Generalized synchronization mediated by a flat coupling between structurally nonequivalent chaotic systems.

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.

Optimal placement of sensor and actuator for controlling low-dimensional chaotic systems based on global modeling.

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.

Sensor Effects in LCL-Type Grid-Connected Shunt Active Filters Control Using Higher-Order Sliding Mode Control Techniques.

The effects of measuring devices/sensors on improving the power quality (PQ) of electric networks are studied in this paper. In this context, improving the performance of an LCL-type grid connected to a three-phase three-wire shunt active filter (SAF) in the presence of voltage perturbations is studied. In order to ensure the high-quality performance of LCL-SAF in the presence of voltage perturbations, the robust continuous second-order sliding mode controller (2-SMC), including twisting and super-twisting controllers, and continuous higher-order sliding mode controller (C-HOSMC)-based approaches are employed. These controllers, whose outputs are processed by pulse-width modulation (PWM), allow minimization of the phase shift and prevent the generation of discontinuous chattering commands, which can severely damage the VSI components. Moreover, an integration of a generalized instantaneous power identification algorithm with an advanced phase locked loop (PLL) was proposed and experimentally tested to validate the effective performances of SAF under severe perturbations. Additionally, the studied approaches were tested via simulations taking into account a conventional nonlinear industrial load in a real textile factory environment, using measurements provided by power quality analyzers. Finally, the effects of the measuring devices, including the current and voltage sensors, on the accuracy and reliability of the SAF and, consequently, on the PQ of the electric power grid were studied via simulations and experimentally. The results of this study support the validity of the recently published patent.

Optimal flatness placement of sensors and actuators for controlling chaotic systems.

Controlling chaotic systems is very often investigated by using empirical laws, without taking advantage of the structure of the governing equations. There are two concepts, observability and controllability, which are inherited from control theory, for selecting the best placement of sensors and actuators. These two concepts can be combined (extended) into flatness, which provides the conditions to fulfill for designing a feedback linearization or another classical control law for which the system is always fully observable and fully controllable. We here design feedback linearization control laws using flatness for the three popular chaotic systems, namely, the Rössler, the driven van der Pol, and the Hénon-Heiles systems. As developed during the last two decades for observability, symbolic controllability coefficients and symbolic flatness coefficients are introduced here and their meanings are tested with numerical simulations. We show that the control law works for every initial condition when the symbolic flatness coefficient is equal to 1.

Observation and sliding mode observer for nonlinear fractional-order system with unknown input.

The main purpose of this paper is twofold. First, the observability and the left invertibility properties and the observable canonical form for nonlinear fractional-order systems are introduced. By using a transformation, we show that these properties can be deduced from an equivalent nonlinear integer-order system. Second, a step by step sliding mode observer for fault detection and estimation in nonlinear fractional-order systems is proposed. Starting with a chained fractional-order integrators form, a step by step first-order sliding mode observer is designed. The finite time convergence of the observer is established by using Lyapunov stability theory. A numerical example is given to illustrate the performance of the proposed approach.

Influence of the singular manifold of nonobservable states in reconstructing chaotic attractors.

It is known that the reconstructed phase portrait of a given system strongly depends on the choice of the observable. In particular, the ability to obtain a global model from a time series strongly depends on the observability provided by the measured variable. Such a dependency results from (i) the existence of a singular observability manifold, M(s)(obs), for which the coordinate transformation between R(m) and the reconstructed space is not defined and (ii) how often the trajectory visits the neighborhood U(M(s)(obs)) of M(s)(obs). In order to clarify how these aspects contribute to the observability coefficients, we introduce the probability of visits of M(s)(obs) and the relative time spent in U(M(s)(obs)) to construct a new coefficient. Combined with the symbolic observability coefficients previously introduced [Letellier and Aguirre, Phys. Rev. E 79, 066210 (2009)] (only taking into account the existence of M(s)(obs)), this new coefficient helps to determine the specific role played by the location of M(s)(obs) with respect to the attractor, in phase portrait reconstruction and in any analysis technique.

Automatic determination of the number of targets present when using the time reversal operator.

Acoustical time reversal mirrors have been shown to provide a highly accurate means of studying and focusing on acoustical sources. The DORT method is a derivation of the time reversal process, which allows for focusing on multiple targets. An important step in this process is the determination of the number of targets or sources present. This is achieved by examining the eigenvalues of the time reversal operator (TRO). The number of significant eigenvalues is then chosen as the number of sources present. However, as mentioned in [N. Mordant, C. Prada, and M. Fink, J. Acoust. Soc. Am. 105, 2634-2642 (1999) and C. Prada, M. Tanter, and M. Fink, in Proceedings of the IEEE Symposium, 1997, pp. 679-683], factors such as low signal to noise ratio (SNR), small data sample, array configuration and the target location may result in the eigenvalues corresponding to the targets no longer being distinguishable from the background noise eigenvalues. This paper proposes a robust method of automatically determining the number of targets even in the presence of a small number of snapshots. For white Gaussian noise, the profile of the ordered eigenvalues is seen to fit an exponential law. The observed eigenvalues are then compared to this model and a mismatch is detected between the observed profile and the noise-only model. The index of the mismatch gives the number of scatterers present.