Spectral analysis of non-linear systems involving square-law operations

This paper contains a number of useful theoretical formulas to analyze the frequency domain properties of Gaussian input data passing through non-linear square-law systems. Special bispectral density functions are defined and applied that are functions of a single variable. From measurements of input data and output data only, results are obtained to identify the separate frequency response functions for two models of linear systems in parallel with non-linear square-law systems. Non-linear coherence functions are defined from these models which determine the proportion of the output spectrum due to the non-linear operations. Together with ordinary coherence functions, a measured output spectrum for these models can be decomposed into three components representing the linear operations, the non-linear operations, and the remaining uncorrelated noise effects. This material indicates also how to analyze other types of non-linear models by employing similar techniques.     

FREQUENCY DOMAIN ARX MODEL AND MULTI-HARMONIC FRF ESTIMATORS FOR NON-LINEAR DYNAMIC SYSTEMS

Non-linear dynamic systems respond at frequencies other than the excitation frequency; however, standard frequency response function estimators for linear systems do not accommodate this harmonic distortion. A new multi-harmonic frequency response function estimator that utilizes discrete frequency models for non-linear systems is introduced here. The multi-harmonic estimator relates the frequency response at each frequency to the input and output spectra within a given frequency band in the same way that autoregressive exogenous input models relate inputs and outputs at particular samples in the time domain. Overdetermined, least-mean-squares calculations are used to minimize model error throughout a frequency band rather than at a single frequency as in the corresponding linear estimators. The resulting multi-harmonic frequency response function models are non-parametric (e.g., vary with amplitude) when linear functions are used and parametric when non-linear functions are used. A new sensitive indicator for experimentally characterizing non-linearity is introduced.     

A computational algorithm for computing nonlinear auditory frequency selectivity.

Computational algorithms that mimic the response of the basilar membrane must be capable of reproducing a range of complex features that are characteristic of the animal observations. These include complex input output functions that are nonlinear near the site's best frequency, but linear elsewhere. This nonlinearity is critical when using the output of the algorithm as the input to models of inner hair cell function and subsequent auditory-nerve models of low- and high-spontaneous rate fibers. We present an algorithm that uses two processing units operating in parallel: one linear and the other compressively nonlinear. The output from the algorithm is the sum of the outputs of the linear and nonlinear processing units. Input to the algorithm is stapes motion and output represents basilar membrane motion. The algorithm is evaluated against published chinchilla and guinea pig observations of basilar membrane and Reissner's membrane motion made using laser velocimetry. The algorithm simulates both quantitatively and qualitatively, differences in input/output functions among three different sites along the cochlear partition. It also simulates quantitatively and qualitatively a range of phenomena including isovelocity functions, phase response, two-tone suppression, impulse response, and distortion products. The algorithm is potentially suitable for development as a bank of filters, for use in more comprehensive models of the peripheral auditory system.     

Estimation of the impulse response of ultra-wide-band systems

The time and frequency algorithms of impulse response recovery in linear ultra-wide-band systems by a finite set of digital data presenting the input and output signals are proposed. The results of approbation of these algorithms by numerical simulation in the presence of additive Gaussian noise in the output signal are given. It is shown that the systems with noise-like signals at the input have the largest noise immunity in the estimation of impulse response. Institute for High-Current Electronics of the Russian Academy of Sciences, Tomsk, Russia. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 41, No. 9, pp. 1195–1206, September, 1998.     

Statistical errors for non-linear system measurements involving square-law operations

This paper develops normalized random error formulas for special bispectra estimates and associated frequency response function estimates in finite memory square-law systems. Error formulas are also derived for output spectrum estimates from these non-linear systems and for associated non-linear coherence functions. These formulas are useful to evaluate such measured non-linear results as well as to design experimental programs.     

A time-varying identification method for mixed response measurements

The proper orthogonal decomposition is a method that may be applied to linear and nonlinear structures for extracting important information from a measured structural response. This method is often applied for model reduction of linear and nonlinear systems and has been applied recently for time-varying system identification. Although methods have previously been developed to identify time-varying models for simple linear and nonlinear structures using the proper orthogonal decomposition of a measured structural response, the application of these methods has been limited to cases where the excitation is either an initial condition or an applied load but not a combination of the two. This paper presents a method for combining previously published proper orthogonal decomposition-based identification techniques for strictly free or strictly forced systems to identify predictive models for a system when only mixed response data are available, i.e. response data resulting from initial conditions and loads that are applied together. This method extends the applicability of the previous proper orthogonal decomposition-based identification techniques to operational data acquired outside of a controlled laboratory setting. The method is applied to response data generated by finite element models of simple linear time-invariant, time-varying, and nonlinear beams and the strengths and weaknesses of the method are discussed.     

Variational Bayesian identification and prediction of stochastic nonlinear dynamic causal models

In this paper, we describe a general variational Bayesian approach for approximate inference on nonlinear stochastic dynamic models. This scheme extends established approximate inference on hidden-states to cover: (i) nonlinear evolution and observation functions, (ii) unknown parameters and (precision) hyperparameters and (iii) model comparison and prediction under uncertainty. Model identification or inversion entails the estimation of the marginal likelihood or evidence of a model. This difficult integration problem can be finessed by optimising a free-energy bound on the evidence using results from variational calculus. This yields a deterministic update scheme that optimises an approximation to the posterior density on the unknown model variables. We derive such a variational Bayesian scheme in the context of nonlinear stochastic dynamic hierarchical models, for both model identification and time-series prediction. The computational complexity of the scheme is comparable to that of an extended Kalman filter, which is critical when inverting high dimensional models or long time-series. Using Monte-Carlo simulations, we assess the estimation efficiency of this variational Bayesian approach using three stochastic variants of chaotic dynamic systems. We also demonstrate the model comparison capabilities of the method, its self-consistency and its predictive power.     

A damage identification technique based on embedded sensitivity analysis and optimization processes

A vibration based structural damage identification method, using embedded sensitivity functions and optimization algorithms, is discussed in this work. The embedded sensitivity technique requires only measured or calculated frequency response functions to obtain the sensitivity of system responses to each component parameter. Therefore, this sensitivity analysis technique can be effectively used for the damage identification process. Optimization techniques are used to minimize the difference between the measured frequency response functions of the damaged structure and those calculated from the baseline system using embedded sensitivity functions. The amount of damage can be quantified directly in engineering units as changes in stiffness, damping, or mass. Various factors in the optimization process and structural dynamics are studied to enhance the performance and robustness of the damage identification process. This study shows that the proposed technique can improve the accuracy of damage identification with less than 2 percent error of estimation.     

System identification from multiple input/output data

For stationary random or transient data representing multicorrelated (multicoherent) input/output data occurring in physical systems, iterative computational algorithms are developed to identify the frequency response functions of optimum constant parameter linear systems connecting this data. Results are obtained from Fourier transform methods and optimum least-squares prediction techniques by changing original arbitrary input records into ordered sets of conditioned input records. These procedures provide the basis for efficient digital computer analysis of general multiple input/output problems. The Appendix contains useful error analysis results for the optimum frequency response estimates determined from measured data.     

Nonlinear identification of NMR spin systems by adaptive filtering

In this paper, we present two new methods for identifying NMR spin systems. These methods are based on nonlinear adaptive filtering. The spin system is assumed to be time-invariant with memory. The first method uses a truncated discrete Volterra series to describe the nonlinear relationship between excitation (input) and system response (output). First-, second-, and third-order kernels of this series are estimated employing the least mean square (LMS) algorithm. Three parallel filters can then model the NMR spin system so that its output is no more than simple sum of three convolution products between combinations of the input signal and filters coefficients. It is also shown that the contribution of the Volterra second-order term to the total system response is neglected compared with the contributions of the first- and the third-order terms. In the second identification method, the output signal is related to the input signal through a recursive nonlinear difference equation with constant coefficients. The LMS algorithm is used again to estimate the equation coefficients. The two methods are validated with a simulated NMR system model based on Bloch equations. The results and the performances of these methods are analyzed and compared. It is shown that our methods permit a simple identification of NMR spin systems. The field of applications of this study is promising in the optimization of NMR signal detection, especially in the cases of low signal-to-noise ratios where optimum signal filtering and analysis must be performed.     

A gradient-free adaptation method for nonlinear active noise control

Active Noise Control (ANC) problems are often affected by nonlinear effects, such as saturation and distortion of microphones and loudspeakers. Nonlinear models and specific adaptation algorithms must be employed to properly account for these effects. The nonlinear structure of the problem complicates the application of gradient-based Least Mean Squares (LMS) algorithms, due to the fact that exact gradient calculation requires executing nonlinear recursive filtering operations, which pose computational and stability issues. One favored solution to this problem consists in neglecting recursive terms in the gradient calculation, an approximation which is not always without consequences on the convergence performance. Besides, an efficient application of nonlinear models cannot avoid some form of model structure selection, to avoid the well-known effects of overparametrization and to reduce the computational load on-line. Unfortunately, the standard ANC setting configures an indirect identification problem, due to the presence of the secondary path in the control loop. In the nonlinear case, this destroys the linear regression structure of the problem even if the control filter is linear-in-the-parameters, thereby making it impossible to apply the many existing model selection methods for linear regression problems. A simple and computationally wise low demanding approach is here proposed for parameter estimation and model structure selection that provides an answer to the mentioned issues. The proposed method avoids altogether the use of the error gradient and relies on direct cost function evaluations. A virtualization scheme is used to assess the accuracy improvements when the model is subject to parametric or structural modifications, without directly affecting the control performance. Several simulation examples are discussed to show the effectiveness of the proposed algorithms.     

The effects of nonlinearity on the output frequency response of a passive engine mount

In this paper, the new concept of output frequency-response function (OFRF) that was derived recently by the authors from the Volterra-series theory of nonlinear systems is briefly introduced. An effective algorithm is proposed to determine the monomials in the OFRF-based representation of the output frequency response of nonlinear systems. The results are then used to analyze the output frequency response of a passive engine mount. Important conclusions regarding the effects of system nonlinearity on the output frequency-response behaviors of the engine mount are reached via theoretical analysis and verified by simulation studies. These conclusions are of significant importance for the analysis and design of vibration isolators such as engine mounts in practice.     

Identifying nonlinear biomechanical models by multicriteria analysis

In this study, the methodology developed by Srdjevic and Cveticanin (International Journal of Industrial Ergonomics 34 (2004) 307–318) for the nonbiased (objective) parameter identification of the linear biomechanical model exposed to vertical vibrations is extended to the identification of n-degree of freedom (DOF) nonlinear biomechanical models. The dynamic performance of the n-DOF nonlinear model is described in terms of response functions in the frequency domain, such as the driving-point mechanical impedance and seat-to-head transmissibility function. For randomly generated parameters of the model, nonlinear equations of motion are solved using the Runge–Kutta method. The appropriate data transformation from the time-to-frequency domain is performed by a discrete Fourier transformation. Squared deviations of the response functions from the target values are used as the model performance evaluation criteria, thus shifting the problem into the multicriteria framework. The objective weights of criteria are obtained by applying the Shannon entropy concept. The suggested methodology is programmed in Pascal and tested on a 4-DOF nonlinear lumped parameter biomechanical model. The identification process over the 2000 generated sets of parameters lasts less than 20 s. The model response obtained with the imbedded identified parameters correlates well with the target values, therefore, justifying the use of the underlying concept and the mathematical instruments and numerical tools applied. It should be noted that the identified nonlinear model has an improved accuracy of the biomechanical response compared to the accuracy of a linear model.     

Forecasting with imperfect models, dynamically constrained inverse problems, and gradient descent algorithms

This paper is part of a larger study investigating the meaning of, and appropriate procedures for, forecasting with imperfect models. (In the author’s opinion there is currently no satisfactory general theory and practice for doing so with complex nonlinear systems.) The focus of this paper is on initialisation of the forecast. At the heart of every forecasting scheme there is an inverse problem that translates observations of reality into an initial state, or ensemble of states, of the model. Inverse problems are divided into two classes depending on whether the underlying model of reality is that of a stochastic process or a (deterministic) dynamical process. The two classes have quite different formulations of their inverse problems and consequent solutions methods. This paper considers dynamical process models and their inverse problems, which will be referred to as the dynamically constrained inverse problem (DCIP) line. The interpretation and solutions of the DCIP line are investigated and new algorithms for solving them are presented. The new algorithms are modifications of classical gradient descent algorithms. The new algorithms are applied to a low-dimensional chaotic system and a high-dimensional operational weather forecasting model. Our examination of DCIP shows that gradient descent algorithms are an effective way of solving the inverse problem for complex nonlinear system given an imperfect dynamical model.     

Intraband discrete breathers in disordered nonlinear systems. I. Delocalization

The existence of Discrete Breathers or DBs (also called Intrinsic Localized Modes or ILMs) and multibreathers, is investigated in a simple one-dimensional chain of random anharmonic oscillators with quartic potentials coupled by springs. When the breather frequency is outside and above the linearized (phonon) spectrum, the existence theorems and numerical methods previously used in periodic nonlinear models for finding time-periodic and spatially localized solutions, hold identically in random nonlinear systems. These solutions are extraband discrete breathers (EDBs). When the frequencies penetrate inside the linearized spectrum, the existence theorems do not hold. Our numerical investigations demonstrate that the strict continuation of (localized) EDBs as intraband discrete breathers (IDBs) is impossible because of cascades of bifurcations generating many discontinuities. A detailed analysis of these bifurcations for small systems with increasing sizes, shows that only a relatively small subset of the spatially extended multibreathers can be strictly continued while their frequency varies inside the phonon spectrum. We propose an ansatz for finding the coding sequences of these solutions and continuing safely these multibreathers in finite systems of any size. This continuation ends at a lower limit frequency where the solution annihilates through a bifurcation with another multibreather. A smaller subset of these multibreather solutions can be continued to amplitude zero and become linear localized modes at this limit. Conversely, any linear localized mode can be continued when increasing its frequency as an extended multibreather. Extrapolation of these results to infinite systems yields the main conclusion of this first part which is that nonlinearity in disordered systems (with localized eigenmodes only) restores their capability of energy transportation by generating infinitely many spatially extended time-periodic solutions. This approach yields mainly spatially extended solutions, except sometimes at their bifurcation points. In the second part of this work, which is presented in our next article, we develop an accurate method for calculating in situ localized IDBs.     

Identifiability and observability analysis for experimental design in nonlinear dynamical models

Dynamical models of cellular processes promise to yield new insights into the underlying systems and their biological interpretation. The processes are usually nonlinear, high dimensional, and time-resolved experimental data of the processes are sparse. Therefore, parameter estimation faces the challenges of structural and practical nonidentifiability. Nonidentifiability of parameters induces nonobservability of trajectories, reducing the predictive power of the model. We will discuss a generic approach for nonlinear models that allows for identifiability and observability analysis by means of a realistic example from systems biology. The results will be utilized to design new experiments that enhance model predictiveness, illustrating the iterative cycle between modeling and experimentation in systems biology.     

A central spectrum model: a synthesis of auditory-nerve timing and place cues in monaural communication of frequency spectrum

A probabilistic psychophysical model for monaural communication from the auditory nerve to the brain is given in the form of a tonotopic display of stimulus spectrum, termed central spectrum. The model builds upon prior research demonstrating the potential of neural timing cues from the auditory nerve for conveying information on complex spectra, and was designed to meet the quantified demands of the psychophysics of frequency measurement. The central spectrum magnitude at each frequency is determined by the response of the auditory-nerve fibre with characteristic frequency matching that frequency. An interval histogram from each fiber is passed through a filter matched to the characteristic frequency of the fiber. This output versus characteristic frequency defines the central spectrum. Detailed analysis demonstrates that efficient probabilistic processing of the central spectrum described known psychophysical properties of frequency measurement in discrimination and periodicity pitch experiments. Psychophysical models based upon the central spectrum model followed by optimum probabilistic pattern recognition are potentially relevant for predicting human communication limits in response to arbitrary sounds of speech and music.     

Unscented Kalman filter and its nonlinear application for tracking a moving target

The extended Kalman filter (EKF) is probably the most widely used estimation algorithm for nonlinear systems. However, more than 40 years of experience in the estimation community has shown that is difficult to implement, difficult to tune, and only reliable for systems that are almost linear on the time scale of the updates. To overcome these limitations, this paper proposes the unscented Kalman filter (UKF). And the algorithms of the FEKF, SEKF and UKF are given. Furthermore, the state models and measurement models of a target are set up. For comparison purpose, the three algorithms is simulated for the target tracking, and the algorithm performance is analyzed and compared by the simulation results of FEKF, SEKF and UKF. Numerical results demonstrate that FEKF and UKF give almost identical results while the estimates of SEKF are clearly worse. The UKF is easier to implement, avoiding Jacobian and Hessian matrices computation.     

Nonparametric identification of asymmetric nonlinear vibration systems with the Hilbert transform

The objective of the paper is presenting a simple and more accurate technique for precise identification of nonlinear elastic force functions acting in asymmetric vibration systems. The identification procedure based on the Hilbert transform is a nonparametric one; it does not require a priori information about the system structure or its parameters. The examples of the identification of asymmetric classic vibration nonlinear models – the Helmholtz and the double-well Duffing oscillators – are investigated.     

Neural network models of sound localization based on directional filtering by the pinna.

Three-layer neural-network functions were developed to transform spectral representations of pinna-filtered stimuli at the input to a space-mapped representation of sound-source direction at the output. The inputs are modeled after transfer functions of the external ear of the cat; the output is modeled on the spatial sensitivity of superior colliculus neurons. Network solutions are obtained by backpropagation and by a method that enforces uniform task distribution in the hidden layer of the model. Solutions are characterized using bandlimited inputs to study the relative strength of potential sound localization cues in various frequency regions. This analysis suggests that the frequency region containing the first spectral notch (5-18 kHz) provides the best localization cues. Response properties of model neurons were studied using input patterns modeled after auditory nerve response profiles to pure tones at various frequencies and sound levels. The response properties of hidden layer model neurons resemble cochlear nucleus types III and IV and their composites. Neurons in both hidden and output layers show the properties of spectral notch detectors. Although neural networks have limitations as models of real neural systems, the results illustrate how they can provide insight into the computation of complex transformations in the nervous system.