Recursive identification for EIV ARMAX systems

The input uk and output yk of the multivariate ARMAX system A（x）yk = B（z）uk ＋ C（z）wk are observed with noises： uk^ob△=uk ＋ εk^u and yk^ob △=yk＋ εk^y, where εk^u and εk^y denote the observation noises. Such kind of systems are called errors-in-variables （EIV） systems. In the paper, recursive algorithms based on observations are proposed for estimating coefficients of A（z）, B（z）, C（z）, and the covariance matrix Rw of wk without requiring higher than the second order statistics. The algorithms are convenient for computation and are proved to converge to the system coefficients under reasonable conditions. An illustrative example is provided, and the simulation results are shown to be consistent with the theoretical analysis.

RECURSIVE SYSTEM IDENTIFICATION

Most of existing methods in system identification with possible exception of those for linear systems are off-line in nature, and hence are nonrecursive. This paper demonstrates the recent progress in recursive system identification. The recursive identification algorithms are presented not only for linear systems （multivariate ARMAX systems） but also for nonlinear systems such as the Hammerstein and Wiener systems, and the nonlinear ARX systems. The estimates generated by the algorithms are online updated and converge a.s. to the true values as time tends to infinity.

Process data compression based on recursive identification of nonuniformly sampled systems

A recursive identification method is proposed to obtain continuous-time state-space models in systems with nonuniformly sampled （NUS） data. Due to the nonuniform sampling feature, the time interval from one recursion step to the next varies and the parameter is always updated partially at each step. Furthermore, this identification method is applied to form a combined data compression method in NUS processes. The data to be compressed are first classified with respect to a series of potentially existing （possibly time-varying） models, and then modeled by the NUS identification method. The model parameters are stored instead of the identification output data, which makes the first compression. Subsequently, as the second step, the conventional swinging door trending method is carried out on the data from the first step. Numeric results from simulation as well as practical data are given, showing the effectiveness of the proposed identification method and fold increase of compression ratio achieved by the combined data compression method.

Recursive modeling and online identification of lithium-ion batteries for electric vehicle applications

For safe and reliable operation of lithium-ion batteries in electric vehicles,the real-time monitoring of their internal states is important.The purpose of our study is to find an easily implementable,online identification method for lithium-ion batteries in electric vehicles.In this article,we propose an equivalent circuit model structure.Based on the model structure we derive the recursive mathematical description.The recursive extended least square algorithm is introduced to estimate the model parameters online.The accuracy and robustness are validated through experiments and simulations.Real-road driving cycle experiment shows that the proposed online identification method can achieve acceptable accuracy with the maximum error of less than 5.52%.In addition,it is proved that the proposed method can also be used to estimate the real-time SOH and SOC of the batteries.

Recursive least squares identification for piecewise affine Hammerstein models

Purpose-The purpose of this paper is to probe the recursive identification of piecewise affine Hammerstein models directly by using input-output data.To explain the identification process of a parametric piecewise affine nonlinear function,the authors prove that the inverse function corresponding to the given piecewise affine nonlinear function is also an equivalent piecewise affine form.Based on this equivalent property,during the detailed identification process with respect to piecewise affine function and linear dynamical system,three recursive least squares methods are proposed to identify those unknown parameters under the probabilistic description or bounded property of noise.Design/methodology/approach-First,the basic recursive least squares method is used to identify those unknown parameters under the probabilistic description of noise.Second,multi-innovation recursive least squares method is proposed to improve the efficiency lacked in basic recursive least squares method.Third,to relax the strict probabilistic description on noise,the authors provide a projection algorithm with a dead zone in the presence of bounded noise and analyze its two properties.Findings-Based on complex mathematical derivation,the inverse function of a given piecewise affine nonlinear function is also an equivalent piecewise affine form.As the least squares method is suited under one condition that the considered noise may be a zero mean random signal,a projection algorithm with a dead zone in the presence of bounded noise can enhance the robustness in the parameter update equation.Originality/value-To the best knowledge of the authors,this is the first attempt at identifying piecewise affine Hammerstein models,which combine a piecewise affine function and a linear dynamical system.In the presence of bounded noise,the modified recursive least squares methods are efficient in identifying two kinds of unknown parameters,so that the common set membership method can be replaced by the proposed methods.

IDENTIFICATION FOR WIENER SYSTEMS WITH INTERNAL NOISE

This paper considers identification of Wiener systems for which the internal variables and output are corrupted by noises. When the internal noise is a sequence of independent and identically distributed （lid） Gaussian random variables, by the Weierstrass transformation （WT） the system under consideration turns to be a Wiener system without internal noise. The nonlinear part of the latter is nothing else than the WT of the nonlinear function of the original system, while the linear subsystem is the same for both systems before and after WT. Under reasonable conditions, the recursive identification algorithms are proposed for the transformed Wiener system, and strong consistency for the estimates is established. By using the inverse WT the nonparametric estimates for the nonlinearity of the original system are derived, and they are strongly consistent if the nonlinearity in the original system is a polynomial, Similar results also hold in the case where the internal noise is non-Gaussian. Simulation results are fully consistent with the theoretical analysis.

Identification of Wiener systems with nonlinearity being piecewise-linear function

Identification of the Wiener system with the nonlinear block being a piecewiselinear function is considered in the paper, generalizing the results given by H. E. Chen to the case of noisy observation. Recursive algorithms are given for estimating all unknown parameters contained in the system, and their strong consistency is proved. The estimation method is similar to that used by H. E. Chen for Hammerstein systems with the same nonlinearity. However, the assumption imposed by H. E. Chen on the availability of an upper bound for the nonsmooth points of the piecewise-linear function has been removed in this paper with the help of designing an additional algorithm for estimating the upper bound.

Performance analysis of stochastic gradient algorithms under weak conditions

By using the stochastic martingale theory, convergence properties of stochastic gradient （SG） identification algorithms are studied under weak conditions. The analysis indicates that the parameter estimates by the SG algorithms consistently converge to the true parameters, as long as the information vector is persistently exciting （i.e., the data product moment matrix has a bounded condition number） and that the process noises are zero mean and uncorrelated. These results remove the strict assumptions, made in existing references, that the noise variances and high-order moments exist, and the processes are stationary and ergodic and the strong persis- tent excitation condition holds. This contribution greatly relaxes the convergence conditions of stochastic gradient algorithms. The simulation results with bounded and unbounded noise variances confirm the convergence conclusions proposed.

Optimal Choice of Weighting Factors in Adaptive Linear Quadratic Control

Most of the processes in the industry have nonlinear behavior. Control of such processes with conventional control methods could lead to unstable, suboptimal, etc., results. On the other hand, the adaptive control is a technique widely used for controlling of nonlinear systems. The approach here is based on the recursive identification of the external linear model as a linear representation of the originally nonlinear system. The controller then reacts to the change of the working point or disturbances which could occur by the change of the parameters, structure, etc. The polynomial synthesis together with the linear quadratic(LQ) approach is employed here for the controller synthesis. These techniques satisfy basic control requirements such as the stability, the reference signal tracking and the disturbance attenuation. Resulted controller could be tuned with the choice of weighting factors in LQ approach. This work investigates the effect of these factors on control results. Proposed methods are tested on the mathematical model of the isothermal continuous stirred-tank reactor and simulated results are also verified on the real model of the continuous stirred tank reactor.