Block-oriented Nonlinear Systems Research Articles

Adaptive Regulation of Block-Oriented Nonlinear Systems Using Binary Sensors With Applications to Automotive Engine Control

In this article, adaptive regulation of block-oriented nonlinear systems, i.e., Hammerstein and Wiener systems, with binary-valued measurements of the regulation errors is considered. Compared with the classical framework for stochastic adaptive control, the new feature here is that only binary-valued observations of regulation errors are available to the controller. An adaptive regulator based on the stochastic approximation algorithm is proposed and it is proved that the regulator is optimal in the sense that it minimizes the long-run average of the squared regulation errors almost surely. Numerical examples as well as real applications of the proposed algorithms to automotive engine control are given.

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.

A Nonlinear Model for Online Identifying a High-Speed Bidirectional DC Motor

The modeling system is a process to define the real physical system mathematically, and the input/output data are responsible for configuring the relation between them as a mathematical model. Most of the actual systems have nonlinear performance, and this nonlinear behavior is the inherent feature for those systems; Mechatronic systems are not an exception. Transforming the electrical energy to mechanical one or vice versa has not been done entirely. There are usually losses as heat, or due to reverse mechanical, electrical, or magnetic energy, takes irregular shapes, and they are concerned as the significant resource of that nonlinear behavior. The article introduces a nonlinear online Identification of a high-speed bidirectional DC motor with dead zone and Coulomb friction effect, which represent a primary nonlinear source, as well as viscosity forces. The Wiener block-oriented nonlinear system with neural networks are implemented to identify the nonlinear dynamic, mechatronic system. Online identification is adopted using the recursive weighted least squares(RWLS) method, which depends on the current and (to some extent) previous data. The identification fitness is found for various configurations with different polynomial orders, and the best model fitness is obtained about 98% according to normalized root mean square criterion for a third order polynomial.

Volterra series identification and its applications in structural identification of nonlinear block-oriented systems

This paper considers the identification of a Volterra system and its applications in structural identification of nonlinear block-oriented models. Any order of the Volterra output is estimated separately via multilevel excitations and optimizations. Then, each order of the Volterra kernels is estimated independently with improved accuracy. Finally, relationships between the first and the second order Volterra kernel functions of block-oriented models are exploited to determine the structures of nonlinear block-oriented systems. The simulation studies verify the effectiveness of the proposed Volterra series identification method and the structure identification method for nonlinear block-oriented systems.

Block-oriented system identification for nonlinear modeling of all-solid-state Li-ion battery technology

The high energy density characteristic of the all-solid-state battery technology makes it one of the promising competitors of current liquid electrolyte-based lithium-ion batteries. However, due to the low ionic conductivity of solid materials and existing issues in active material-solid electrolyte interfaces, all-solid-state batteries show strong nonlinear behavior when the amplitude of input current is relatively high. In this paper, we have developed a methodology based on block-oriented nonlinear systems that can detect, quantize, and model the battery behavior, accurately. For this purpose, two solid-state coin cells, which have different ionic conductivity at the cathode interface are manufactured and excited with multisine input current. The Best Linear Approximation (BLA) method has been used to separate linear frequency response function (FRF) from nonlinear distortion. Then, a Hammerstein-Wiener system has been developed and parametrized to model the nonlinear distortions caused by ionic conductivity variation of the solid electrolyte at the cathode layer. Furthermore, the developed model has been utilized to detect possible faults in the electrode-electrolyte interface based on the nonlinearity level of the measured voltage.

Parameter Identification of Block-Oriented Nonlinear Systems in the Frequency Domain

A problem of parameter identification of nonlinear dynamic systems of manufacturing processes on the set of block-oriented models in the frequency domain is considered. Method of parameter identification in steady state at the input sinusoidal influences is proposed. The solution of the problem of parameter identification is reduced to the solution of the systems of algebraic equations by using the Fourier approximation. The parameters estimations are received by the least squares method. The identification method is investigated by theoretical analysis and computer modelling.

Interval variable step-size spline adaptive filter for the identification of nonlinear block-oriented system

In order to improve the convergence speed of the nonlinear spline adaptive filter (SAF) in the identification of block-oriented systems, an interval variable step-size algorithm is proposed. Traditional SAF algorithm uses constant step size during iteration, leading to a contradiction between convergence speed and steady-state accuracy. In this paper, a new kind of variable step-size algorithm is proposed, fully considering the particularity of spline interpolation in the nonlinear part of the block-oriented model. The step size of each interpolation interval is independent from that of other intervals, and it is dominated by the correlated squared error which is evaluated by an exponential-weighted averaging (EWA) process. In this paper, the independent step size in each interpolation interval is also updated through an EWA process of the correlated error. The effects of the parameters on the convergence performance of the proposed strategy have been theoretically analyzed and verified by simulations. Finally, some numerical simulations have confirmed that the proposed interval variable step-size approach can significantly improve the convergence speed as well as reduce the steady-state error compared with the traditional SAF and the existing variable step-size SAF algorithms.

Recursive identification of block-oriented nonlinear systems in the presence of outliers

The paper considers the recursive identification of nonlinear (Hammerstein) model in the presence of outliers. In this model the nonlinear part is a polynomial of a known order in the input and linear part is described as an ARX (auto - regressive model with exogenous inputs) model. It is assumed that there is a priori information about a disturbance distribution in the form of a class of distributions. Owing that fact it is possible to use robust statistics in the sense of Huber. But Huber`s loss function is only first order differentiable. It follows that second order methods (Newton – Raphson methodology) cannot be used. The problem is avoided by analysis of the structure of least favourable distribution. It is shown that robust recursive algorithm derivation can be set in frame of l1−l2 – norm estimation problem. The main contributions of the paper are: (i) it is shown that the robust identification belongs to l1−l2 – norm estimation problem; (ii) the derivation of the new robust recursive algorithm; (iii) it is shown that robust recursive algorithm is switching algorithm (collection of linear least squares and sign algorithm) and that fact opens the new research opportunities.

Parameter Identification of Nonlinear Systems with Time-delay Based on the Multi-innovation Stochastic Gradient Algorithm

This paper considers the parameter identification problem of block-oriented Hammerstein nonlinear systems with time-delay. Firstly, we adopt the data filtering technique to transform the identification model so that all the parameters will be separated in the resulting identification model which has no redundant parameters. Secondly, a multi-innovation stochastic gradient algorithm is used to estimate the system parameters. The proposed method has high computational efficiency and good accuracy. Simulation results are presented to demonstrate the effectiveness of the proposed algorithm.

Identification of nonlinear block-oriented systems with backlash and saturation

Abstract A new approach to modeling and identification of discrete-time nonlinear dynamic systems with input backlash and output saturation nonlinearities is presented. The proposed three-block cascade mathematical model results from successive applications of the key-term separation principle. This provides special nonlinear model description that is linear in parameters. An iterative technique with internal variable estimation is proposed for estimation of all the model parameters based on measured input/output data and minimizing the least-squares criterion. Illustrative example of cascade system identification with backlash and saturation is included.

Structure Identification of Continuous-Time Block-Oriented Nonlinear Systems in the Frequency Domain

Abstract A problem of structure identification of nonlinear dynamic systems on the set of continuous block-oriented models with nonlinear elements in the form of polynomial functions of finite degree and stable linear elements in the frequency domain is considered. Method of structure identification in steady state based on the observation of the system’s input and output variables at the input sinusoidal influences is proposed. The problem of the structure identification is considered according to the classical definition of identification of Zadeh. The structure definition criterion of the model is developed. The identification method is investigated by theoretical analysis and computer modelling.

Multi-innovation Stochastic Gradient Algorithms for Input Nonlinear Time-Varying Systems Based on the Line Search Strategy

Block-oriented nonlinear systems have attracted a considerable attention for their flexible structure and practicability. This study proposes a novel multi-innovation stochastic gradient (MISG) algorithm to address the identification problem in input nonlinear systems. This involves applying the inexact line search strategy to determine an appropriate convergence factor at each recursive step. The proposed algorithm tracks the nonlinear system dynamics faster than the conventional MISG algorithm. It is therefore suitable for online identification and can be applied to nonlinear time-varying systems. The concept of auxiliary model identification is also adopted for dealing with unmeasurable variables. The effectiveness of the proposed algorithm is verified through simulated examples.

Gradient‐based iterative parameter estimation for bilinear‐in‐parameter systems using the model decomposition technique

The parameter estimation issues of a block-oriented non-linear system that is bilinear in the parameters are studied, i.e. the bilinear-in-parameter system. Using the model decomposition technique, the bilinear-in-parameter model is decomposed into two fictitious submodels: one containing the unknown parameters in the non-linear block and the other containing the unknown parameters in the linear dynamic one and the noise model. Then a gradient-based iterative algorithm is proposed to estimate all the unknown parameters by formulating and minimising two criterion functions. The stochastic gradient algorithms are provided for comparison. The simulation results indicate that the proposed iterative algorithm can give higher parameter estimation accuracy than the stochastic gradient algorithms.

Identification of block-oriented nonlinear systems starting from linear approximations: A survey

Block-oriented nonlinear models are popular in nonlinear system identification because of their advantages of being simple to understand and easy to use. Many different identification approaches were developed over the years to estimate the parameters of a wide range of block-oriented nonlinear models. One class of these approaches uses linear approximations to initialize the identification algorithm. The best linear approximation framework and the ϵ-approximation framework, or equivalent frameworks, allow the user to extract important information about the system, guide the user in selecting good candidate model structures and orders, and prove to be a good starting point for nonlinear system identification algorithms. This paper gives an overview of the different block-oriented nonlinear models that can be identified using linear approximations, and of the identification algorithms that have been developed in the past. A non-exhaustive overview of the most important other block-oriented nonlinear system identification approaches is also provided throughout this paper.

Iterative identification of block-oriented nonlinear systems based on biconvex optimization

We investigate the identification of a class of block-oriented nonlinear systems which is represented by a common model in this paper. Then identifying the common model is formulated as a biconvex optimization problem. Based on this, a normalized alterative convex search (NACS) algorithm is proposed under a given arbitrary nonzero initial condition. It is shown that we only need to find the unique partial optimum point of a biconvex cost function in order to obtain its global minimum point. Thus, the convergence property of the proposed algorithm is established under arbitrary nonzero initial conditions. By applying the results to Hammerstein–Wiener systems with an invertible nonlinear function, the long-standing problem on the convergence of iteratively identifying such systems under arbitrary nonzero initial conditions is also now solved.

Structure discrimination in block-oriented models using linear approximations: A theoretic framework

In this paper we show that it is possible to retrieve structural information about complex block-oriented nonlinear systems, starting from linear approximations of the nonlinear system around different setpoints. The key idea is to monitor the movements of the poles and zeros of the linearized models and to reduce the number of candidate models on the basis of these observations. Besides the well known open loop single branch Wiener-, Hammerstein-, and Wiener–Hammerstein systems, we also cover a number of more general structures like parallel (multi branch) Wiener–Hammerstein models, and closed loop block oriented models, including linear fractional representation (LFR) models.

Hankel matrices for system identification

The coefficients of a linear system, even if it is a part of a block-oriented nonlinear system, normally satisfy some linear algebraic equations via Hankel matrices composed of impulse responses or correlation functions. In order to determine or to estimate the coefficients of a linear system it is important to require the associated Hankel matrix be of row-full-rank. The paper first discusses the equivalent conditions for identifiability of the system. Then, it is shown that the row-full-rank of the Hankel matrix composed of impulse responses is equivalent to identifiability of the system. Finally, for the row-full-rank of the Hankel matrix composed of correlation functions, the necessary and sufficient conditions are presented, which appear slightly stronger than the identifiability condition. In comparison with existing results, here the minimum phase condition is no longer required for the case where the dimension of the system input and output is the same, though the paper does not make such a dimensional restriction.

Truncation order and its effect in a class of nonlinear systems

Fundamental results are established for truncation order and its effect in the Volterra class of nonlinear systems. These results provide a significant insight into the analytical relationships among truncation order, truncation error, and estimation error for the output spectra of nonlinear systems, and also lead to the development of novel methods for choosing the truncation order and excitation magnitudes for accurate determination of nonlinear frequency response components. These results should be important in the kernel estimation and frequency domain analysis of nonlinear systems on the basis of the Volterra series, structure and order selection of block-oriented nonlinear systems with polynomial nonlinearity. The results also provide a much improved version of the multi-level excitation method for system analysis and identification.

Iterative Method in the Identification of Block-Oriented Systems Based on Biconvex Optimization

In this paper, we investigate the identification of the class of block-oriented nonlinear systems presented by Li et al. [2011] by using an iterative method. Firstly a common model is proposed to represent such block-oriented systems. Then identifying the common model is formulated as a biconvex optimization problem. Based on this, a normalized alterative convex search (NACS) algorithm is proposed under a given arbitrary nonzero initial condition. It is shown that we only need to find the unique partial optimum point of a biconvex cost function in the formulated optimization problem in order to obtain its global minimum point. Thus, the convergence property of the proposed algorithm is established under arbitrary nonzero initial conditions. The approach presented in this paper provides a unified framework for the identification of block-oriented systems.

Mixed parametric-nonparametric identification of Hammerstein and Wiener systems - a survey

The paper surveys the ideas of cooperation between parametric and nonparametric (kernel-based) algorithms of nonlinear block-oriented system identification. Various strategies are proposed, dependently on the system structure, number of data and the prior knowledge. The estimates are consistent and their rates of convergence are presented. The aim of the paper is to show some recent results in the field in a systematic, ordered way.