Static Nonlinear Part Research Articles

MISO Nonlinear Gauss-Hammerstein Model Identification of Ultrasonic Motor

A novel two-input nonlinear Hammerstein model of ultrasonic motor is proposed to provide the basis for the design and analysis of motor’s multivariable control system. The static nonlinear part of the model is designed as Gauss function to describe the nonlinear characteristics of ultrasonic motor more accurately. Differential evolution algorithm is used to identify the model parameters, and the algorithm is improved to ensure the correctness and validity of the modeling process. The debugging method of optimization algorithm parameter value is also proposed to ensure the robustness of the algorithm. The comparison between the experimental data and the calculated results of the model indicates that the established model has higher precision and the modeling method is effective.

Explicit nonlinear predictive control algorithms for Laguerre filter and sparse least square support vector machine-based Wiener model

In this paper, three computationally proficient model predictive control (MPC) algorithms for least square support vector machine (LSSVM)-based Wiener model are described. A Wiener model with Laguerre filter as dynamic linear part and LSSVM approximator as nonlinear static part is considered. Even though having excellent approximation abilities, LSSVM suffers from lack of sparseness. A pruning algorithm for LSSVM model is proposed and its comparison is made with classical pruning algorithm. The proposed pruning algorithm is able to remove 99% of support vectors with no remarkable drop in modelling accuracy. Using pruned Wiener model, three computationally efficient MPC algorithms are described. In the first algorithm, linearization of Wiener model is performed at every sampling interval and therefore control vector is determined by carrying out a quadratic optimization task. In the second algorithm, control signal is determined by an explicit control law and parameters of this control law are computed by performing lower-upper (LU) factorization of a matrix and solving linear equations without any online optimization. In the third algorithm, the parameters of explicit control law are calculated directly by another LSSVM approximator, which is trained offline. The advantages and effectiveness of proposed methods are demonstrated on the benchmark pH neutralization reactor. The control performance and computational efficiency of proposed algorithms are compared with computationally complex nonlinear MPC, which repeats a nonlinear optimization task at every sampling interval. The impact of pruning on model accuracy, computational efficiency and control accuracy is also discussed.

Parameter Estimation of Neuro-Fuzzy Wiener Model With Colored Noise Using Separable Signals

This paper considers a neuro-fuzzy based identification problem for Wiener model with controlled autoregressive moving average noise. The separable signal is applied to decouple the dynamic linear part and the static nonlinear part, and the correlation analysis method is adopted to estimate the parameters of the linear part. To improve the convergence rate of generalized extended stochastic gradient (GESG) algorithm, a generalized extended stochastic gradient algorithm with a forgetting factor is derived for estimating the parameters of the nonlinear part and the parameters of noise model. Examples results verify the effectiveness of the proposed method.

A novel fuzzy Wiener-based nonlinear modelling for engineering applications

This study proposes a novel fuzzy Wiener structure for identifying engineering systems. The proposed model has a cascade structure; a nonlinear static part preceded by a linear dynamic part. The nonlinear static part is represented by an interval type-2 fuzzy Takagi–Sugeno–Kang (IT2TSK) system in which the antecedents of the rules are described by interval type-2 fuzzy sets (IT2FSs) and a TSK-type system describes the consequents. An autoregressive moving average (ARMA) model is developed for representing the linear dynamic part. The proposed structure parameters including the ARMA, antecedent and consequent parameters are updated using the Lyapunov function to ensure the model stability. The simulation results confirm that the proposed Wiener structure can successfully model nonlinear engineering applications in the existence of system uncertainties and noisy measurement data. Moreover, the proposed model consistently achieves higher fitness (FIT) and smaller root mean square error (RMSE) values than other existing schemes.

Modeling and Discrete-Time Terminal Sliding Mode Control of a DEAP Actuator with Rate-Dependent Hysteresis Nonlinearity

Dielectric electro-active polymer (DEAP) materials, also called artificial muscle, are a kind of EAP smart materials with extraordinary strains up to 30% at a high driving voltage. However, the asymmetric rate-dependent hysteresis is a barrier for trajectory tracking control of DEAP actuators. To overcome the barrier, in this paper, a Hammerstein model is established for the asymmetric rate-dependent hysteresis of a DEAP actuator first, in which a modified Prandtl-Ishlinskii (MPI) model is used to represent the static hysteresis nonlinear part, and an autoregressive with exogenous inputs (ARX) model is used to represent the linear dynamic part. Applying Levenberg-Marquardt (LM) algorithm identifies the parameters of the Hammerstein model. Then, based on the MPI model, an inverse hysteresis compensator is obtained to compensate the hysteresis behavior. Finally, a compound controller consisting of the hysteresis compensator and a novel discrete-time terminal sliding mode controller (DTSMC) without state observer is proposed to achieve the high-precision trajectory tracking control. Stability analysis of the closed-loop system is verified by using Lyapunov stability theorem. Experimental results based on a DEAP actuator show that the proposed controller has better tracking control performance compared with a conventional discrete-time sliding mode controller (DSMC).

Parameter estimation of Hammerstein–Wiener nonlinear system with noise using special test signals

In this paper, an identification procedure of Hammerstein–Wiener nonlinear system with process noise using special test signals is presented. Special test signals that contain separable signal and uniformly random multi-step signal are employed to separate the identification problems of the linear dynamic part and the output static nonlinear part from that of the input static nonlinear part, and then correlation analysis method is applied to estimate the parameters of the output static nonlinear part and linear dynamic part. Moreover, a filter is embedded to form extended Hammerstein–Wiener system to calculate the noise correlation functions by the information of zeros and poles of the extended system, further an error compensation term which consists of the noise correlation functions is added to least square estimation to compensate the error caused by process noise. Therefore, the unbiased estimation of model parameters can be derived by error compensation based recursive least square method. Simulation results demonstrate that proposed approach can effectively identify parameters of Hammerstein–Wiener nonlinear system in the presence of process noise.

Recursive identification of multiple-input single-output fractional-order Hammerstein model with time delay

This paper deals with identification of the continuous-time Hammerstein systems with time delay using Genetic Algorithm (GA) combined with the Recursive Least-Squares (RLS) method. This model consists of the Radial Basis Function Neural Network (RBFNN) as its nonlinear static part and fractional order transfer function as its dynamic linear part. The fractional orders are identified by GA with an innovative strategy called Modified Genetic Algorithm (MGA). The main innovative idea is the selection and transferring the best characteristics or properties to the next generation. On the other hand, the centers and widths and the weighting parameters of the RBFNN and the transfer function coefficients of the linear dynamic part are updated by the RLS method. Simulation results are applied to illustrate the proposed method accuracy.

Block-Structured Models Composed of Nonlinear Fuzzy Dynamic and Static Parts – a Case Study

Block-Structured Models Composed of Nonlinear Fuzzy Dynamic and Static Parts – a Case Study

Nonlinear system identification of fractional Wiener models

The Wiener model is a case of block-oriented model, which is suitable for modeling a large class of nonlinear systems. It consists of the cascade of two parts, where a linear dynamic part is followed by a static nonlinear part. On the other hand, fractional order models have gained increasing interest over the last decades due to their better representation of some physical phenomena. In this paper, the fractional Wiener system identification is aimed. The polynomial nonlinear fractional state space equations are used to describe the nonlinear model. Thus, the extension of an output error method (Levenberg Marquard algorithm) is developed to estimate the fractional Wiener system. The efficiency of the identification method is evaluated and confirmed by simulation examples.

Correlation analysis method based SISO neuro-fuzzy Wiener model

A novel identification algorithm for neuro-fuzzy based single-input-single-output (SISO) Wiener model with colored noises is presented in this paper. The separable signal is adopted to identify the Wiener model, leading to the identification problem of the linear part separated from nonlinear counterpart. Then, the correlation analysis method can be employed for identification of linear part. Moreover, in the presence of random signal, the least square method based parameters estimation algorithm of static nonlinear part are proposed to avoid the impact of colored noise. As a result, proposed method can circumvent the problem of initialization and convergence of the model parameters encountered by the existing iterative algorithms used for identification of Wiener model. Examples are used to verify the effectiveness of the proposed method.

Identification method of neuro‐fuzzy‐based Hammerstein model with coloured noise

In this study, neuro-fuzzy-based identification procedure for Hammerstein model with coloured noise is presented. Separable signal is used to realise the decoupling of the identification of dynamic linear part from that of static non-linear part, and then correlation analysis method is adopted to identify the parameters of the linear part. Furthermore, by combining multi-innovation and gradient search theory, multi-innovation-based extended stochastic gradient approach is derived for improving the parameters estimation accuracy of the non-linear part and the noise model. In addition, the convergence analysis in the martingale theory illustrates that the parameter estimation error will converge to zero under the persistent excitation condition. Finally, two simulation results demonstrate that the proposed approach has high identification accuracy and good robustness to the disturbance of coloured noise.

A Wiener-Type Dynamic Neural Network Approach to the Modeling of Nonlinear Microwave Devices

A new Wiener-type dynamic neural network (DNN) approach for nonlinear device modeling is proposed in this paper. The proposed analytical formulation of Wiener-type DNN structure consists of a cascade of a simplified linear dynamic part and a nonlinear static part. The simplified linear dynamic equations are obtained by vector fitting enhancing the efficiency of the model. A new formulation and new sensitivity analysis technique is derived to train the proposed Wiener-type DNN with dc, small- and large-signal data. A new gradient-based training algorithm is also formulated to speed up the training of proposed Wiener-type DNN model. The proposed Wiener-type DNN model can be trained to be accurate relative to device data. Furthermore, the proposed Wiener-type DNN provides enhanced convergence properties over existing neural network approaches such as time delay neural network (TDNN) and TDNN with extrapolation. Application examples on modeling GaAs metal-semiconductor-field-effect transistor, GaAs high-electron mobility transistor (HEMT), and a real <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$2 \times 50$ </tex-math></inline-formula> gatewidth GaAs pseudomorphic HEMT are presented. The use of Wiener-type DNN model in harmonic balance simulations demonstrates that the Wiener-type DNN is a robust method for developing models for different types of microwave devices. These models could be implemented into circuit simulators conveniently.

Neuro-fuzzy based identification method for Hammerstein output error model with colored noise

In this paper, a neuro-fuzzy based identification procedure for Hammerstein output error model with colored noise is presented. Separable signal is used to realize the decoupling of the identification of dynamic linear part from that of static nonlinear part, and then correlation analysis method is adopted to identify the parameters of the linear part. Next, a filter is embedded to form extended Hammerstein model to calculate the noise correlation function by the information of zeros and poles of the extended model. The correlation functions which consist of the noise correlation function are applied to compensate the bias caused by colored noise. As a result, the parameters of the nonlinear part can be identified through recursive least square method. Examples results illustrate that the proposed approach has high identification accuracy and good robustness to the disturbance of colored noise.

Modeling and Control of Wiener Systems Using Multiple Models and Neural Networks: Application to a Simulated pH Process

This paper describes an efficient multiple models control scheme based on a bank of neural Wiener models and its application. Each Wiener model consists of a linear dynamic part in connection with a nonlinear static part which is approximated by a radial basis function (RBF) neural network. The control algorithm is developed from a Clarke performance index and can copy with unstable zero-dynamics problems. An extended system that is associated with the criterion function is defined for system parameters identification and control law design. After establishing multiple neural Wiener models for several distinct operation regions, a robust switching mechanism is introduced for the selection of the best controller. The overall scheme is applied to a simulated pH neutralization process. The set-point tracking, disturbance rejection, and robustness performance of this scheme are compared to several alternatives.

Dynamic Modeling and Very Short-term Prediction of Wind Power Output Using Box-Cox Transformation

We propose a statistical modeling method of wind power output for very short-term prediction. The modeling method with a nonlinear model has cascade structure composed of two parts. One is a linear dynamic part that is driven by a Gaussian white noise and described by an autoregressive model. The other is a nonlinear static part that is driven by the output of the linear part. This nonlinear part is designed for output distribution matching: we shape the distribution of the model output to match with that of the wind power output. The constructed model is utilized for one-step ahead prediction of the wind power output. Furthermore, we study the relation between the prediction accuracy and the prediction horizon.

Online identification of non-linear dynamic systems by Wiener model using subspace method and neural networks

This paper presents a method for online identification of non-linear dynamic systems using the Wiener model. For the linear dynamic part the subspace identification method with the multivariable output-error state-space algorithm is employed, whereas for the non-linear static part the multi-layer perceptron neural network with Levenberg–Marquardt algorithm is used. The stability and convergence of the proposed method is shown using the Lyapunov direct method and the region solution of the linear matrix inequality (LMI) approach. The proposed method is tested by simulations performed on the continuous stirred tank reactor (CSTR) plant, which is presented by non-linear differential equations. Moreover, the method is applied on the input–output data that are obtained from a practical system of the CSTR plant as well as the pH neutralization plant. The results show significant improvements in online identification of the non-linear dynamic systems compared with the recently reported methods in literature.

Performance assessment for life extending control of steam turbine based on polynomial method

Damage modeling and life extending control for steam turbine are studied in this work. Taking the rolling process of a 300MW steam turbine unit as an example, this paper proposes a control performance benchmark for life extending controller. And a polynomial control performance assessment method is developed. Nonlinear low cycle fatigue damage model is decomposed to static nonlinear part and dynamic linear part. This mathematical model represents a class of damage assessment problems. A numerical simulation using practical industrial data validates the contribution of present work.

Novel Wiener models with a time-delayed nonlinear block and their identification

Popular Wiener models are a combination of dominant linear behavior and static nonlinear characteristics. However, they are less accurate in dealing with problems with strong nonlinear time-delayed effects. Hence, a group of novel nonlinear models based on Wiener models are proposed in this study. The static nonlinear part of general Wiener models is replaced by a time-delayed nonlinear block. Thus, two novel Wiener models with a quasi-dynamic nonlinear block are constructed, called the Wiener-QDN models. The autoregressive with exogenous input (ARX) model is utilized to identify the dynamic linear block, and the radial basis function neural network is used to build the nonlinear part of Wiener and the Wiener-QDN models. Results reveal the effectiveness of these models in nonlinear system identification. The first test cases are systems governed by simple equations. The Wiener-QDN models can identify these systems with or without measurement noise. These models are further applied in unsteady aerodynamic modeling cases. Test cases of a NACA 0012 airfoil pitching in transonic flow indicate that both linearity and nonlinearity are well captured by the proposed Wiener-QDN models.

Research on Nonlinear Commutating Characteristics of Valve-Controlled Motor System Based on Hammerstein Model

Aiming at the low accuracy problem of linear model of Valve-controlled motor system in conditions of frequent start-stop and commutation, the nonlinear characteristics of the main components of valve-controlled motor system are analyzed, and non-linear mathematical models and simulation models are established to study the impact on the system of each nonlinear characteristic separately. Hammerstein model consists of a static nonlinear part and dynamic linear part connected in series is adopted, and the non-linear characteristics are introduced to the system. The multi-frequency sinusoidal input signal of test is used as the input of Hammerstein simulation model. And the simulation results are compared with test results and other model output, the results proved that the nonlinear characteristic impacts of the motor valve controlled system really exist, and Hammerstein model can describe nonlinear characteristics of the system accurately.

Correlation analysis based MIMO neuro-fuzzy Hammerstein model with noises

A novel identification algorithm for neuro-fuzzy based MIMO Hammerstein system with noises by using the correlation analysis method is presented in this paper. A special test signal that contains independent separable signals and uniformly random multi-step signal is adopted to identify the MIMO Hammerstein system, resulting in the identification problem of the linear model separated from that of nonlinear part. As a result, it can circumvent the problem of initialization and convergence of the model parameters encountered by the existing iterative algorithms used for identification of MIMO Hammerstein model. Moreover, least square method based parameter identification algorithms of dynamic linear part and static nonlinear part are proposed to avoid the influence of noise. Examples are used to illustrate the effectiveness of the proposed method.