Recursive identification of continuous-time Hammerstein systems

A continuous-time Hammerstein system is identified. The characteristic of its nonlinear subsystem and the impulse response of the dynamic parts are estimated from observations taken at input and output of the whole system. All algorithms are of the on-line type. Their convergence is shown. Results of simulation examples are also presented.     

Recursive identification of errors-in-variables Wiener systems

This paper considers the recursive identification of errors-in-variables (EIV) Wiener systems composed of a linear dynamic system followed by a static nonlinearity. Both the system input and output are observed with additive noises being ARMA processes with unknown coefficients. By a stochastic approximation incorporated with the deconvolution kernel functions, the recursive algorithms are proposed for estimating the coefficients of the linear subsystem and for the values of the nonlinear function. All the estimates are proved to converge to the true values with probability one. A simulation example is given to verify the theoretical analysis.     

Recursive identification for Wiener systems using Gaussian inputs

The recursive algorithms are given for identifying the single‐input single‐output Wiener system which consists of a moving average type linear subsystem followed by a static nonparametric nonlinearity. The input is defined to be a sequence of mutually independent Gaussian random variables. The estimates for coefficients of the linear subsystem as well as for f(v) at any v are proved to converge to the true values with probability one. A numerical example is given, justifying the theoretical analysis. Copyright © 2008 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

Recursive identification of bilinear systems

Methods of identifying bilinear systems from recorded input-output data are discussed in this article. A short survey of the existing literature on the topic is given. ‘Standard’ methods from linear systems identification, such as least squares, extended least squares, recursive prediction error and instrumental variable methods are transferred to bilinear, input-output model structures and tested in simulation. Special attention is paid to problems of stabilizing the model predictor, and it is shown how a time-varying Kalman filter and associated parameter estimation algorithm can deal with this problem.     

Recursive identification of overparametrized systems

A recursive identification algorithm based on extended least squares is proposed to deal with the contingency of overparametrization. The algorithm is relatively simple compared to those involving online order determination, being based on adaptively introducing suitable excitation into the algorithm to avoid ill-conditioning. In the case of extended-least-squares-based adaptive estimation, then the regressors are appropriately stochastically perturbed. The algorithm is shown to converge to a uniquely defined signal model with any pole-zero cancellations at the origin     

Recursive identification for Wiener model with discontinuous piece-wise linear function

This paper deals with identification of Wiener systems with nonlinearity being a discontinuous piece-wise linear function. Recursive estimation algorithms are proposed to estimate six unknown parameters contained in the nonlinearity and all unknown coefficients of the linear subsystem by using the iid Gaussian inputs. The estimates are proved to converge to the corresponding true values with probability one. A numerical example is given to justify the obtained theoretical results.     

Recursive identification of switched ARX systems

We consider the problem of recursively identifying the parameters of a deterministic discrete-time Switched Auto-Regressive eXogenous (SARX) model, under the assumption that the number of models, the model orders and the mode sequence are unknown. The key to our approach is to view the identification of multiple ARX models as the identification of a single, though more complex, lifted dynamical model built by applying a polynomial embedding to the input/output data. We show that the dynamics of this lifted model do not depend on the value of the discrete state or the switching mechanism, and are linear on the so-called hybrid model parameters. Therefore, one can identify the parameters of the lifted model using a standard recursive identifier applied to the embedded input/output data. The estimated hybrid model parameters are then used to build a polynomial whose derivatives at a regressor give an estimate of the parameters of the ARX model generating that regressor. The estimated ARX model parameters are shown to converge exponentially to their true values under a suitable persistence of excitation condition on a projection of the embedded input/output data. Such a condition is a natural generalization of the well known result for ARX models. Although our algorithm is designed for perfect input/output data, our experiments also evaluate its performance as a function of the level of noise for different choices of the number of models and model orders. We also present an application to temporal video segmentation.     

Recursive identification for multivariate errors-in-variables systems

The recursive algorithm is given for estimating matrix coefficients of the multivariate errors-in-variables (EIV) systems. It is shown that under mild conditions the estimate given by the algorithm converges to a limit belonging to the solution set of the Yule-Walker equation satisfied by the true coefficients of the system. The sufficient conditions guaranteeing the uniqueness of the solution to the Yule-Walker equation are given. In this case the estimate provided by the recursive algorithm is strongly consistent.     

Recursive identification for EIV ARMAX systems

The input u _k and output y _k of the multivariate ARMAX system A(z)y _k = B(z)u _k + C(z)w _k are observed with noises: u _k ^ob ≜ u _k + ε _k ^u and y _k ^ob ≜ y _k + ε _k ^y , where ε _k ^u and ε _k ^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 w _k 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.     

On-line identification and optimal control of continuous-time systems

In this paper, a moving algorithm for on-line identification of continuous-time systems is developed. With the proposed algorithm, the observed input–output data can be directly used to estimate the system parameters without any numerical pre-processing, and by means of a recursive formula the estimates can be updated step by step without repeatedly computing the matrix inversion. In this way, the use of both computer memory and computing time can be reduced. Besides, the computations are simple and straightforward. From the moving identification algorithm, a linear moving model can be obtained to represent the control systems. The on-line optimal control algorithm is also developed via the linear moving model. A slider-crank motion control system is used to illustrate that the proposed on-line identification and optimal control algorithms can give satisfactory results.     

Recursive identification of linear and non-linear systems

The paper develops recursive techniques for off-line identification of linear and nonlinear systems. It is shown that if the system is linear and time invariant, impulse response characterization of the system coupled with an orthogonal series approximation can be utilized for the purpose stated above. The techniques of adaptive Kalman filtering are shown to be applicable, which besides permitting recursive evaluation of the coefficients, lead to a number of important advantages. In the second part of the study, the proposed method is extended for recursive identification of a class of non-linear systems which can be represented as a cascade combination of a linear dynamical system and a non-linear zero memory system. The method of Volterra series representation of such systems is utilized. Results are illustrated through numerical examples in each case.     

MINLIP for the identification of monotone Wiener systems

This paper studies the MINLIP estimator for the identification of Wiener systems consisting of a sequence of a linear FIR dynamical model, and a monotonically increasing (or decreasing) static function. Given T observations, this algorithm boils down to solving a convex quadratic program with O(T) variables and inequality constraints, implementing an inference technique which is based entirely on model complexity control, a technique which is dubbed as Einstein’s razor.² This technique is proven to yield, here, almost consistent estimates in case the given data is suitably rich (local Persistently Exciting of sufficient order), and if the samples obey a ‘true’ monotone Wiener system satisfying some regularity conditions. It is then indicated how to extend the method in order to cope with noisy data, and empirical evidence is given in order to support the claim of efficiency.     

Strong consistence of recursive identification for Wiener systems

The paper concerns identification of the Wiener system consisting of a linear subsystem followed by a static nonlinearity f(·) with no invertibility and structure assumption. Recursive estimates are given for coefficients of the linear subsystem and for the value f(v) at any fixed v. The main contribution of the paper consists in establishing convergence with probability one of the proposed algorithms to the true values. This probably is the first strong consistency result for this kind of Wiener systems. A numerical example is given, which justifies the theoretical analysis.     

Recursive identification for dynamic systems with backlash

A recursive algorithm for identification of nonlinear dynamic systems with backlash is proposed in this paper. In this method, the backlash, which is a non‐smooth function, is decomposed into a combination of a group of piecewise linearized models so that all the parameters of the backlash can be estimated separately. Moreover, the model of the backlash is embedded into a Hammerstein‐type model. Thus, a pseudo‐Hammerstein model with backlash is constructed. The estimation of the parameters for such a non‐smooth nonlinear system can be implemented through a so‐called recursive general identification algorithm (RGIA). Then, the corresponding convergence analysis of the RGIA for the model with backlash is also investigated. After that, two examples are presented to show the performance of the proposed method. Copyright © 2009 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

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     

Parameter identification of continuous-time systems using iterative learning control

This paper presents a system identification technique for continuous-time state-space system using the iterative learning control. The transfer function parameters are regarded as functions with respect to the state-space parameters which will be identified. The relationship between the state-space parameters and the response error is explicitly derived. An update law of the state-space parameters is proposed so as to improve the convergence speed. The effectiveness of the proposed identification technique is demonstrated by numerical examples.     

Online identification of continuous-time systems with unknown time delay

In this note, we present a recursive algorithm for online identification of systems with unknown time delay. The proposed algorithm can be interpreted as an approximate nonlinear least-squares, corresponding to modified normalized or un-normalized least-squares when the normalizing factor /spl gamma/>0 or /spl gamma/=0, respectively. Both algorithms are essentially extensions to general least-squares methodology. Simulation studies demonstrate the characteristics and performance of these algorithms.     

Subspace identification for continuous-time stochastic systems via distribution-based approach

This paper presents a novel approach for system identification of continuous-time stochastic state space models from random input-output continuous data. The approach is based on the introduction of random distribution theory in describing (higher) time derivatives of stochastic processes, and the input-output algebraic relationship is derived which is treated in the time-domain. The efficacy of the approach is examined by comparing with other approaches employing the filters.