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.     

Application of evolving Takagi–Sugeno fuzzy model to nonlinear system identification

In this paper, a new encoding scheme is presented for learning the Takagi–Sugeno (T–S) fuzzy model from data by genetic algorithms (GAs). In the proposed encoding scheme, the rule structure (selection of rules and number of rules), the input structure (selection of inputs and number of inputs), and the antecedent membership function (MF) parameters of the T–S fuzzy model are all represented in one chromosome and evolved together such that the optimisation of rule structure, input structure, and MF parameters can be achieved simultaneously. The performance of the developed evolving T–S fuzzy model is first validated by studying the benchmark Box–Jenkins nonlinear system identification problem and nonlinear plant modelling problem, and comparing the obtained results with other existing results. Then, it is applied to approximate the forward and inverse dynamic behaviours of a magneto-rheological (MR) damper of which identification problem is significantly difficult due to its inherently hysteretic and highly nonlinear dynamics. It is shown by the validation applications that the developed evolving T–S fuzzy model can identify the nonlinear system satisfactorily with acceptable number of rules and appropriate inputs.     

Black box model identification of nonlinear input–output models: A Wiener–Hammerstein benchmark

This work analyzes the performance of several black box nonlinear model identification techniques for input–output models with polynomial nonlinearities on a benchmark identification problem. The case study, proposed in Schoukens, Suykens, and Ljung (2008), concerns a nonlinear SISO electronic system with a Wiener–Hammerstein structure, originally documented in Vandersteen (1997). The objective being the obtainment of an accurate simulation model, capable of replicating the dynamic behavior of the system without using past measured output data, various output-error approaches have been tested and compared with more standard equation-error techniques. The provided analysis shows that excellent modeling performance can be obtained with these methods even without explicitly taking into account the block structure of the nonlinear system.     

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.     

Critical thermodynamic evaluation of oxide systems relevant to fuel ashes and slags,Part 4: Sodium oxide–potassium oxide–silica

A thermochemical assessment was performed for the system K₂O–Na₂O–SiO₂. The modified associate species model was applied to the ternary liquid in the system. All binary subsystems remained unchanged. The new databank was used for the representation of the phase equilibria in the ternary system including the quasi-binary sections of the ternary diagram. The calculated phase relations are in good agreement with the experimental data. The phase equilibria in the experimentally uninvestigated region near the alkali oxide edge are proposed as extrapolations using the new databank.     

Initialization of nonlinear state-space models applied to the Wiener–Hammerstein benchmark

In this work a new initialization scheme for nonlinear state-space models is applied to the problem of identifying a Wiener–Hammerstein system on the basis of a set of real data. The proposed approach combines ideas from the statistical learning community with classic system identification methods. The results on the benchmark data are discussed and compared to the ones obtained by other related methods.     

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.     

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.     

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.     

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.     

A Scheil–Gulliver model dedicated to the solidification of steel

Modelling of solidification is of industrial and theoretical relevance. An accurate estimation of the actual liquidus and solidus temperatures leads to significant improvements in quality and efficiency of steel production as well as substantially reduces the energy consumption and the ecological footprint.An optimisation of the Scheil–Gulliver model for solidification with the aim to predict the solidus temperature of steels (Scheil–Gulliver for Steel, SGS) is presented. The SGS model allows an easy and accurate simulation of the solidification interval using software based on the CALPHAD approach. Based only on the steel composition, the model consistently choses between full equilibrium for ferrite and partial redistribution of alloying elements (Scheil–Gulliver approach) for austenite. The predictions of the model were compared to differential thermal analysis (DTA) measurements of industrial heats, which represent a wide range of compositions. The agreement of the data calculated with the SGS model with the values measured by DTA represents an improvement compared to existing models.     

Application of permutation genetic algorithm for sequential model building–model validation design of experiments

The work presented in this paper is motivated by a complex multivariate engineering problem associated with engine mapping experiments, which require efficient design of experiments (DoE) strategies to minimise expensive testing. The paper describes the development and evaluation of a Permutation Genetic Algorithm (PermGA) to enable an exploration-based sequential DoE strategy for complex real-life engineering problems. A known PermGA was implemented to generate uniform OLH DoEs, and substantially extended to support generation of model building–model validation (MB–MV) sequences, by generating optimal infill sets of test points as OLH DoEs that preserve good space-filling and projection properties for the merged MB + MV test plan. The algorithm was further extended to address issues with non-orthogonal design spaces, which is a common problem in engineering applications. The effectiveness of the PermGA algorithm for the MB–MV OLH DoE sequence was evaluated through a theoretical benchmark problem based on the Six-Hump-Camel-Back function, as well as the Gasoline Direct Injection engine steady-state engine mapping problem that motivated this research. The case studies show that the algorithm is effective in delivering quasi-orthogonal space-filling DoEs with good properties even after several MB–MV iterations, while the improvement in model adequacy and accuracy can be monitored by the engineering analyst. The practical importance of this work, demonstrated through the engine case study, is that significant reduction in the effort and cost of testing can be achieved.     

An hp finite element adaptive scheme to solve the Laplace model for fluid–solid vibrations

In this paper we introduce an hp finite element method to solve a two-dimensional fluid–structure spectral problem. This problem arises from the computation of the vibration modes of a bundle of parallel tubes immersed in an incompressible fluid. We prove the convergence of the method and a priori error estimates for the eigenfunctions and the eigenvalues. We define an a posteriori error estimator of the residual type which can be computed locally from the approximate eigenpair. We show its reliability and efficiency by proving that the estimator is equivalent to the energy norm of the error up to higher order terms, the equivalence constant of the efficiency estimate being suboptimal in that it depends on the polynomial degree. We present an hp adaptive algorithm and several numerical tests which show the performance of the scheme, including some numerical evidence of exponential convergence.     

Application of the Box–Behnken design to the optimization of process parameters in foam cup molding

Currently, foam molding technologies are widely adopted for most bra styles, which demonstrate the incomparable advantages in the contemporary intimate apparel industry. The determination of proper molding conditions, such as molding temperatures and length of time on the basis of cup sizes and styles, is crucial in achieving the required cup shape with high stability, which is regarded as the most challenging part of the molded bra making process. To determine the optimal process parameter settings, numerous process trials are generally required to evaluate the molding variables and their interactions. This study proposes a novel systematic methodology to identify the optimal molding process parameters based on design of experiment (DOE) and a parameterization-based remesh method to evaluate the 3D shape conformity of molded cups. By solving the regression equation obtained from a Box–Behnken design (BBD) and analyzing the response surface plots, the results prove that molding temperature has greater influence than the length of the dwell time on the 3D shape conformity of molded cups. The optimal molding conditions can be determined for the cup depths of different sized mold heads, which are validated by the experimental results.     

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.     

Comments on “the parameter identification of a population model of China”

The parametric model used by Zhu and Wan for expressing survival rate as a function of age can be replaced by a simpler model without discernable loss of accuracy. The parameters of this simpler model are subject to easy interpretation.     

Development and Application of the Cauchy–Poisson Method to Layer Elastodynamics and the Timoshenko Equation

We consider a generalization of the Cauchy–Poisson method to an n-dimensional Euclidean space and its application to the construction of hyperbolic approximations. In Euclidean space, constraints on derivatives are introduced. The principle of hyperbolic degeneracy in terms of parameters is formulated and its implementation in the form of necessary and sufficient conditions is given. As the particular case of a four-dimensional space with preserving operators up to the sixth order a generalized hyperbolic equation is obtained for bending vibrations of plates with coefficients dependent only on the Poisson number. As special cases, this equation includes all the well-known Bernoulli–Euler, Kirchhoff, Rayleigh, and Timoshenko equations. As a development of Maxwell’s and Einstein’s research on the propagation of perturbations with finite velocity in a continuous medium, Tymoshenko’s non-trivial construction of the equation for bending vibrations of a beam is noted.