Theory and application of the modulating function method—I. Review and theory of the method and theory of the spline-type modulating functions

Preisig, Heinz A.; Rippin, D.W.T.

doi:10.1016/0098-1354(93)80001-4

Cited by 101 publications

(47 citation statements)

References 23 publications

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“…In cases where a model of the parameter's functional ity with time and process variables is previously deter mined, the embedded parameters belonging to the function can be estimated, provided the differential equation remains linear in parameters (Unbehauen & Rao, 1990;Preisig & Rippin, 1993a). Such an approach is limited to fixed and predetermined functional forms of time dependencies.…”

Section: Modulating Functions Methodsmentioning

confidence: 99%

“…The method was first suggested by Shinbrot (1957) as a means of parameter estimation in nonlinear dynamic systems by converting the original differential equations into a set of algebraic equations. Several versions of the method are currently in use with a variety of modulat ing functions, including spline type functions (Maletinsky, 1979;Preisig & Rippin, 1993a), Hermite functions (Takaya, 1968;Jalili, Jordan & Mackie, 1992), Poisson moment functionals (Saha & Rao, 1982), Hartley modulating functions (Patra & Unbe hauen, 1995;Daniel-Berhe & Unbehauen, 1998), sinu soidal functions (Shinbrot, 1957;Pearson & Lee, 1985;Benhadj-Braiek & Rotella, 1990;Co & Ydstie, 1990;Co & Ungarala, 1997) and wavelets (Schoenwald, 1993;Carrier & Stephanopoulos, 1998).…”

Section: Introductionmentioning

confidence: 99%

“…Time-varying parameters have been dealt with previ ously by assuming a functional form for the time dependency of the parameter and obtaining the esti mates of the constant coefficients in the functional form (Puchkov & Chinayev, 1973;Benhadj-Braiek & Rotella, 1990;Preisig & Rippin, 1993a). The modulating func tions method is essentially an off-line method that is implemented on a batch of time series data to estimate constant parameters.…”

Section: Introductionmentioning

confidence: 99%

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Time-varying system identification using modulating functions and spline models with application to bio-processes

Ungarala

2000

Computers & Chemical Engineering

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Section: Modulating Functions Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Time-varying system identification using modulating functions and spline models with application to bio-processes

Ungarala

2000

Computers & Chemical Engineering

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“…Hence, this method takes advantage from the low-pass filtering property of modulating functions integrals [31,29]. It was used to parameter identification for nonlinear systems, time-varying systems and noisy sinusoidal signals [32,33,31].…”

Section: Introductionmentioning

confidence: 99%

Fractional order differentiation by integration: an application to fractional linear systems

Liu

Laleg‐Kirati

Gibaru

2013

IFAC Proceedings Volumes

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In this article, we propose a robust method to compute the output of a fractional linear system defined through a linear fractional differential equation (FDE) with timevarying coefficients, where the input can be noisy. We firstly introduce an estimator of the fractional derivative of an unknown signal, which is defined by an integral formula obtained by calculating the fractional derivative of a truncated Jacobi polynomial series expansion. We then approximate the FDE by applying to each fractional derivative this formal algebraic integral estimator. Consequently, the fractional derivatives of the solution are applied on the used Jacobi polynomials and then we need to identify the unknown coefficients of the truncated series expansion of the solution. Modulating functions method is used to estimate these coefficients by solving a linear system issued from the approximated FDE and some initial conditions. A numerical result is given to confirm the reliability of the proposed method.

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“…Using the derivatives of the data in system identification has been studied, especially in the identification of continuous time models (Brewer, Barenco, Callard, Hubank, & Stark, 2008;Preisig & Rippin, 1993;Schmidt & Lipson, 2009). However, as far as the authors are aware this study is the first time the weak derivatives have been combined with the least squares criterion to build a completely new metric for the prediction errors and which uses the new metric to improve the model structure detection in non-linear system identification.…”

Section: Introductionmentioning

confidence: 99%

Ultra-Orthogonal Forward Regression Algorithms for the Identification of Non-Linear Dynamic Systems

Guo

Billings

et al. 2016

Neurocomputing

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A new Ultra Least Squares (ULS) criterion is introduced for system identification. Unlike the standard least squares criterion which is based on the Euclidean norm of the residuals, the new ULS criterion is derived from the Sobolev space norm. The new criterion measures not only the discrepancy between the observed signals and the model prediction but also the discrepancy between the associated weak derivatives of the observed and the model signals. The new ULS criterion possesses a clear physical interpretation and is easy to implement. Based on this, a new Ultra Orthogonal Forward Regression (UOFR) algorithm is introduced for nonlinear system identification, which includes converting a least squares regression problem into the associated ultra least squares problem and solving the ultra least squares problem using the orthogonal forward regression method. Numerical simulations show that the new UOFR algorithm can significantly improve the performance of the classic OFR algorithm.Key words: orthogonal forward regression, system identification, ultra least squares, ultra orthogonal forward regression, ultra orthogonal least squares. IntroductionSystem identification plays a more and more important role in revealing the unknown mechanisms and rules underlying complex phenomena (Schmidt & Lipson, 2009). System identification includes the detection of the model structure and estimation of the associated parameters. A system identification problem can often be thought of as an optimization problem where the optimal model is searched from a large predefined candidate model dictionary given a criterion. The criterion is 2 used to evaluate the performance of each model by measuring the discrepancy between the observed data and the model predictions. The candidate model dictionary is often chosen to be large enough to include the unknown correct model. Hence an exhaustive search algorithm is often infeasible in these kinds of applications because of the large solution space. Even an evolutionary algorithm which can greatly reduce the search process can still be very computationally intensive.Hence an algorithm which can efficiently find the optimal solution is desired. However, a fast algorithm often dictates an optimal substructure; otherwise the search may converge to a suboptimal solution. Many efforts have been made to improve the search process under a certain specific loss function or performance index, for example, the simulated annealing algorithm, particle swarm optimisation, and so on. In this paper, a different and new methodology will be introduced.Instead of improving the search method, a new and effective criterion will be introduced to describe the objective of the regression more accurately. Under the new criterion, the solution space has a better structure and a fast algorithm is more likely to find the optimal solution.System identification aims to identify a model from observed data based on a criterion. A good criterion results in not only better parameter estimation but also a good search pa...

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Theory and application of the modulating function method—I. Review and theory of the method and theory of the spline-type modulating functions

Cited by 101 publications

References 23 publications

Time-varying system identification using modulating functions and spline models with application to bio-processes

Time-varying system identification using modulating functions and spline models with application to bio-processes

Fractional order differentiation by integration: an application to fractional linear systems

Ultra-Orthogonal Forward Regression Algorithms for the Identification of Non-Linear Dynamic Systems

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