Nonlinear State Monitoring With Karhunen-Loe`ve-Transform

2004 ◽  
Author(s):  
Philipp Glo¨smann ◽  
Edwin Kreuzer
Keyword(s):  
Nonlinear Systems ◽  
Coherent Structures ◽  
Nonlinear Dynamical Systems ◽  
Short Review ◽  
Mathematical Concept ◽  
Random Processes ◽  
State Monitoring ◽  
Process Data ◽  
Nonlinear Dynamical ◽  
Nonlinear State

Nonlinear dynamical systems often appear to have uncorrelated output. This characteristic leads to the idea of analyzing the dynamics of nonlinear systems with methods developed for random processes. The Karhunen-Loe`ve-Transform (KLT) was designed to detect coherent structures in random process data. It can also be applied for state monitoring of complex systems. This paper gives a short review on the mathematical concept of the KLT and discusses an approach to characterize the dynamics of nonlinear systems based on experimental data.

Carathe´odory Controllability Criterion for Nonlinear Dynamical Systems

1978 ◽  
Vol 100 (3) ◽  
pp. 209-213 ◽  
Author(s):  
G. Langholz ◽  
M. Sokolov
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Nonlinear Systems ◽  
Control Theory ◽  
Linear Systems ◽  
Nonlinear Dynamical Systems ◽  
Nonlinear Dynamical ◽  
Modern Control Theory ◽  
Open Question ◽  
Modern Control

The question of whether a system is controllable or not is of prime importance in modern control theory and has been actively researched in recent years. While it is a solved problem for linear systems, it is still an open question when dealing with bilinear and nonlinear systems. In this paper, a controllability criterion is established based on a theorem by Carathe´odory. By associating a given dynamical system with a certain Pfaffian equation, it is argued that the system is controllable (uncontrollable) if its associated Pfaffian form is nonintegrable (integrable).

PREDICTIVE CONTROL OF NONLINEAR SYSTEMS BASED ON IDENTIFICATION BY BACKPROPAGATION NETWORKS

1994 ◽  
Vol 05 (04) ◽  
pp. 335-344 ◽  
Author(s):  
JIANBIN HAO ◽  
JOOS VANDEWALLE ◽  
SHAOHUA TAN
Keyword(s):  
Nonlinear Systems ◽  
Predictive Control ◽  
Neural Control ◽  
Nonlinear Dynamical Systems ◽  
Universal Approximation ◽  
Backpropagation Algorithm ◽  
Nonlinear Dynamical ◽  
Control Scheme ◽  
One Step ◽  
Simulation Results

Using the property of universal approximation of multilayer perceptron neural network, a class of discrete nonlinear dynamical systems are modeled by a perceptron with two hidden layers. A backpropagation algorithm is then used to train the model to identify the nonlinear systems to a desired level of accuracy. Based on the identified model, a one-step-ahead predictive control scheme is proposed in which the future control inputs are obtained through some nonlinear optimization process. Making use of the online learning properties of neural networks, the predictive control scheme is further developed into an adaptive one which is robust to the incompleteness of identification. Simulation results show that this neural control scheme works well even for some very complicated nonlinear systems.

scholarly journals Exact Asymptotic Stability Analysis and Region-of-Attraction Estimation for Nonlinear Systems

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
Min Wu ◽  
Zhengfeng Yang ◽  
Wang Lin
Keyword(s):  
Stability Analysis ◽  
Nonlinear Systems ◽  
Asymptotic Stability ◽  
Lyapunov Function ◽  
Lyapunov Functions ◽  
Nonlinear Dynamical Systems ◽  
Region Of Attraction ◽  
Nonlinear Dynamical ◽  
Recovery Techniques ◽  
Regions Of Attraction

We address the problem of asymptotic stability and region-of-attraction analysis of nonlinear dynamical systems. A hybrid symbolic-numeric method is presented to compute exact Lyapunov functions and exact estimates of regions of attraction of nonlinear systems efficiently. A numerical Lyapunov function and an estimate of region of attraction can be obtained by solving an (bilinear) SOS programming via BMI solver, then the modified Newton refinement and rational vector recovery techniques are applied to obtain exact Lyapunov functions and verified estimates of regions of attraction with rational coefficients. Experiments on some benchmarks are given to illustrate the efficiency of our algorithm.

scholarly journals Dynamical Movement Primitives: Learning Attractor Models for Motor Behaviors

2013 ◽  
Vol 25 (2) ◽  
pp. 328-373 ◽  
Author(s):  
Auke Jan Ijspeert ◽  
Jun Nakanishi ◽  
Heiko Hoffmann ◽  
Peter Pastor ◽  
Stefan Schaal
Keyword(s):  
Motor Control ◽  
Dynamical Systems ◽  
Nonlinear Systems ◽  
Nonlinear Dynamical Systems ◽  
Emergent Behavior ◽  
Nonlinear Dynamical ◽  
Movement Primitives ◽  
Model Complex ◽  
Linear Differential ◽  
Goal Directed Behavior

Nonlinear dynamical systems have been used in many disciplines to model complex behaviors, including biological motor control, robotics, perception, economics, traffic prediction, and neuroscience. While often the unexpected emergent behavior of nonlinear systems is the focus of investigations, it is of equal importance to create goal-directed behavior (e.g., stable locomotion from a system of coupled oscillators under perceptual guidance). Modeling goal-directed behavior with nonlinear systems is, however, rather difficult due to the parameter sensitivity of these systems, their complex phase transitions in response to subtle parameter changes, and the difficulty of analyzing and predicting their long-term behavior; intuition and time-consuming parameter tuning play a major role. This letter presents and reviews dynamical movement primitives, a line of research for modeling attractor behaviors of autonomous nonlinear dynamical systems with the help of statistical learning techniques. The essence of our approach is to start with a simple dynamical system, such as a set of linear differential equations, and transform those into a weakly nonlinear system with prescribed attractor dynamics by means of a learnable autonomous forcing term. Both point attractors and limit cycle attractors of almost arbitrary complexity can be generated. We explain the design principle of our approach and evaluate its properties in several example applications in motor control and robotics.

NONLINEAR TIME SEQUENCE ANALYSIS

1991 ◽  
Vol 01 (03) ◽  
pp. 521-547 ◽  
Author(s):  
PETER GRASSBERGER ◽  
THOMAS SCHREIBER ◽  
CARSTEN SCHAFFRATH
Keyword(s):  
Dynamical Systems ◽  
Sequence Analysis ◽  
Lyapunov Exponents ◽  
Spectral Methods ◽  
Nonlinear Dynamical Systems ◽  
Short Review ◽  
Time Sequence ◽  
Nonlinear Dynamical ◽  
Time Sequences ◽  
Time Sequence Analysis

We review several aspects of the analysis of time sequences, and concentrate on recent methods using concepts from the theory of nonlinear dynamical systems. In particular, we discuss problems in estimating attractor dimensions, entropies, and Lyapunov exponents, in reducing noise and in forecasting. For completeness and since we want to stress connections to more traditional (mostly spectrum-based) methods, we also give a short review of spectral methods.

Nonlinear Dynamical Systems, Their Stability, and Chaos

2014 ◽  
Vol 66 (2) ◽  
Author(s):  
Amol Marathe ◽  
Rama Govindarajan
Keyword(s):  
Fluid Flow ◽  
Dynamical Systems ◽  
Nonlinear Systems ◽  
Fluid Mechanics ◽  
Fixed Points ◽  
Chaotic Systems ◽  
Nonlinear Dynamical Systems ◽  
Not Given ◽  
Nonlinear Dynamical ◽  
Detailed Explanation

This introduction to nonlinear systems is written for students of fluid mechanics, so connections are made throughout the text to familiar fluid flow systems. The aim is to present how nonlinear systems are qualitatively different from linear and to outline some simple procedures by which an understanding of nonlinear systems may be attempted. Considerable attention is paid to linear systems in the vicinity of fixed points, and it is discussed why this is relevant for nonlinear systems. A detailed explanation of chaos is not given, but a flavor of chaotic systems is presented. The focus is on physical understanding and not on mathematical rigor.

Signal Based Methods to Extract A Priori Information for the Identification of Nonlinear Systems

2001 ◽  
Author(s):  
Jens-Uwe Bruns ◽  
Karl Popp
Keyword(s):  
Nonlinear Systems ◽  
Nonlinear Dynamical Systems ◽  
A Priori ◽  
A Priori Information ◽  
Nonlinear Dynamical ◽  
Additional Information ◽  
System States ◽  
Priori Information ◽  
System Order

Abstract The identification of nonlinear dynamical systems still is a non-trivial procedure. Signal based methods that retrieve basic information about the system prior to detailed identification can provide valuable assistance in this task. In the paper the detection and characterization of nonlinearities as well as the estimation of the system order are discussed. Regarding the first topic, a new method is presented that is based on the Method of Internal Harmonics Cross-Correlation by Dimentberg and Sokolov but provides additional information about the nature of the nonlinearity and uses nonlinear instead of linear correlation. Concerning the second topic, a method is presented that, in contrast to most existing methods for nonlinear systems, incorporates a random input signal to promote the excitation of all system states. Both methods are illustrated with numerical examples.

scholarly journals Symmetries and itineracy in nonlinear systems with many degrees of freedom

2001 ◽  
Vol 24 (5) ◽  
pp. 813-813 ◽  
Author(s):  
Michael Breakspear ◽  
Karl Friston
Keyword(s):  
Dynamical Systems ◽  
Nonlinear Systems ◽  
Brain Function ◽  
Degrees Of Freedom ◽  
Nonlinear Dynamical Systems ◽  
Research Direction ◽  
Potential Contribution ◽  
Nonlinear Dynamical

Tsuda examines the potential contribution of nonlinear dynamical systems, with many degrees of freedom, to understanding brain function. We offer suggestions concerning symmetry and transients to strengthen the physiological motivation and theoretical consistency of this novel research direction: Symmetry plays a fundamental role, theoretically and in relation to real brains. We also highlight a distinction between chaotic “transience” and “itineracy.”

The Response Statistics, Jump and Bifurcation of Nonlinear Dynamical Systems Subjected to White Noise and Combined Sinusoidal and White Noise Excitation

2007 ◽  
Author(s):  
Pankaj Kumar ◽  
S. Narayanan
Keyword(s):  
Nonlinear Systems ◽  
White Noise ◽  
Nonlinear Dynamical Systems ◽  
Gaussian White Noise ◽  
Nonlinear Dynamical ◽  
White Noise Excitation ◽  
Bladed Disk Assembly ◽  
Noise Frequency ◽  
System Nonlinearity ◽  
Non Gaussian

The prediction of the response of nonlinear systems subjected to stochastic parametric, narrowband and wideband or coloured external excitation is of importance in the field of structural and rotor dynamics. The transitional probability density function (pdf) for the random response of nonlinear systems under white or coloured noise excitation (delta-correlated) is governed by both the forward Fokker-Planck (FP) and backward Kolmogorov equations. This paper presents efficient numerical solution of the FP equation for the pdf of response for general nonlinear systems subjected to external white noise and combined sinusoidal and white noise excitation. The effect of intensity of white noise, frequency and amplitude of sinusoidal excitation and level of system nonlinearity on the non-Gaussian nature of response caused by the system nonlinearity are investigated. Stochastic behaviours like stability, jump, bifurcation are examined as the system parameters change. The finite element (FE) scheme is used to solve the FP equation and obtain the statistics of a two degree-of-freedom linear system representative of the vibration of gas turbine tip-shrouded bladed disk assembly subjected to Gaussian white noise excitation as an illustrative example.

Periodic Flows to Chaos Based on Discrete Implicit Mappings of Continuous Nonlinear Systems

2015 ◽  
Vol 25 (03) ◽  
pp. 1550044 ◽  
Author(s):  
Albert C. J. Luo
Keyword(s):  
Time Delay ◽  
Dynamical Systems ◽  
Nonlinear Systems ◽  
Nonlinear Dynamical Systems ◽  
Duffing Oscillator ◽  
Single Step ◽  
Nonlinear Dynamical ◽  
Periodic Flows ◽  
Mapping Structures ◽  
Stability And Bifurcations

This paper presents a semi-analytical method for periodic flows in continuous nonlinear dynamical systems. For the semi-analytical approach, differential equations of nonlinear dynamical systems are discretized to obtain implicit maps, and a mapping structure based on the implicit maps is employed for a periodic flow. From mapping structures, periodic flows in nonlinear dynamical systems are predicted analytically and the corresponding stability and bifurcations of the periodic flows are determined through the eigenvalue analysis. The periodic flows predicted by the single-step implicit maps are discussed first, and the periodic flows predicted by the multistep implicit maps are also presented. Periodic flows in time-delay nonlinear dynamical systems are discussed by the single-step and multistep implicit maps. The time-delay nodes in discretization of time-delay nonlinear systems were treated by both an interpolation and a direct integration. Based on the discrete nodes of periodic flows in nonlinear dynamical systems with/without time-delay, the discrete Fourier series responses of periodic flows are presented. To demonstrate the methodology, the bifurcation tree of period-1 motion to chaos in a Duffing oscillator is presented as a sampled problem. The method presented in this paper can be applied to nonlinear dynamical systems, which cannot be solved directly by analytical methods.

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