Hammerstein Structure Research Articles

Nonlinear RF impairments modeling with OFDM symbol-based parallel Wiener–Hammerstein structure

In the paper, a novel orthogonal frequency division multiplexing (OFDM) symbol-based parallel Wiener–Hammerstein structure is proposed to deal with the problem of nonlinear radio frequency (RF) impairments modeling for a cyclic prefix OFDM (CP-OFDM) waveform in telecommunication systems. The aforementioned structure is an equivalent linear-in-parameters representation of the parallel Wiener–Hammerstein structure in the considered problem. The structure is formally derived enabling the usage of identification algorithms compared with the artificial neural network (ANN) approach. The solution is compliant with current telecommunication standards.

An Adaptive Hammerstein Model for FES-Induced Torque Prediction Based on Variable Forgetting Factor Recursive Least Squares Algorithm.

Modeling the muscle response to functional electrical stimulation (FES) is an important step during model-based FES control system design. The Hammerstein structure is widely used in simulating this nonlinear biomechanical response. However, a fixed relationship cannot cope well with the time-varying property of muscles and muscle fatigue. In this paper, we proposed an adaptive Hammerstein model to predict ankle joint torque induced by electrical stimulation, which used variable forgetting factor recursive least squares (VFFRLS) method to update the model parameters. To validate the proposed model, ten healthy individuals were recruited for short-duration FES experiments, ten for long-duration FES experiments, and three stroke patients for both. The isometric ankle dorsiflexion torque induced by FES was measured, and then the test performance of the fixed-parameter Hammerstein model, the adaptive Hammerstein model based on fixed forgetting factor recursive least squares (FFFRLS) and the adaptive Hammerstein model based on VFFRLS was compared. The goodness of fit, root mean square error, peak error and success rate were applied to evaluate the accuracy and stability of the model. The results indicate a significant improvement in both the accuracy and stability of the proposed adaptive model compared to the fixed-parameter model and the adaptive model based on FFFRLS. The proposed adaptive model enhances the ability of the model to cope with muscle changes.

Precision positioning control of wind tunnel centrosome

AbstractThe centrosome plays a crucial role in regulating the flow field of a continuous wind tunnel, and its positioning precision directly impacts the control accuracy of the Mach number. A high‐precision positioning control algorithm for the wind tunnel centrosome is proposed based on a model predictive control framework. Since that the centrosome is a typical non‐linear system with a Hammerstein structure comprising an input backlash and a linear dynamic block in series, a parameterized least‐squares identification method is constructed based on an adaptive forgetting factor strategy and a centrosome position prediction model with input backlash is established. To counteract the weakening effect of the backlash non‐linear block on the control signal, a compensator is constructed to compensate for the non‐linear input block. Considering that the input undetectable disturbance signal in the actuator may lead to unpredictable responses, an intermediate state observer is designed and the stable convergence of the observer algorithm is also proved. Finally, the closed‐loop stability of the proposed control algorithm is analyzed by constructing a Lyapunov function. Experimental testing verifies the superiority and applicability of the proposed algorithm.

High-Performance Flux Tracking Controller for Reluctance Actuator

To meet the ever-increasing demand for next-generation lithography machines, the actuator plays an important role in the achievement of high acceleration of the wafer stage. However, the voice coil motor, which is widely used in high-precision positioning systems, is reaching its physical limits. To tackle this problem, a novel way to design the actuator using the magnetoresistance effect is argued due to the high force densities. However, the strong nonlinearity limits its application in the nan-positioning system. In particular, the hysteresis is coupled with eddy effects and displacement, which lead to a rate-dependent and displacement-dependent hysteresis effect in the reluctance actuator. In this paper, a Hammerstein structure is used to model the rate-dependent reluctance actuator. At the same time, the displacement-dependent of the model is regarded as the interference with the system. Additionally, a control strategy combining inverse model compensation and the disturbance observer-based discrete sliding mode control was proposed, which can effectively suppress the hysteresis effect. It is worthy pointing out that the nonlinear system is transformed into a linear system with inversion bias and disturbance by the inverse model compensation. What is more, the sliding mode controller based on the disturbance observer is designed to deal with the unmodeled dynamics, displacement disturbances, and model identification errors in linear systems. Thus, the tracking performance and robustness to external disturbances of the system are improved. The simulation results show that it is superior to the PI controller combined with an inverse compensator and even to the discrete sliding mode controller connected with inverse compensator, confirming the effectiveness of the novel control method in alleviating hysteresis.

A Hierarchical Identification Method for Lithium-Ion Battery SOC Based on the Hammerstein Model

Two-input one-output Hammerstein model consists of two parallel nonlinear static blocks followed by a linear dynamic part. By using Hammerstein structure to map relation between a battery State of Charge (SOC) and its terminal voltage/current, a hierarchical stochastic gradient algorithm is studied to estimate parameters of Hammerstein SOC model, so as to predict battery SOC. Firstly, the Hammerstein model is transformed into a bilinear parameter system with the least number of required parameters. Then, a hierarchical stochastic gradient algorithm with a forgetting factor is used to update the two sets of model parameters of the bilinear parameter system, so as to realize SOC estimation. Furthermore, the experiment platform of lithium-ion battery was built and the data of the urban dynamometer driving schedule (UDDS) profile and the Los Angeles 92 (LA92) profile were collected. Finally, the MATLAB simulation results show that the proposed parameter optimized method based Hammerstein model has the advantages of fast convergence speed and high SOC estimation accuracy.

Design and on-site implementation of an off-grid marine current powered hydrogen production system

In this work, an off-grid hydrogen production system powered by marine current energy was studied, which employed a horizontal axis marine current turbine (HAMCT) and a polymer electrolyte membrane (PEM) electrolyzer fed with ultrapure water. The fluid kinetic energy of the marine current will be captured by the turbine and finally stored as hydrogen energy through the electrolysis reaction in the electrolyzer. It is important to fully understand the characteristics of the electrolyzer for the stable and efficient operation of the system. Here, a dynamic model of PEM electrolyzers was developed, which is based on the Hammerstein structure. The particle swarm optimization (PSO) method and the least squares method were used to fit the static part and the dynamic part of the model, respectively. The experimental validation shows enough precision for engineering applications and the ability to characterize the transient behavior of the electrolyzer. Faraday’s efficiency of the stack was modeled using an empirical formula. The simulation of the proposed system was then carried out using the measured current velocity data as input. The results demonstrate that the system achieved the designed operating performance with the power coefficient of 0.42 and the estimated average energy conversion efficiency from marine current-to-hydrogen of 16.4%. Then, the sea trial was conducted in Zhoushan Archipelago. The power coefficient and the average energy conversion efficiency were found to be 0.35 and 11.9% respectively, with a decrease compared to the simulated results, which was attributed to the idealization of the simulation model and the degradation of the PEM electrolyzer. The performance degradation of the PEM electrolyzer throughout experiments and its effects were discussed. The principle and feasibility of the marine current-hydrogen system were successfully demonstrated.

Nonlinear Hammerstein System Identification: A Novel Application of Marine Predator Optimization Using the Key Term Separation Technique

The mathematical modelling and optimization of nonlinear problems arising in diversified engineering applications is an area of great interest. The Hammerstein structure is widely used in the modelling of various nonlinear processes found in a range of applications. This study investigates the parameter optimization of the nonlinear Hammerstein model using the abilities of the marine predator algorithm (MPA) and the key term separation technique. MPA is a population-based metaheuristic inspired by the behavior of predators for catching prey, and utilizes Brownian/Levy movement for predicting the optimal interaction between predator and prey. A detailed analysis of MPA is conducted to verify the accurate and robust behavior of the optimization scheme for nonlinear Hammerstein model identification.

Performance Improvement in Fast Extremum Seeking With Adaptive Phase Compensator and High-Gain Optimizer

In this article, an adaptive phase compensator and a high-gain optimizer are proposed for the fast extremum-seeking (ES) scheme acting on a Hammerstein plant. Unlike the widely applied three-time scale tuning, a time scale-independent tuning is achieved by restricting the plant to have a Hammerstein structure and elevating the adaptation of the control input which makes it as fast as the dither signal. Albeit the existed fast ES schemes utilize a high-frequency sinusoidal dither in order to achieve fast minimization of the plant’s static nonlinearity, the phase shift of the plant could not be neglected as the frequency is high and the ES may fail to reach the extreme values if the shift is large enough. Thus, an effective adaptive frequency varying phase compensation method is introduced to deal with the phase shift under the assumptions in this article. Moreover, a high-gain optimizer is presented to remove the restriction that the adaptation gain has to be small and could still guarantee the stability properties, which makes the proposed scheme more practical. The effectiveness of the proposed scheme is illustrated by numerical simulations.

Multimuscle Functional-Electrical-Stimulation-Based Wrist Tremor Suppression Using Repetitive Control

Wrist motion is produced by a group of muscles acting in a coordinated way. However, existing functional electrical stimulation (FES)-based wrist tremor suppression methods just stimulate one pair of muscles, which can limit the tremor suppression performance and cause muscle fatigue. To address these problems, this article proposes a multimuscle FES-based wrist tremor suppression method by fully considering the properties of wrist motion. First, with the consideration of the mainly involved two pairs of muscles in wrist flexion and extension motion, a multimuscle wrist musculoskeletal model with a Hammerstein structure is developed, and the parameters are identified. Then, a feedback repetitive controller combined with a feedforward linearization controller is proposed for tremor suppression. A frequency-modified inverse repetitive control algorithm and a gradient-based repetitive control algorithm are put forward to regulate the FES level properly. Finally, experiments on both unimpaired subjects and intention tremor patients verify that compared to the existing single-muscle-pair FES-based methods, the proposed methods can substantially improve the performance of tremor suppression and effectively reduce the level of electrical stimulation significantly, thereby reducing muscle fatigue.

Coupling Modeling and Adaptive Control for Piezoelectric-Actuated Positioning Stage

In this paper, a nonlinear coupling model with hysteresis, dynamics, and creep is proposed to describe accurately the complex characteristics of piezoelectric-actuated positioning stage, where a classic Hammerstein model in series with a fractional-order model is given. The fractional-order model is presented to express the nonlinear creep characteristics. Firstly, the Hammerstein structure model is composed of two blocks, where the former block is the classical PI model to describe the static hysteresis effects, and the latter block is the second-order discrete transfer function model to characterize the dynamic characteristics. In addition, the parameters of the coupling model are identified. Secondly, based on the built model, the inverse of fractional-order model and the inverse of PI model are implemented as the feedforward compensations, and an adaptive control is designed to adjust the tracking performance of the whole system. Finally, the effectiveness of the proposed coupling model and controllers are verified by the piezoelectric-actuated positioning experiment stage. Experimental results show that the established coupling model can accurately characterize the hysteresis, dynamics, and creep properties of the stage. Also, the results show that the tracking error is less than 0.8% at low frequency and mixed frequency.

Multiperiodic Repetitive Control for Functional Electrical Stimulation-Based Wrist Tremor Suppression

Intention tremor refers to the rhythmic and involuntary contraction and relaxation of muscles with movement toward a target, which is a common sequela of multiple sclerosis and usually occurs in the distal joints of the upper limb. Functional electrical stimulation (FES) is feasible for tremor suppression because of its fewer side effects, low cost, and portability. Most existing FES-based design methods assume that tremor is a single-frequency signal, though it is multifrequency in reality. The idealized simplification will limit the performance of tremor suppression. To address the problem, this article proposes an FES-based multiperiodic repetitive control (MP-RC) scheme to suppress multiple frequency wrist tremors. First, a nonlinear wrist musculoskeletal model with a Hammerstein structure is established. Then, a control strategy combining the model inverse linearization control and MP-RC is proposed for tremor suppression. A frequency-modified inverse RC algorithm and a gradient-based RC algorithm are developed to regulate the FES level. Finally, comparative experiments on four unimpaired participants and an intention tremor patient are conducted to validate the effectiveness of the proposed control schemes. Experimental results show that the MP-RC scheme can suppress tremors by up to 90.52%. Compared with the traditional filter-based feedback controller and the single-periodic repetitive controller, the proposed multiperiodic repetitive controller can achieve an average of 26% and 16% improvement, respectively, in tremor suppression, demonstrating the advantages of the proposed design.

A kernel-based identification approach for a class of nonlinear systems with quantized output data

In this paper, we consider the problem of identification for a class of nonlinear systems with Hammerstein structure, which is a combination of some basis functions for the static nonlinearity and finite impulse response models for the linear block under quantized output data. Following the empirical Bayes method, we model the impulse response of the system using a Gaussian process whose covariance matrix depends on some suitable kernels. Then, we compute the minimum mean-square error estimation of the impulse response via a special case of Markov Chain Monte Carlo sampling approach, namely, Gibbs sampler. The estimation of the impulse response depends on the kernel hyperparameters, the noise variance and the coefficients of the static nonlinearity. These unknowns are estimated by the marginal likelihood maximization, which is solved via the expectation-conditional maximization algorithm combining the Gibbs sampler. The expectation-conditional maximization method is a variation of the standard expectation-maximization method for solving maximum-likelihood problems. Numerical simulations show the effectiveness of the proposed scheme.

Model Predictive Control With a Cascaded Hammerstein Neural Network of a Wind Turbine Providing Frequency Containment Reserve

This article presents an application of neural network-based Model Predictive Control (MPC) to improve the wind turbine control system’s performance in providing frequency control ancillary services to the grid. A closed-loop Hammerstein structure is used to approximate the behavior of a 5MW floating offshore wind turbine with a Permanent Magnet Synchronous Generator (PMSG). The multilayer perceptron neural networks estimate the aerodynamic behavior of the nonlinear steady-state part, and the linear AutoRegressive with Exogenous input (ARX) is applied to identify the linear time-invariant dynamic part. Using the specific structure of the Cascade Hammerstein design simplifies the online linearization at each operating point. The proposed algorithm evades the necessity of nonlinear optimization and uses quadratic programming to obtain control actions. Eventually, the proposed control design provides a fast and stable response to the grid frequency variations with optimal pitch and torque cooperation. The performance of the MPC is compared with the gain-scheduled proportional-integral (PI) controller. Results demonstrate the effectiveness of the designed control system in providing Frequency Containment Reserve (FCR) and frequency regulation in the future of power systems.

Low-Complexity Adaptive Frequency-Domain Nonlinear Equalization for Analog RoF Mobile Fronthaul Using FFT/IFFT-Assisted Channel Aggregation

Addressing the nonlinearity induced signal degradation is of critical importance for harvesting the transparent benefits of an analog radio over fiber (A-RoF) architecture, to support mobile fronthaul (MFH) with high spectral efficiency and reduced latency. Here, for linearizing the low-latency A-RoF MFH using digital-domain fast Fourier transform /inverse fast Fourier transform (FFT/IFFT) assisted channel aggregation/de-aggregation (CA/DA), we propose and experimentally demonstrate a compatible adaptive frequency-domain (FD) nonlinear equalization scheme. This scheme is characterized by a much lower computational complexity compared with the popular time-domain Volterra nonlinear equalizer while delivering a comparable performance. The proposed equalization scheme is implemented using the FD two-box cascaded Hammerstein structure. It enables simplified coefficients estimation, computationally efficient equalization, FFT/IFFT calculation reusing in channel DA, and shared legacy one-tap FD linear equalization in the OFDM system. Experimental validations of the proposed nonlinear equalization approach are carried out via typical electroabsorption modulator (EAM) and Mach-Zehnder modulator (MZM) based intensity modulation and direct detection (IMDD) A-RoF links. Cost-effective fiber-optic (with EAM) and optical up-conversion enabled V-band millimeter-wave (mm-wave) fiber-wireless (with MZM) transmissions are demonstrated. Experimental results show that the proposed scheme can effectively improve the performances of both EAM and MZM based 10-km fiber-optic A-RoF MFH links, in terms of the (>8 dB) reduction of adjacent channel leakage ratio, (>10 dB) improvement of error-vector magnitude (EVM) and (>5 dB) enhancement of input intermediate-frequency (IF) power range, which retains an acceptable (below 8 %) EVM performance for the 64-QAM modulation. The input IF signal aggregates 8 125-MHz 64-QAM OFDM channels. Furthermore, the proposed scheme enables the transmission of a 5-Gbit/s aggregated IF 64-QAM signal over a 10-km fiber link and 1.2-m 60-GHz wireless channel, wherein a >12 dB improved input power margin is observed.

Model Predictive Control Based on the Generalized Bouc-Wen Model for Piezoelectric Actuators in Robotic Hand With Only Position Measurements

Hysteresis nonlinearity is known to typify piezoelectric actuators and systems. If not appropriately accounted for in the control law, it can lead to the closed-loop’s performance loss or even instability. Several studies have proposed various techniques to design control laws with explicit consideration of this nonlinearity. However, they are limited to symmetrical hysteresis, and their efficiency is no longer ensured in the face of asymmetrical ones. This letter proposes to model the strong asymmetrical hysteresis in a piezoelectric actuator devoted to robotic hands using the generalized Bouc-Wen model. A Hammerstein structure is employed to also account for its dynamics. Then, we propose an output feedback control law consisting of nonlinear control and a nonlinear model predictive control. To get an estimate of the hysteresis signal used for feedback, we design a nonlinear observer. The observer and controls together stabilize the closed-loop output. The efficiency of the proposed control technique is finally demonstrated through several simulations.

State-Space Modeling of Viscous Unsteady Aerodynamic Loads

In this paper, we summarize our recently developed viscous unsteady theory, which couples potential flow with the triple-deck boundary-layer theory. This approach provides a viscous extension of potential-flow unsteady aerodynamics. As such, a Reynolds-number-dependent transfer function is determined for unsteady lift. We then use the Wiener–Hammerstein structure to develop a finite-dimensional approximation of such an infinite-dimensional theory, presenting it in a state-space model. This novel nonlinear state-space model of viscous unsteady aerodynamic loads is expected to serve aerodynamicists better than the classical Theodorsen’s model because it captures viscous effects (that is, Reynolds number dependence) as well as nonlinearity and additional lag in the lift dynamics; it also allows simulation of arbitrary time-varying airfoil motions (not necessarily harmonic). Moreover, being in a state-space form makes it quite convenient for simulation and coupling with structural dynamics to perform aeroelasticity, flight dynamics analysis, and control design. We then develop a linearization of such a model, which enables analytical results. Subsequently, we derive an analytical representation of the viscous lift frequency response function: an explicit function of both the frequency and Reynolds number. We also develop a state-space model of the linearized response. We finally simulate the nonlinear and linear models to a nonharmonic small-amplitude pitching maneuver at a Reynolds number of 100,000 and compare the resulting lift and pitching moment with those obtained from potential flow; this is in reference to relatively higher-fidelity computations of the unsteady Reynolds-averaged Navier–Stokes equations.

Steady-state real-time optimization using transient measurements in the absence of a dynamic mechanistic model: A framework of HRTO integrated with Adaptive Self-Optimizing IHMPC

In processes with slow dynamics or subjected to frequent disturbances, the detection of a steady-state operation might be rare, which hinders the application of the classic RTO. To overcome this issue, the Hybrid RTO (HRTO) is a proposition where a dynamic observer substitutes the Steady-State Detection (SSD) and the static parameter estimation steps. Recent progress has been made in HRTO frameworks. Still, all of them start from the assumption that a reliable mechanistic dynamic model is available, which is not true in many applications. In fact, the development of a dynamic process model is not considered in most typical RTO design projects. Therefore, the present work proposes an HRTO framework aiming, mainly, to overcome this first assumption. We present a complete control framework where the basic assumption is that a reliable mechanistic dynamic model is not available, but a static process model is at hand. The HRTO is made possible by using an approximate dynamic model that takes advantage of the static process model and identified linear dynamics from the plant in a Hammerstein structure. Three models are proposed based on the Hammerstein structure. These models are used in the dynamic observer, a proposed variation of an Extended Kalman Filter (EKF), and as the controller’s internal model. The economic objectives are introduced into the control layer by the self-optimizing variables in an adaptive infinite-horizon MPC formulation. The framework presents full model compatibility between the observer, controller, and optimizer layers. A validation case study is developed in the Williams–Otto reactor with two proposed EKF tunings. The open-loop results provide evidence that the proposed Hammerstein structure is adequate as an approximate dynamic model, preserving the observability characteristics of the static model. The closed-loop results show that our framework outperforms the classic RTO structure in economic return, especially in the transient regions. Moreover, the proposed approach has a very natural way of handling active constraint changes. This effect is shown in the results at the moment that a constraint becomes active. The economic performance of the RTO is impaired compared to our proposed HRTO scheme. Finally, the average computational cost of each iteration of our framework is twice as fast as the classic RTO framework, which suggests the potential applicability of the proposed methodology on an industrial scale.

The Loewner framework for nonlinear identification and reduction of Hammerstein cascaded dynamical systems

AbstractWe present an algorithm for data‐driven identification and reduction of nonlinear cascaded systems with Hammerstein structure. The proposed algorithm relies on the Loewner framework (LF) which constitutes a non‐intrusive algorithm for identification and reduction of dynamical systems based on interpolation. We address the following problem: the actuator (control input) enters a static nonlinear block. Then, this processed signal is used as an input for a linear time‐invariant system (LTI). Additionally, it is considered that the orders of the linear transfer function and of the static nonlinearity are not a priori known.

A Novel Hammerstein Model for Nonlinear Networked Systems Based on an Interval Type-2 Fuzzy Takagi–Sugeno–Kang System

In this article, a novel Hammerstein structure is proposed for nonlinear networked systems based on an interval type-2 Takagi–Sugeno–Kang (IT2TSK) fuzzy system. The proposed approach is designed based on the general Hammerstein form, where an autoregressive moving average and an IT2TSK structures are designed as the linear energetic and nonlinear static components, respectively. The consequents of the nonlinear subsystem are characterized by a TSK-type system while the antecedents are characterized by interval type-2 fuzzy sets. The structure of the nonlinear subsystem is learned online based on the type-2 fuzzy clustering. Two new updating algorithms are introduced based on the Lyapunov theorem for learning the proposed model parameters and adaptive learning rates to assure the model stability and the parameter fast convergence. To illustrate the proposed model robustness, a test is performed under networked environment with time-varying delay and packet losses. The simulation results prove a higher performance for the proposed model than that of compared models.

A nonlinear distributed model predictive scheme for systems based on Hammerstein model

A distributed model predictive control (DMPC) scheme for systems based on the Hammerstein structure is proposed in this work. To deal with the nonlinearity of the Hammerstein model, the DMPC problem is reduced to an optimization problem on the intermediate variable and, subsequently, the inverse of the nonlinear block is considered to find the corresponding control inputs. Also, a sub-optimality bound of the method is presented. To illustrate our approach, we consider a nonlinear system controlled in a distributed manner by two agents that exchange information and make cooperative decisions. The efficiency of the proposed DMPC is demonstrated through a simulation example where it is compared to the corresponding centralized approach.