Initializing Wiener-Hammerstein Models Based on Partitioning of the Best Linear Approximation

This paper describes a new algorithm for initializing and estimating Wiener- Hammerstein models. The algorithm makes use of the best linear model of the system which is split in all possible ways into two linear sub-models. For all possible splits, a Wiener- Hammerstein model is initialized which means that a nonlinearity is introduced in between the two sub-models. The linear parameters of this nonlinearity can be estimated using leastsquares. All initialized models can then be ranked with respect to their fit. Typically, one is only interested in the best one, for which all parameters are fitted using prediction error minimization. The paper explains the algorithm and the consistency of the initialization is stated. Computational aspects are investigated, showing that in most realistic cases, the number of splits of the initial linear model remains low enough to make the algorithm useful. The algorithm is illustrated on an example where it is shown that the initialization is a tool to avoid many local minima

Identification of multiple-input single-output Hammerstein models using Bezier curves and Bernstein polynomials

AbstractThis paper considers the implementation of Bezier–Bernstein polynomials and the Levenberg–Marquart algorithm for identifying multiple-input single-output (MISO) Hammerstein models consisting of nonlinear static functions followed by a linear dynamical subsystem. The nonlinear static functions are approximated by the means of Bezier curves and Bernstein basis functions. The identification method is based on a hybrid scheme including the inverse de Casteljau algorithm, the least squares method, and the Levenberg–Marquart (LM) algorithm. Furthermore, results based on the proposed scheme are given which demonstrate substantial identification performance

B-spline neural networks based PID controller for Hammerstein systems

A new PID tuning and controller approach is introduced for Hammerstein systems based on input/output data. A B-spline neural network is used to model the nonlinear static function in the Hammerstein system. The control signal is composed of a PID controller together with a correction term. In order to update the control signal, the multi-step ahead predictions of the Hammerstein system based on the B-spline neural networks and the associated Jacobians matrix are calculated using the De Boor algorithms including both the functional and derivative recursions. A numerical example is utilized to demonstrate the efficacy of the proposed approaches

Identification scheme for fractional Hammerstein Models with the delayed Haar Wavelet

The parameter identification of a nonlinear Hammerstein-type process is likely to be complex and challenging due to the existence of significant nonlinearity at the input side. In this paper, a new parameter identification strategy for a block-oriented Hammerstein process is proposed using the Haar wavelet operational matrix (HWOM). To determine all the parameters in the Hammerstein model, a special input excitation is utilized to separate the identification problem of the linear subsystem from the complete nonlinear process. During the first test period, a simple step response data is utilized to estimate the linear subsystem dynamics. Then, the overall system response to sinusoidal input is used to estimate nonlinearity in the process. A single-pole fractional order transfer function with time delay is used to model the linear subsystem. In order to reduce the mathematical complexity resulting from the fractional derivatives of signals, a HWOM based algebraic approach is developed. The proposed method is proven to be simple and robust in the presence of measurement noises. The numerical study illustrates the efficiency of the proposed modeling technique through four different nonlinear processes and results are compared with existing methods

Identification of nonlinear interconnected systems

This thesis was submitted for the degree of Master of Philosophy and awarded by Brunel University.In this work we address the problem of identifying a discrete-time nonlinear system composed of a linear dynamical system connected to a static nonlinear component. We use linear fractional representation to provide a united framework for the identification of two classes of such systems. The first class consists of discrete-time systems consists of a linear time invariant system connected to a continuous nonlinear static component. The identification problem of estimating the unknown parameters of the linear system and simultaneously fitting a math order spline to the nonlinear data is addressed. A simple and tractable algorithm based on the separable least squares method is proposed for estimating the parameters of the linear and the nonlinear components. We also provide a sufficient condition on data for consistency of the identification algorithm. Numerical examples illustrate the performance of the algorithm. Further, we examine a second class of systems that may involve a nonlinear static element of a more complex structure. The nonlinearity may not be continuous and is approximated by piecewise a±ne maps defined on different convex polyhedra, which are defined by linear combinations of lagged inputs and outputs. An iterative identification procedure is proposed, which alternates the estimation of the linear and the nonlinear subsystems. Standard identification techniques are applied to the linear subsystem, whereas recently developed piecewise affine system identification techniques are employed for the estimation of the nonlinear component. Numerical examples show that the proposed procedure is able to successfully profit from the knowledge of the interconnection structure, in comparison with a direct black box identification of the piecewise a±ne system.Funding was obtained as a Marie Curie Early Stage Researcher Training fellowship, under the NET-ACE project (MEST-CT-2004-6724)

A combined B-Spline-Neural-Network and ARX Model for Online Identi cation of Nonlinear Dynamic Actuation Systems

This paper presents a block oriented nonlinear dynamic model suitable for online identi cation.The model has the well known Hammerstein architecture where as a novelty the nonlinear static part is represented by a B-spline neural network (BSNN), and the linear static one is formalized by an auto regressive exogenous model (ARX). The model is suitable as a feed-forward control module in combination with a classical feedback controller to regulate velocity and position of pneumatic and hydraulic actuation systems which present non stationary nonlinear dynamics. The adaptation of both the linear and nonlinear parts is taking place simultaneously on a patterby- patter basis by applying a combination of error-driven learning rules and the recursive least squares method. This allows to decrease the amount of computation needed to identify the model's parameters and therefore makes the technique suitable for real time applications. The model was tested with a silver box benchmark and results show that the parameters are converging to a stable value after 1500 samples, equivalent to 7.5s of running time. The comparison with a pure ARX and BSNN model indicates a substantial improvement in terms of the RMS error, while the comparison with alternative non linear dynamic models like the NNOE and NNARX, having the same number of parameters but greater computational complexity, shows comparable performances

A combined B-Spline-Neural-Network and ARX Model for Online Identi cation of Nonlinear Dynamic Actuation Systems

This paper presents a block oriented nonlinear dynamic model suitable for online identi cation.The model has the well known Hammerstein architecture where as a novelty the nonlinear static part is represented by a B-spline neural network (BSNN), and the linear static one is formalized by an auto regressive exogenous model (ARX). The model is suitable as a feed-forward control module in combination with a classical feedback controller to regulate velocity and position of pneumatic and hydraulic actuation systems which present non stationary nonlinear dynamics. The adaptation of both the linear and nonlinear parts is taking place simultaneously on a patterby- patter basis by applying a combination of error-driven learning rules and the recursive least squares method. This allows to decrease the amount of computation needed to identify the model's parameters and therefore makes the technique suitable for real time applications. The model was tested with a silver box benchmark and results show that the parameters are converging to a stable value after 1500 samples, equivalent to 7.5s of running time. The comparison with a pure ARX and BSNN model indicates a substantial improvement in terms of the RMS error, while the comparison with alternative non linear dynamic models like the NNOE and NNARX, having the same number of parameters but greater computational complexity, shows comparable performances