies to a classical analog of the Mandelstam-Tamm versions of the time-energy uncertainty relation. These classical bounds are set by fluctuations in the local stability of state space. To measure these fluctuations, we introduce a definition of the Fisher information in terms of Lyapunov vectors in tangent space, analogous to the quantum Fisher information defined in terms of wave vectors in Hilbert space. This information sets an upper bound on the speed at which classical dynamical systems and their observables, instantaneous Lyapunov exponents and dissipation, evolve. This speed limit applies to systems that are open or closed, conservative or dissipative, actively driven or passively evolving, and directly connects the geometries of phase space and information.

Proceedings of the 24th International Conference on Hybrid Systems: Computation and Control, 2021

This paper studies topological entropy of switched nonlinear systems. We construct a general upper bound for the topological entropy in terms of an average of the asymptotic suprema of the measures of Jacobian matrices of individual modes, weighted by the corresponding active rates. A general lower bound is constructed in terms of an active-rate-weighted average of the asymptotic infima of the traces of these Jacobian matrices. For switched systems with diagonal modes, we construct upper and lower bounds that only depend on the eigenvalues of Jacobian matrices, their relative order among individual modes, and the active rates. For both cases, we also construct more conservative upper bounds that require less information on the switching, with their relations illustrated by numerical examples of a switched Lotka-Volterra ecosystem model.

2011

In this work we investigate the information loss in (nonlinear) dynamical input-output systems and provide some general results. In particular, we present an upper bound on the information loss rate, defined as the (non-negative) difference between the entropy rates of the jointly stationary stochastic processes at the input and output of the system. We further introduce a family of systems with vanishing information loss rate. It is shown that not only linear filters belong to that family, but-under certain circumstances-also finiteprecision implementations of the latter, which typically consist of nonlinear elements.

arXiv (Cornell University), 2023

Fisher information is a lower bound on the uncertainty in the statistical estimation of classical and quantum mechanical parameters. While some deterministic dynamical systems are not subject to random fluctuations, they do still have a form of uncertainty: Infinitesimal perturbations to the initial conditions can grow exponentially in time, a signature of deterministic chaos. As a measure of this uncertainty, we introduce another classical information, specifically for the deterministic dynamics of isolated, closed, or open classical systems not subject to noise. This classical measure of information is defined with Lyapunov vectors in tangent space, making it less akin to the classical Fisher information and more akin to the quantum Fisher information defined with wavevectors in Hilbert space. Our analysis of the local state space structure and linear stability lead to upper and lower bounds on this information, giving it an interpretation as the net stretching action of the flow. Numerical calculations of this information for illustrative mechanical examples show that it depends directly on the phase space curvature and speed of the flow.