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.     

A unified approach for the parametric identification of SISO/MIMO Wiener and Hammerstein systems

Hammerstein and Wiener models are nonlinear representations of systems composed by the coupling of a static nonlinearity N and a linear system L in the form N–L and L–N respectively. These models can represent real processes which made them popular in the last decades. The problem of identifying the static nonlinearity and linear system is not a trivial task, and has attracted a lot of research interest. It has been studied in the available literature either for Hammerstein or Wiener systems, and either in a discrete-time or continuous-time setting. The objective of this paper is to present a unified framework for the identification of these systems that is valid for SISO and MIMO systems, discrete- and continuous-time settings, and with the only a priori knowledge that the system belongs to the set including Wiener and Hammerstein models.     

随机连续系统基于Laguerre多项式逼近的Markov参数估计

先讨论基于Ｌａｇｕｅｒｒｅ多项式逼近、有连续Ｗｉｅｎｅｒ过程扰动的随机连续线性系统最小二乘参数估计，然后讨论Ｗｉｅｎｅｒ过程的Ｌａｇｕｅｒｒｅ多项式逼近值的相关性和最小二乘估计的有偏性，在此基础上，提出无偏一致的Ｍａｒｋｏｖ估计（最小方差估计）算法，仿真结果显示本文方法的有效性．     

Optimal Controls that Maximize the Expectation of First Passage Time

This paper deals with optimal controls that maximize the expectation of first passage time of the state, of a stochastic non-linear system, to the complement of an open and bounded domain. The performance index includes a penalty on the magnitude of the deviation of the first passage time from its expectation. The nonlinear system considered here is subjected to two different kinds of perturbations. The first kind of perturbation is represented by a vector of independent standard Wiener processes and the second kind by a vector of a generalized type of Poisson process.By using a variational approach, necessary conditions on the optimal controls are derived. These conditions are given by a set of four coupled nonlinear partial integro- differential equations. A nonlinear stochastic third-order system is given as a test case, and a numerical method for the computation of its optimal controls, is suggested. The efficiency and applicability of this method are demonstrated with examples.     

Delay-dependent H∞ filtering for discrete singular Markov jump systems with Wiener process and partly unknown transition probabilities

This paper is devoted to the investigation of the delay-dependent H_∞ filtering problem for a class of discrete-time singular Markov jump systems with Wiener process and partly unknown transition probabilities. The class of stochastic singular model under consideration is more general and covers the stochastic singular Markov jump time-varying delay systems with completely known and completely unknown transition probabilities as two special cases. Firstly, based on a stochastic Lyapunov–Krasovskii candidate function and an auxiliary vector function, by employing some appropriate free-weighting matrices, the discretized Jensen inequality and combining them with the structural characteristics of the filtering error system, a set of delay-dependent sufficient conditions are established, which ensure that the filtering error system is stochastically admissible. And then, a singular filter is designed such that the filtering error system is not only regular, causal and stochastically stable, but also satisfy a prescribed H_∞ performance for all time-varying delays no larger than a given upper bound. Furthermore, the sufficient conditions for the solvability of the H_∞ filtering problem are obtained in terms of a new type of Lyapunov–Krasovskii candidate function and a set of linear matrix inequalities. Finally, simulation examples are presented to illustrate the effectiveness of the proposed method in the paper.     

三种综合脉冲星时算法的研究和比较

建立在小波分析基础上的综合脉冲星时算法，能把脉冲星的观测计时残差在小波域分解，提取出不同频率范围的分量，然后用小波方差表征脉冲星在不同频率范围的稳定度来对单脉冲星时进行加权平均，得到综合脉冲星时；脉冲星的计时残差包括了计时参考的原子钟的误差和与脉冲星本身有关的计时误差两部分，用维纳滤波的方法可以将两者进行一定区分，并消除掉估计的参考钟误差，将剩余部分作为计时残差实现对脉冲星计时的综合。实验证明，小波分析和维纳滤波方法比经典的加权算法更好，得到的综合脉冲星时的长期稳定度有了较大提高。     

基于维纳滤波图像复原的设计与实现

像复原的目的是从退化图像中重建原始图像,改善退化图像的视觉质量。维纳滤波能够较好地进行图像恢复。     

Composite fault-tolerant control with disturbance observer for stochastic systems with multiple disturbances

In this paper, a composite fault tolerant control (CFTC) with disturbance observer scheme is considered for a class of stochastic systems with faults and multiple disturbances. The disturbances are divided into two parts. One represents the stochastic disturbance with partial known information which is formulated by an exogenous system. The other is independent Wiener process. A stochastic disturbance observer is designed to estimate exogenous disturbance. To make the first type of disturbance can be rejected and the fault can be diagnosed, a composite fault diagnosis observer with disturbance observer is constructed. Furthermore, a composite fault-tolerant controller is proposed to compensate disturbances and faults. Finally, simulation examples are given to demonstrate the feasibility and effectiveness of the proposed scheme.     

Reinvigorating the Wiener-Hopf technique in the pursuit of understanding processes and materials

The Wiener-Hopf(WH)method was created in 1931,by Norbert Wiener and Eberhard Hopf,to deliver exact solutions to integral equations with convolution-type kernels...     

On the Computation of Functionals Over Sample Paths of a Randomly Rotating Rigid Body

This paper deals with the computation of the values of two functionals which are defined over the sample paths of a randomly rotating rigid body. It is assumed that the body is subjected to two different kinds of perturbation. The first kind of perturbation is represented by the standard Wiener process and the second kind by a homogeneous process with independent increments, finite second-order moments, mean zero and no continuous sample functions. In order to measure quantitatively the stochastic stability of the body's motion, two functionals are defined over its sample paths. It is shown that each of these functionals is a solution to a corresponding partial integro-differential equation. A numerical procedure for the solution of these equations is suggested, and its efficiency and applicability are demonstrated with examples.     

Identification of Wiener–Hammerstein models based on variational bayesian approach in the presence of process noise

This paper addresses the identification of Wiener–Hammerstein (WH) models in the presence of process and measurement noises, which has not been well studied yet in the existing works. To achieve an unbiased estimation, the model parameters are obtained by maximizing the likelihood function, which is solved in the expectation-maximization framework. Due to the difficulty of computing the posterior distributions of the latent variables of WH models, variational Bayes (VB) is used here, and a method for approximating the posterior distributions based on Monte Carlo integral is proposed in VB framework. To the best of our knowledge, it is the first time to use VB approach for WH model identification. Two simulation examples demonstrate the effectiveness of the proposed method. Moreover, the proposed method is used for a WH benchmark problem, and the results show that it improves the identification performance.     

Sliding mode state and parameter identification for linear stochastic systems

This paper presents the sliding mode mean-square and mean-module state filtering and parameter identification problems for linear stochastic systems with unknown parameters over linear observations, where unknown parameters are considered Wiener processes. The original problems are reduced to the sliding mode mean-square and mean-module filtering problems for an extended state vector that incorporates parameters as additional states. The obtained sliding mode filters for the extended state vector also serve as the optimal identifiers for the unknown parameters. Performance of the designed sliding mode mean-square and mean-module state filters and parameter identifiers are verified for both, stable and unstable, linear uncertain systems.     

随机利率下的责任准备金

本文针对按年缴费的终身寿险模型，改进传统的常值利率的准备金模型。考虑突发事件对利率的影响，利用Weiner过程和Poisson过程联合对利息力建模，求出了此时的均衡保费和准备金的表达式，并在此基础上得出了损失变量方差的表达式。     

An application of homomorphic deconvolution to the tomographic restoration of radiographs

This paper derives an optimal homomorphic tomographic filter, to restore tomographic images of blurred radiographs, by taking into account the noise contributed by the other layers. The method is based on the homomorphic deconvolution technique, which is well-known for its performance in image restoration. A computer simulation of the procedure is presented. The results are compared with those obtained by using inverse filtering (3, IEEE Trans on Medical Imaging, Vol. 2, pp. 89–102, 1983) and Wiener filtering (4, Proc. Digitech '84). The paper shows that homomorphic filtering is a suitable and often preferable technique for the tomographic filtering of radiographs.     

一种复小波域维纳滤波结合偏微分方程图像去噪方法

本文提出了一种基于复数小波域上的多方向窗维纳滤波与偏微分方程保持边缘细节相结合的方法。针对小波域维纳滤波的方向性差,本方法先进行复数小波变换,获得六个方向上的图像信号,该六个方向进行方向维纳滤波,对图像进行去噪,以此引导偏微分方程中的扩散函数,实现各项异性进行扩散。实验结果表明,本文所提出的方法的峰值信噪比,以及视觉质量都较复小波去噪或各项异性非线性扩散去噪方法有明显的改善。     

Optimal filtering over linear observations with unknown parameters

This paper presents the optimal filtering and parameter identification problem for linear stochastic systems over linear observations with unknown parameters, where the unknown parameters are considered Wiener processes. The original problem is reduced to the filtering problem for an extended state vector that incorporates parameters as additional states. The resulting filtering system is bilinear in state and linear in observations. The obtained optimal filter for the extended state vector also serves as the optimal identifier for the unknown parameters. Performance of the designed optimal state filter and parameter identifier is verified for both, positive and negative, parameter values.     

Computer ethics: Its birth and its future

This article discusses some``historical milestones' in computer ethics, aswell as two alternative visions of the futureof computer ethics. Topics include theimpressive foundation for computer ethics laiddown by Norbert Wiener in the 1940s and early1950s; the pioneering efforts of Donn Parker,Joseph Weizenbaum and Walter Maner in the1970s; Krystyna Gorniak's hypothesis thatcomputer ethics will evolve into ``globalethics'; and Deborah Johnson's speculation thatcomputer ethics may someday ``disappear'.     

Modified multi-innovation stochastic gradient algorithm for Wiener–Hammerstein systems with backlash

In this paper, the identification of the Wiener–Hammerstein systems with unknown orders linear subsystems and backlash is investigated by using the modified multi-innovation stochastic gradient identification algorithm. In this scheme, in order to facilitate subsequent parameter identification, the orders of linear subsystems are firstly determined by using the determinant ratio approach. To address the multi-innovation length problem in the conventional multi-innovation least squares algorithm, the innovation updating is decomposed into sub-innovations updating through the usage of multi-step updating technique. In the identification procedure, by reframing two auxiliary models, the unknown internal variables are replaced by using the outputs of the corresponding auxiliary model. Furthermore, the convergence analysis of the proposed algorithm has shown that the parameter estimation error can converge to zero. Simulation examples are provided to validate the efficiency of the proposed algorithm.     

Robust identification of Wiener time-delay system with expectation-maximization algorithm

This paper considers the parameter estimation for Wiener time-delay systems with the output data contaminated with outliers. The time-delay and corrupted output data bring great challenges to the parameter estimation problem. The statistical model of the estimation problem is constructed based on the Laplace distribution and the identification problem is formulated in the scheme of the expectation-maximization (EM) algorithm. The negative effect of outliers imposed on the parameter estimation problem is sufficiently suppressed and the unknown time-delay and model parameters can be estimated simultaneously. The simulation example is given to demonstrate the effectiveness of the proposed algorithm.     

Theory of nonlinear systems

This paper gives a general review of the Theory of Nonlinear Systems. In 1960, the author presented a paper “Theory of Nonlinear Control” at the First IFAC Congress at Moscow. Professor Norbert Wiener, who attended this Congress, drew attention to his work on the synthesis and analysis of nonlinear systems in terms of Hermitian polynomials in the Laguerre coefficients of the past of the input.Wiener's original idea was to use white noise as a probe on any nonlinear system. Applying this input to a Laguerre network gives u₁, u₂,…, u_s, and then to a Hermite polynomial generator gives V(α)'s. Applying the same input to the actual nonlinear system gives output c(t). Putting c(t) and V(α)'s through a product averaging device, we get $\text{c(t)V(α)}\text{=}\text{A}{}_{\mathrm{\alpha }}\text{2л}{}^{\mathrm{<mtext>s</mtext><mtext>2</mtext>}}$, where the upper bar denotes time average and A_α's can be considered as characteristic coefficients of the nonlinear system. A desired output z(itt) may replace c(itt) to get a new set of A_α's.The Volterra functional method suggested by Wiener in 1942 has been greatlydeveloped from 1955 to the present. The method involves a multi-dimensional convolution integral with multi- dimensional kernels. The associated multi-dimensional transforms are given by Y.H. Ku and A.A. Wolf (J. Franklin Inst., Vol. 281, pp. 9–26, 1966). Wiener extended the Volterra functionals by forming an orthogonal set of functionals known as G-functionals, using Gaussian white noise as input. Volterra kernels and Wiener kernels can be correlated and form the characteristic functions of nonlinear systems.From an extension of the linear system to the nonlinear system, the input-output crosscorrelation φ_xy can be shown to be equal to the convolution of system impulse response h₁ with the autocorrelation φ_xx. Using the white noise as input, where its power density spectrum is a constant, say, A, the crosscorrelation is given by φ_xy(σ) = Ah₁(σ), while the autocorrelation is φ_xx(τ) = Au(τ). This extension forms the basis of an optimum method for nonlinear system identification. Measurement of kernels can be made through proper circuitry.Parallel to the Volterra series and the Wiener series, another series based on Taylor-Cauchy transforms developed since 1959 are given for comparison. The Taylor-Cauchy transform method can be applied in the analysis of simultaneous nonlinear systems. It is noted that the Volterra functional method and the Taylor-Cauchy transform method give identical final results.A selected Bibliography is appended not only to include other aspects of nonlinear system theory but also to show the wide application of nonlinear system characterization and identification to problems in biology, ecology, physiology, cybernetics, control theory, socio- economic systems, etc.