System Identification

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• Identification of Block-oriented Nonlinear Models
  • Researchers
  • Objectives
  • Introduction
  • Methods based on Orthonormal Bases
  • Subspace Methods
  • References

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• Identification of Block-oriented Nonlinear Models
  • Researchers: J.C. Gómez and E. Baeyens (University of Valladolid, Spain)
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  • Objectives
    The main objectives of this Research Project are:
    • To develop Noniterative Identification Algorithms for Block-oriented Nonlinear Models.
    • To analyze the estimation accuracy (bias and variance errors) of the developed methods.
    • To implement the algorithms in a Matlab environment.
    • To integrate the algorithms to a Toolbox for Identification of Nonlinear Systems using a Graphical User Interfase.
    • To perform a comparative study of the different identification algorithms developed and those available in the literature.
    Two different approaches are being investigated for the developement of the identification algorithms, viz.
    • Identification using Orthonormal Bases
    • Subspace-based identification methods
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  • Introduction
    In the last decades, a considerable amount of research has been carried out on modelling, identification, and control of nonlinear systems. Most dynamical systems can be better represented by nonlinear models, which are able to describe the global behavior of the system over the whole operating range, rather than by linear ones that are only able to approximate the system around a given operating point. One of the most frequently studied classes of nonlinear models are the so called block-oriented models, which consist of the interconnection of Linear Time Invariant (LTI) systems and static (memoryless) nonlinearities. Within this class, three of the more common model structures are:
    • the Hammerstein model, which consists of the cascade connection of a static (memoryless) nonlinearity followed by a LTI system (see for instance [1] for a review on identification of Hammerstein models)
    • the Wiener model, in which the order of the linear and the nonlinear blocks in the cascade connection is reversed (see for instance [2], [4], [5] for different methods for the identification of Wiener models)
    • the Feedback Block-Oriented (FBO) model, which consists of a static nonlinearity in the feedback path around a LTI system (see for instance [6] for an identification algorithm for this type of model).
    These models have been successfully used to represent nonlinear systems in a number of practical applications in the areas of chemical processes [1], biological processes [7], signal processing [29], communications, and control [8].
    Several techniques have been proposed in the literature for the identification of Hammerstein and Wiener models (see for instance [1],[2],[3],[4],[5],[7],[8], [9],[10],[11],[12],[13],[14], and the references therein). Among them, three main approaches can be distinguished. The first one is the traditional iterative algorithm proposed by Narendra and Gallman in [9]. In this algorithm, an appropriate parameterization of the system allows the prediction error to be separately linear in each set of parameters characterizing the linear and the nonlinear parts. The estimation is then carried out by minimizing alternatively with respect to each set of parameters, a quadratic criterion on the prediction errors. An analytical counterexample by Stoica [15] showed that the original algorithm could be divergent in some particular cases. A second approach, based on correlation techniques, is introduced in [10],[11],[12],[13]. This method relies on a separation principle, but with the rather restrictive requirement on the input to be white noise. A more recent approach for the identification of Hammerstein-Wiener systems has been introduced by Bai in [16]. This algorithm is based on least squares estimation (LSE) and singular value decomposition (SVD), however it only applies to the single-input/single-output (SISO) case, and consistency of the estimates can only be assured for the case of the disturbances being white noise, or in the noise-free case.
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  • Methods based on Orthonormal Bases
    Inspired by the work in [16], Gómez et al. [23],[24] developed a noniterative identification algorithm for the identification of Hammerstein and Wiener models, which is also based on LSE and SVD, but that in contrast to the one in [16], it also applies to the multivariable case, it allows a more general representation (using basis functions) of the nonlinearity, and consistency of the estimates can be guaranteed even in the presence of coloured noise. Key on the derivation of these results has been the use of basis functions for the representation of the linear and nonlinear parts of the Hammerstein and Wiener models. The use of Orthonomal Bases in Identification has been the subject of an intense research activity in recent years, as a natural answer to the issue of how to introduce 'a priori' information in the identification of "black box" LTI model structures. It has been shown that choosing the poles of the bases close to the (approximately known) system poles the accuracy of the estimate can be considerably improved (see [17],[18],[19],[20],[21],[22] and the references therein). A detailed review of the use of Orthonormal Bases in Identification of LTI Systems can be found in [17] . Another advantage of using orthonormal bases to model LTI systems is that the input-output equation can be written as a linear regression. As a consequence, a parameter estimate can be obtained in closed form by minimizing a quadratic criterion on the prediction errors (viz, the Least Squares Estimate). In addition, since the regressors only depend on past inputs, the estimate is consistent even if the output is corrupted by coloured noise, under the assumption that the actual system belongs to the model class (i.e., there is no undermodelling). The objective of this part of the Research Project is to extend the results presented in [23], to other block-oriented nonlinear models like the FBO model, and "sandwich" structures (LTI-Nonlinear-LTI), which are also widely used to represent nonlinear systems. In particular, the FBO model allows to represent some phenomena that can not be represented using Hammertein or Wiener models, such as input and output multiplicities. These phenomena appear frequently in chemical processes. From the control design point of view, the main advantage of the method in [23], and its possible extensions to other block-oriented nonlinear models, is its relatively low computational load and numerical robustness, that make it suitable for its use in model-based control design methodologies. In particular, these identification techniques seem to be appropriate for use in combination with Model Predictive Control (MPC) Schemes [8].
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  • Subspace Methods
    Subspace-based State-Space System IDentification (4SID) methods are able to deliver reliable state-space models of multivariable LTI systems directly from input-output data, requiring only a modest computational complexity without the need of (nonlinear) iterative optimization procedures (see for instance the book [25] for a unified description of the different subspace algorithms, and the survey paper [26]). The methods have their origin in state-space realization theory as developed in the sixties [27][28], and the main computational tools employed are QR and SVD decompositions. The objective of this part of the Research Project is to develop Subspace Identification Methods for block-oriented nonlinear models. For the Hammerstein model, the idea is to use any available subspace method to estimate the system matrices of a linear model whose input is a filtered (by the nonlinear functions used to represent the nonlinear static block) version of the actual input to the Hammerstein system, and then to perform a projection (via SVD) in order to estimate separately the nonlinear and the linear matrix coefficients, using the results in [23]. Some preliminary results have been presented in [30].
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  • References
    
    • [1] E. Eskinat and S. H. Johnson and W. L. Luyben. Use of Hammerstein models in identification of nonlinear systems. AIChE Journal, 37(2):255-268, 1991.
    • [2] W. Greblicki. Nonparametric identification of Wiener systems by orthogonal series. IEEE Transactions on Automatic Control, 39(10):2077-2086, October 1994.
    • [3] W. Greblicki and M. Pawlak. Nonparametric identification of Hammerstein systems. IEEE Transactions on Information Theory, 35(2):409-418, March 1989.
    • [4] T. Wigren. Recursive prediction error identification using the nonlinear Wiener model. Automatica, 29(4):1011-1025, 1993.
    • [5] T. Wigren. Convergence analysis of recursive identification algorithms based on the nonlinear Wiener model. IEEE Transactions on Automatic Control, 39(11):2191-2206, November 1994.
    • [6] M. Pottmann and R. K. Pearson. Block-oriented NARMAX models with output multiplicities. AIChE Journal, 44:131-140, 1998.
    • [7] M. J. Korenberg. Identification of biological cascades of linear and static nonlinear systems. Proceedings of the 16th. Midwest Symposium on Circuit Theory, 1973.
    • [8] K.P. Fruzzetti, A. Palazoglu and K.A. McDonald. Nonlinear Model Predictive Control using Hammerstein models. Journal of Process Control, 7(1):31-41, 1997.
    • [9] K. S. Narendra and P. G. Gallman. An Iterative Method for the Identification of Nonlinear Systems Using a Hammerstein Model. IEEE Transactions on Automatic Control, AC-11:546-550, 1966.
    • [10] S.A. Billings and S. Y. Fakhouri. Identification of a class of nonlinear systems using correlation analysis. Proceedings IEE, Corresp., 125:1307, 1978.
    • [11] S.A. Billings and S. Y. Fakhouri. Identification of nonlinear systems using correlation analysis. Proceedings IEE, 125:691, 1978.
    • [12] S.A. Billings. Identification of Nonlinear Systems - A Survey. Proceedings of IEEE, Part D, 127(1):272, 1980.
    • [13] S.A. Billings and S. Y. Fakhouri. Identification of Systems Containing linear Dynamic and Static Nonlinear Elements. Automatica, 18(1):15-26, 1982.
    • [14] M. Boutayeb and M. Darouach. Recursive Identification Method for MISO Wiener-Hammerstein Model. IEEE Transactions on Automatic Control, AC40(2):287-291, 1995.
    • [15] P. Stoica. On the Convergence of an Iterative Algorithm used for Hammerstein System Identification. IEEE Transactions on Automatic Control, AC-26(4):967-969, 1981.
    • [16] E-W. Bai. An Optimal Two-Stage Identification Algorithm for Hammerstein-Wiener Nonlinear Systems. Automatica, 34(3):333-338, 1998.
    • [17] J.C. Gomez. Analysis of Dynamic System Identification using Rational Orthonormal Bases. PhD Thesis, Department of Electrical and Computer Engineering, The University of Newcastle, Australia, August, 1998.
    • [18] B. Ninness and F. Gustafsson. A unifying construction of orthonormal bases for system identification. IEEE Transactions on Automatic Control, AC-42(4):515-521, 1997.
    • [19] B. Wahlberg. System Identification Using Laguerre Models. IEEE Transactions on Automatic Control, AC-36(5):551-562, 1991.
    • [20] B. Wahlberg. System Identification Using Kautz Models. IEEE Transactions on Automatic Control, AC-39(6):1276-1282, 1994.
    • [21] P. M. J. Van den Hof, P. S. C. Heuberger and J. Bokor. System Identification with generalized orthonormal basis functions. Automatica, 31(12):1821-1834, 1995.
    • [22] P. S. C. Heuberger, P. M. J. Van den Hof and O. H. Bosgra. A generalized Orthonormal basis for linear dynamical systems. IEEE Transactions on Automatic Control, AC-40(3):451465, 1995.
    • [23] J.C. Gomez and E. Baeyens. Identification of Multivariable Hammerstein Systems using Rational Orthonormal Bases. In Proceedings of the 39th IEEE Conference on Decision and Control, Sydney, Australia, December 2000.
    • [24] J.C. Gomez and E. Baeyens. Hammerstein and Wiener Model Identification using Rational Orthonormal Bases. Latin American Applied Research, Vol. 33, No. 3, 2003.
    • [25] P. Van Overschee and B. de Moor. Subspace Identification for Linear Systems: Theory, Implementation, Applications. Kluwer Academic Publishers, Boston, 1996.
    • [26] M. Viberg. Subspace-based methods for the identification of linear time-invariant systems. Automatica, 31(12):1835-1851, 1995.
    • [27] B.L. Ho and R.E. Kalman. Effective construction of linear state-variable models from input-output functions. Regelungstechnik, 14(12):545-592, 1966.
    • [28] S.Y. Kung. A new identification method and model reduction algorithm via singular value decomposition. In Proceedings of the 12th Asilomar Conference on Circuits, Systems and Computers, pp. 705-714, Pacific Grove, CA, 1978.
    • [29] J. C. Stapleton and S. C. Bass. Adaptive noise cancellation for a class of nonlinear dynamic reference channels. IEEE Transactions on Circuits and Systems, CAS-32:143- 150, February 1985.
    • [30] J.C. Gomez and E. Baeyens. Subspace Identification of Hammerstein and Wiener Models. In Proceedings of the 15th IFAC World Congress, Barcelona, Spain, July 2002.
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