NLFXLMS and THF‐NLFXLMS Algorithms for Wiener‐Hammerstein Nonlinear Active Noise Control

Nonlinear Filtered‐X LMS (NLFXLMS) is an indirect adaptive control algorithm for nonlinear active noise control (NANC) system. The algorithm has been developed for both Hammerstein and Wiener secondary paths where the nonlinearity is represented by scaled error function (SEF) and tangential hyperbolic function (THF). NLFXLMS algorithm is limited in practical application because the degree of nonlinearity has to be known in advance. This limitation leads to the development of the THF‐NLFXLMS algorithm where the degree of nonlinearity is estimated by modelling the secondary path. In this work, the NLFXLMS and THF‐NLFXLMS are extended to Wiener‐Hammerstein system. The performance of the proposed Wiener‐Hammerstein THF‐NLFXLMS is compared with NLFXLMS algorithm which is considered as the benchmark and second order Volterra algorithm of comparable computational complexity. Simulation results show that the THF‐NLFXLMS has a similar performance to NLFXLMS and outperforms the second order Volterra algorithm as the system becomes more nonlinear.     

Particle swarm optimization based nonlinear active noise control under saturation nonlinearity

In this paper, a method is proposed to overcome the saturation non-linearity linked to the microphones and loudspeakers of active noise control (ANC) system. The reference microphone gets saturated when the acoustic noise at the source increases beyond the dynamic limits of the microphone. When the controller tries to drive the loudspeaker system beyond its dynamic limits, the saturation nonlinearity is also introduced into the system. The secondary path which is generally estimated with a low level auxiliary noise by a linear transfer function does not model such saturation nonlinearity. Therefore, the filtered-x least mean square (FXLMS) algorithm fails to perform when the noise level is increased. For alleviating the saturation nonlinearity effect a nonlinear functional expansion based ANC algorithm is proposed where the particle swarm optimization (PSO) algorithm is suitably applied to tune the parameters of a filter bank based functional link artificial neural network (FLANN) structure, named as PSO based nonlinear structure (PSO-NLS) algorithm. The proposed algorithm does not require any computation of secondary path estimate filtering unlike other conventional gradient based algorithms and hence has got computational advantage. The computer simulation experiments show its superior performance compared to the FXLMS, filtered-s LMS and genetic algorithms under saturation present at both at secondary and reference paths. The paper also includes a sensitivity analysis to study the effect of different parameters on ANC performance.     

Robust sampled‐data model predictive control for networked systems with time‐varying delay

In this paper, the problem of sampled‐data model predictive control (MPC) is investigated for linear networked control systems with both input delay and input saturation. The delay‐induced nonlinearity is overapproximatively modeled as a polytopic inclusion. The nonlinear behavior of input saturation is expressed as a convex polytope. The resulting closed‐loop systems are represented as linear systems with polytopic and additive norm‐bounded uncertainties. The aim is to determine a robust MPC controller that asymptotically stabilizes the uncertain system at the origin with a certain level of quadratic performance. The effectiveness of the proposed algorithm is demonstrated by a numerical example.     

Output error convergence of adaptive filters with compensation foroutput nonlinearities

Output error convergence of a Wiener model-based nonlinear stochastic gradient algorithm is analyzed. The normalized scheme estimates the parameters of a linear finite impulse response model in cascade with a known output nonlinearity. The algorithm can be interpreted as a normalized least mean square algorithm with compensation for an output nonlinearity. Linearizing inversion of the nonlinearity is not utilized. Global output error convergence is then proved, provided that the nonlinearity is monotone (not strictly monotone), and provided that a previously observed mechanism resulting in deadlock does not occur. The algorithm and the analysis include important practical cases like sensor saturation and dead zones that must be excluded when global parametric convergence is studied     

Identification of IIR Wiener systems with spline nonlinearities that have variable knots

An algorithm is developed for the identification of Wiener systems, linear dynamic elements followed by static nonlinearities. In this case, the linear element is modeled using a recursive digital filter, while the static nonlinearity is represented by a spline of arbitrary but fixed degree. The primary contribution in this note is the use of variable knot splines, which allow for the use of splines with relatively few knot points, in the context of Wiener system identification. The model output is shown to be nonlinear in the filter parameters and in the knot points, but linear in the remaining spline parameters. Thus, a separable least squares algorithm is used to estimate the model parameters. Monte-Carlo simulations are used to compare the performance of the algorithm identifying models with linear and cubic spline nonlinearities, with a similar technique using polynomial nonlinearities.     

Convergence analysis of recursive identification algorithms basedon the nonlinear Wiener model

Recursive identification algorithms, based on the nonlinear Wiener model, are presented. A recursive identification algorithm is first derived from a general parameterization of the Wiener model, using a stochastic approximation framework. Local and global convergence of this algorithm can be tied to the stability properties of an associated differential equation. Since inversion is not utilized, noninvertible static nonlinearities can be handled, which allows a treatment of, for example, saturating sensors and blind adaptation problems. Gauss-Newton and stochastic gradient algorithms for the situation where the static nonlinearity is known are then suggested in the single-input/single-output case. The proposed methods can outperform conventional linearizing inversion of the nonlinearity when measurement disturbances affect the output signal. For FIR (finite impulse response) models, it is also proved that global convergence of the schemes is tied to sector conditions on the static nonlinearity. In particular, global convergence of the stochastic gradient method is obtained, provided that the nonlinearity is strictly monotone. The local analysis, performed for IIR (infinite impulse response) models, illustrates the importance of the amplitude contents of the exciting signals     

A unifying view of wiener and volterra theory and polynomial kernel regression

Volterra and Wiener series are perhaps the best-understood nonlinear system representations in signal processing. Although both approaches have enjoyed a certain popularity in the past, their application has been limited to rather low-dimensional and weakly nonlinear systems due to the exponential growth of the number of terms that have to be estimated. We show that Volterra and Wiener series can be represented implicitly as elements of a reproducing kernel Hilbert space by using polynomial kernels. The estimation complexity of the implicit representation is linear in the input dimensionality and independent of the degree of nonlinearity. Experiments show performance advantages in terms of convergence, interpretability, and system sizes that can be handled.     

A recursive hierarchical parametric estimation algorithm for nonlinear systems described by Wiener‐Hammerstein models

In this paper, a recursive hierarchical parametric estimation (RHPE) algorithm is proposed for stochastic nonlinear systems which can be described by Wiener‐Hammerstein (W‐H) mathematical models. The formulation of parameters estimation problem is based on the prediction error approach and the gradient techniques. The convergence analysis of the developed RHPE algorithm is derived using stochastic gradient‐based theory. Wiener‐Hammerstein hydraulic process is treated to prove the efficiency of the proposed approach.     

Self‐tuning minimum variance control of linear systems followed by saturation nonlinearities in a noisy frame

The aim of the present paper was to increase the efficiency of self‐tuning minimum variance (MV) control of linear systems followed by the so‐called hard nonlinearities. To this end, an approach based on reordering of observations to be processed for the reconstruction of an unmeasurable internal intermediate signal, which acts between a linear dynamic time‐invariant (LTI) system and a static nonlinear block of the closed loop Wiener system with a saturation nonlinearity in an output, has been developed. The technique based on the ordinary least squares and on data partition is used for the internal signal extraction. The results of numerical simulation, identification, and self‐tuning MV control as well as generalized MV control of the second‐order discrete‐time closed loop LTI system with the saturation nonlinearity are given by the computer. Copyright © 2012 John Wiley & Sons, Ltd.     

Minimum variance lower bound ratio based nonlinearity measure for closed loop systems

Uninterrupted wide range operations are salient features of the process industries. Characteristics including nonlinearity, time-delay, and inertia, of industrial control loops are always significantly changed with the working conditions. To qualify the nonlinear degree, a nonlinearity measure based on the minimum variance lower bound ratio is presented in this paper. This measure belongs to the data-driven class, and can be applied to Hammerstein structure, Wiener structure and Wiener–Hammerstein structure, whether the controller is linear or nonlinear. The effectiveness and consistency of this measure are illustrated through five simulation tests. An industrial case in the fossil fuel power generation process is studied to confirm the practicability of the proposed measure.     

Recursive identification for Hammerstein–Wiener systems with dead-zone input nonlinearity

A new recursive algorithm is proposed for the identification of a special form of Hammerstein–Wiener system with dead-zone nonlinearity input block. The direct motivation of this work is to implement on-line control strategies on this kind of system to produce adaptive control algorithms. With the parameterization model of the Hammerstein–Wiener system, a special form of model estimation error is defined; and then its approximate formula is given for the following derivation. Based on these, a recursive identification algorithm is established that aims at minimizing the sum of the squared parameter estimation errors. The conditions of uniform convergence are obtained from the property analysis of the proposed algorithm and an adaptive setting method for a weighted factor in the algorithm is given, which enhances the convergence of the proposed algorithm. This algorithm can also be used for the identification of the Hammerstein systems with dead-zone nonlinearity input block. Three simulation examples show the validity of this algorithm.     

Subspace based approaches for Wiener system identification

We consider the problem of Wiener system identification in this note. A Wiener system consists of a linear time invariant block followed by a memoryless nonlinearity. By modeling the inverse of the memoryless nonlinearity as a linear combination of known nonlinear basis functions, we develop two subspace based approaches, namely an alternating projection algorithm and a minimum norm method, to solve for the Wiener system parameters. Based on computer simulations, the algorithms are shown to be robust in the presence of modeling error and noise.     

Behavioral modeling for concurrent dual‐band power amplifiers using 2D hammerstein/wiener models

Behavioral modeling for the concurrent dual‐band power amplifier (PA) is a critical problem in practical applications. The nonlinear distortion in the concurrent dual‐band PA is quite different from that in the conventional single‐band PA. This article analyzes the nonlinearities in the concurrent dual‐band PA and reveals that both input signals in the dual bands are important for the behavioral modeling. The 2D Hammerstein model and 2D Wiener model are proposed for the first time for the concurrent dual‐band PA. They are extended versions of conventional Hammerstein and Wiener structures used in the single‐band PA by including the cross‐band intermodulation in the static nonlinearity block. The proposed 2D models require much less coefficients than the original work of the 2D‐DPD model. Experiments were carried out for an 880 MHz/1960 MHz concurrent dual‐band Doherty PA to demonstrate the effectiveness of the proposed models. The results clearly show that less than ?40 dB normalized mean square errors (NMSEs) are obtained in the dual bands in the behavioral modeling. © 2012 Wiley Periodicals, Inc. Int J RF and Microwave CAE 23: 646–654, 2013.     

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.     

一种基于Wiener模型的非线性预测控制算法

针对一类Wiener模型描述的非线性系统,提出了一种改进的非线性预测控制算法.该算法利用Laguerre函数描述Wiener模型动态线性部分的控制信号，将预测控制中在预测时域内优化求解未来控制输入序列转化为优化求解一组无记忆的Laguerre系数,以减少优化所需的计算量.利用静态模糊模型来逼近Wiener模型的非线性部分,将非线性预测控制优化问题转化为线性预测控制优化问题,克服了求控制输入时解非线性方程的困难,进而推导出了预测控制输入的解析式.CSTR过程的仿真结果表明了本文算法的有效性和可行性.     

基于头脑风暴优化算法的Wiener模型参数辨识

Wiener模型是一种典型的模块化非线性模型,广泛应用于工业过程控制领域.由于其结构的非线性,参数辨识无法直接得到解析解.为此,将Wiener模型的参数估计转化为带约束的非线性优化问题,以头脑风暴优化(BSO)算法并行搜索该问题的最优解,并以搜索过程中的反馈信息调整BSO算法的变异过程,以改进算法的收敛速度和辨识精度.数值仿真和工业数据验证了所提算法的有效性.     

A robust evolutionary feedforward active noise control system using Wilcoxon norm and particle swarm optimization algorithm

The conventional filtered-x least mean square (FxLMS) algorithm commonly employed for active noise control (ANC) is sensitive to disturbances acquired by the error microphone and yields poor performance in such scenario. To circumvent this problem, in this paper, a Wilcoxon FxLMS (WFxLMS) algorithm is proposed and used in the design of an efficient ANC which is robust to outliers in the secondary path and immune to burst noise acquired by the error microphone. It is demonstrated through simulation study that under such situation the proposed algorithm outperforms the traditional FxLMS algorithm. A particle swarm optimization (PSO) algorithm based robust ANC system, which does not require the modeling of the secondary path is also derived in the paper. Improved performance of the robust evolutionary ANC system over L2 norm based evolutionary ANC system is also shown.     

Identification of time-varying pH processes using sinusoidal signals

This paper presents an approach to the identification of time-varying, nonlinear pH processes based on the Wiener model structure. The algorithm produces an on-line estimate of the titration curve, where the shape of this static nonlinearity changes as a result of changes in the weak-species concentration and/or composition of the process feed stream. The identification method is based on the recursive least-squares algorithm, a frequency sampling filter model of the linear dynamics and a polynomial representation of the inverse static nonlinearity. A sinusoidal signal for the control reagent flow rate is used to generate the input-output data along with a method for automatically adjusting the input mean level to ensure that the titration curve is identified in the pH operating region of interest. Experimental results obtained from a pH process are presented to illustrate the performance of the proposed approach. An application of these results to a pH control problem is outlined.     

Application of Wiener model predictive control (WMPC) to an industrial C2-splitter

Dual composition control of a high-purity distillation column is recognized as an industrially important, yet notoriously difficult control problem. It is proposed, however, that Wiener models, consisting of a linear dynamic element followed in series by a static nonlinear element, are ideal for representing this and several other nonlinear processes. They are relatively simple models requiring little more effort in development than a standard linear step response model, yet offer superior characterization of systems with highly nonlinear gains. Wiener models may be incorporated into MPC schemes in a unique way that effectively removes the nonlinearity from the control problem, preserving many of the favorable properties of linear MPC, especially in the analysis of stability. In this paper, Wiener model predictive control is applied to an industrial C2-splitter at the Orica Olefines plant with promising results.