Hammerstein Model Research Articles

Parameter identification of fractional order Hammerstein model with two-stage piecewise nonlinearity based on iterative algorithms

This paper discusses identification of the fractional order Hammerstein model with colored noise. The static part of the Hammerstein model is a two-stage piecewise nonlinearity, while the dynamic part is a CARMA structure. We deduce the identification expression of the model through the definition of the Grünwald–Letnikov fractional differential. Then, two iterative algorithms are adopted to identify the unknown parameters. One of them is the Levenberg–Marquardt iterative algorithm, and the other is the particle swarm optimization iterative algorithm. The two algorithms are extended from the traditional integer order system identification field to the fractional order nonlinear colored noise system identification field. A numerical example and a case study of servo system identification are respectively presented to demonstrate the feasibility of the identification algorithms. It can be seen that the estimation errors of these two algorithms are relatively small, which reflects their good identification effect.

Tracking necessary condition of optimality by a data-driven solution combining steady-state and transient data

One of the difficulties in practical implementations of the classic Real-Time Optimization (RTO) strategy is the integration between optimization and control layers, mainly due to the differences between the models used in each layer, which may result in unreachable setpoints coming from optimization to the control layer. In this context, Economic Model Predictive Control (EMPC) is a strategy where optimization and control problems are solved simultaneously. However, this strategy is based on the assumption that a nonlinear dynamic model is available, which may not be valid. Also, when considering a first-principles nonlinear model, the computational cost and convergence may be relevant issues. The present work presents an RTO framework based on an EMPC structure considering a Hammerstein model for the plant. This modeling approach can be applied even in the absence of first-principles models. The proposed EMPC considers the minimization of the economic objective function gradient calculated through a steady-state model based on a Gaussian Process. This strategy was applied to the Willians–Otto Reactor benchmark and presented superior results than the classic RTO and Hybrid RTO (H-RTO) approaches in closed-loop and a lower average iteration time than these other approaches.

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.

Nonlinear Modeling and Control of Polyvinyl Chloride (PVC) Gel Actuators

Polyvinyl chloride (PVC) gels represent a novel type of electroactive polymer actuators with a number of appealing characteristics, including low cost, high compliance, large strain, medium-to-high stress output, fast response, and thermal stability. Despite their vast potential in a variety of applications, modeling and control of nonlinear dynamics of PVC actuators has received little attention. In this article, we first present a data-driven approach to modeling nonlinear dynamics of PVC gel actuators. A Hammerstein model, consisting of a nonlinear module cascaded with linear dynamics, is proposed to capture the pronounced dependence of the voltage input–displacement output frequency response on the input amplitude and bias. A control scheme is then designed based on the model, where an inverse compensator for the nonlinear element is combined with a PID feedback controller. A disturbance observer is further introduced for the estimation and rejection of the influences from imperfect inverse compensation and model uncertainties. Experimental results are presented to support the efficacy of the proposed modeling and control approach. In particular, for a number of reference trajectories, the proposed control scheme results in over 80% reduction of tracking error in comparison with a well-tuned PID controller.

Theoretical and experimental investigation study of discrete time rate‑dependent hysteresis modeling and adaptive vibration control for smart flexible beam with MFC actuators

A new rate‑dependent hysteresis modeling and identification method, and a new adaptive filtering control algorithm is proposed to solve the hysteresis nonlinearity identification problem and adaptive parameter auto-tuning problem for vibration control of smart flexible beam employing MFC actuators. A Hammerstein model is proposed by composing asymmetric Bouc-Wen model and ARX model in series to for secondary path modelling to model the MFC actuator hysteresis and mechanical hysteresis of the flexible beam. To improve the convergence speed and performance for complex parameter identification of the Hammerstein model, an improved Jaya algorithm named hybrid Jaya is proposed which combines the mutation strategy of two kinds of DE algorithms as the individual update method of the proposed hybrid Jaya. To balance the convergence speed and steady-state error for active vibration control performance, a new variable tap-length and step-size FXLMS algorithm named FX-VSSFT-LMS algorithm is proposed and the adaptive parameters are auto-tuned by our proposed Hybrid Jaya algorithm. To verify the proposed Hammerstein model and adaptive filtering algorithm, a cantilever beam experimental platform with MFC actuators is constructed. Hybrid Jaya algorithm and Jaya algorithm were used to identify the Hammerstein model and Bouc-wen model of MFC actuator, and the improved effects of hybrid Jaya algorithm and Hammerstein model were compared and verified. The FX-VSSFT-LMS algorithm is compared with FX-VSSLMS algorithm with variable step size and FX-FTLMS algorithm with variable tap-length on the MFC cantilever beam. The experimental comparison proves that the Hammerstein model identified by the proposed Hybrid Jaya algorithm is effective and the FX-VSSFT-LMS algorithm has better convergence performance than FX-VSSLMS and FX-FTLMS algorithm.

Force Tracking Control of Functional Electrical Stimulation via Hybrid Active Disturbance Rejection Control

Stroke is a worldwide disease with a high incidence rate. After surviving a stroke, most patients are left with impaired upper or lower limb. Muscle force training is vital for stroke patients to recover limb function and improve their quality of life. This paper proposes a force tracking control method for upper limb based on functional electrical stimulation (FES), which is a promising rehabilitation approach. A modified Hammerstein model is proposed to describe the nonlinear dynamics of biceps brachii, which consists of a nonlinear mapping function, linear dynamics and time delay component to represent the biochemical process of muscle contraction. A quick model identification method is presented based on the least square algorithm. To deal with the variation of muscle dynamics, a hybrid active disturbance rejection control (ADRC) is proposed to estimate and compensate for the model uncertainty and unmeasured disturbances. The parameter tuning process is given. In the end, the performance of the proposed methods is verified via simulations and experiments. Compared with the Proportional integral derivative controller (PID) method, the proposed methods could suppress the model uncertainty and improve the tracking precision.

Parameter learning for the nonlinear system described by Hammerstein model with output disturbance

AbstractA novel parameter learning scheme using multi‐signal processing is developed that aims at estimating parameters of the Hammerstein nonlinear model with output disturbance in this paper. The Hammerstein nonlinear model consists of a static nonlinear block and a dynamic linear block, and the multi‐signals are devised to estimate separately the nonlinear block parameters and the linear block parameters; the parameter estimation procedure is greatly simplified. Firstly, in view of the input–output data of separable signals, the linear block parameters are computed through correlation analysis method, thereby the influence of output noise is effectively handled. In addition, model error probability density function technology is employed to estimate the nonlinear block parameters with the help of measurable input–output data of random signals, which not only controls the space state distribution of model error but also makes error distribution tends to normal distribution. The simulation results demonstrate that the developed approach obtains high learning accuracy and small modeling error, which verifies the effectiveness of the developed approach.

Identification of a Practical Digital Twin for Simulation of Machine Tools

A practical digital twin for machine tools is proposed in this study. The proposed digital twin is capable of time-domain simulation of machine tools and consists of a controller model, machining process model, and machine dynamic model. To predict the quality of the machined surface after the finishing processes, a precise dynamic model is required. The developed dynamic model consists of an interaction force model, vibration model, and friction force model. A linear auto regressive with exogenous inputs (ARX) model is adopted for the interaction and vibration models. Based on a systematic analysis of the disturbance forces of the machine tool, the friction characteristics are extracted to a displacement-dependent friction model and velocity-dependent friction model. A nonlinear Hammerstein model is adopted to identify the friction. Online identification systems based on the recursive least-squares (RLS) method are developed and tested for each model.

Data-driven state estimation toward blade individual load-reduction of wind turbines1

As wind turbines tend to be build larger every year, the importance of load-reducing controls to damp the rotor blade oscillations increases. For a control algorithm to perform blade individual load-reducing strategies, online information about each rotor blade’s deflection is needed. An approach for wind turbine state estimation with special attention to the rotor blade deflections is given using a Hammerstein model driven by simulation data in a Kalman Filter (KF). The Hammerstein’s linear submodel is mainly derived as time discrete state space system via least squares, while nonlinear effects are defined in terms of their proportionality. The model noise definition required by the KF is calculated by the system model’s one-step prediction error. The simulation results show the estimate to have a standard deviation of less than 2 % in edge and 9% in flap-wise direction when using an appropriate sensor configuration. Using state-of-the-art sensors, standard deviations of less than 4% and 25% respectively are achieved. The approach is particularly attractive due to the low level of tuning required. The usage of the state estimation in a closed control loop as well as the Hammerstein model in a model-based control system offer room for promising further investigations.

Identification of nonlinear discrete systems using a new Hammerstein model with Volterra neural network

Identification of nonlinear discrete systems using a new Hammerstein model with Volterra neural network

Gas turbine system identification using a bilayer equilibrium manifold expansion model

PurposeThis paper aims to establish a multi-input equilibrium manifold expansion (EME) model for gas turbine (GT). It proposes that the extension of model input dimension is realized based on similarity theory and affine structure in the framework of single-input EME model. The study aims to expand the scope of application of the EME model so that it can be used for the control or fault diagnosis of GTs.Design/methodology/approachIn this paper, the concepts of corrected equilibrium manifold expansion (CEME) model and multi-cell equilibrium manifold expansion (MEME) model are first proposed. This paper uses theoretical analysis and simulation experiments to demonstrate the effectiveness of the bilayer equilibrium manifold expansion (BEME) model, which is a combination of the CEME and the MEME models. Simulation experiments include confirmatory experiments and comparative experiments.FindingsThe paper provides a new sight into building a multiple-input EME (MI-EME) model for GTs. The proposed method can build an accurate and robust MI-EME model that has superior performance compared with widely used nonlinear models including Wiener model (WM), Hammerstein model (HM), Hammerstein–Wiener model (HWM) and nonlinear autoregressive with exogenous inputs (NARX) network model. In terms of accuracy, the maximum error percentage of the proposed model is just 1.309%, far less than WM, HM and HWM. In terms of the stability of model calculation, the range of the mean error percentage of the proposed model is just a quarter of that of NARX network model.Originality/valueThe paper fulfills the construction of a novel multi-input nonlinear model, which has laid a foundation for the follow-up research of model-based GT fault detection and isolation or GT control.

Wiener–Hammerstein nonlinear system identification using spectral analysis

AbstractIn this paper, a new approach is developed for the identification of Wiener–Hammerstein model structures. Unlike several other papers, the transfer functions of linear elements are of unknown structure. Indeed, the linear subsystems are not necessarily parametric or FIR. Furthermore, the function of nonlinear block can be nonparametric and noninvertible. The main challenge in identifying a Wiener–Hammerstein system lies in the separation of the dynamics over the input and the output LTI elements. Presently, this issue does not arise, without any conditions being imposed on the frequency band. Then, a two‐stage identification method is suggested. Firstly, the system is excited by a set of constant inputs to capture the system nonlinearity. In the second stage, an identification approach based on the spectral analysis and using periodic input signals is developed to determine the linear elements parameters. In the present method, very interesting concepts are used, like the Fourier decomposition, the frequency approach and the spectrum analysis. Simulations demonstrate that the proposed identification method is both effective and efficient.

Output feedback model predictive control for Hammerstein model with bounded disturbance

This paper proposes two model predictive control (MPC) methods for a Hammerstein model with both unmeasurable state and bounded disturbance. First, a dynamic output feedback MPC is designed such that the dynamics of the state estimation error is decoupled from that of the estimated state. Hence, the separation principle holds and a part of the controller parameters can be designed off-line. Second, a case is considered where Hammerstein nonlinearity is not exactly inverted, that is, a reserved static nonlinearity exists in the closed-loop system. This reserved nonlinearity is merged into a polytopic description in combination with the linear dynamic part. The control move is then parameterized as a feedback law followed by the approximate inverse of Hammerstein nonlinearity. The recursive feasibility and closed-loop stability of both approaches are guaranteed. Numerical examples are given in order to show effectiveness of the proposed approaches.

Performance Analysis of Orthogonal Gradient Sign Algorithm Using Spline-based Hammerstein Model for Smart Application

This paper presents a spline-based Hammerstein model for adaptive filtering based on a sign algorithm with the normalised orthogonal gradient algorithm. Spline-based Hammerstein architecture consists of an interpolation spline-based adaptive lookup table in the part of nonlinear filter and an adaptive finite impulse response filter used in the part of linear filter. Hammerstein spline adaptive filter (HSAF) is a nonlinear filter for the nonlinear systems among the advantages in the low computational cost and high performance. An adaptive lookup table and spline control points are determined and derived with the orthogonal gradient-based mechanism. Performance analysis in terms of convergence properties and mean square analysis based on the mean square error (MSE) constraint are proven by using the Taylor series expansion of the estimation error in the form of the excess MSE. Experimental results indicate the robust performance of the proposed algorithm can provide the better performance than the other models based on the conventional least mean square Hammerstein spline adaptive filtering algorithm.

MILM hybrid identification method of fractional order neural-fuzzy Hammerstein model

MILM hybrid identification method of fractional order neural-fuzzy Hammerstein model

NONLINEAR DYNAMIC CALIBRATION AND CORRECTION OF ACCELERATION SENSOR BASED ON ADAPTIVE NEURAL NETWORK

With the development of industrial technology, acceleration sensors have a wide range of applications. The application environment has strict requirements on acceleration sensor, which needs it to get accurate input and output response in complexity. According to the Hammerstein model, this paper studies the dynamic nonlinear relationship between the output voltage and the input acceleration of the acceleration sensor that can be divided into static nonlinear component and dynamic linear component. We combine the least square method with adaptive neural network to calculate the parameters of static nonlinear and dynamic linear components. The least square method is used to improve the training performance of the network and avoid the network falling into the local minimum of the traditional neural network. Experimental results show that compared with other methods, the proposed method has the advantages of less training steps and strong approximation ability, and the algorithm is less affected by external noise. This method can realize nonlinear system identification of acceleration sensor and provide reliable basis for compensation.

Metaheuristics algorithms to identify nonlinear Hammerstein model: a decade survey

Metaheuristics have been acknowledged as an effective solution for many difficult issues related to optimization. The metaheuristics, especially swarm’s intelligence and evolutionary computing algorithms, have gained popularity within a short time over the past two decades. Various metaheuristics algorithms are being introduced on an annual basis and applications that are more new are gradually being discovered. This paper presents a survey for the years 2011-2021 on multiple metaheuristics algorithms, particularly swarm and evolutionary algorithms, to identify a nonlinear block-oriented model called the Hammerstein model, mainly because such model has garnered much interest amidst researchers to identify nonlinear systems. Besides introducing a complete survey on the various population-based algorithms to identify the Hammerstein model, this paper also investigated some empirically verified actual process plants results. As such, this article serves as a guideline on the fundamentals of identifying nonlinear block-oriented models for new practitioners, apart from presenting a comprehensive summary of cutting-edge trends within the context of this topic area.

Recursive Identification for MIMO Fractional-Order Hammerstein Model Based on AIAGS

In this paper, adaptive immune algorithm based on a global search strategy (AIAGS) and auxiliary model recursive least square method (AMRLS) are used to identify the multiple-input multiple-output fractional-order Hammerstein model. The model’s nonlinear parameters, linear parameters, and fractional order are unknown. The identification step is to use AIAGS to find the initial values of model coefficients and order at first, then bring the initial values into AMRLS to identify the coefficients and order of the model in turn. The expression of the linear block is the transfer function of the differential equation. By changing the stimulation function of the original algorithm, adopting the global search strategy before the local search strategy in the mutation operation, and adopting the parallel mechanism, AIAGS further strengthens the original algorithm’s optimization ability. The experimental results show that the proposed method is effective.

Discrete indirect adaptive sliding mode control for uncertain Hammerstein nonlinear systems

The robustness issue of uncertain nonlinear systems’ control has attracted the attention of numerous researchers. In this paper, we propose three techniques to deal with the uncertain Hammerstein nonlinear model. First, a discrete sliding mode control (SMC) is developed, which is based on converting the original nonlinear system into a linearized one in the vicinity of the operating region using Taylor series expansion. However, the presence of relatively high nonlinearities and parameter variations leads to the deterioration of the desired performances. In order to overcome these problems and to improve the performance of classical SMC, we propose two solutions. The first one is based on the synthesis of a discrete SMC, taking into account the presence of nonlinearity. The second solution is a new discrete adaptive SMC for input–output Hammerstein model. In order to show the effectiveness of the proposed controllers, a detailed robustness analysis is clearly developed. Simulation examples are reported at the end of the paper.

Improving the modelling efficiency of Hammerstein system using Kalman filter and its parameters optimised using social mimic algorithm: Application to heating and cascade water tanks

Identification of Hammerstein systems finds extensive applications in solving the stability and design issues of non-linear dynamic systems. Hence, in this paper, a first attempt is made to solve the parameter estimation problem of Hammerstein non-linear systems by using the evolutionary optimisation algorithm (EOA) coupled Kalman filter (KF). The standard KF produces the best possible optimal states for the particular state estimation problem based on its properly tuned parameters θ^0−, P0−, φ, Q and R termed as initial state, initial error covariance, state transition, process noise covariance and measurement noise covariance, respectively. These parameters must be concurrently tuned to achieve a stable filtering operation. The manual and existing adaptive tuning methods do not guarantee optimality, convergence and also require a large amount of data. Moreover, the adaptive methods tune only a few statistics (Q and R) simultaneously. To defeat these foreknown problems, in the proposed method, initially, a recently established EOA named social mimic optimisation (SMO) algorithm is employed to get the global optimised KF parameters along with an efficient cost function that has been derived based on the model output for the concurrent achievement of all the KF parameters. Finally, the standard KF technique identifies the estimated parameters of the Hammerstein model with those optimised KF parameters. Various numerical and experimental test results demonstrate that the proposed SMO-based KF approach offers high accuracy in estimating the parameters along with the good convergence speed compared to the standard KF, other presented EOA-based KF approaches and recently reported techniques.