Variance-error quantification for identified poles and zeros

This paper deals with quantification of noise induced errors in identified discrete-time models of causal linear time-invariant systems, where the model error is described by the asymptotic (in data length) variance of the estimated poles and zeros. The main conclusion is that there is a fundamental difference in the accuracy of the estimates depending on whether the zeros and poles lie inside or outside the unit circle. As the model order goes to infinity, the asymptotic variance approaches a finite limit for estimates of zeros and poles having magnitude larger than one, but for zeros and poles strictly inside the unit circle the asymptotic variance grows exponentially with the model order. We analyze how the variance of poles and zeros is affected by model order, model structure and input excitation. We treat general black-box model structures including ARMAX and Box–Jenkins models.     

Adaptive control of hammerstein systems with unknown input nonlinearity and partially modeled linear dynamics

We numerically investigate that an adaptive control law achieves internal model principle control in the presence of plant input nonlinearities. We focus on retrospective cost adaptive control (RCAC) applied to Hammerstein systems with unknown input nonlinearity and limited modeling of the linear dynamics. The goal is to determine whether the control law achieves the correct gain and phase shift for internal stability along with asymptotic command following and disturbance rejection.     

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.     

Blind maximum likelihood identification of Hammerstein systems

This paper is about the identification of discrete-time Hammerstein systems from output measurements only (blind identification). Assuming that the unobserved input is white Gaussian noise, that the static nonlinearity is invertible, and that the output is observed without errors, a Gaussian maximum likelihood estimator is constructed. Its asymptotic properties are analyzed and the Cramér–Rao lower bound is calculated. In practice, the latter can be computed accurately without using the strong law of large numbers. A two-step procedure is described that allows to find high quality initial estimates to start up the iterative Gauss–Newton based optimization scheme. The paper includes the illustration of the method on a simulation example. A theoretical analysis demonstrates that additive output measurement noise introduces a bias that is proportional to the variance of that additive, unmodeled noise source. The simulations support this result, and show that this bias is insignificant beyond a certain Signal-to-Noise Ratio (40 dB in the example).     

On the accuracy in errors-in-variables identification compared to prediction-error identification

Errors-in-variables estimation problems for single-input–single-output systems with Gaussian signals are considered in this contribution. It is shown that the Fisher information matrix is monotonically increasing as a function of the input noise variance when the noise spectrum at the input is known and the corresponding noise variance is estimated. Furthermore, it is shown that Whittle’s formula for the Fisher information matrix can be represented as a Gramian and this is used to provide a geometric representation of the asymptotic covariance matrix for asymptotically efficient estimators. Finally, the asymptotic covariance of the parameter estimates for the system dynamics is compared for the two cases: (i) when the model includes white measurement noise on the input and the variance of the noise is estimated, and (ii) when the model includes only measurement noise on the output. In both cases, asymptotically efficient estimators are assumed. An explicit expression for the difference is derived when the underlying system is subject only to measurement noise on the output.     

Quantification of the variance of estimated transfer functions in the presence of undermodeling

We study the effect of undermodeling on the parameter variance for prediction error time-domain identification in open loop. We consider linear time-invariant discrete time single-input-single-output systems with known noise model. We examine asymptotic expressions for the variance for large number of data. This quantity depends in general on the fourth order statistical properties of the applied input. However, we establish a sufficient condition under which the asymptotic variance depends on the input power spectrum only. For this case, we deliver exact expressions. For a stochastic input the undermodeling contributes to the parameter variance due to the correlation between the prediction errors and its gradients, while for a deterministic input it has no influence. As an additional contribution, we investigate the parameter variance under the assumptions of the stochastic embedding procedure.     

A genetic approach to the identification of linear dynamical systems with static nonlinearities

This paper investigates the use of genetic algorithms in the identification of linear systems with static nonlinearitites. Linear systems with static nonlinearities at the input known as the Hammerstein model, and linear systems with static nonlinearities at the output known as the Wiener model are considered in this paper. The parameters of the Hammerstein and the Wiener models are estimated using genetic algorithms from the input-output data by minimizing the error between the true model output and the identified model output. Using genetic algorithms, the Hammerstein and the Wiener models with known nonlinearity structure and unknown parameters can be identified. Moreover, systems with non-minimum phase characteristics can be identified. Extensive simulations have been used to study the convergence properties of the proposed scheme. Simulation examples are included to demonstrate the effectiveness and robustness of the proposed identification scheme.     

A multi-variate Hammerstein model for processes with input directionality

A new formulation of a block-structured model based on the Hammerstein operator is presented for the identification of multi-variate systems with input directionality. In contrast to the existing formulations for multi-variate Hammerstein models, the proposed structure offers the possibility to independently model the dynamic and nonlinear characteristics of the system and at the same time preserves the possibility to use the new efficient algorithms developed for the identification of single input Hammerstein models. Further, the formulation allows for a representation of arbitrary static nonlinear coupling of input variables with a considerably lower amount of parameters compared to existing formulations. The new model structure is applied to the identification of a fluid catalytic cracking (FCC) unit and significantly outperforms all previous multi-variate Hammerstein model structures by reducing the prediction error by over 50%.     

Identification of MIMO Hammerstein models using least squares support vector machines

This paper studies a method for the identification of Hammerstein models based on least squares support vector machines (LS-SVMs). The technique allows for the determination of the memoryless static nonlinearity as well as the estimation of the model parameters of the dynamic ARX part. This is done by applying the equivalent of Bai's overparameterization method for identification of Hammerstein systems in an LS-SVM context. The SISO as well as the MIMO identification cases are elaborated. The technique can lead to significant improvements with respect to classical overparameterization methods as illustrated in a number of examples. Another important advantage is that no stringent assumptions on the nature of the nonlinearity need to be imposed except for a certain degree of smoothness.     

Convergence of fixed‐point iteration for the identification of Hammerstein and Wiener systems

Convergence property of the iterative algorithm for Hammerstein or Wiener systems is generally hard to establish because of the existence the unmeasurable internal variables in such systems. In this paper, a fixed‐point iteration is introduced to identifying both Hammerstein and Wiener systems with a unified algorithm. This newly proposed estimation algorithm gives consistent estimates under arbitrary nonzero initial conditions. In addition, the errors of the estimates are established as functions of the noise variance, and thus how the noise affects the quality of parameter estimates for a finite number of data points is made clear. Copyright © 2012 John Wiley & Sons, Ltd.     

Modeling and parameter identification of systems with multisegmentpiecewise-linear characteristics

This paper deals with the modeling and parameter identification of nonlinear systems having multi-segment piecewise-linear characteristics. The decomposition of the corresponding mapping provides a new form of multi-segment nonlinearity representation, leading to an output equation where all the parameters to be estimated are separated. Hence, an iterative method with internal variable estimation can be applied for parameter identification using input/output data records. The only required a-priori knowledge of the nonlinear characteristic represents the limits for the domain partition. The proposed model of given static nonlinearity is also incorporated into the Hammerstein model. Examples of parameter identification for static and dynamic systems with multi-segment piecewise-linear characteristics are presented     

Recursive identification of Hammerstein systems with dead-zone nonlinearity in the presence of bounded noise

The existing identification algorithms for Hammerstein systems with dead-zone nonlinearity are restricted by the noise-free condition or the stochastic noise assumption. Inspired by the practical bounded noise assumption, an improved recursive identification algorithm for Hammerstein systems with dead-zone nonlinearity is proposed. Based on the system parametric model, the algorithm is derived by minimising the feasible parameter membership set. The convergence conditions are analysed, and the adaptive weighting factor and the adaptive covariance matrix are introduced to improve the convergence. The validity of this algorithm is demonstrated by two numerical examples, including a practical DC motor case.     

Asymptotic variance of closed-loop subspace identification methods

In this paper, the asymptotic properties of a version of the "innovation estimation" algorithm by Qin and Ljung as well as of a version of the "whitening filter" based algorithm introduced by Jansson are studied. Expressions for the asymptotic error as the sum of a "bias" term plus a "variance" term are given. The analysis is performed under rather mild assumptions on the spectrum of the joint input-output process; however, in order to avoid unnecessary complications, the asymptotic variance formulas are computed explicitly only for finite memory systems, i.e., of the ARX type. This assumption could be removed at the price of some technical complications; the simulation results confirm that when the past horizon is large enough (as compared to the predictor dynamics) the asymptotic expressions provide a good approximation of the asymptotic variance also for ARMAX systems.     

基于分段线性化Hammerstein模型的自校正控制器设计

许多实际系统可以表示为不连续非线性块状结构模型,其不连续非线性部分常采用符号函数参数化,该处理方法适用于递推参数辨识,但自适应控制器的设计较为困难。鉴于此,针对一类含有不连续非线性环节的Hammerstein模型,采用一系列线性分段函数参数化不连续非线性环节,提出自校正控制方法。根据线性分段函数的逆函数特性,求解自适应控制律。理论分析证明了闭环系统的稳定性,仿真结果验证了所提出方法的有效性。     

非线性分离法广义预测控制稳态特性分析

对具有输入饱和约束和Harnmerstein非线性的系统,采用“非线性分离法广义预测控制(GPC)”策略,即采用线性GPC时先不考虑Hammerstein非线性,然后采用解非线性代数方程的方法处理该非线性。根据处理饱和约束和解方程的不同顺序,可得到两种“两步法GPC”和一种“非线性移去法GPC”,分析了这些方法的稳态特性,并通过仿真进行了验证。     

Recursive identification of Hammerstein systems with discontinuous nonlinearities containing dead-zones

This note deals with the recursive parameter identification of Hammerstein systems with discontinuous nonlinearities, i.e., two-segment piecewise-linear with dead-zones and preloads. A special form of the Hammerstein model with this type of nonlinearity is incorporated into the recursive least squares identification scheme supplemented with the estimation of model internal variables. The proposed method is illustrated by examples.     

Asymptotic normality analysis of the estimation error of steady-state models for industrial processes: SISO case

The asymptotic normality of the estimation error of steady-state models for industrial processes is investigated under quite mild conditions. The estimate is formed from the estimated parameters of an approximate linear model which is strongly consistent with the steady-state gain of slow time-varying linear SISO systems. In the parameter estimation, the weighted least-squares method is employed. The input signal (the system set point) is the usual step change din the optimization procedure. The rate of convergence is given. The stationarity and the distribution of the stochastic process are not demanded. It is also worth mentioning that, under some acceptable conditions, robustness to the structure of the approximate linear model is achieved. A simulation study shows that, for limited length of the sampled data, the best choice for the structure of approximate models as regards estimation precision is dependent upon the realization of the stochastic noise.     

Identification of Hammerstein models for control using ASYM

Identification of single-input single-output Hammerstein models is studied in this work. The basic idea here is to extend the recently developed asymptotic method (ASYM) of linear model identification to include input non-linearity in the model set. First identification test design will be discussed. In parameter estimation, prediction error criterion is used in order to maintain consistence when the process is operating in closed-loop. A relaxation iteration scheme is proposed by making use of a model structure in which the error is bilinear in the parameters. The order of the linear part and nonlinear part are determined by looking at an output error related criterion which is control-relevant. The frequency domain upper error bound of the linear part will be derived and used for model validation. Simulation study will be used to illustrate the method and comparisons with other methods are also given.     

Block pulse function approach for the identification of Hammerstein model non-linear systems

Two recursive algorithms based on block pulse functions are presented for identifying continuous Hammerstein models of non-linear systems with (i) a state space model and (ii) an input–output model. Since the continuous non-linear systems are transformed approximately into the corresponding difference equations via block pulse functions, these recursive estimation algorithms can easily be obtained using a derivation similar to that of the discrete-time models expressed by difference equations. Both algorithms derived here are simple and straightforward, and can easily be implemented on-line. As discussed in this paper, these algorithms can also be extended to the identification of certain continuous non-linear systems with a feedback loop or with time delays. The illustrative examples show that these recursive algorithms give satisfactory results for the identification problems of certain continuous non-linear systems.     

State estimation for nonlinear systems under model uncertainties: a class of sliding-mode observers

This paper deals with the classical problem of state estimation, considering partially unknown, nonlinear systems with noise measurements. Estimation of both, state variables and unstructured uncertain term, are performed simultaneously. In order to transform the measured disturbance into system disturbance, an alternative system representation is proposed, which lead a more advantageous observer structure. The observer proposed contains a proportional-type contribution and a sliding term for the measurement of error, which provides robustness against noisy measurements and model uncertainties. Convergence analysis of the estimation methodology proposed is performed, analysing the equation of the dynamics of the estimation error; it is shown that the observer exhibits asymptotic convergence. Estimation of monomer concentration, average molecular weight, polydispersity and filtering of temperature in a batch stirred polymerization reactor illustrates the good performance of the observer proposed.