Estimating model parameters by chaos synchronization

Global modeling of complex data series using the term-ranking approach and its application to voice synthesis

A term-ranking approach is proposed to globally model the underlying dynamics of a chaotic series. The basic idea of this approach is to rank candidate bases before they are used to construct the global model. The ranked bases are involved in the global model one by one in a sequence from high to low until the best model is found. Simulations show that the model obtained by the term-ranking approach has a much longer prediction time, but fewer coefficients, than the widely used standard model. The proposed approach is also successfully applied to coding and synthesis of chaoslike voice data, showing promise for its use with truly noisy experimental data.

Detection of dynamical structure from short and noisy chaotic series

A method is proposed to detect the dynamical structure hiding behind complex chaotic series by comparing prediction performance of trial functions. This method is valid even when the original system is contaminated with noise or only a relatively short data series is recorded. Using this method, the dynamical structure of the Gray-Scott model is detected from its rich spatiotemporal patterns. Finally, this method is successfully applied to the experimental data of Chua's circuit, which promises its potential of the application to realistic systems.

Structure identification based on steady-state control: experimental results and applications

We report experimental results on structure identification of nonlinear systems by a steady-state control method. The idea underlying the method is to drive the nonlinear system to steady state by applying a suitable feedback control input. It turns out experimentally that this control-based structure identification method can be used for some applications, such as estimation of initial conditions and state variables of nonlinear systems and structure identification of some special elements. Two attractors of the Chua oscillator are presented to illustrate the reliability of the suggested techniques under the hypotheses of measurable state variables and physical access to the system for implementing the proportional feedback.

Parameter and state estimation of experimental chaotic systems using synchronization

We examine the use of synchronization as a mechanism for extracting parameter and state information from experimental systems. We focus on important aspects of this problem that have received little attention previously and we explore them using experiments and simulations with the chaotic Colpitts oscillator as an example system. We explore the impact of model imperfection on the ability to extract valid information from an experimental system. We compare two optimization methods: an initial value method and a constrained method. Each of these involves coupling the model equations to the experimental data in order to regularize the chaotic motions on the synchronization manifold. We explore both time-dependent and time-independent coupling and discuss the use of periodic impulse coupling. We also examine both optimized and fixed (or manually adjusted) coupling. For the case of an optimized time-dependent coupling function u(t) we find a robust structure which includes sharp peaks and intervals where it is zero. This structure shows a strong correlation with the location in phase space and appears to depend on noise, imperfections of the model, and the Lyapunov direction vectors. For time-independent coupling we find the counterintuitive result that often the optimal rms error in fitting the model to the data initially increases with coupling strength. Comparison of this result with that obtained using simulated data may provide one measure of model imperfection. The constrained method with time-dependent coupling appears to have benefits in synchronizing long data sets with minimal impact, while the initial value method with time-independent coupling tends to be substantially faster, more flexible, and easier to use. We also describe a method of coupling which is useful for sparse experimental data sets. Our use of the Colpitts oscillator allows us to explore in detail the case of a system with one positive Lyapunov exponent. The methods we explored are easily extended to driven systems such as neurons with time-dependent injected current. They are expected to be of value in nonchaotic systems as well. Software is available on request.

Adaptive scheme for synchronization-based multiparameter estimation from a single chaotic time series and its applications

Chaos-synchronization-based multiparameter estimation of a multiply delayed feedback system is investigated. We propose an adaptive method that can estimate all the parameters of the response system using the driving signal only. In the past few years, various methods have been developed for estimation of multiparameters of a chaotic system but most of them require more than one time series to estimate all the parameters of a chaotic or hyperchaotic system. The proposed method requires only a single chaotic time series to estimate all the parameters. A sufficient condition for synchronization is derived and it is shown that the numerical results well support the analytic calculations. The synchronized system has applications in cryptographic encoding for digital and analog signals, which is shown with an example.

Chaotic component obscured by strong periodicity in voice production system

The effect of glottal aerodynamics in producing the nonlinear characteristics of voice is investigated by comparing the outputs of the asymmetric composite model and the two-mass model. The two-mass model assumes the glottal airflow to be laminar, nonviscous, and incompressible. In this model, when the asymmetric factor is decreased from 0.65 to 0.35, only 1:1 and 1:2 modes are detectable. However, with the same parameters, four vibratory modes (1:1, 1:2, 2:4, 2:6) are found in the asymmetric composite model using the Navier-Stokes equations to describe the complex aerodynamics in the glottis. Moreover, the amplitude of the waveform is modulated by a small-amplitude noiselike series. The nonlinear detection method reveals that this noiselike modulation is not random, but rather it is deterministic chaos. This result agrees with the phenomenon often seen in voice, in which the voice signal is strongly periodic but modulated by a small-amplitude chaotic component. The only difference between the two-mass model and the composite model is in their descriptions of glottal airflow. Therefore, the complex aerodynamic characteristics of glottal airflow could be important in generating the nonlinear dynamic behavior of voice production, including bifurcation and a small-amplitude chaotic component obscured by strong periodicity.

Estimating parameters by autosynchronization with dynamics restrictions

We suggest a general approach to parameter estimation using autosynchronization with some restrictions on system dynamics. This parameter identification method can be extended to estimate parameters from a scalar time series. Furthermore, we propose an average filter method to suppress the influence of noise on parameter estimation. Some limits and extensions of the autosynchronization method are given as well. Several examples are presented to illustrate all methods suggested.

Parameter identification technique for uncertain chaotic systems using state feedback and steady-state analysis

A technique is introduced for identifying uncertain and/or unknown parameters of chaotic dynamical systems via using simple state feedback. The proposed technique is based on bringing the system into a stable steady state and then solving for the unknown parameters using a simple algebraic method that requires access to the complete or partial states of the system depending on the dynamical model of the chaotic system. The choice of the state feedback is optimized in terms of practicality and causality via employing a single feedback signal and tuning the feedback gain to ensure both stability and identifiability. The case when only a single scalar time series of one of the states is available is also considered and it is demonstrated that a synchronization-based state observer can be augmented to the state feedback to address this problem. A detailed case study using the Lorenz system is used to exemplify the suggested technique. In addition, both the Rössler and Chua systems are examined as possible candidates for utilizing the proposed methodology when partial identification of the unknown parameters is considered. Finally, the dependence of the proposed technique on the structure of the chaotic dynamical model and the operating conditions is discussed and its advantages and limitations are highlighted via comparing it with other methods reported in the literature.

Chaos synchronization and parameter estimation from a scalar output signal

We propose an observer-based approach for chaos synchronization and parameter estimation from a scalar output signal. To begin with, we use geometric control to transform the master system into a standard form with zero dynamics. Then we construct a slaver to synchronize with the master using a combination of slide mode control and linear feedback control. Within a finite time, partial synchronization is realized, which further results in complete synchronization as time tends to infinity. Even if there exists model uncertainty in the slaver, we can also estimate the unknown model parameter by a simple adaptive rule.

Estimating system parameters from chaotic time series with synchronization optimized by a genetic algorithm

A method is proposed to estimate system parameters by optimizing synchronization with a genetic algorithm. This method can effectively find the parameter values of a chaotic system with a rugged parameter landscape. Furthermore, even the parameters of a 200-dimensional coupled-map-lattice spatiotemporal chaotic system can be extracted from a scalar time series. Finally, a Chua's circuit experiment shows the capacity of this method to estimate multiple parameters of real systems.

Multiparameter estimation using only a chaotic time series and its applications

An important extension to the techniques of synchronization-based parameter estimation is presented. Based on adaptive chaos synchronization, several methods are proposed to dynamically estimate multiple parameters using only a scalar chaotic time series. In comparison with previous schemes, the presented methods decrease the cost of parameter estimation and are more applicable in practice. Numerical examples are used to demonstrate the effectiveness and robustness of the presented methods. As an example application, an implementation of multichannel digital communication is proposed, where multiparameter modulation is used to simultaneously transmit more than one digital message. From a theoretical perspective, such an encoding increases the difficulty to directly read out the message from the transmitted signal and decreases the implementation cost.

Extracting Physiologically Relevant Parameters of Vocal Folds From High-Speed Video Image Series

In this paper, a new method is proposed to extract the physiologically relevant parameters of the vocal fold mathematic model including masses, spring constants and damper constants from high-speed video (HSV) image series. This method uses a genetic algorithm to optimize the model parameters until the model and the realistic vocal folds have similar dynamic behavior. Numerical experiments theoretically test the validity of the proposed parameter estimation method. Then the validated method is applied to extract the physiologically relevant parameters from the glottal area series measured by HSV in an excised larynx model. With the estimated parameters, the vocal fold model accurately describes the vibration of the observed vocal folds. Further studies show that the proposed parameter estimation method can successfully detect the increase of longitudinal tension due to the vocal fold elongation from the glottal area signal. These results imply the potential clinical application of this method in inspecting the tissue properties of vocal fold.

Theoretical and experimental studies of parameter estimation based on chaos feedback synchronization

In this paper, we perform theoretical and experimental studies of parameter estimation based on chaos feedback synchronization. The Chua chaotic system is applied as the original system, and its time series of a single variable is applied as a feedback variable to drive a simulative system. The simplex method is employed to minimize the synchronization error in the simulative system to iteratively approach the original system parameters. Furthermore, the effects of feedback function, feedback weight, and noise are investigated. The parameter estimation method based on feedback synchronization shows its robustness to noise perturbation. Finally, the circuit experiment of parameter estimation is implemented in order to examine its practical applicability. The original Chua circuit parameters can be well approached from its recorded time series of a single variable. The results show the parameter estimation method based on feedback synchronization may be capable of estimating real system parameters from a chaotic time series.

Parameter estimation of an asymmetric vocal-fold system from glottal area time series using chaos synchronization

In this paper, we apply an iterative parameter adaption scheme based on chaos synchronization to estimate system parameters of the asymmetric vocal folds from glottal area time series. The original asymmetric vocal-fold system associated with recurrent laryngeal paralysis shows chaotic vibrations with positive Lyapunov exponents. Aperiodic glottal area time series from the original system will be applied as the feedback variable coupling the simulative and the original vocal-fold systems. The parameter adaption technique based on chaos synchronization is employed to manipulate the simulative system parameters. The chaotic vibrations, system parameters, and the bifurcation diagram of the original vocal-fold system can be exactly reproduced in the simulative system, and the two chaotic systems can be synchronized. Furthermore, the effects of noise, sampling rate, and equation difference due to nonlinear spring terms on vocal-fold parameter estimations are investigated. Despite large noise perturbations, large equation differences, and low sampling rate, the parameter adaption scheme can effectively estimate the original vocal-fold system parameters. This study provides a theoretical base to apply chaos synchronization to estimate the vocal-fold system parameters from the glottal area data and show its potential application in laryngeal physiology.

Adaptive approximation method for joint parameter estimation and identical synchronization of chaotic systems

We introduce a numerical approximation method for estimating an unknown parameter of a (primary) chaotic system which is partially observed through a scalar time series. Specifically, we show that the recursive minimization of a suitably designed cost function that involves the dynamic state of a fully observed (secondary) system and the observed time series can lead to the identical synchronization of the two systems and the accurate estimation of the unknown parameter. The salient feature of the proposed technique is that the only external input to the secondary system is the unknown parameter which needs to be adjusted. We present numerical examples for the Lorenz system which show how our algorithm can be considerably faster than some previously proposed methods.

Spatiotemporal chaos in excised larynx vibrations

Spatiotemporal chaos in excised larynx vibrations is reported using high-speed digital imaging. Spatiotemporal correlation and eigenmode analyses are applied to describe the spatiotemporal dynamics of the vocal fold vibrations and to investigate the effects of subglottal pressure. High subglottal pressures cause spatiotemporal chaos with decreased spatiotemporal correlation and increased entropy in the vocal fold vibrations. Spatiotemporal analysis shows a valuable biomedical application in investigating the spatiotemporal chaotic dynamics of the vocal fold system.

Simulations of temporal patterns of oral airflow in men and women using a two-mass model of the vocal folds under dynamic control

In this study we use a low-dimensional laryngeal model to reproduce temporal variations in oral airflow produced by speakers in the vicinity of an abduction gesture. It attempts to characterize these temporal patterns in terms of biomechanical parameters such as glottal area, vocal fold stiffness, subglottal pressure, and gender differences in laryngeal dimensions. A two-mass model of the vocal folds coupled to a two-tube approximation of the vocal tract is fitted to oral airflow records measured in men and women during the production of /aha/ utterances, using the subglottal pressure, glottal width, and Q factor as control parameters. The results show that the model is capable of reproducing the airflow records with good approximation. A nonlinear damping characteristics is needed, to reproduce the flow variation at glottal abduction. Devoicing is achieved by the combined action of vocal fold abduction, the decrease of subglottal pressure, and the increase of vocal fold tension. In general, the female larynx has a more restricted region of vocal fold oscillation than the male one. This would explain the more frequent devoicing in glottal abduction-adduction gestures for /h/ in running speech by women, compared to men.

Synchronized pseudorandom systems and their application to speech communication

An approach to the synchronization of pseudorandom systems is proposed and applied to secure speech communication. The encoding signal produced by the pseudorandom synchronization scheme passes the random test, and shows much more complex dynamics, better random properties, and greater sensitivity to parameter mismatches than that produced by the active-passive decomposition scheme. Also, two coupled pseudorandom systems can be exactly synchronized despite their different initial states or seeds. Pseudorandom encoding and synchronization may yield great security in communication.