C. Letellier, S. Mangiarotti, L. Minati, Mattia Frasca & J.-P. Barbot
Optimal placement of sensor and actuator for controlling low-dimensional chaotic systems based on global modeling
Chaos, 33, 013140, 2023. Online

- Abstract
Controlling chaos is fundamental in many applications, and for this reason, many techniques have been proposed to address this problem. Here, we propose a strategy based on an optimal placement of the sensor and actuator providing global observability of the state space and global controllability to any desired state. The first of these two conditions enables the derivation of a model of the system by using a global modeling technique. In turn, this permits the use of feedback linearization for designing the control law based on the equations of the obtained model and providing a zero-flat system. The procedure is applied to three case studies, including two piecewise linear circuits, namely, the Carroll circuit and the Chua circuit whose governing equations are approximated by a continuous global model. The sensitivity of the procedure to the time constant of the dynamics is also discussed.

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, 29 (9-12), 1341-1368, 2023.

- Abstract
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.

- Christophe Letellier, Irene Sendi˝a-Nadal, I. Leyva & Jean-Pierre Barbot,
Generalized synchronization mediated by a flat coupling between structurally nonequivalent chaotic systems,
Chaos, 33 (9) : 093117, 2023. Online

- Abstract
Synchronization of chaotic systems is usually investigated for structurally equivalent systems typically coupled through linear diffusive functions. Here, we focus on a particular type of coupling borrowed from a nonlinear control theory and based on the optimal placement of a sensor—a device measuring the chosen variable—and an actuator—a device applying the actuating (control) signal to a variable’s derivative—in the response system, leading to the so-called flat control law. We aim to investigate the dynamics produced by a response system that is flat coupled to a drive system and to determine the degree of generalized synchronization between them using statistical and topological arguments. The general use of a flat control law for getting generalized synchronization is discussed.

Christophe Letellier, Eduardo M. A. M. Mendes & Jean-Marc Malasoma
Lorenz-like systems and Lorenz-like attractors : definition, examples and equivalences
Physical Reivew E, 108, 044209, 2023. Online

- Abstract
Since the early 1970s, numerous systems exhibiting an algebraic structure resembling that of the 1963 Lorenz system have been proposed. These systems have occasionally yielded the same attractor as the Lorenz system, while in other cases, they have not. Conversely, some systems that are evidently distinct from the Lorenz system, particularly in terms of symmetry, have resulted in attractors that bear a resemblance to the Lorenz attractor. In this paper, we put forward a definition for Lorenz-like systems and Lorenz-like attractors. The former definition is based on the algebraic structure of the governing equations, while the latter relies on topological characterization. Our analysis encompasses over 20 explicitly examined chaotic systems.

Ludovico Minati ⋅ Mattia Frasca ⋅ Pedro A. Valdes-Sosa ⋅ Jean-Pierre Barbot ⋅ Christophe Letellier
Flatness-based real-time control of experimental analog chaotic oscillators
Chaos, Solitons & Fractals, 177, 114274, 2023. Online

- Abstract In the control of non-linear dynamics, the notion of flatness provides a systematic framework for analyzing the observability and controllability of a system. Several successful applications of flatness-based control (in brief, flat control) have been demonstrated, but, to date, the control of chaos using this approach has been obtained only numerically. Here, for the first time, this issue is addressed through a systematic experimental investigation of two canonical chaotic systems, namely, the R÷ssler system and the Saito circuit, realized in the form of analog electronic oscillators. These differ in the types of non-linearity and associated dynamics, as well as their observability and controllability. The corresponding flat control laws, including a homogeneous law, are derived and implemented, using suitable numerical reconstructions of the high-order derivatives, in real-time on a microcontroller interfaced with the analog circuits. Albeit with some limitations, viable control is attained over a wide range of settings, and the influences of the device non-idealities are analyzed in detail. These initial results suggest that, besides chaos suppression in engineering applications from vehicle stabilization to cardiology, flat chaos control could probably also be applied towards obtaining desired dynamical and synchronization states in large- scale physical models of complex systems.

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