Parameter Identification of Lorenz System with Incomplete Information: Case of One Unknown Function

2021 ◽  
Vol 55 ◽  
pp. 82-102
Author(s):  
Michael Y. Shatalov ◽  
Samuel A. Surulere ◽  
Lilies M. Phadime ◽  
Phumezile Kama
Keyword(s):  
Parameter Identification ◽  
Incomplete Information ◽  
Lorenz System ◽  
Classical Method ◽  
Original System ◽  
Parametric Identification ◽  
Unknown Parameters ◽  
Novel Method ◽  
The Lorenz System ◽  
Lagrange’S Multipliers

Inverse problem of the Lorenz system parametric identification is considered in the case of incomplete information about solutions of the system. In the present paper, it is assumed that only two solutions of the system from three are known in different combinations. The problem of the parameter identification of the system is solved by means of elimination of unknown functions from the original system. The obtained system of equations has the same order as the original one, but contains the unknown original parameters in new combinations. Sometimes, the number of new unknown parameters is higher than number of the original unknowns. In this case, the method of the constrained least squares minimization is used in the special formulation, developed by the authors. This novel formulation exploits linearity of the system with respect to the new unknown parameters, by means of which the number of nonlinear equations becomes equal to the number of the constraints between the new parameters. Two methods of the constraint minimization are considered: the classical method of Lagrange’s multipliers and a novel method of the auxiliary parameters. Numerical simulations demonstrate effectiveness of the algorithms.

Parameter Identification of Chaotic Systems by a Novel Dual Particle Swarm Optimization

2016 ◽  
Vol 26 (02) ◽  
pp. 1650024 ◽  
Author(s):  
Yunxiang Jiang ◽  
Francis C. M. Lau ◽  
Shiyuan Wang ◽  
Chi K. Tse
Keyword(s):  
Particle Swarm Optimization ◽  
Parameter Identification ◽  
Chaotic Systems ◽  
Particle Swarm ◽  
Original System ◽  
Pso Algorithm ◽  
Search Range ◽  
Swarm Optimization ◽  
Estimation System ◽  
The Lorenz System

In this paper, we propose a dual particle swarm optimization (PSO) algorithm for parameter identification of chaotic systems. We also consider altering the search range of individual particles adaptively according to their objective function value. We consider both noiseless and noisy channels between the original system and the estimation system. Finally, we verify the effectiveness of the proposed dual PSO method by estimating the parameters of the Lorenz system using two different data acquisition schemes. Simulation results show that the proposed method always outperforms the traditional PSO algorithm.

Parameter Identification of Lorenz System with Incomplete Information: Case of One Known and Two Unknown Functions

2021 ◽  
Vol 55 ◽  
pp. 103-121
Author(s):  
Michael Y. Shatalov ◽  
Samuel A. Surulere ◽  
Lilies M. Phadime ◽  
Thomson T. Mthombeni
Keyword(s):  
Parameter Identification ◽  
Lorenz System ◽  
Initial Time ◽  
Accurate Estimation ◽  
Lorenz Attractor ◽  
Time Interval ◽  
Problem Solution ◽  
Unknown Parameters ◽  
Random Components ◽  
Discrete Values

In the present paper, which is the continuation of the previous one, the problem of parameter identification of the Lorenz system is solved in assumption that only one of three functions is known at discrete time instants on finite time initial time interval. Two other functions are assumed to be unknown. The regular methods of guess values determination of the unknown parameters are developed. They are based on the Lagrange multiplier and auxiliary parameters approaches. A novel method of initial value problem solution is proposed in which the abovementioned guess values are used for more accurate estimation of the system parameters. It is demonstrated that the proposed IVP method simultaneously solves three different tasks: the problem of function interpolation from its discrete values on the initial time interval; the problem of unknown functions reconstruction on the same time interval, and the problem of extrapolation of all functions on limited time interval. It is also shown that the proposed method reconstructs the Lorenz attractor from limited data volume and data including random components.

Synchronization of Quadratic Chaotic Systems Based on Simultaneous Estimation of Nonlinear Dynamics

2018 ◽  
Vol 13 (8) ◽  
Author(s):  
Amin Zarei ◽  
Saeed Tavakoli
Keyword(s):  
Nonlinear Dynamics ◽  
Chaotic Systems ◽  
Lorenz System ◽  
Simultaneous Estimation ◽  
Control Law ◽  
Adaptive Mechanism ◽  
Lyapunov Theorem ◽  
Unknown Parameters ◽  
The Lorenz System ◽  
Synchronization Scheme

To synchronize quadratic chaotic systems, a synchronization scheme based on simultaneous estimation of nonlinear dynamics (SEND) is presented in this paper. To estimate quadratic terms, a compensator including Jacobian matrices in the proposed master–slave schematic is considered. According to the proposed control law and Lyapunov theorem, the asymptotic convergence of synchronization error to zero is proved. To identify unknown parameters, an adaptive mechanism is also used. Finally, a number of numerical simulations are provided for the Lorenz system and a memristor-based chaotic system to verify the proposed method.

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 Software Sensor to Enhance Online Parametric Identification for Nonlinear Closed-Loop Systems for Robotic Applications

Sensors ◽  
2021 ◽  
Vol 21 (11) ◽  
pp. 3653
Author(s):  
Lilia Sidhom ◽  
Ines Chihi ◽  
Ernest Nlandu Kamavuako
Keyword(s):  
Degrees Of Freedom ◽  
Closed Loop ◽  
Sliding Mode ◽  
Robot Manipulator ◽  
Parametric Identification ◽  
Recursive Least Squares ◽  
Unknown Parameters ◽  
Closed Loop Identification ◽  
Input Variables ◽  
Robotic Applications

This paper proposes an online direct closed-loop identification method based on a new dynamic sliding mode technique for robotic applications. The estimated parameters are obtained by minimizing the prediction error with respect to the vector of unknown parameters. The estimation step requires knowledge of the actual input and output of the system, as well as the successive estimate of the output derivatives. Therefore, a special robust differentiator based on higher-order sliding modes with a dynamic gain is defined. A proof of convergence is given for the robust differentiator. The dynamic parameters are estimated using the recursive least squares algorithm by the solution of a system model that is obtained from sampled positions along the closed-loop trajectory. An experimental validation is given for a 2 Degrees Of Freedom (2-DOF) robot manipulator, where direct and cross-validations are carried out. A comparative analysis is detailed to evaluate the algorithm’s effectiveness and reliability. Its performance is demonstrated by a better-quality torque prediction compared to other differentiators recently proposed in the literature. The experimental results highlight that the differentiator design strongly influences the online parametric identification and, thus, the prediction of system input variables.

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.

3D Printing — The Basins of Tristability in the Lorenz System

2017 ◽  
Vol 27 (08) ◽  
pp. 1750128 ◽  
Author(s):  
Anda Xiong ◽  
Julien C. Sprott ◽  
Jingxuan Lyu ◽  
Xilu Wang
Keyword(s):  
Lorenz System ◽  
Initial Conditions ◽  
Equilibrium Points ◽  
Basins Of Attraction ◽  
Symmetric Pair ◽  
State Space Approach ◽  
3D Printers ◽  
The Lorenz System ◽  
Almost All ◽  
Space Approach

The famous Lorenz system is studied and analyzed for a particular set of parameters originally proposed by Lorenz. With those parameters, the system has a single globally attracting strange attractor, meaning that almost all initial conditions in its 3D state space approach the attractor as time advances. However, with a slight change in one of the parameters, the chaotic attractor coexists with a symmetric pair of stable equilibrium points, and the resulting tri-stable system has three intertwined basins of attraction. The advent of 3D printers now makes it possible to visualize the topology of such basins of attraction as the results presented here illustrate.

A systematic identification approach for biaxial piezoelectric stage with coupled Duhem-type hysteresis

2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Qun Chen ◽  
Zong-Xiao Yang ◽  
Zhumu Fu
Keyword(s):  
Parameter Identification ◽  
High Efficiency ◽  
Full Range ◽  
Unknown Parameters ◽  
Content Type ◽  
De Algorithm ◽  
Complicated Model ◽  
Identification Approach ◽  
Cross Axis ◽  
Systematic Identification

Purpose The problem of parameter identification for biaxial piezoelectric stages is still a challenging task because of the existing hysteresis, dynamics and cross-axis coupling. This study aims to find an accurate and systematic approach to tackle this problem. Design/methodology/approach First, a dual-input and dual-output (DIDO) model with Duhem-type hysteresis is proposed to depict the dynamic behavior of the biaxial piezoelectric stage. Then, a systematic identification approach based on a modified differential evolution (DE) algorithm is proposed to identify the unknown parameters of the Duhem-type DIDO model for a biaxial piezostage. The randomness and parallelism of the modified DE algorithm guarantee its high efficiency. Findings The experimental results show that the characteristics of the biaxial piezoelectric stage can be identified with adequate accuracy based on the input–output data, and the peak-valley errors account for 2.8% of the full range in the X direction and 1.5% in the Y direction. The attained results validated the correctness and effectiveness of the presented identification method. Originality/value The classical DE algorithm has many adjustment parameters, which increases the inconvenience and difficulty of using in practice. The parameter identification of Duhem-type DIDO piezoelectric model is rarely studied in detail and its successful application based on DE algorithm on a biaxial piezostage is hitherto unexplored. To close this gap, this work proposed a modified DE-based systematic identification approach. It not only can identify this complicated model with more parameters, but also has little tuning parameters and thus is easy to use.

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