I. Sendińa-Nadal & C. Letellier
Observability analysis and state reconstruction for networks of nonlinear systems
Chaos, 32 (8), 083109, 2022. Online
We address the problem of retrieving the full state of a network of Rössler systems from the knowledge of the actual state of a limited set of nodes. The selection of the nodes where sensors are placed is carried out in a hierarchical way through a procedure based on graphical and symbolic observability approaches applied to pairs of coupled dynamical systems. By using a map directly obtained from the governing equations, we design a nonlinear network reconstructor which is able to unfold the state of the non measured nodes with working accuracy. For sparse networks, the number of sensors scales with half the network size and node reconstruction errors are lower in networks with heterogeneous degree distributions. The method performs well even in the presence of parameter mismatch and non-coherent dynamics, and for dynamical systems with completely different algebraic structure like the Hindmarsch-Rose, therefore, we expect it to be useful for designing robust network control laws.
G. D. Charó, C. Letellier & D. Sciamarella
Templex : a bridge between homologies and templates for chaotic attractors
Chaos, 32 (8), 083108, 2022. Online
The theory of homologies introduces cell complexes to provide an algebraic description of spaces up to topological equivalence. Attractors in state space can be studied using Branched Manifold Analysis through Homologies : this strategy constructs a cell complex from a cloud of points in state space and uses homology groups to characterize its topology. The approach, however, does not consider the action of the flow on the cell complex. The procedure is here extended to take this fundamental property into account, as done with templates. The goal is achieved endowing the cell complex with a directed graph that prescribes the flow direction between its highest dimensional cells. The tandem of cell complex and directed graph, baptized templex, is shown to allow for a sophisticated characterization of chaotic attractors and for an accurate classification of them. The cases of a few well-known chaotic attractors are investigated — namely the spiral and funnel Rössler attractors, the Lorenz attractor, the Burke and Shaw attractor and a four-dimensional system. A link is established with their description in terms of templates.
L. A. Aguirre, F. B. Freitas & C. Letellier
Numerical interpretation of controllability coefficients in nonlinear dynamics
Communications in Nonlinear Science and Numerical Simulation, 116, 106875, 2023. Online
As its dual, observability, the controllability of controllable systems actuated in different ways can be quantified by coefficients which can be computed either by a numerical approach or a symbolic one. If the interpretation of observability coefficients is rather straightforward, this is not the case for controllability. This paper, after proposing slight but important adjustments to the computation of numerical controllability coefficients puts forward a numerical framework that can be used to provide some clue as how to interpret the role played by numerical and symbolical controllability coefficients. Using four different chaotic systems, it is showed how controllability may depend on the derivative to which the control law is applied and how this may be related to the corresponding coefficients. In practice could help to decide where to place actuators. The developed framework has two backbones : the power needed to achieve control and the rate of success of the control, the latter is not considered in the surveyed literature. Our results show that numerical and symbolic coefficients quantify different aspects of the problem and confirm how controllability is difficult to interpret in practice without any consideration for observability.
C. Letellier, L. Minati & J.-P. Barbot
Optimal placement of sensor and actuator for controlling the piecewise linear Chua circuit via a discretized controller
Journal of Difference Equations and Applications, in press
Controlling the dynamics of chaotic systems is a task which is often addressed in an empirical way, particularly for placing sensors and actuators. Here, we show that selecting the measured variable and placing the actuator can be guided by considering the observability and controllability symbolic coefficients and applying the notion of flatness. This approach is here demonstrated on the piecewise linear Chua circuit, whose specific features are leveraged in constructing a discretized controller with a switch mechanism and optimally placed sensor and actuator. The feedback linearization is compared to a homogeneous and a passivity-based control laws, the flat control laws being more efficient than the others. It is thus shown that the proposed flat control law by feedback linearization is very efficient. The continuous time and discretized Chua circuit, governed by differential and difference equations, respectively, are treated. Most likely, these results could be extendable to a large group of natural and experimental systems.