×

Gaussian process regression for the estimation of generalized frequency response functions. (English) Zbl 1429.93234

Summary: Bayesian learning techniques have recently garnered significant attention in the system identification community. Originally introduced for low variance estimation of linear impulse response models, the concept has since been extended to the nonlinear setting for Volterra series estimation in the time domain. In this paper, we approach the estimation of nonlinear systems from a frequency domain perspective, where the Volterra series has a representation comprised of generalized frequency response functions (GFRFs). Inspired by techniques developed for the linear frequency domain case, the GFRFs are modelled as real/complex Gaussian processes with prior covariances related to the time domain characteristics of the corresponding Volterra series. A Gaussian process regression method is developed for the case of periodic excitations, and numerical examples demonstrate the efficacy of the proposed method, as well as its advantage over time domain methods in the case of band-limited excitations.

MSC:

93C80 Frequency-response methods in control theory
93B30 System identification
93C10 Nonlinear systems in control theory
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Birpoutsoukis, G.; Csurcsia, P. Z.; Schoukens, J., Efficient multidimensional regularization for Volterra series estimation, Mechanical Systems and Signal Processing, 104, 896-914 (2018)
[2] Birpoutsoukis, G.; Marconato, A.; Lataire, J.; Schoukens, J., Regularized nonparametric Volterra kernel estimation, Automatica, 82, 324-327 (2017) · Zbl 1372.93189
[3] Boyd, S.; Tang, Y.; Chua, L., Measuring volterra kernels, IEEE Transactions on Circuits and Systems, 30, 8, 571-577 (1983) · Zbl 0518.93058
[4] Chen, T.; Ljung, L., Implementation of algorithms for tuning parameters in regularized least squares problems in system identification, Automatica, 49, 2213-2220 (2013) · Zbl 1364.93825
[5] Chen, T.; Ohlsson, H.; Goodwin, G.; Ljung, L., Kernel selection in linear system identification - Part II: a classical perspective, (Proc. of CDC-ECC (2011)), 4326-4331
[6] Evans, C.; Rees, D.; Jones, L.; Weiss, M., Periodic signals for measuring nonlinear Volterra kernels, IEEE Transactions on Instrumentation and Measurement, 45, 2, 362-371 (1996)
[7] Franz, M. O.; Schölkopf, B., A unifying view of Wiener and Volterra theory and polynomial kernel regression, Neural Computation, 18, 12, 3097-3118 (2006) · Zbl 1127.94316
[8] George, D., Continuous Nonlinear Systems (1959), MIT RLE Technical Report
[9] Lang, Z. Q.; Billings, S. A., Output frequencies of nonlinear systems, International Journal of Control, 67, 5, 713-730 (1997) · Zbl 0886.93042
[10] Lataire, J.; Chen, T., Transfer function and transient estimation by Gaussian process regression in the frequency domain, Automatica, 72, 217-229 (2016) · Zbl 1344.93103
[11] Nemeth, J. G.; Kollar, I.; Schoukens, J., Identification of volterra kernels using interpolation, IEEE Transactions on Instrumentation and Measurement, 51, 4, 770-775 (2002)
[12] Pillonetto, G.; De Nicolao, G., A new kernel-based approach for linear system identification, Automatica, 46, 1, 81-93 (2010) · Zbl 1214.93116
[13] Rasmussen, C. E.; Williams, C. K.I., Gaussian Processes for Machine Learning (2006), MIT Press · Zbl 1177.68165
[14] Rijlaarsdam, D.; Nuij, P.; Schoukens, J.; Steinbuch, M., A comparative overview of frequency domain methods for nonlinear systems, Mechatronics, 42, Supplement C, 11-24 (2017)
[15] Schetzen, M., The Volterra and Wiener theories of nonlinear systems (1980), John Wiley and Sons · Zbl 0501.93002
[16] Stoddard, J. G.; Welsh, J. S., Frequency domain estimation of parallel Hammerstein systems using Gaussian process regression, (Proceedings of the 18th IFAC Symposium on System Identification (2018)), 1014-1019
[17] Stoddard, J. G.; Welsh, J. S.; Hjalmarsson, H., EM-based hyperparameter optimization for regularized Volterra kernel estimation, IEEE Control Systems Letters, 1, 2, 388-393 (2017)
[18] Westwick, D. T.; Kearney, R. E., Identification of Nonlinear Physiological Systems (2003), John Wiley and Sons
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.