Identification of discrete Hammerstein systems by the Fourier series regression estimate

The identification of a single-input, single-output (SISO) discrete Hammerstein system is studied. Such a system consists of a non-linear memoryless subsystem followed by a dynamic, linear subsystem. The parameters of the dynamic, linear subsystem are identified by a correlation method and the Newton-Gauss method. The main results concern the identification of the non-linear, memoryless subsystem. No conditions are imposed on the functional form of the non-linear subsystem, recovering the non-linear using the Fourier series regression estimate. The density-free pointwise convergence Of the estimate is proved, that is.algorithm converges for all input densities The rate of pointwise convergence is obtained for smooth input densities and for non-linearities of Lipschitz type.Globle convergence and its rate are also studied for a large class of non-linearities and input densities     

Hammerstein system identification by non-parametric regression estimation

A discrete-time, multiple-input non-linear Hammerstein system is identified. The dynamical subsystem is recovered using the standard correlation method. The main results concern estimation of the non-linear memoryless subsystem. No conditions concerning the functional form of the transform characteristic of the subsystem are made and an algorithm for estimation of the characteristic is given. The algorithm is simply a non-parametric kernel estimate of the regression function calculated from the dependent data. It is shown that the algorithm converges to the characteristic of the subsystem in the pointwise as well as the global sense. For sufficiently smooth characteristics, the rate of convergence is o(n^-1/(2+d in probability, where d is the dimension of the input variable.     

Continuous-time Wiener system identification

A continuous-time Wiener system is identified. The system consists of a linear dynamic subsystem and a memoryless nonlinear one connected in a cascade. The input signal is a stationary white Gaussian random process. The system is disturbed by stationary white random Gaussian noise. Both subsystems are identified from input-output observations taken at the input and output of the whole system. The a priori information is very small and, therefore, resulting identification problems are nonparametric. The impulse impulse of the linear part is recovered by a correlation method, while the nonlinear characteristic is estimated with the help of the nonparametric kernel regression method. The authors prove convergence of the proposed identification algorithms and examine their convergence rates     

Continuous-time Hammerstein system identification

A continuous-time Hammerstein system, i.e., a system consisting of a nonlinear memoryless subsystem followed by a linear dynamic one, is identified. The system is driven and disturbed by white random signals. The a priori information about both subsystems is nonparametric, which means that functional forms of both the nonlinear characteristic and the impulse response of the dynamic subsystem are unknown. An algorithm to estimate the nonlinearity is presented and its pointwise convergence to the true characteristic is shown. The impulse response of the dynamic part is recovered with a correlation method. The algorithms are computationally independent. Results of a simulation example are given     

Nonparametric identification of Wiener systems by orthogonal series

A Wiener system, i.e., a system comprising a linear dynamic and a nonlinear memoryless subsystems connected in a cascade, is identified. Both the input signal and disturbance are random, white, and Gaussian. The unknown nonlinear characteristic is strictly monotonous and differentiable and, therefore, the problem of its recovering from input-output observations of the whole system is nonparametric. It is shown that the inverse of the characteristic is a regression function and a class of orthogonal series nonparametric estimates recovering the regression is proposed and analyzed. The estimates apply the trigonometric, Legendre, and Hermite orthogonal functions. Pointwise consistency of all the algorithms is shown. Under some additional smoothness restrictions, the rates of their convergence are examined and compared. An algorithm to identify the impulse response of the linear subsystem is proposed     

Strong consistency of recursive identification for Hammerstein systems with discontinuous piecewise-linear memoryless block

This note deals with identification of Hammerstein systems with discontinuous piecewise-linear memoryless block followed by a linear subsystem. Recursive algorithms are proposed for estimating coefficients of the linear subsystem and six unknown parameters contained in the nonlinear static block. By taking a sequence of iid random variables with uniform distribution to serve as the system input, strong consistency is proved for all estimates given in the note. The theoretical results are verified by computer simulation.     

Identification of a linear system observed through zero-memory non-linearity

The paper deals with the identification problem for the Wiener-type system, where the memoryless output non-linearity is known but is not necessarily one-to-one or even monotonic. First, the deterministic identifiability test for the linear dynamic segment is derived assuming that the respective segment outputs are accessible for direct measurements. Next, using the results by Masry and Cambanis (1980), a least-squares type parameter estimation algorithm for linear dynamics is proposed for the case where only the noise-corrupted outputs of the overall non-linear tandem can be gained. A convergence with probability one is shown of the parameter estimate to the true value of the system parameter vector and some computational aspects of the proposed algorithm are discussed.     

Identification of discrete Hammerstein systems using kernel regression estimates

In this note a discrete-time Hammerstein system is identified. The weighting function of the dynamical subsystem is recovered by the correlation method. The main results concern estimation of the nonlinear memoryless subsystem. No conditions concerning functional form of the transform characteristic of the subsystem are made and an algorithm for estimation of the characteristic is presented.The algorithm is a nonparametric kernel estimate of regression functions calculated from dependent data. It is shown that the algorithm converges to the characteristic as the number of observations tend to infinity. For sufficiently smooth characteristics, the rate of convergence isO(n^{-2/5})in probability.     

A KYP lemma and invariance principle for systems with multiple hysteresis non-linearities

Absolute stability criteria for systems with multiple hysteresis non-linearities are given in this paper. It is shown that the stability guarantee is achieved with a simple two part test on the linear subsystem. If the linear subsystem satisfies a particular linear matrix inequality and a simple residue condition, then, as is proven, the non-linear system will be asymptotically stable. The main stability theorem is developed using a combination of passivity, Lyapunov and Popov stability theories to show that the state describing the linear system dynamics must converge to an equilibrium position of the non-linear closed loop system. The invariant sets that contain all such possible equilibrium points are described in detail for several common types of hystereses. The class of non-linearities covered by the analysis is very general and includes multiple slope-restricted memoryless non-linearities as a special case. Simple numerical examples are used to demonstrate the effectiveness of the new analysis in comparison to other recent results, and graphically illustrate state asymptotic stability.     

应用遗传算法辨识Hammerstein模型

基于遗传算法,提出了一种辨识Hammerstein模型的方法,该方法能够克服有色观测噪声的污染,获得非线性静态环节参数和线性动态环节参数的无偏估计,并与Hammerstein模型的MSLS辨识方法进行了比较,仿真结果说明了该方法的有效性。     

An integro-differential formula on the Wiener kernels and its application to sandwich system identification

It is well known that a nonlinear system with a white Gaussian noise input can be characterized in terms of kernels using the celebrated Wiener theory. In a practical use of the method, however, one may encounter difficulty in obtaining higher order kernels except for the first few because of, for instance, the excessive computational requirement. In this paper, we give an integro-differential formula on the kernels and as its application, an algorithm to identify a cascade system of a linear, a memoryless nonlinear, and linear subsystems, which we call a sandwich system as a whole. According to the formula, kernels up to the second order for different power levels of the input noise are required to identify the subsystems. Impulse response functions of the two linear subsystems can be obtained under appropriate normalization conditions, while the nonlinear subsystem is estimated in the form of a truncated Hermite polynomial expansion. As illustrated examples, two such systems are identified using the algorithm.     

Weak convergence results for sequential regression in memoryless systems†

Weak convergence results are obtained for a sequential regression algorithm that arises in the identification of nonlinear, memoryless systems and the adaptive design of moving average filters. The algorithm is shown to be weakly consistent if the system input is a wide-sense stationary sequence of order four that satisfies certain covariance and fourth-cumulant conditions. The conditions are essentially asymptotic independence requirements that permit one to relax the (usually required) strict independence requirements on the input data.     

Stochastic approximation in nonparametric identification of Hammerstein systems

Derived from the idea of stochastic approximation, recursive algorithms to identify a Hammerstein system are presented. Two of them recover the characteristic of the nonlinear memoryless subsystem, while the third one estimates the impulse response of the linear dynamic part. The a priori information about both subsystems is nonparametric. Consistency in quadratic mean is shown, and the convergence rate is examined. Results of numerical simulation are also presented.     

Recursive estimation in piecewise affine systems using parameter identifiers and concurrent learning

Piecewise affine systems constitute a popular framework for the approximation of non-linear systems and the modelling of hybrid systems. This paper addresses the recursive subsystem estimation in continuous-time piecewise affine systems. Parameter identifiers are extended from continuous-time state-space models to piecewise linear and piecewise affine systems. The convergence rate of the presented identifiers is improved further using concurrent learning, which makes concurrent use of current and recorded measurements. In concurrent learning, assumptions on persistence of excitation are replaced by the less restrictive linear independence of the recorded data. The introduction of memory, however, reduces the tracking ability of concurrent learning because errors in the recorded measurements prevent convergence to the true parameters. In order to overcome this limitation, an algorithm is proposed to detect and remove erroneous measurements at run-time and thereby restore the tracking ability. Detailed examples are included to validate the proposed methods numerically.     

Genetic algorithms in norm-optimal linear and non-linear iterative learning control

In this paper it is analysed whether or not it is possible to apply the norm-optimal iterative learning control algorithm to non-linear plant models. As a new theoretical result it is shown that if the non-linear plant meets a certain technical invertibility condition, the sequence of tracking errors generated by the norm-optimal algorithm will converge geometrically to zero. However, due to the non-linear nature of the plant, it is typically impossible to calculate analytically the sequence of input functions produced by the norm-optimal algorithm. Therefore it is proposed that genetic algorithms can be used as a computational tool to calculate the sequence of norm-optimal inputs. The proposed approach benefits from the design of a low-pass FIR filter. This filter successfully removes unwanted high frequency components of the input signal, which are generated by the genetic algorithm method due to the random nature of the genetic algorithm search. Simulations are used to illustrate the performance of this new approach, and they demonstrate good results in terms of convergence speed and tracking of the reference signal regardless of the nature of the plant.     

Identification of non-linear characteristics of a class of block-oriented non-linear systems via Daubechies wavelet-based models

The paper deals with the identification of non-linear characteristics of a class of block-oriented dynamical systems. The systems are driven by random stationary white processes (i.i.d. random input sequences) and disturbed by a zero-mean stationary, white or coloured, random noise. The prior knowledge about non-linear characteristics is non-parametric excluding implementation of standard parametric identification methods. To recover non-linearities, a class of Daubechies wavelet-based models using only input-output measurement data is introduced and their accuracy is investigated in the global MISE error sense. It is shown that the proposed models converge with a growing collection of data to the true non-linear characteristics (or their versions), provided that the complexity of the models is appropriately fitted to the number of measurements. Suitable rules for optimum model size selection, maximizing the convergence speed, are given and the asymptotic rate of convergence of the MISE error for optimum models is established. It is shown that in some circumstances the rate is the best possible that can be achieved in non-parametric inference. We also show that the convergence conditions and the asymptotic rate of convergence are insensitive to the correlation of the noise and are the same for known and unknown input probability density function (assumed to exist). The theory is illustrated by simulation examples.     

Distributed predictive control: A non-cooperative algorithm with neighbor-to-neighbor communication for linear systems

This paper presents a novel Distributed Predictive Control (DPC) algorithm for linear discrete-time systems. This method enjoys the following properties: (i) state and input constraints can be considered; (ii) under mild assumptions, convergence of the closed loop control system is proved; (iii) it is not necessary for each subsystem to know the dynamical models of the other subsystems; (iv) the transmission of information is limited, in that each subsystem only needs the reference trajectories of the state variables of its neighbors. A simulation example is reported to illustrate the main characteristics and performance of the algorithm.     

Structural classification of non-linear systems by input and output measurements

It is desirable to know the internal structure of a non-linear system that is regarded as a 'black box'. The structural classification of the 'black box' is made from input and output relations. In this paper, the necessary conditions of structural classification are developed by the Volterra and Wiener theories and correlation analysis. A class of non-linear models in cascade is described by cross-correlation functions and Volterra and Wiener kernels. These conditions are first discussed in the frequency domain. Then the formulae of the conditions for structural identification in the time domain are obtained. Further, relations between structural information and the kernels are developed. Simulations are carried out that verify the authors' methods of structural testing. Next, some formulae for discriminating the feedback non-linear system's structure are newly developed. Furthermore, it is shown that the linear subsystem transfer functions and the non-linear subsystem parameters for these feedback non-linear systems can be estimated. Finally, the validity and sufficiency of the conditions for structural testing are discussed.     

Global stabilization of a PVTOL aircraft model with bounded inputs

We propose a global stabilizing control design for the planar vertical takeoff and landing (PVTOL) aircraft, with bounded inputs. The approach is based on the use of non-linear combinations of linear saturation functions bounding the thrust input and the rolling moment to arbitrary saturation limits. We provide global convergence of the state to the origin, using a relatively simple algorithm.     

一种信号适度失真的宽线性SAEC模型

现有的立体声回声抵消器是一个实变量双输入双输出的装置,其结构复杂不易实现。宽线性模型的引入,提供了一种复变量单输入单输出的装置来替代实变量双输入双输出装置,其优点是只需处理一个复变量的输出信号而不是两个实变量输出信号,而且能通过复变量输入信号的相位和幅值分别调控声音的立体感和音质。利用输入信号适度失真的方法降低两个信号之间的相关性以解决因滤波而产生的非唯一性问题。把宽线性模型和失真信号应用到仿射投影算法中,通过仿真验证改进方法的误差性能和收敛速度。结果表明改进的方法具有误差小和收敛快的特点,因此宽线性SAEC模型更有优势。