Static Nonlinearity Research Articles

Fixed point iteration‐based subspace identification of Hammerstein state‐space models

In this study, a fixed point iteration-based subspace identification method is proposed for Hammerstein state-space systems. The original system is decomposed into two subsystems with fewer parameters based on the hierarchical identification principle. Each subsystem is related directly to either the linear dynamics or the static non-linearity. A two-stage least-squares-based iterative method is then implemented to separately estimate the coefficients of the non-linear subsystem and the extended Markov parameters of the linear subsystem. The linear subsystem parameters are extracted from the identified extended Markov parameters using a singular value decomposition based method. Convergence analysis of the proposed method is established using fixed point theory, which shows that the proposed method gives consistent estimates under arbitrary non-zero initial conditions. Simulation results are included to show the performance of the proposed method.

Stable Process Approach to Analysis of Systems Under Heavy-Tailed Noise: Modeling and Stochastic Linearization

The Wiener process has provided a lot of practically useful mathematical tools to model stochastic noise in many applications. However, this framework is not enough for modeling extremal events, since many statistical properties of dynamical systems driven by the Wiener process are inevitably Gaussian. The goal of this work is to develop a framework that can represent a heavy-tailed distribution without losing the advantages of the Wiener process. To this end, we investigate models based on stable processes (this term “stable” has nothing to do with “dynamical stability”) and clarify their fundamental properties. In addition, we propose a method for stochastic linearization, which enables us to approximately linearize static nonlinearities in feedback systems under heavy-tailed noise, and analyze the resulting error theoretically. The proposed method is applied to assessing wind power fluctuation to show the practical usefulness.

SSA optimized digital pre-distorter for compensating non-linear distortion in high power amplifier

We applied the stochastic salp swarm algorithm (SSA) to design high power amplifier (PA) and digital pre-distorter (DPD) using generalized memory polynomial model. This algorithm has high exploitation and convergence speed to solve the non-linear coefficient of memory polynomial. We considered a single carrier WCDMA input for the static non-linearity of the memory based high power amplifier. Various simulations have been conducted to validate the performance of SSA over PA and DPD using different memory depths and degrees combinations which demonstrates that the proposed approach is an effective solution for linearity of high generation wideband transmitters. The performance of SSA is compared with particle swarm optimization and shows the superiority of SSA in terms of lower adjacent channel power ratio, error vector magnitude, normalized mean square error and modeling error. The implementation of SSA on PA and DPD has been done separately and then cascaded to generate a resultant linear output.

Phase Locking of Auditory-Nerve Fibers Reveals Stereotyped Distortions and an Exponential Transfer Function with a Level-Dependent Slope.

Phase locking of auditory-nerve-fiber (ANF) responses to the fine structure of acoustic stimuli is a hallmark of the auditory system's temporal precision and is important for many aspects of hearing. Period histograms from phase-locked ANF responses to low-frequency tones exhibit spike-rate and temporal asymmetries, but otherwise retain an approximately sinusoidal shape as stimulus level increases, even beyond the level at which the mean spike rate saturates. This is intriguing because apical cochlear mechanical vibrations show little compression, and mechanoelectrical transduction in the receptor cells is thought to obey a static sigmoidal nonlinearity, which might be expected to produce peak clipping at moderate and high stimulus levels. Here we analyze phase-locked responses of ANFs from cats of both sexes. We show that the lack of peak clipping is due neither to ANF refractoriness nor to spike-rate adaptation on time scales longer than the stimulus period. We demonstrate that the relationship between instantaneous pressure and instantaneous rate is well described by an exponential function whose slope decreases with increasing stimulus level. Relatively stereotyped harmonic distortions in the input to the exponential can account for the temporal asymmetry of the period histograms, including peak splitting. We show that the model accounts for published membrane-potential waveforms when assuming a power-of-three, but not a power-of-one, relationship to exocytosis. Finally, we demonstrate the relationship between the exponential transfer functions and the sigmoidal pseudotransducer functions obtained in the literature by plotting the maxima and minima of the voltage responses against the maxima and minima of the stimuli.SIGNIFICANCE STATEMENT Phase locking of auditory-nerve-fiber responses to the temporal fine structure of acoustic stimuli is important for many aspects of hearing, but the mechanisms underlying phase locking are not fully understood. Intriguingly, period histograms retain an approximately sinusoidal shape across sound levels, even when the mean rate has saturated. We find that neither refractoriness nor spike-rate adaptation is responsible for this behavior. Instead, the peripheral auditory system operates as though it contains an exponential transfer function whose slope changes with stimulus level. The underlying mechanism is distinct from the comparatively weak cochlear mechanical compression in the cochlear apex, and likely resides in the receptor cells.

Statistics-Based Approach for Blind Post-Compensation of Modulator’s Imperfections and Power Amplifier Nonlinearity

Power amplifier (PA) nonlinearity and in-phase and quadrature-phase (I/Q) imbalance are major concerns for wireless transmitters. In this paper, we present a new closed-form expression for the probability density function (PDF) of I and Q components in the presence of transmitter’s impairments and propose a blind post-compensation approach for the mitigation of these impairments. These impairments include static PA nonlinearity and frequency-independent I/Q imbalance. The accuracy of the analytical PDF is evaluated using Kullback–Leibler divergence and Hellinger square distance. Simulation results show a reasonable correspondence between the derived PDF and non-parametric kernel density estimation-based PDF. After a closed-form PDF is obtained, higher order statistics-based method is used to estimate PA nonlinearity in the presence of I/Q impairments. Finally, a maximum-likelihood estimation of I/Q imbalance parameters is obtained using the analytical PDF. Simulation results show a normalized mean-squared error (NMSE) of around −40 dB and an adjacent channel power ratio of around −53 dBc, along with an error vector magnitude (EVM) of around 1%, for a 3-MHz local thermal equilibrium signal. Using laboratory measurements, an NMSE of around −35 dB and an EVM of 1.5% are achieved.

Stability and convergence properties of forced infinite-dimensional discrete-time Lur'e systems

ABSTRACT Incremental stability and convergence properties for forced, infinite-dimensional, discrete-time Lur'e systems are addressed. Lur'e systems have a linear and nonlinear component and arise as the feedback interconnection of a linear control system and a static nonlinearity. Discrete-time Lur'e systems arise in, for example, sampled-data control and integro-difference models. We provide conditions, reminiscent of classical absolute stability criteria, which are sufficient for a range of incremental stability properties and input-to-state stability (ISS). Consequences of our results include sufficient conditions for the converging-input converging-state (CICS) property, and convergence to periodic solutions under periodic forcing.

High-bandwidth fine tracking system for optical communication with double closed-loop control method

A fine tracking system is crucial for maintaining the accuracy of the optical communication terminals that aim at each other. To ensure the reliability of the communication link, the fine tracking system requires high bandwidth to mitigate the effect of arrival angle fluctuation caused by atmospheric turbulence. Traditionally, the fine tracking system includes only a single feedback loop; a high bandwidth is obtained by increasing the high gain of the fine tracking system, which usually suffers considerably from the time delay engendered by sampling and data processing, the hysteresis nonlinearity of a fast steering mirror (FSM), and the limitations of dynamic response of FSM. To track the beacon in real time and with high precision, a pioneering control method is presented in our paper, namely, double closed-loop control (DCC), which performs better in a tracking system compared with a traditional single-feedback loop. In the inner feedback loop, the response of FSM is measured by a strain gauge sensor (SGS) and used as the inner feedback signal. Thus, by co-operating with the outer CCD-based feedback loop, a DCC scheme is proposed for the fine tracking system. With the SGS signal, the inner loop controller is designed to obtain a rapid response without overshooting; meanwhile, the hysteresis nonlinearity is diminished. Experimental results indicate that the static hysteresis nonlinearity of FSM is reduced from 15.6% to 1.4% by an inner feedback loop, and the dynamic response and stability of FSM is greatly improved, thereby simplifying the outer loop controller design. Then, with the SGS signal, the time delay of the outer loop can be compensated accurately with a predicted signal compensation method. The experimental results show that the −3 dB error rejection bandwidth is increased from 76 to 85 Hz, and the coupling efficiency in our optical communication system is improved by 16.87% after using the DCC fine tracking method. These results indicate that the DCC method can effectively achieve the goal of fast and accurate tracking for optical communications systems.

Recursive Parsimonious Subspace Identification for Closed-Loop Hammerstein Nonlinear Systems

In this paper, a recursive closed-loop subspace identification method for Hammerstein nonlinear systems is proposed. To reduce the number of unknown parameters to be identified, the original hybrid system is decomposed as two parsimonious subsystems, with each subsystem being related directly to either the linear dynamics or the static nonlinearity. To avoid redundant computations, a recursive least-squares (RLS) algorithm is established for identifying the common terms in the two parsimonious subsystems, while another two RLS algorithms are established to estimate the coefficients of the nonlinear subsystem and the predictor Markov parameters of the linear subsystem, respectively. Subsequently, the system matrices of the linear subsystem are retrieved from the identified predictor Markov parameters. The convergence of the proposed method is analyzed. Two illustrative examples are shown to demonstrate the effectiveness and merit of the proposed method.

Quantifying static and dynamic stiffness anisotropy and nonlinearity in finely laminated shales: Experimental measurement and modeling

Sedimentary rocks contain layers and a wide range of microstructures that may produce mechanical complexities including dynamic and quasistatic stiffness anisotropy and nonlinearity. However, most applications in geophysics and geomechanics disregard these mechanical complexities, which can lead to significant error and uncertainty in rock properties and may increase the risk associated with cost-intensive drilling and completions operations in shales. We have conducted simultaneous triaxial stress tests and ultrasonic wave propagation monitoring to measure and model stiffness anisotropy and nonlinearity of Mancos Shale plugs with varying bedding orientations. Results highlight the need for different sets of nonlinear coefficients to describe different stress loading paths, in which isotropic loading exhibits larger increases in stiffness for a given change in mean stress (and strain) than deviatoric loading. The vertical transverse isotropic (VTI) nonlinear model helps to account for the appreciable anisotropy and nonlinearity of Mancos samples, in which the dynamic Young’s moduli [Formula: see text] are more than 25% higher than [Formula: see text] and [Formula: see text] increases by approximately 35% during deviatoric stress loading. Measured static moduli are typically less than 50% of their dynamic equivalent and exhibit separate anisotropic and nonlinear relationships. Therefore, we have developed anisotropic stress-dependent dynamic-static transforms to estimate the static moduli from the nonlinear VTI model. Although heterogeneity and discontinuities cause samples to deviate from VTI symmetry, our modified dynamic-static transforms provide an excellent fit to the experimentally measured Young’s moduli and Poisson’s ratios. Post-test X-ray micro-CT imaging evidences the impact of sample layering and heterogeneity on rock failure and failure geometry. Bedding planes can act as preferential failure planes, whereas layering-induced mechanical stratigraphy can cause fractures to reorient due to changes in lithology. Our combined experimental, modeling, and imaging results provide insight into the complex deformational and failure behavior of shales. The analysis and results also highlight the need to consider the elastic and plastic deformations in shales.

Control issues of a fast positioner intended for deflection of a high power laser beam

This paper discusses some crucial aspects of control of a fast positioner intended for positioning of a mirror reflecting high power laser beam. Positioner is driven by a two linear movement piezo actuators acting on a common flexible flat spring bending it in two perpendicular directions. The reflector is fixed to this spring. Clearly, the actuator is a dynamic oscillating system. This makes control of positioner demanding. In order to overcome the effects of static nonlinearity, hysteresis and their time evolution, there were proposed two independent closed loop positioning systems each governing each own axis. The simulation investigations allowed for determining the suboptimal structure and parameters of controllers. Two different control loops were investigated and discussed. The simulation experiments shown, that the most efficient controller is able to “mask” the actuator hysteresis while introducing appropriate dynamic damping and stiffness to the controlled system. The concept of the positioner controller was described and some chosen results of laboratory experiments were presented. Finally, the assessment of the control quality is given together with some general conclusions closing the paper.

Method of Restoration of Input Signals of Nonlinear Dynamic Object with Destributed Parameters

The article deals with the method of signal restoration at the input of a nonlinear dynamic object with distributed parameters. To describe these objects, a universal mathematical model in the form of a Volterra integro-degree series has been chosen. The problem of signal restoration is reduced to the problem of solving the Volterra polynomial equation of the first kind. The numerical implementation of such models is suggested to be carried out using quadrature methods, in particular, the method of trapezoids. In order to increase the stability of the solution in the presence of noise interference in the input data, it is suggested to use the differential regularization operator, which allows the incorrectly set task to be transformed into a class of correct ones. The possibility of applying such an approach is studied in solving the Volterra polynomial integral equation of the second order type, which describes nonlinear dynamic objects with quadratic nonlinearity. The computational formulas for solving this type of equations are given in the article. The received nonlinear second-order algebraic equations after approximation of the initial equation by integral sums are solved by iterative methods with initial approximation in the form of a pre-calculated radical. The developed algorithms are implemented as software modules in the Matlab, with the help of which a number of computational experiments have been carried out. As an example, non-linear dynamic objects that contain static non-linearity of the second order and dynamic links that are typical for objects with distributed parameters have been chosen. Such links are: a semi-integral link, an attenuation link (semi-delay) and a semi-inertial link. On the basis of applying the equivalent transformations, the macromodels of objects with distributed parameters have been obtained in the form of a polynomial integral Volterra equation of the I kind of the second order with the kernels that describe the above-mentioned components. The results of the computational experiments, presented in the form of graphs, showed that the suggested approach can be effectively used in restoration of signals at the input of nonlinear dynamic objects with distributed parameters.

Distributed Hammerstein Modeling for Cross-Coupling Effect of Multiaxis Piezoelectric Micropositioning Stages

Hysteresis modeling is interesting yet challenging for piezoelectric actuated systems, which are often used in micro/nano scale measurement and manufacturing equipments. However, due to its complexity, few efforts have been devoted to characterizing cross-coupling hysteresis effect of multiaxis piezoelectric micropositioning stages. To this end, a distributed Hammerstein model, composed of a cascaded connection of a static nonlinearity and a dynamic linearity, is proposed in this paper to approximate the nonlinear spatial/temporal cross-coupling effect. This model outperforms conventional piezo models, such as the Preisach model. Meanwhile, theoretical analysis is provided to guarantee the convergence of the proposed Hammerstein model. Finally, extensive experiments are conducted to verify the superiority of the proposed modeling method.

A new Hammerstein model control strategy: feedback stabilization and stability analysis

This paper treats the identification, the control and the stability analysis of the discrete Hammerstein structure which is composed by a static nonlinearity gain associated with a linear dynamic subsystem. Its parameters are obtained from the subspace identification using the multivariable output error state space algorithm. The stabilization of this class of systems is generally leaded by a nonlinear control law, including an exact inverse of the static nonlinearity. Then, it is possible to use a linear state feedback controller to stabilize the Hammerstein system in closed-loop and to track the reference input. Thus, the performances of this method become limited when the considered static nonlinearity is approximately invertible. In this context, an approximate inverse of this component is investigated. Furthermore, based on the Lyapunov method, the stability conditions are obtained by solving a set of linear matrix inequalities constraints. The case of a coupled two-tank system is considered in order to illustrate the validity of the proposed approach.

Hysteresis nonlinearity modeling and linearization approach for Envelope Tracking Power Amplifiers in wireless systems

Novel Dual-Input Single-Output (DISO) behavioral modeling and linearization techniques for Envelope Tracking Power Amplifiers (ET PAs) in wireless application are presented in this paper. The proposed modeling approach applies the Hammerstein structure for the Amplitude-to-Phase (AM/PM) conversion. An extension of the Saleh AM/PM model is also proposed to model the static nonlinearity in the ET PA AM/PM conversion. This paper presents a new linearization technique for ET PAs by inverting the proposed ET PA model and using it as a Digital Predistortion (DPD). The measurement results show that both the proposed model and DPD can clearly outperform both the original Saleh model and the DPD based on the Saleh model inversion. Compared to the state-of-the-art behavioral models for ET PAs, the proposed model and DPD can offer an improved complexity-accuracy trade-off thanks to the lower number of coefficients.

Missing-Code-Occurrence Probability Calibration Technique for DAC Nonlinearity With Supply and Reference Circuit Analysis in a SAR ADC

This paper reports a calibration scheme that can be used to correct the inter-stage gain error in two-stage analog-to-digital converters (ADCs). We measure the static nonlinearity at the raw ADC outputs and compensate it through a feedback path to adjust the gain in the analog domain, which avoids the dynamic range loss in the conventional pure digital calibrations. The proposed scheme detects the error via the missing-code-occurrence probability (MCOP), which is insensitive to the comparator offset and has better immunity to its noise. Besides the gain error, we discuss other DAC linearity issues, such as the implementations of the reference buffer and the low-dropout (LDO) regulator. To verify the MCOP calibration, we design a 12-bit successive-approximation-register ADC, employing the bridge-DAC, in 65-nm CMOS. We integrate the ADC operating at 100 MS/s with the MCOP calibration, the reference buffers, and the LDO. The measurement results demonstrate the linearity enhancement after the proposed calibration, where we improve the signal-to-noise-and-distortion ratio to 61.38 dB, leading to a FoM of 16.7 fJ/conversion step at the Nyquist input frequency.

A simplified model of a fueldraulic actuation system with application to load estimation

In this work, a simplified model of the compressor variable stator vane fueldraulic actuation system of a jet engine is presented. The actuation system is a sub-assembly of the engine’s hydro-mechanical unit. A unique characteristic of the actuator is an internal cooling flow which prevents the overheating of fuel. It is shown that the effect of the cooling flow is well represented by a static input nonlinearity. The resulting model is of the Hammerstein structure. It is then shown that the model can be used for the estimation of the actuator’s external load. The results are validated using an accurate real system simulator.

Computational modelling of salamander retinal ganglion cells using machine learning approaches

Artificial vision using computational models that can mimic biological vision is an area of ongoing research. One of the main themes within this research is the study of the retina and in particular, retinal ganglion cells which are responsible for encoding the visual stimuli. A common approach to modelling the internal processes of retinal ganglion cells is the use of a linear – non-linear cascade model, which models the cell’s response using a linear filter followed by a static non-linearity. However, the resulting model is generally restrictive as it is often a poor estimator of the neuron’s response. In this paper we present an alternative to the linear – non-linear model by modelling retinal ganglion cells using a number of machine learning techniques which have a proven track record for learning complex non-linearities in many different domains. A comparison of the model predicted spike rate shows that the machine learning models perform better than the standard linear – non-linear approach in the case of temporal white noise stimuli.

Analysis of Lur’e dominant systems in the frequency domain

Frequency domain analysis of linear time-invariant (LTI) systems in feedback with static nonlinearities is a classical and fruitful topic of nonlinear systems theory. We generalize this approach beyond equilibrium stability analysis with the aim of characterizing feedback systems whose asymptotic behavior is low dimensional. We illustrate the theory with a generalization of the circle criterion for the analysis of multistable and oscillatory Lur’e feedback systems.

Modelling and identification of nonlinear cascade systems with backlash input and static output nonlinearities

ABSTRACTA new approach to the parameter identification of nonlinear dynamic systems using cascade models with nonlinear dynamic, linear dynamic and nonlinear static blocks is presented. Application of the key-term separation principle provides special expressions for the corresponding nonlinear model description that are linear in parameters. A least-squares-based iterative technique is proposed allowing estimation of all the model parameters based on measured input/output data. Illustrative examples of nonlinear cascade systems identification with input backlash and nonlinear static output characteristics are included.

Structure selection based on interval predictor model for recovering static non‐linearities from chaotic data

This study introduces a method of structure selection based on interval predictor model (IPM) and sum of squares formulation. The main contribution is to provide polynomial identified models that can recover static non-linearities from chaotic data. Moreover, the dynamical behaviour of the identified models is also examined in the structure selection by considering convex combinations of the polynomial functions that describe the IPM. Numerical experiments contemplating non-linear maps borrowed from the literature are presented to illustrate the potential and efficacy of the proposed approach.