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.

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.

CONTROL OF THE DIFFERENTIALLY FLAT LORENZ SYSTEM

2001 ◽  
Vol 11 (07) ◽  
pp. 1989-1996 ◽  
Author(s):  
JIN MAN JOO ◽  
JIN BAE PARK
Keyword(s):  
Equilibrium State ◽  
Lorenz System ◽  
Degree Of Freedom ◽  
Design Approach ◽  
State Trajectory ◽  
System A ◽  
Full State ◽  
Tracking Controller ◽  
The Lorenz System ◽  
Two Degree Of Freedom

This paper presents an approach for the control of the Lorenz system. We first show that the controlled Lorenz system is differentially flat and then compute the flat output of the Lorenz system. A two degree of freedom design approach is proposed such that the generation of full state feasible trajectory incorporates with the design of a tracking controller via the flat output. The stabilization of an equilibrium state and the tracking of a feasible state trajectory are illustrated.

Multistability in the Lorenz System: A Broken Butterfly

2014 ◽  
Vol 24 (10) ◽  
pp. 1450131 ◽  
Author(s):  
Chunbiao Li ◽  
Julien Clinton Sprott
Keyword(s):  
Parameter Space ◽  
Limit Cycles ◽  
Lorenz System ◽  
Dynamical Behavior ◽  
Symmetric Pair ◽  
Physical Systems ◽  
System A ◽  
Plausible Model ◽  
Fluid Convection ◽  
The Lorenz System

In this paper, the dynamical behavior of the Lorenz system is examined in a previously unexplored region of parameter space, in particular, where r is zero and b is negative. For certain values of the parameters, the classic butterfly attractor is broken into a symmetric pair of strange attractors, or it shrinks into a small attractor basin intermingled with the basins of a symmetric pair of limit cycles, which means that the system is bistable or tristable under certain conditions. Although the resulting system is no longer a plausible model of fluid convection, it may have application to other physical systems.

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.

scholarly journals Lorenz Type Behaviors in the Dynamics of Laser Produced Plasma

Symmetry ◽  
2019 ◽  
Vol 11 (9) ◽  
pp. 1135 ◽  
Author(s):  
Stefan Andrei Irimiciuc ◽  
Florin Enescu ◽  
Andrei Agop ◽  
Maricel Agop
Keyword(s):  
Experimental Data ◽  
Lorenz System ◽  
Integral Form ◽  
Diagnostic Techniques ◽  
Oscillatory Behavior ◽  
Mathematical Representation ◽  
System A ◽  
Theoretical Predictions ◽  
Fractal Space ◽  
The Lorenz System

An innovative theoretical model is developed on the backbone of a classical Lorenz system. A mathematical representation of a differential Lorenz system is transposed into a fractal space and reduced to an integral form. In such a conjecture, the Lorenz variables will operate simultaneously on two manifolds, generating two transformation groups, one corresponding to the space coordinates transformation and another one to the scale resolution transformation. Since these groups are isomorphs various types isometries become functional. The Lorenz system was further adapted to describe the dynamics of ejected particles as a result of laser matter interaction in a fractal paradigm. The simulations were focused on the dynamics of charged particles, and showcase the presence of current oscillations, a heterogenous velocity distribution and multi-structuring at different interaction scales. The theoretical predictions were compared with the experimental data acquired with noninvasive diagnostic techniques. The experimental data confirm the multi-structure scenario and the oscillatory behavior predicted by the mathematical model.

scholarly journals Oscillating forcings and new regimes in the Lorenz system: a four-lobe attractor

2012 ◽  
Vol 19 (3) ◽  
pp. 315-322 ◽  
Author(s):  
V. Pelino ◽  
F. Maimone ◽  
A. Pasini
Keyword(s):  
Lorenz System ◽  
Climate Dynamics ◽  
System A ◽  
The Lorenz System

Abstract. It has been shown that forced Lorenz models generally maintain their two-lobe structure, just giving rise to changes in the occurrence of their regimes. Here, using the richness of a unified formalism for Kolmogorov-Lorenz systems, we show that introducing oscillating forcings can lead to the birth of new regimes and to a four-lobe attractor. Analogies within a climate dynamics framework are mentioned.

Robust Control of the Chaotic Lorenz System

1997 ◽  
Vol 07 (12) ◽  
pp. 2847-2854 ◽  
Author(s):  
Henning Lenz ◽  
Dragan Obradovic
Keyword(s):  
Feedback Linearization ◽  
Feedback Loop ◽  
Lorenz System ◽  
Input State ◽  
Model Uncertainties ◽  
Partial Linearization ◽  
Lyapunov’S Second Method ◽  
The Lorenz System ◽  
The Stability ◽  
Classical Control

This paper presents a global approach for controlling the Lorenz system. The basic idea is to partially cancel the nonlinear cross-coupling terms such that the stability of the resulting system can be guaranteed by sequentially proving the stability of each individual state. The method combines ideas from feedback linearization, classical control theory, and Lyapunov's second method. Robust behavior with respect to model uncertainties in the feedback loop is proven. The performance of partial linearization compared to input-state linearization is illustrated on tracking of several trajectories including a periodic orbit and a steady state.

Sign in / Sign up

Export Citation Format

Share Document