Identification of Wiener–Hammerstein systems by a nonparametric separation of the best linear approximation

Wiener–Hammerstein models are flexible, well known and often studied. The main challenge in identifying a Wiener–Hammerstein model is to distinguish the linear time invariant (LTI) blocks at the front and the back. This paper presents a nonparametric approach to separate the front and back dynamics starting from the best linear approximation (BLA). Next, the nonparametric estimates of the LTI blocks in the model can be parametrized, taking into account a phase shift degeneration. Once the dynamics are known, the estimation of the static nonlinearity boils down to a simple linear least squares problem. The consistency of the proposed approach is discussed and the method is validated on the Wiener–Hammerstein benchmark that was presented at the IFAC SYSID conference in 2009.     

Identification of Wiener–Hammerstein models: Two algorithms based on the best split of a linear model applied to the SYSID'09 benchmark problem

This paper describes the identification of Wiener–Hammerstein models and two recently suggested algorithms are applied to the SYSID'09 benchmark data. The most difficult step in the identification process of such block-oriented models is to generate good initial values for the linear dynamic blocks so that local minima are avoided. Both of the considered algorithms obtain good initial estimates by using the best linear approximation (BLA) which can easily be estimated from data. Given the BLA, the two algorithms differ in the way the dynamics are separated into two linear parts. The first algorithm simply considers all possible splits of the dynamics. Each of the splits is used to initialize one Wiener–Hammerstein model using linear least-squares and the best performing model is selected. In the second algorithm, both linear blocks are initialized with the entire BLA model using basis function expansions of the poles and zeros of the BLA. This gives over-parameterized linear blocks and their order is decreased in a model reduction step. Both algorithms are explained and their properties are discussed. They both give good, comparable models on the benchmark data.     

Adaptive control of Hammerstein–Wiener nonlinear systems

The Hammerstein–Wiener model is a block-oriented model, having a linear dynamic block sandwiched by two static nonlinear blocks. This note develops an adaptive controller for a special form of Hammerstein–Wiener nonlinear systems which are parameterized by the key-term separation principle. The adaptive control law and recursive parameter estimation are updated by the use of internal variable estimations. By modeling the errors due to the estimation of internal variables, we establish convergence and stability properties. Theoretical results show that parameter estimation convergence and closed-loop system stability can be guaranteed under sufficient condition. From a qualitative analysis of the sufficient condition, we introduce an adaptive weighted factor to improve the performance of the adaptive controller. Numerical examples are given to confirm the results in this paper.     

Least-Squares Support Vector Machines for the identification of Wiener–Hammerstein systems

This paper considers the identification of Wiener–Hammerstein systems using Least-Squares Support Vector Machines based models. The power of fully black-box NARX-type models is evaluated and compared with models incorporating information about the structure of the systems. For the NARX models it is shown how to extend the kernel-based estimator to large data sets. For the structured model the emphasis is on preserving the convexity of the estimation problem through a suitable relaxation of the original problem. To develop an empirical understanding of the implications of the different model design choices, all considered models are compared on an artificial system under a number of different experimental conditions. The obtained results are then validated on the Wiener–Hammerstein benchmark data set and the final models are presented. It is illustrated that black-box models are a suitable technique for the identification of Wiener–Hammerstein systems. The incorporation of structural information results in significant improvements in modeling performance.     

A subspace-based identification of Wiener–Hammerstein benchmark model

This paper develops a subspace-based method of identifying the Wiener–Hammerstein system, where a nonlinearity is sandwiched by two linear subsystems. First, a state space model of the best linear approximation of it is identified by using a subspace identification method and the poles of the best linear model are allocated between two linear subsystems by a state transformation. Unknown system parameters and coefficients of a basis function expansion of the nonlinearity are estimated by using the separable least-squares for all possible allocations of poles, so that there is a possibility that many iterative minimization problems should be solved. Finally, the best Wiener–Hammerstein system that yields the minimum mean square error is selected. Numerical results for a benchmark model show the applicability of the proposed method.     

Identification of Wiener–Hammerstein benchmark model via rank minimization

In this paper, a Wiener–Hammerstein system identification problem is formulated as a semidefinite programming (SDP) problem which provides a sub-optimal solution for a rank minimization problem. In the proposed identification method, the first linear dynamic system, the static nonlinear function, and the second linear dynamic system are parameterized as an FIR model, a polynomial function, and a rational transfer function respectively. Subsequently the optimization problem is formulated by using the over-parameterization technique and an iterative approach is proposed to update two unmeasurable intermediate signals. For the modeling of static nonlinearity, the monotonically non-deceasing condition was applied to limit the number of possible selections for intermediate signals. At each step of iteration, the over-parametrized parameters are estimated and then system parameters are separated by using a singular value decomposition (SVD). The proposed method is applied to the benchmark problem and the estimation result shows the effectiveness of the proposed algorithm.     

Fast extremum-seeking for Wiener–Hammerstein plants

This paper considers an observer-based extremum-seeking (ES) scheme acting on a Wiener–Hammerstein (WH) plant. The ES scheme utilises a high-frequency sinusoidal dither in order to achieve fast minimisation of the plant’s static nonlinearity. Unlike the prevailing analyses of “fast” ES schemes acting on WH plants, the presented result is semi-global and applicable to plants with fairly general, unknown nonlinearities. Design of the ES scheme requires knowledge of the relative orders of both the input and output dynamics of the plant. Two different tunings of the ES parameters are considered: one which forces the ES dynamics to be fast, and one which is less restrictive. It is shown that approximate knowledge of the phase of the plant’s input dynamics is required if the ES scheme is to use the less restrictive tuning. For both tunings, it is shown how the ES scheme may be designed in order to simultaneously achieve arbitrarily fast and accurate minimisation of the static nonlinearity from an arbitrarily large set of initial conditions. Moreover, the result shows that a WH plant with unstable input dynamics can, in fact, be stabilised by the ES scheme.     

Discrete-time extremum-seeking for Wiener–Hammerstein plants

This paper presents a new formulation of extremum-seeking control for discrete-time Wiener–Hammerstein plants. A novel method of analysis using Linear Parameter-Varying (LPV) system theory demonstrates semi-global stability of the control scheme. Assuming only limited plant knowledge, the stability result ensures convergence of the plant output in steady state to a point in an arbitrarily small neighbourhood of the extremum, for appropriately chosen controller parameters. The behaviour of the control scheme is analysed on a simple simulated system, prior to being implemented on an internal combustion engine. Experiments demonstrate how the scheme is able to maximise engine output torque in the presence of an uncertain fuel composition by modifying the spark timing.     

Piece wise linear least–squares approximation of planar curves

Two algorithms for solving the piecewise linear least–squares approximation problem of plane curves are presented. The first is for the case when the L ₂ residual (error) norm in any segment is not to exceed a pre–assigned value. The second algorithm is for the case when the number of segments is given and a (balanced) L ₂ residual norm solution is required. The given curve is first digitized and either algorithm is then applied to the discrete points. For each segment, we obtain the upper triangular matrix R in the QR factorization of the (augmented) coefficient matrix of the resulting system of linear equations. The least–squares solutions are calculated in terms of the R (and Q) matrices. The algorithms then work in an iterative manner by updating the least–squares solutions for the segments via up dating the R matrices. The calculation requires as little computational effort as possible. Numerical results and comments are given. This, in a way, is a tutorial paper.     

Finite element approximation of the Cahn–Hilliard equation on surfaces

In this paper, we consider the phase separation on general surfaces by solving the nonlinear Cahn–Hilliard equation using a finite element method. A fully discrete approximation scheme is introduced, and we establish a priori estimates for the discrete solution that does not rely on any knowledge of the exact solution beyond the initial time. This in turn leads to convergence and optimal error estimates of the discretization scheme. Numerical examples are also provided to substantiate the theoretical results.     

Generation of concatenated Greenberger–Horne–Zeilinger-type entangled coherent state based on linear optics

The concatenated Greenberger–Horne–Zeilinger (C-GHZ) state is a new type of multipartite entangled state, which has potential application in future quantum information. In this paper, we propose a protocol of constructing arbitrary C-GHZ entangled state approximatively. Different from previous protocols, each logic qubit is encoded in the coherent state. This protocol is based on the linear optics, which is feasible in experimental technology. This protocol may be useful in quantum information based on the C-GHZ state.     

Recursive identification for multi-input–multi-output Hammerstein–Wiener system

In this paper, an identification method based on the recursive auxiliary variable least squares algorithm is proposed for a multi-input–multi-output Hammerstein–Wiener system with process noise. In the proposed identification method, the system is converted into the multivariate regression form under the condition that the nonlinear block in the output part is invertible. Then, the auxiliary variable is constructed, the parameters of the regression equations are identified, and the system parameter matrices can be obtained by matrix composition of the parameter product matrix. A theoretical analysis showed that the proposed method has uniform convergence when the process noise is white and has a finite variance. The effectiveness of the proposed method is validated through the experiments.     

Stability robustness bounds for linear state–space models with structured uncertainty based on ellipsoidal set-theoretic approach

This paper is concerned with the problem of robust stability of linear dynamic systems with structured uncertainty by means of ellipsoidal set-theoretic approach. In this paper, the uncertainty in the physical parameters is expressed in terms of an ellipsoidal set in appropriate vector space. Two ellipsoidal set-theoretic approaches are presented for giving sufficient conditions for robust stability property of the systems with structured uncertainty. The bound produced by the ellipsoidal extension function theorem is shown to be less conservative than the one predicted by the Lagrange multiplier method. In order to introduce the ellipsoidal extension function theorem, in Appendix A of this paper, we try to present the theory of ellipsoidal algebra, following the thought of interval analysis. First of all, we give the concept of ellipsoidal numbers and define their arithmetic operations. Based on them, we finally introduce ellipsoidal vectors and ellipsoidal functions. In terms of the inclusion monotonic property of ellipsoidal functions, we present and prove the ellipsoidal extension function theorem.     

Adaptive control of stochastic Hammerstein–Wiener nonlinear systems with measurement noise

This paper deals with the adaptive control of a class of stochastic Hammerstein–Wiener nonlinear systems with measurement noise. Despite the fundamental progress achieved so far, a general theory framework about adaptive control of Hammerstein–Wiener models is still absent. Such situation is mainly due to the lack of an appropriate parameterisation model. To this end, this paper presents a novel parameterisation model that is to replace unmeasurable internal variables with their estimations. Then, the adaptive control algorithm to be applied is derived on the basis of self-tuning control. In addition, due to the use of the internal variable estimations, the stability and convergence properties are different from the self-tuning control. Our aim, in theoretical analysis, is to discover what limitations are in using the estimations instead of the true values in a control algorithm. Representative numerical examples are given and the simulation results verify the theoretical analysis.     

A nonlinear study on the interfacial instabilities in electro-osmotic flows based on the Debye–Hückel approximation

Interfacial instabilities in an electro-osmotic micro-film flow are studied by deriving an evolution equation for the local film thickness and subsequent numerical integrations. The free-surface electro-osmotic flow has an inherent instability of the long-wave type, which generates corrugations on the film surface. These corrugations may critically affect the transport characteristics of the flow, and deserve a nonlinear analysis based on conservation laws. It is shown that the electro-osmotic instability can cause severe local depression of the film even in the absence of the van der Waals attraction between the film surface and the substrate. The electrical double layer (EDL) then may be penetrated by the film surface, and film rupture can occur, resulting in loss of the electro-osmotic driving force. Since the Debye–Hückel approximation used becomes inadequate as the film thins locally to a nano-scale, quantitative analysis of the incipient rupture reported would require a fully coupled system for fluid flow, ionic concentration, and electric field.     

Wiener–Hammerstein system identification – an evolutionary approach

The problem of identifying parametric Wiener–Hammerstein (WH) systems is addressed within the evolutionary optimisation context. Specifically, a hybrid culture identification method is developed that involves model structure adaptation using genetic recombination and model parameter learning using particle swarm optimisation. The method enjoys three interesting features: (1) the risk of premature convergence of model parameter estimates to local optima is significantly reduced, due to the constantly maintained diversity of model candidates; (2) no prior knowledge is needed except for upper bounds on the system structure indices; (3) the method is fully autonomous as no interaction is needed with the user during the optimum search process. The performances of the proposed method will be illustrated and compared to alternative methods using a well-established WH benchmark.     

A fractional approach to identify Wiener–Hammerstein systems

Block-oriented nonlinear models are appealing due to their simplicity and parsimony. Existing methods to identify the Wiener–Hammerstein model suffer from one or several drawbacks. This paper shows that it is possible to generate initial estimates in an alternative way. A fractional model parameterization is the key to the success of this approach. Advantages are that no more than two iterative optimizations are needed and that large model orders can be handled. As illustrated through a simulation example and experimental benchmark data, it gives superior initial estimates and comparable optimized results.     

Uniquely connecting frequency domain representations of given order polynomial Wiener–Hammerstein systems

The notion of frequency response functions has been generalized to nonlinear systems in several ways. However, a relation between different approaches has not yet been established. In this paper, frequency domain representations for nonlinear systems are uniquely connected for a class of nonlinear systems. Specifically, by means of novel analytical results, the generalized frequency response function (GFRF) and the higher order sinusoidal input describing function (HOSIDF) for polynomial Wiener–Hammerstein systems are explicitly related, assuming the linear dynamics are known. Necessary and sufficient conditions for this relation to exist and results on the uniqueness and equivalence of the HOSIDF and GFRF are provided. Finally, this yields an efficient computational procedure for computing the GFRF from the HOSIDF and vice versa.     

Stability and passivity preserving Petrov–Galerkin approximation of linear infinite-dimensional systems

This contribution presents two approximation methods for linear infinite-dimensional systems that ensure the preservation of stability and passivity. The first approach allows one to approximate internal source free infinite-dimensional systems such that the resulting approximation is a port-controlled Hamiltonian system with dissipation. The second method deals with the class of systems that are not required to have conjugated outputs but only a dissipative system operator. It yields approximations with a dissipative system matrix for which bounds of their stability margin are provided. Both approaches are based on a state space formulation of the infinite-dimensional system. This makes it possible to use the Petrov–Galerkin approximation whose free parameters are partly used for achieving the structure preservation. Since still free parameters remain, further application specific objectives, such as, e.g., moment matching, can be achieved. Both approaches are applied to the approximation of an Euler–Bernoulli beam.