Linear Dynamic Block Research Articles

Nonparametric Modeling of Mutually Ambiguous Mappings

This paper considers the problem of approximating a function from observations in the case when the object under study contains mutually ambiguous characteristics in its description. This formulation is essential in identifying and controlling systems of the Wiener and Hammerstein class, in which nonlinear processes can be represented as a sequential connection of linear dynamic and nonlinear inertia-free blocks. Often elements such as hysteresis loop, backlash and others are used as nonlinear blocks. These elements are mutually ambiguous mappings. In connection with the transition to automated digital production, which is controlled in real time by intelligent systems rather than humans, nonlinear dynamic processes are found everywhere. The complexity of solving the problem lies in the lack of sufficient a priori information about the parametric structure of the model of the process under study. The paper proposes some modifications of the nonparametric estimation of the regression function, allowing for the modeling of mutually ambiguous mappings. Some fragments of numerical studies are presented that show acceptable results in terms of reconstruction accuracy.

Identification methodology for MIMO Hammerstein nonlinear model with process noise

In this paper, we present a methodology for identifying the multi-input–multi-output (MIMO) Hammerstein nonlinear model under colored noise. The Hammerstein model presented comprises neural fuzzy models (NFMs) as its static nonlinear block and rational transfer functions (RTF) model as its dynamic linear block. The hybrid signals consisting of separable signals and random signals are utilized to deal with the MIMO Hammerstein model identification issue, and the separable signals to implement separation identification of MIMO Hammerstein model is introduced, that is, the two blocks are separately identified. First, parameters of the linear block are estimated applying correlation function-based least squares method in the presence of measurable input–output of Gaussian signals, which can efficiently weaken the process noise interference. Second, estimate of noise parameters vector is introduced to solve the unknown noise vector in the information matrix, then a recursive extended least squares method is developed for identifying parameters of nonlinear block and colored noise model based on available input–output of random signals. The validity and precision of the presented methodology are demonstrated applying a numerical simulation and a nonlinear process, and it is known from the research results that compared with existing identification techniques, the methodology utilized achieved higher identification accuracy.

Parameter learning for Wiener systems with time‐delay state‐space model

AbstractThis paper discusses a novel scheme for learning the Wiener output error nonlinear system with time‐delay state‐space model. In the Wiener system, the dynamic linear block is approximated by time‐delay state‐space model, and the static nonlinear block is established using neural fuzzy network. Combined signals designed including separable signal and random signal are devoted to achieving parameters separation learning of the Wiener system, that is, the two blocks are learned independently. Firstly, using the properties of shift operator and transforming state‐space model with time‐delay into a representation with input and output, then linear dynamic block parameters are learned by the virtue of correlation analysis method in the condition of Gaussian signals. Moreover, a recursive extended least squares estimation is carried out to learn parameters of static nonlinear block and colored noise model under the condition of random signals. The efficiency and accuracy of proposed scheme are confirmed on experiment results of a numerical simulation and a typical practical nonlinear process, and experimental simulation results demonstrate that the learning scheme proposed obtains good learning precision.

Estimation of wiener nonlinear systems with measurement noises utilizing correlation analysis and Kalman filter

AbstractThis paper is concerned with parameter estimation of Wiener systems with measurement noises employing correlation analysis method and adaptive Kalman filter. The presented Wiener system consists of two series blocks, that is, a dynamic block represented by auto‐regressive moving average (ARMA) model, and static nonlinear block established by neural fuzzy model. Aim at estimating separately the two blocks, the separable signals are introduced. First, applying the separable signals to decouple the identification of linear dynamic block from that of static nonlinear block, then ARMA model parameters are estimated employing correlation function‐based least squares principle. Moreover, aiming at handle with error caused by colored measurement noise, adaptive Kalman filter technique and cluster method are introduced to estimate parameter of the nonlinear block and noises model, enhancing parameter estimation precision. The accuracy and applicability of estimated scheme presented are verified through numerical simulation and nonlinear process, the results demonstrate that it is feasible for estimating the Wiener systems in the presence of colored measurement noises.

Parameters estimation for the Hammerstein‐Wiener models with colored noise based on hybrid signals

SummaryA three‐stage estimation approach of the Hammerstein‐Wiener model with colored noise using hybrid signals is considered in this article. The Hammerstein‐Wiener model where a linear dynamic block is embedded between two static nonlinear elements, in which two nonlinear elements represented by two independent neural fuzzy models and a linear element represented by autoregressive exogenous model. The designed hybrid signals that consist of separable signals and random signals are devoted to estimating independently. First, the characteristics of separable signals in the action of the static nonlinear element are analyzed, then the output nonlinear element parameters are estimated utilizing two groups of separable signals with a multiple. Moreover, least squares based on correlation analysis method is applied to estimate linear element using one set of separable signals, which handles the interference of colored process noise. Finally, the recursive extended least squares algorithm is derived to estimate the input nonlinear element and the autoregressive moving average noise model, which improves parameters estimation accuracy owing to estimating colored noise model parameters in recursive estimate process. The feasibility of the presented estimation technique is demonstrated by an illustrative simulation example and a practical nonlinear process.

Neural Network-Based Hammerstein Model Identification of a Lab-Scale Batch Reactor.

This paper focuses on two types of neural network-based Hammerstein model identification methods for the acrylamide polymerization reaction of a batch reactor process. The first neural-based identification type formulates the weights of the multilayer network directly as parameters of the nonlinear static and linear dynamic blocks of the Hammerstein model and trains the weights using a gradient-based backpropagation algorithm. In the second identification type, the nonlinear static block of the Hammerstein model is framed as a single hidden-layer feedforward network and both nonlinear and linear block parameters are trained using an extreme learning machine, where the training procedure is exempted from gradient calculation. The primary focus of the paper is neural-based model identification of a complex nonlinear system, which facilitates ease of linear/nonlinear controller design with good learning speed and less computations. A future work toward the machine learning-based nonlinear model predictive controller implementation using the Jetson Orin Nano board is also described.

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.

Fast Nonlinear Predictive Control Using Classical and Parallel Wiener Models: A Comparison for a Neutralization Reactor Process

The Wiener model, composed of a linear dynamical block and a nonlinear static one connected in series, is frequently used for prediction in Model Predictive Control (MPC) algorithms. The parallel structure is an extension of the classical Wiener model; it is expected to offer better modeling accuracy and increase the MPC control quality. This work discusses the benefits of using the parallel Wiener model in MPC. It has three objectives. Firstly, it describes a fast MPC algorithm in which parallel Wiener models are used for online prediction. In the presented approach, sophisticated trajectory linearization is performed online, which leads to computationally fast quadratic optimization. The second objective of this work is to study the influence of the model structure on modeling accuracy. The well-known neutralization benchmark process is considered. It is shown that the parallel Wiener models in the open-loop mode generate significantly fewer errors than the classical structure. This work’s third objective is to validate the efficiency of parallel Wiener models in closed-loop MPC. For the neutralization process, it is demonstrated that parallel models demonstrate better control quality using various indicators, but the difference between the classical and parallel models is not significant.

Exponential excitations for effective identification of Wiener system

The paper considers the problem of nonparametric estimation of nonlinear characteristic in the FIR Wiener system. Methods proposed so far have suffered from the so-called ‘curse of dimensionality’ and their practical applications have been limited to the very short impulse responses of the linear component. In the proposed approach, the class of (stochastically initialised) repeated exponential input excitations is introduced, and a modified kernel-based estimator is presented. It is shown that the rate of convergence to the genuine system characteristic is significantly improved and does not depend on the length of the impulse response of the linear dynamic block.

Identification of series–parallel systems composed of linear and nonlinear blocks

AbstractMost works on system identification of block‐oriented nonlinear systems were devoted to Wiener and Hammerstein systems. This article is focused on a more general, and so more complex, nonlinear model structure. Specifically, the system under study is a series connection of two subsystems each one being constituted of linear dynamic block and a nonlinear static block coupled in parallel. Clearly, this system structure generalizes those of Wiener and Hammerstein systems. Interestingly, the linear blocks can be parametric or not and are allowed to be of unknown structure. We develop a two‐stage frequency‐type identification method that provides accurate estimates of the all system parts. The proposed identification method enjoys the simplicity and the consistency of developed estimators. Furthermore, the parameters of linear dynamic blocks can be easily decoupled without any further experiments. The performances of the proposed identification method are highlighted by several simulation results. As perspectives, one or the two nonlinearities can be considered non‐static, for example, of backlash type.

Correlation analysis-based parameter learning of Hammerstein nonlinear systems with output noise

In this paper, a novel correlation analysis-based parameter learning scheme for Hammerstein nonlinear systems with output noise is presented. The developed Hammerstein system contains a static nonlinear block approximated by neural fuzzy network and a linear dynamic block modeled by transfer function, and parameter separation learning of the nonlinear block and linear block are realized by using combined signals. Firstly, based on the input and output of the separable signal, the correlation analysis algorithm is applied to estimate linear block parameters, thereby the interference of moving average noise is dramatically handled. Moreover, to improve parameter estimation precision of the learned Hammerstein system, multi-innovation theory and data filtering technology are introduced, and a data filtering-based multi-innovation stochastic gradient learning scheme is implemented for jointly estimating nonlinear block parameters and noise model using random signals. Studies with a numerical simulation and a practical nonlinear process demonstrate the efficiency. In the numerical simulation, when the noise to signal ratio is less than 33.37%, estimate error of the linear block parameters using least squares method is less than 0.0796. In contrast, estimate error of the proposed method is less than 0.0338. For the modeling ability of the nonlinear block, the proposed method has an improvement of 67.07% than the MI-ESG method with innovation length of six. With regard to practical nonlinear process, when the concentration set value is 0.1, the rise time of the proposed method is 0.021 h, while the rise time of MPC and traditional PI controller and is 0.055 h and 0.143 h, respectively.

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 MIMO Wiener-type Koopman models for data-driven model reduction using deep learning

We use Koopman theory to develop a data-driven nonlinear model reduction and identification strategy for multiple-input multiple-output (MIMO) input-affine dynamical systems. While the present literature has focused on linear and bilinear Koopman models, we derive and use a Wiener-type Koopman formulation. We discuss that the Wiener structure is particularly suitable for model reduction, and can be naturally derived from Koopman theory. Moreover, the Wiener block-structure unifies the mathematical simplicity of linear dynamical blocks and the accuracy of bilinear dynamics. We present a Koopman deep-learning strategy combining autoencoders and linear dynamics that generates low-order surrogate models of MIMO Wiener type. In three case studies, we apply our framework for identification and reduction of a system with input multiplicity, a chemical reactor and a high-purity distillation column. We compare the prediction performance of the identified Wiener models to linear and bilinear Koopman models. We observe the highest accuracy and strongest model reduction capabilities of low-order Wiener-type Koopman models, making them promising for control.

Recursive Identification for Fractional Order Hammerstein Model Based on ADELS

This paper deals with the identification of the fractional order Hammerstein model by using proposed adaptive differential evolution with the Local search strategy (ADELS) algorithm with the steepest descent method and the overparameterization based auxiliary model recursive least squares (OAMRLS) algorithm. The parameters of the static nonlinear block and the dynamic linear block of the model are all unknown, including the fractional order. The initial value of the parameter is obtained by the proposed ADELS algorithm. The main innovation of ADELS is to adaptively generate the next generation based on the fitness function value within the population through scoring rules and introduce Chebyshev mapping into the newly generated population for local search. Based on the steepest descent method, the fractional order identification using initial values is derived. The remaining parameters are derived through the OAMRLS algorithm. With the initial value obtained by ADELS, the identification result of the algorithm is more accurate. The simulation results illustrate the significance of the proposed algorithm.

A novel learning algorithm of the neuro-fuzzy based Hammerstein–Wiener model corrupted by process noise

The Hammerstein–Wiener model is a nonlinear system with three blocks where a dynamic linear block is sandwiched between two static nonlinear blocks. For parameter learning of the Hammerstein–Wiener model, the synchronous parameter learning methods are proposed to learn the model parameters by constructing hybrid model of the three series block, such as over parameterization method, subspace method and maximum likelihood method. It should be pointed out that the aforementioned methods appeared the product term of model parameters in the process of parameter learning, and parameter separation method is further adopted to separate hybrid parameters, which increases the complexity of parameter learning. To address this issue, a novel three-stage parameter learning method of the neuro-fuzzy based Hammerstein–Wiener model corrupted by process noise using combined signals is developed in this paper. The combined signals are designed to completely separate the parameter learning issues of the static input nonlinear block, the linear dynamic block and the static output nonlinear block, which effectively simplifies the process of parameter learning of the Hammerstein–Wiener model. Parameter learning of the Hammerstein–Wiener model are summarized into the following three aspects: The first one is to learn the output static nonlinear block parameters using two sets of separable signals with different sizes. The second one is to estimate the linear dynamic block parameters by means of the correlation analysis method, the unmeasurable intermediate variable information problem is effectively handled. The final one is to determine the parameters of the static input nonlinear block and the moving average noise model using recursive extended least square scheme. The simulation results are presented to illustrate that the proposed learning approach yields high learning accuracy and good robustness for the Hammerstein–Wiener model corrupted by process noise.

Modeling and simulating dynamics of lithium-ion batteries using block-oriented models with piecewise linear static nonlinearity

The article deals with the research of the efficiency of modelling the dynamics of voltage change in lithium-ion rechargeable batteries in charging/discharging modes using nonlinear block-oriented systems. Drawing on experimental data, a structural and parametric identification of the Hammerstein, Wiener and Hammerstein-Wiener models with a polynomial structure of the linear dynamic block and piecewise linear static nonlinearities was performed. It has been established that the best modelling accuracy was ensured by using the Hammerstein-Wiener system with a linear model having the 6th order of the numerator and denominator polynomials and an input delay of 3 samples. It showed 15.67% and 6.2% higher accuracy compared to the Wiener and Hammerstein systems, respectively. The application of those models in battery management systems will make it possible to improve the control quality for battery assemblies of solar and wind power plants in the context of the variable nature of the charging/discharging processes due to the variability of weather conditions and fluctuations in power consumption during a 24-hour period. This will ensure a wider introduction of renewable power generation into existing power systems, which is currently the leading way to ensure sustainable development of the energy sector.

Data‐driven Learning Algorithm of Neural Fuzzy Based Hammerstein‐Wiener System

A novel data‐driven learning approach of nonlinear system represented by neural fuzzy Hammerstein‐Wiener model is presented. The Hammerstein‐Wiener system has two static nonlinear blocks represented by two independent neural fuzzy models surrounding a dynamic linear block described by finite impulse response model. The multisignal theory is designed for employing Hammerstein‐Wiener system to separate parameter learning issues. To begin with, the output nonlinearity parameters are learned utilizing separable signal with different amplitudes. Furthermore, correlation analysis method is implemented for estimating linear block parameters using separable signal inputs and outputs; thereby, the interference of process noise is effectively handled. Finally, multi‐innovation learning technology is introduced to improve system learning accuracy, and then, multi‐innovation extended stochastic gradient algorithm is obtained for optimizing input nonlinearity and noise model using multi‐innovation technique and gradient search method. The simulation results display that presented data‐driven learning approach has the availability of learning Hammerstein‐Wiener system.

Robust stabilization and tracking control schemes for disturbed multi-input multi-output Hammerstein model in presence of approximate polynomial nonlinearities

This article presents robust stabilization and tracking control problems for multi-input multi-output Hammerstein model with external disturbances. This model is characterized by static nonlinear elements followed by a linear dynamic block. Moreover, the unknown parameters of the identified mathematical model are estimated using the multivariable output error state space subspace algorithm. Unlike the general control strategy that used the nonlinearity inversion method, the nonlinearities are supposed not bijective. In this context, inverse nonlinear functions of polynomial structure are suggested in this article. Furthermore, the composition of the static nonlinear elements and their approximate inverses in series with the linear dynamic block are then decomposed into a set of linear parts using the Takagi–Sugeno fuzzy representation. Consequently, new sufficient stability conditions with decay rate and disturbance attenuation using the [Formula: see text] criterion and linear matrix inequality tools are discussed. Finally, simulation studies are provided to illustrate the merit of our purpose.

Modeling of low density polyethylene tubular reactor using nonlinear block-oriented model

Abstract In this paper, nonlinear block-oriented models, namely Neural Wiener (NW) and Neural Hammerstein (NH) model, are developed to simulate Low density polyethylene (LDPE) tubular reactor process. The desired simulated outputs of the reactor model are LDPE polymer conversion and melt flow index (MFI). Generally, a nonlinear block-oriented model consists of dynamic linear block and static nonlinear element, which are identified using nonlinear optimization techniques. A modified Neural Wiener (M−NW) with additional input scheme is also considered during the model development stage. In order to generate data for the model identification process, a steady state model of the LDPE polymerization process is developed using Aspen Plus and then converted into dynamic state using Aspen Dynamic software. Based on the model validation results, M−NW has outperformed the other two models with the coefficient of determination (R2) of 0.993 for polymer conversion and 0.986 for MFI results. Moreover, the simulation results for NW and NH models have shown a comparable performance in modeling the polymer conversion. However, the NW model has performed better in predicting MFI with R2 0.984 compared to than the NH model with R2 0.955. Thus, based on the simulation results, the application of nonlinear block-oriented models such as the M−NW and NW model in simulating the LDPE polymerization process is well justified. Such models can be further implemented in model-based control scheme or as soft sensor.

Real-Time Optimization of a CO Preferential Oxidation Reactor Temperature with Extremum Seeking Control Techniques.

A hydrogen (H2)-rich gas mixture is used as the fuel of the proton exchange membrane fuel cell (PEMFC). A small amount of the carbon monoxide (CO) gas in the gas mixture can significantly deactivate the catalyst of the PEMFC, resulting in a reduction in the efficiency of power generation. Preferential oxidation is used to reduce the CO concentration less than 10 ppm in the gas mixture. It has an optimal reaction temperature at which the reaction shows the minimum exit CO concentration with minimum consumption of H2. This optimal temperature continuously changes under the varying conditions of operation and catalyst deactivation. In this study, two modified extremum seeking control (ESC) methods were proposed to continuously seek and maintain this optimal reaction temperature, guaranteeing CO concentration under 10 ppm even under time-varying conditions. The proposed methods have a smaller number of design parameters than the conventional extremum seeking approaches so that tuning the ESC method is much easier, more intuitive, and efficient. In addition, the proposed method can additionally use the secondary measurement to improve the performance of the ESC method by removing the possibility that the modeling error of the linear dynamic block can deteriorate the accuracy of calculating the gradient of the nonlinear static block. The experimental results confirmed that the proposed methods can track the optimal temperature within a short time compared to the conventional approach while successfully maintaining the CO concentration below 10 ppm.