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.     

Revisiting Hammerstein system identification through the Two-Stage Algorithm for bilinear parameter estimation

The Two-Stage Algorithm (TSA) has been extensively used and adapted for the identification of Hammerstein systems. It is essentially based on a particular formulation of Hammerstein systems in the form of bilinearly parameterized linear regressions. This paper has been motivated by a somewhat contradictory fact: though the optimality of the TSA has been established by Bai in 1998 only in the case of some special weighting matrices, the unweighted TSA is usually used in practice. It is shown in this paper that the unweighted TSA indeed gives the optimal solution of the weighted nonlinear least squares problem formulated with a particular weighting matrix. This provides a theoretical justification of the unweighted TSA, and also leads to a generalization of the obtained result to the case of colored noise with noise whitening. Numerical examples of identification of Hammerstein systems are presented to validate the theoretical analysis.     

Approximation of the Feasible Parameter Set in worst-case identification of Hammerstein models

The estimation of the Feasible Parameter Set (FPS) for Hammerstein models in a worst-case setting is considered. A bounding procedure is determined both for polytopic and ellipsoidic uncertainties. It consists in the projection of the FPS of the extended parameter vector onto suitable subspaces and in the solution of convex optimization problems which provide Uncertainties Intervals of the model parameters. The bounds obtained are tighter than in the previous approaches.     

Iterative identification of block-oriented nonlinear systems based on biconvex optimization

We investigate the identification of a class of block-oriented nonlinear systems which is represented by a common model in this paper. Then identifying the common model is formulated as a biconvex optimization problem. Based on this, a normalized alterative convex search (NACS) algorithm is proposed under a given arbitrary nonzero initial condition. It is shown that we only need to find the unique partial optimum point of a biconvex cost function in order to obtain its global minimum point. Thus, the convergence property of the proposed algorithm is established under arbitrary nonzero initial conditions. By applying the results to Hammerstein–Wiener systems with an invertible nonlinear function, the long-standing problem on the convergence of iteratively identifying such systems under arbitrary nonzero initial conditions is also now solved.     

Identification of an additive nonlinear system and its applications in generalized Hammerstein models

An identification algorithm is developed for a class of nonlinear systems that are multi-input and multi-output in an additive form. The convergence results are achieved and its applications to identification of a generalized Hammerstein system is also discussed.     

Fixed point theorem of generalized quasi-contractive mapping in cone metric space

In this paper, we introduce the generalized quasi-contractive mapping f in a cone metric space (X,d). f is called a generalized quasi-contractive if there is a real λ∈[0,1) such that for all x,y∈X,

d(fx,fy)≤λs     

On linear models for nonlinear systems

Best linear time-invariant (LTI) approximations are analysed for several interesting classes of discrete nonlinear time-invariant systems. These include nonlinear finite impulse response systems and a class of nonsmooth systems called bi-gain systems. The Fréchet derivative of a smooth nonlinear system is studied as a potential good LTI model candidate. The Fréchet derivative is determined for nonlinear finite memory systems and for a class of Wiener systems. Most of the concrete results are derived in an ?_∞ signal setting. Applications to linear controller design, to identification of linear models and to estimation of the size of the unmodelled dynamics are discussed.     

Synthetic detection of change point and outliers in bilinear time series models

This paper proposes a procedure of synthetic detection for the location of a change point and outliers in bilinear time series models with a change after an unknown time point. Based on Bayesian framework, we first derive the conditional posterior distribution of the change point and from that distribution estimate the position of the change point. Then we use these results to detect the outliers in the time series before and after that change point via Gibbs sampler algorithm. Our simulation studies show that the proposed procedure is effective.     

The complexity of fixed point models of trust in distributed networks

In this paper we consider the complexity of some problems arising in a fixed point model of trust in large-scale distributed systems, based on the notion of trust structures introduced by Carbone, Nielsen and Sassone; a set of trust levels with two distinct partial orderings. In the trust model, a global trust state exists as the least fixed point of a collection of local policy functions of nodes in the network.     

System identification of nonlinear state-space models

This paper is concerned with the parameter estimation of a general class of nonlinear dynamic systems in state-space form. More specifically, a Maximum Likelihood (ML) framework is employed and an Expectation Maximisation (EM) algorithm is derived to compute these ML estimates. The Expectation (E) step involves solving a nonlinear state estimation problem, where the smoothed estimates of the states are required. This problem lends itself perfectly to the particle smoother, which provides arbitrarily good estimates. The maximisation (M) step is solved using standard techniques from numerical optimisation theory. Simulation examples demonstrate the efficacy of our proposed solution.     

Multivariate point process models for response times in multiprogrammed systems

We consider the formulation of marked multivariate point process models for job response times in multiprogrammed computer systems. Complementing queueing network representation of the structure of the system to be modeled, the particularR-process (Response time process) model we propose permits representation of resource contention, facilitates the incorporation of realistic workload characteristics into system performance predictions, and can reproduce inhomogeneities observed in running systems. Specification of the structure of theR-process model is conditional on workload marks; this effectively separates the difficult problem of formal representation of workload characteristics from the overall problem of response time prediction. To illustrate these ideas, an application to database management systems is considered. Evidence of the predictive capability of theR-process model, based on statistical analysis of response time data from an IMS system, is also given.