scholarly journals Data-Rate Constrained Observers of Nonlinear Systems

Entropy ◽  
2019 ◽  
Vol 21 (3) ◽  
pp. 282 ◽  
Author(s):  
Quentin Voortman ◽  
Alexander Pogromsky ◽  
Alexey Matveev ◽  
Henk Nijmeijer
Keyword(s):  
Dynamical System ◽  
Communication Channel ◽  
Lorenz System ◽  
Upper Bounds ◽  
Data Rate ◽  
The State ◽  
Box Dimension ◽  
Lyapunov Dimension ◽  
Remote Location ◽  
The Lorenz System

In this paper, the design of a data-rate constrained observer for a dynamical system is presented. This observer is designed to function both in discrete time and continuous time. The system is connected to a remote location via a communication channel which can transmit limited amounts of data per unit of time. The objective of the observer is to provide estimates of the state at the remote location through messages that are sent via the channel. The observer is designed such that it is robust toward losses in the communication channel. Upper bounds on the required communication rate to implement the observer are provided in terms of the upper box dimension of the state space and an upper bound on the largest singular value of the system’s Jacobian. Results that provide an analytical bound on the required minimum communication rate are then presented. These bounds are obtained by using the Lyapunov dimension of the dynamical system rather than the upper box dimension in the rate. The observer is tested through simulations for the Lozi map and the Lorenz system. For the Lozi map, the Lyapunov dimension is computed. For both systems, the theoretical bounds on the communication rate are compared to the simulated rates.

Feedback Linearization of the Lorenz System: Stabilization and Tracking Control

2002 ◽  
Author(s):  
L. G. Crespo ◽  
J. Q. Sun
Keyword(s):  
Feedback Linearization ◽  
Lorenz System ◽  
The State ◽  
Differential Flatness ◽  
State Transformation ◽  
System A ◽  
Excellent Control ◽  
High Control ◽  
The Lorenz System ◽  
Control Study

This paper presents a control study of the Lorenz system by using feedback linearization. The effects of the state transformation on the dynamics of the Lorenz system are studied first. Then, composite controllers are developed for both stabilization and tracking of the system. The controls are designed to overcome the barrier in controllability imposed by the state transformation. The transition through the manifold defined by such a singularity is achieved by inducing the chaotic response within a boundary layer that contains the singularity. Outside this region, a conventional feedback nonlinear control is applied. In this fashion, the authority of the control is enlarged to the whole state space and the need for high control efforts is substantially mitigated. Tracking problems that involve single and cooperative objectives are studied by using the differential flatness property of the system. A good understanding of the system dynamics proves to be invaluable in the design of better controls. In all numerical examples, the proposed approach led to excellent control performances.

CHAOS OR TURBULENCE?

1992 ◽  
Vol 02 (04) ◽  
pp. 1005-1009 ◽  
Author(s):  
RAY BROWN ◽  
LEON O. CHUA
Keyword(s):  
Dynamical System ◽  
Lorenz System ◽  
Three Dimensional ◽  
New Technique ◽  
Chaotic Dynamical System ◽  
The Lorenz System

We present a new three-dimensional autonomous chaotic dynamical system that appears to have a closer relationship to turbulence than the Lorenz system. We have developed this system using the new technique of dynamical synthesis.

UNMASKING A MODULATED CHAOTIC COMMUNICATIONS SCHEME

1996 ◽  
Vol 06 (02) ◽  
pp. 367-375 ◽  
Author(s):  
KEVIN M. SHORT
Keyword(s):  
Dynamical System ◽  
Phase Space ◽  
Lorenz System ◽  
Sinusoidal Signal ◽  
Chaotic Communication ◽  
Communication Scheme ◽  
Transmitted Signal ◽  
Chaotic Communications ◽  
The Lorenz System ◽  
Perfect Synchrony

This paper will address the problem of unmasking a new chaotic communication scheme using synchronizing circuits, where the Lorenz system is modulated by the message and the x-coordinate of the modulated system is added to the message and transmitted to the receiver. The receiver is driven into perfect synchrony with the transmitter even in the presence of the message, and since the message becomes part of the dynamics it provides very little distortion to the phase space of the dynamical system. However, this paper will demonstrate that it is still possible to extract a sinusoidal message from the transmitted signal. It will also be shown that it is possible to extract the sinusoidal signal solely from the x-coordinate, without secondarily adding back the message sinusoid before transmission. The message extraction is also shown to work for simple frequency-modulated and phase-modulated message signals. The modulated communication scheme does effectively nullify a multi-step unmasking technique which had been somewhat successful when applied to chaotic communication schemes which employed additive message signals.

On the Feedback Linearization of the Lorenz System

2004 ◽  
Vol 10 (1) ◽  
pp. 85-100 ◽  
Author(s):  
L.G. Crespo ◽  
J.Q. Sun
Keyword(s):  
Boundary Layer ◽  
Rayleigh Number ◽  
Feedback Linearization ◽  
Lorenz System ◽  
Control Variable ◽  
The State ◽  
Differential Flatness ◽  
State Transformation ◽  
The Lorenz System ◽  
Control Study

In this paper we present a control study of the Lorenz system via feedback linearization using the Rayleigh number as a control variable. The effects of the state transformation on the dynamics of the system are studied first. Then, composite controls are derived for both stabilization and tracking problems. The transition through the manifold where the state transformation is singular and the system is insensitive to the control is achieved by inducing the natural chaotic response of the system within a boundary layer. Outside the boundary layer, the control designed via feedback linearization is applied. Tracking problems that involve single and cooperative objectives are studied by using differential flatness. A good understanding of the system dynamics proves to be invaluable in the design of better controls.

scholarly journals The Lorenz system: hidden boundary of practical stability and the Lyapunov dimension

2020 ◽  
Vol 102 (2) ◽  
pp. 713-732 ◽  
Author(s):  
N. V. Kuznetsov ◽  
T. N. Mokaev ◽  
O. A. Kuznetsova ◽  
E. V. Kudryashova
Keyword(s):  
Global Stability ◽  
Lorenz System ◽  
Basin Of Attraction ◽  
Small Time ◽  
Delayed Feedback Control ◽  
Control Technique ◽  
Lyapunov Dimension ◽  
Time Intervals ◽  
The Lorenz System ◽  
Time Delayed Feedback

AbstractOn the example of the famous Lorenz system, the difficulties and opportunities of reliable numerical analysis of chaotic dynamical systems are discussed in this article. For the Lorenz system, the boundaries of global stability are estimated and the difficulties of numerically studying the birth of self-excited and hidden attractors, caused by the loss of global stability, are discussed. The problem of reliable numerical computation of the finite-time Lyapunov dimension along the trajectories over large time intervals is discussed. Estimating the Lyapunov dimension of attractors via the Pyragas time-delayed feedback control technique and the Leonov method is demonstrated. Taking into account the problems of reliable numerical experiments in the context of the shadowing and hyperbolicity theories, experiments are carried out on small time intervals and for trajectories on a grid of initial points in the attractor’s basin of attraction.

Dynamics, Synchronization and Electronic Implementations of a New Lorenz-like Chaotic System with Nonhyperbolic Equilibria

2019 ◽  
Vol 29 (14) ◽  
pp. 1950197 ◽  
Author(s):  
P. D. Kamdem Kuate ◽  
Qiang Lai ◽  
Hilaire Fotsin
Keyword(s):  
Chaotic System ◽  
Chaotic Systems ◽  
Lorenz System ◽  
Chaotic Behavior ◽  
Piecewise Linear ◽  
Period Doubling ◽  
Adaptive Synchronization ◽  
The Novel ◽  
Algebraic Equations ◽  
The Lorenz System

The Lorenz system has attracted increasing attention on the issue of its simplification in order to produce the simplest three-dimensional chaotic systems suitable for secure information processing. Meanwhile, Sprott’s work on elegant chaos has revealed a set of 19 chaotic systems all described by simple algebraic equations. This paper presents a new piecewise-linear chaotic system emerging from the simplification of the Lorenz system combined with the elegance of Sprott systems. Unlike the majority, the new system is a non-Shilnikov chaotic system with two nonhyperbolic equilibria. It is multiplier-free, variable-boostable and exclusively based on absolute value and signum nonlinearities. The use of familiar tools such as Lyapunov exponents spectra, bifurcation diagrams, frequency power spectra as well as Poincaré map help to demonstrate its chaotic behavior. The novel system exhibits inverse period doubling bifurcations and multistability. It has only five terms, one bifurcation parameter and a total amplitude controller. These features allow a simple and low cost electronic implementation. The adaptive synchronization of the novel system is investigated and the corresponding electronic circuit is presented to confirm its feasibility.

scholarly journals Event-Triggering State and Fault Estimation for a Class of Nonlinear Systems Subject to Sensor Saturations

Sensors ◽  
2021 ◽  
Vol 21 (4) ◽  
pp. 1242
Author(s):  
Cong Huang ◽  
Bo Shen ◽  
Lei Zou ◽  
Yuxuan Shen
Keyword(s):  
Nonlinear Systems ◽  
Difference Equations ◽  
Estimation Error ◽  
Upper Bounds ◽  
The State ◽  
Fault Estimation ◽  
Triggering Mechanism ◽  
Transmission Frequency ◽  
Current Transmission ◽  
Event Triggering

This paper is concerned with the state and fault estimation issue for nonlinear systems with sensor saturations and fault signals. For the sake of avoiding the communication burden, an event-triggering protocol is utilized to govern the transmission frequency of the measurements from the sensor to its corresponding recursive estimator. Under the event-triggering mechanism (ETM), the current transmission is released only when the relative error of measurements is bigger than a prescribed threshold. The objective of this paper is to design an event-triggering recursive state and fault estimator such that the estimation error covariances for the state and fault are both guaranteed with upper bounds and subsequently derive the gain matrices minimizing such upper bounds, relying on the solutions to a set of difference equations. Finally, two experimental examples are given to validate the effectiveness of the designed algorithm.

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