Static Nonlinear Function Research Articles

Online identification of Hammerstein systems with B‐spline networks

SummaryNonlinear systems widely exist in real‐word applications and the research for these systems has enjoyed a long and fruitful history, including the system identification community. However, the modeling for nonlinear systems is often quite challenging and still remains many unresolved questions. This article considers the online identification issue of Hammerstein systems, whose nonlinear static function is modeled by a B‐spline network. First, the identification model of the studied system is constructed using the bilinear parameter decomposition model. Second, the online recursive algorithms are proposed to find the estimates using the moving data window and the particle swarm optimization procedure, and show that these estimates converge to their true values with a low computational burden. Numerical examples are also given to test the effectiveness of the proposed algorithms.

Compensating torque ripples in a coarse pointing mechanism for free-space optical communication: A Gaussian process repetitive control approach

Actuators that require commutation algorithms, such as the switched reluctance motor (SRM) considered in this paper and employed in the coarse pointing assembly (CPA) for free-space optical communication, often have torque-ripple disturbances that are periodic in the commutation-angle domain that deteriorate the positioning performance. The aim of this paper is to model the torque ripple as a Gaussian Process (GP) in the commutation-angle domain and consequently compensate for it at arbitrary velocity. The approach employs repetitive control (RC) at a constant velocity. A spatial GP with a periodic kernel is trained using data that is obtained from the RC step resulting in a static non-linear function for compensation at arbitrary velocity. Stability conditions are provided for both steps. The approach is successfully applied to a CPA prototype to improve the tracking performance for laser communication, where the torque ripple is compensated at arbitrary velocity.

A 2D-LUT Scheme Design for Complex-Valued Spline Adaptive Filter

This brief presents a two dimensional Look-Up-Table (2D-LUT) scheme for complex-valued (CV) spline adaptive filter. The key idea of this structure is to model a complex-valued nonlinear static function by implementing the Carstesian product of two univariate cubic uniform B-spline curves. Therefore, the real and imaginary components of the system complex-valued output can be readily achieved hinging on two different sets of spline control point respectively, each of which is stored in a 2D-LUT separately. In addition, as an extension of the real-valued (RV) sign spline adaptive algorithm, the CV counterpart is also derived to update both the weight coefficients of the linear subsystem and spline control points associated with 2D-LUTs. Simulations on the identification of RF nonlinear power amplifier (PA) show that the proposed scheme outperforms the existing methods in terms of estimation accuracy against impulsive noises, as well as lower computational complexity.

Identification of MISO Hammerstein system using sparse multiple kernel-based hierarchical mixture prior and variational Bayesian inference

The Hammerstein model is a cascade composition of a static memoryless nonlinear function followed by a linear time-invariant dynamical subsystem, which is capable of modeling a wide range of nonlinear dynamical systems. Model structural parameter selection (including the model order and the nonlinearity order) and sparse representation of the static nonlinear function are two issues that receive increasing interests in Hammerstein system identification. In this paper, a novel Bayesian sparse multiple kernel-based identification method (BSMKM) for multiple-input single-output (MISO) Hammerstein system is proposed to handle those issues, where the basis-function model and the finite impulse response model are used to represent the nonlinear part and the linear part respectively. Firstly, in order to jointly realize the model parameter estimation, the sparse representation of static nonlinear function (the nonlinearity order selection can also be realized indirectly) and the model order selection of linear dynamical system, a hierarchical prior distribution is constructed based on Gaussian scale mixture model and sparse multiple kernel, which can characterize both inter-group sparsity and intra-group correlation structure. Then, a full Bayesian method based on variational Bayesian inference is proposed to estimate all unknown model parameters, including finite impulse response coefficients, hyperparameters and noise variance. Finally, the performance of the proposed BSMKM identification method is evaluated by numerical experiments using both simulation and real data.

Comparison between an exact and a heuristic neural mass model with second-order synapses

Neural mass models (NMMs) are designed to reproduce the collective dynamics of neuronal populations. A common framework for NMMs assumes heuristically that the output firing rate of a neural population can be described by a static nonlinear transfer function (NMM1). However, a recent exact mean-field theory for quadratic integrate-and-fire (QIF) neurons challenges this view by showing that the mean firing rate is not a static function of the neuronal state but follows two coupled nonlinear differential equations (NMM2). Here we analyze and compare these two descriptions in the presence of second-order synaptic dynamics. First, we derive the mathematical equivalence between the two models in the infinitely slow synapse limit, i.e., we show that NMM1 is an approximation of NMM2 in this regime. Next, we evaluate the applicability of this limit in the context of realistic physiological parameter values by analyzing the dynamics of models with inhibitory or excitatory synapses. We show that NMM1 fails to reproduce important dynamical features of the exact model, such as the self-sustained oscillations of an inhibitory interneuron QIF network. Furthermore, in the exact model but not in the limit one, stimulation of a pyramidal cell population induces resonant oscillatory activity whose peak frequency and amplitude increase with the self-coupling gain and the external excitatory input. This may play a role in the enhanced response of densely connected networks to weak uniform inputs, such as the electric fields produced by noninvasive brain stimulation.

Perturbation-based pH control systems for buffer and equivalence points

The dynamics of the pH process is well-described by the Wiener model, in which a nonlinear static function follows a linear dynamic subsystem. Gain scheduling applied to the process output, or equivalently to the controller gain, has to be used to keep the pH process at a given set point under the severe static nonlinearity of the titration curve. The present study investigates two control problems of keeping the pH process on the buffer and equivalence points. The proposed method uses extremum seeking control techniques to find the first and second derivatives with continuous process input perturbations. It finds and maintains process operating points where the second derivatives of pH with respect to the process input changes are zero, the local points of minimum and maximum slopes. The performances of the proposed method are verified with simulation and experiment.

Hybrid CFD/low-order modeling of thermoacoustic limit cycle oscillations in can-annular configurations

We propose a hybrid strategy for modeling non-linear thermoacoustic phenomena, e.g. limit-cycle (LC) oscillations, in can-annular combustion systems. The suggested model structure comprises a compressible CFD simulation limited to the burner/flame zone of one single can, coupled to a low-order model (LOM) representing the remaining combustor. In order to employ the suggested strategy for modeling non-linear phenomena like LC oscillations, the LOM must capture non-linear flame dynamics in the cans, which are not resolved by CFD. Instead of identifying such non-linear flame models in preliminary simulations, we aim at learning the non-linear dynamics “on-the-fly”, while simulating the self-excited system under consideration. Based on the observation of flame dynamics in the CFD domain, the parameters of the employed non-linear models are estimated during run time. The present study reveals that block-oriented models, which comprise a linear dynamic part followed by a static non-linear function, are well suited for this purpose. The proposed hybrid model is applied to a laminar can-annular combustor. Results agree well with the monolithic CFD simulation of the entire combustor, while the computational cost is drastically reduced. The employed flame models, whose parameters are identified during the simulation of the self-excited LC oscillation, represent well the relevant non-linear dynamics of the considered flame.

An integrated control of front in-wheel motors and rear electronic limited slip differential for high-speed cornering performance

This paper presents an integrated control of in-wheel motor (IWM) and electronic limited slip differential (eLSD) for high-speed cornering performance. The proposed algorithm is designed to improve the handling performance near the limits of handling. The proposed controller consists of a supervisor, upper-level controller, and lower-level controller. First, the supervisor determines a target motion based on the yaw rate reference with a target understeer gradient. The target understeer gradient is devised to improve the lateral stability with in-wheel motor control based on a nonlinear static map. The yaw rate reference is designed based on the target understeer gradient to track the yaw reference with eLSD control. Second, the upper-level controller calculates the desired yaw moments for IWM and eLSD to generate the target motion. Third, the lower-level controller converts the desired yaw moment to the actuator torque commands for IWMs and eLSD. The tire friction limits are estimated based on the tire model and friction circle model to prevent tire saturation by limiting the torque inputs. The proposed algorithm has been investigated via both simulations and vehicle tests. The performance of the integrated control was compared with those of individual control and uncontrolled case in the simulation study. The vehicle tests have been performed using a rear wheel drive vehicle equipped with two front IWMs and eLSD in the rear axle. The vehicle test has been conducted at a racing track to show that the proposed algorithm can improve the lateral stability near the limits of handling.

Dual-Input Slope Seeking Control of Continuous Micro-Algae Cultures with Experimental Validation

This paper investigates the application of adaptive slope-seeking strategies to dual-input single output dynamic processes. While the classical objective of extremum seeking control is to drive a process performance index to its optimum, this paper also considers slope seeking, which allows driving the performance index to a desired level (which is thus sub-optimal). Moreover, the consideration of more than one input signal allows minimizing the input energy thanks to the degrees of freedom offered by the additional inputs. The actual process is assumed to be locally approachable by a Hammerstein model, combining a nonlinear static map with a linear dynamic model. The proposed strategy is based on the interplay of three components: (i) a recursive estimation algorithm providing the model parameters and the performance index gradient, (ii) a slope generator using the static map parameter estimates to convert the performance index setpoint into slope setpoints, and (iii) an adaptive controller driving the process to the desired setpoint. The performance of the slope strategy is assessed in simulation in an application example related to lipid productivity optimization in continuous cultures of micro-algae by acting on both the incident light intensity and the dilution rate. It is also validated in experimental studies where biomass production in a continuous photo-bioreactor is targeted.

Optimized Inverse Nonlinear Function-Based Wiener Model Predictive Control for Nonlinear Systems

In model predictive control applications, the static nonlinear function of the Wiener model is an obstacle to forming a quadratic cost function. That case results in suboptimal control actions. As is known, a Wiener model predictive controller makes predictions employing the linear block of the Wiener model, and so its control solution is optimal due to the quadratic cost function. If the actual process output is used as a parameter in the empirical model, the inverse of the nonlinear function cannot be obtained as the roots of the nonlinear function in an offline manner, and the Wiener model predictive controller cannot be developed. Having regard to those situations, this paper introduces an offset-free Wiener model-based predictive controller with optimized inverse nonlinear function for the perturbed neutralization processes. Here, to facilitate accounts in the prediction horizon, an autoregressive with an external input model is preferred as the linear element of the Wiener model, and the least-squares-based optimization method is proposed to get the inverse of the nonlinear function. In this way, the model correction term can be employed to achieve an offset-free controller just as employed in the linear model predictive controller. Finally, the control performance of the developed controller is confirmed via comparative MATLAB/Simulink studies.

A Blended Active Detection Strategy for False Data Injection Attacks in Cyber-Physical Systems

In recent years, different solutions have been proposed to detect advanced stealthy cyber-attacks against networked control systems. In this article, we propose a blended detection scheme that properly leverages and combines two existing detection ideas, namely, watermarking and moving target . In particular, a watermarked signal and a nonlinear static auxiliary function are combined to both limit the attacker's disclosure resources and obtain an unidentifiable moving target. The proposed scheme is capable of detecting a broad class of false data injection attacks, including zero-dynamics, replay, and covert attacks. Moreover, it is shown that the proposed approach mitigates the drawbacks of standard moving target and watermarking defense strategies. Finally, an extensive simulation study is reported to contrast the proposed detector with recent competitor schemes and provide tangible evidence of the effectiveness of the proposed solution.

Detecting Weak Signals by Using Memristor-Involved Chua’s Circuit and Verification in Experimental Platform

Since the traditional detection methods cannot accurately detect, determine and extract weak signals, the extreme sensitivity of chaotic systems to initial values is used for weak signal detection using a memristor-based chaotic system. Then, in order to find out all kinds of static nonlinear circuits suitable for Chua’s circuit with identical parameters, a comparative research platform is designed to generate five kinds of nonlinearity by taking advantage of the active short-circuit line method using the memristor-involved chaotic Chua’s circuit. The comparative research platform consists of three parts: a linear circuit unit, multiple nonlinear static function circuits and a nonlinear characteristic curve measurement unit connected by an electronic switch. By pressing the space bar, the switch between the active short-circuit line and the physical short-circuit line can be realized. The diffeomorphism between them is proved by comparing the memristive nonlinearity shape and the trilinear amplitude limiting the nonlinearity in the chaotic systems. Accordingly, hardware circuit experiments are carried out to verify the effectiveness and feasibility of the comparative research platform with various nonlinearity for Chua’s circuit. A good agreement is shown between the numerical simulations and the experimental results.

One-shot set-membership identification of Generalized Hammerstein–Wiener systems

In this paper we consider the identification of Generalized Hammerstein–Wiener systems in the presence of bounded noise affecting both the input and the output sequences. The considered class of systems is characterized by the cascade connection of Hammerstein building blocks, where the nonlinear static input map is a polynomial, which is not assumed to be invertible. The problem of computing the parameter uncertainty intervals is formulated in terms of a set of suitable semialgebraic optimization problems, where both the parameter to be estimated and the inner unmeasurable signals connecting all the subsystems are considered as decision variables. The effectiveness of the proposed algorithm is shown by means of a numerical simulation example. The presented approach is also applied to the problem of identifying the mathematical model of a JFET amplifier from a set of experimentally collected data.

Mem-models as building blocks for simulation and identification of hysteretic systems

In this study, mem-springs and mem-dashpots from a newly introduced family of mem-models are used as fundamental building blocks in hysteresis modeling. The usefulness of such assemblies of mem-models is investigated for both simulation and system identification. First, numerical simulations demonstrate the general capability of these models to describe strain ratcheting behaviors. Next, system identification is addressed by extending the concepts of mem-springs to include linear and nonlinear springs and those of mem-dashpots to include linear and nonlinear dashpots. A reconfigurable device made of steel and/or shape memory alloy wires and wire ropes provides a fitting test for the proposed mem-model-based family. A system identification procedure corroborated by physical insights is proposed and the results are validated using physics-based analysis. Multilayer feedforward neural networks are used for static nonlinear function approximation. The model class and system identification procedure proposed here are shown to extract similarities and dissimilarities among different configurations of the device by quantifying the spring and damping effects.

Quantifying the Nonlinear Interaction in the Nervous System Based on Phase-Locked Amplitude Relationship.

This paper introduces the Cross-frequency Amplitude Transfer Function (CATF), a model-free method for quantifying nonlinear stimulus-response interaction based on phase-locked amplitude relationship. The CATF estimates the amplitude transfer from input frequencies at stimulation signal to their harmonics/intermodulation at the response signal. We first verified the performance of CATF in simulation tests with systems containing a static nonlinear function and a linear dynamic, i.e., Hammerstein and Wiener systems. We then applied the CATF to investigate the second-order nonlinear amplitude transfer in the human proprioceptive system from the periphery to the cortex. The simulation demonstrated that the CATF is a general method, which can well quantify nonlinear stimulus-response amplitude transfer for different orders of nonlinearity in Wiener or Hammerstein system configurations. Applied to the human proprioceptive system, we found a complicated nonlinear system behavior with substantial amplitude transfer from the periphery stimulation to cortical response signals in the alpha band. This complicated system behavior may be associated with the nonlinear behavior of the muscle spindle and the dynamic interaction in the thalamocortical radiation. This paper provides a new tool to identify nonlinear interaction in the nervous system. The results provide novel insight of nonlinear dynamics in the human proprioceptive system.

Control-oriented model reduction for a class of hyperbolic systems with application to managed pressure drilling

This paper presents a model reduction approach for systems of hyperbolic partial differential equations (PDEs) with nonlinear boundary conditions. These systems can be decomposed into a feedback interconnection of a linear hyperbolic subsystem and a static nonlinear mapping. This structure motivates us to reduce the overall model complexity by only reducing the linear subsystem (the PDE part). We show that the linear PDE subsystem can effectively be approximated by a cascaded structure of systems of continuous time difference equations (CTDEs) and ordinary differential equations (ODEs), where the CTDE captures the infinite-dimensional nature of the PDE model. These systems are constructed by adapting an interpolation method based on frequency-domain data. Models in the form of hyperbolic PDEs with nonlinear boundary conditions are for example encountered in managed pressure drilling (MPD). The proposed technique is verified by application to such an MPD model.

An efficient model-error model update strategy for multi-stage NMPC with model-error model

Multi-stage NMPC with model-error model (MS-MEM) handles structural plant-model mismatch present in the nominal model of the plant in a non-conservative fashion. A model-error model (MEM) that consists of a stable linear time-invariant dynamics and a static time-variant nonlinear mapping is built using the past data such that it captures the unmodeled dynamics of the plant. The scenario tree is built for the nominal and for the extreme realizations of the plant obtained using the nominal model and the model-error model, and a multi-stage decision problem is formulated. In this paper, we propose an efficient strategy to update the model-error model present in the MS-MEM approach if new measurements invalidate the model-error model. The advantages of the proposed scheme over the previous approach where only the gain of the linear model is updated are demonstrated for a continuous stirred tank reactor (CSTR) benchmark example.

Parametric Identification of Nonlinear Fractional Hammerstein Models

In this paper, a system identification method for continuous fractional-order Hammerstein models is proposed. A block structured nonlinear system constituting a static nonlinear block followed by a fractional-order linear dynamic system is considered. The fractional differential operator is represented through the generalized operational matrix of block pulse functions to reduce computational complexity. A special test signal is developed to isolate the identification of the nonlinear static function from that of the fractional-order linear dynamic system. The merit of the proposed technique is indicated by concurrent identification of the fractional order with linear system coefficients, algebraic representation of the immeasurable nonlinear static function output, and permitting use of non-iterative procedures for identification of the nonlinearity. The efficacy of the proposed method is exhibited through simulation at various signal-to-noise ratios.

Hierarchical least squares identification for feedback nonlinear equation-error systems

Because of complex structures, the identification of nonlinear systems is very difficult, especially for closed-loop nonlinear systems (i.e., feedback nonlinear systems). This paper considers the parameter identification of a feedback nonlinear system where the forward channel is a controlled autoregressive model and the feedback channel is a static nonlinear function. Using the hierarchical identification principle decomposes a feedback nonlinear system into two subsystems, one contains the parameters of the linear dynamic block and the other contains the parameters of the nonlinear static block. A hierarchical least squares algorithm and a recursive least squares algorithm are presented for feedback nonlinear systems. The proposed algorithms are simple in principle and easy to implement on-line.

Machine learning–based piecewise affine model of wind turbines during maximum power point tracking

AbstractIn this paper, a discrete‐time piecewise affine (PWA) model of a wind turbine during Maximum Power Point Tracking (MPPT) region is identified. A clustering‐based identification method is utilized to create PWA maps for nonlinear aerodynamic torque and thrust force functions. This method exploits the combined use of clustering, pattern recognition, and parameter identification techniques. The well‐known K‐means clustering method is employed along with a perceptron‐based multiclassifier for pattern recognition and the least squared technique for parameter estimation. The identified maps are approximated the nonlinear static functions of the dynamic model of the wind turbine. Characteristics of a 5‐MW wind turbine are considered and the resulting model, which consists of 25 subregions is compared with the nonlinear dynamic model. Two test cases are studied in order to validate the presented model. Simulation results demonstrate the effectiveness and accuracy of the PWA model such that the response of the identified PWA model is fitted well to the nonlinear one. The PWA model identified in this paper can be widely used for advanced control systems design and long‐term performance and security assessment of the power grid.