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Simultaneous estimation of the input and output frequencies of nonlinear systems. (English) Zbl 1149.93344

Summary: The simultaneous estimation of the input and output frequencies of nonlinear systems is considered. As the output frequencies are generated from the input frequencies, and are integer combinations of these frequencies, it is shown in this paper that the simultaneous estimation of both the input and output frequencies can therefore be formulated as a constrained estimation problem. First, the constrained Cramér-Rao lower bound, an important general property of any unbiased estimator, is derived. The procedure and algorithm for estimating the input and output frequencies are devised based on the periodogram method. Numerical examples are presented to illustrate the performance and implementation of the proposed estimation procedure and algorithm.

MSC:

93E10 Estimation and detection in stochastic control theory
93C10 Nonlinear systems in control theory

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[1] Chua, L. O.; Ushida, A., Algorithms for computing almost periodic steady-state response of nonlinear systems to multiple input frequencies, IEEE Transactions on Circuits and Systems, CAS-28, 953-971 (1981) · Zbl 0484.93027
[2] Chen, C. T., Signals and systems (2004), Oxford University Press: Oxford University Press New York
[3] Kay, S. M., Fundamentals of statistical signal processing: Estimation theory (1993), Prentice Hall: Prentice Hall Englewood Cliffs, NJ · Zbl 0803.62002
[4] Lagarias, J. C.; Reeds, J. A.; Wright, M. H.; Wright, P. E., Convergence properties of the Nelder-Mead simplex method in low dimensions, SIAM Journal of Optimization, 9, 112-147 (1998) · Zbl 1005.90056
[5] Lang, Z. Q.; Billings, S. A., Evaluation of output frequency response of nonlinear systems under multiple inputs, IEEE Transactions on Circuits and Systems-II: Analog and Digital Signal Processing, 47, 28-38 (2000) · Zbl 0980.93050
[6] Marzetta, T. L., A simple derivation of the constrained multiple parameter Cramer-Rao Bound, IEEE Transactions on on Signal Processing, 41, 2247-2249 (1993) · Zbl 0775.93213
[7] Nelles, O., Nonlinear System Identification (2001), Springer
[8] Quinn, B. G.; Hannan, E. J., The estimation and tracking of frequency (2001), Cambridge Univ. Press: Cambridge Univ. Press Cambridge, UK · Zbl 0969.62060
[9] Rife, D. C.; Boorstyn, R. R., Multiple tone parameter estimation from discrete-time observations, Bell System Technical Journal, 1389-1410 (1976)
[10] Roy, R.; Kailath, T., ESPRIT-Estimation of signal parameter via rotational invariance techniques, IEEE Transactions on Acoustics, Speech, Signal Processing, 37, 984-995 (1989)
[11] Schmidt, R. O., Multiple emitter location and signal parameter estimation, IEEE Transactions on Antennas Propagation, 34, 276-280 (1986)
[12] So, H. C.; Chan, K. W.; Chan, Y. T.; Ho, K. C., Linear prediction approach for efficient frequency estimation of multiple real sinusoids: Algorithms and analyses, IEEE Transactions on Signal Processing, 53, 2290-2305 (2005) · Zbl 1370.94243
[13] Stoica, P.; Moses, R. L.; Friedlander, B.; Soderstrom, T., Maximum likelihood estimation of the parameters of multiple sinusoids from noisy measurements, IEEE Transactions on Acoustics, Speech, Signal Processing, 37, 378-392 (1989)
[14] Stoica, P., & Nehorai, A. (1988). Statistical analysis of two nonlinear least-squares estimators of sine wave parameters in the colored noise case. In Proc. int. conf. acoustics, speech, signal processing: Vol. 4; Stoica, P., & Nehorai, A. (1988). Statistical analysis of two nonlinear least-squares estimators of sine wave parameters in the colored noise case. In Proc. int. conf. acoustics, speech, signal processing: Vol. 4
[15] Stoica, P.; Ng, B. C., On the Cramér-Rao bound under parametric constraints, IEEE Signal Processing Letters, 5, 177-179 (1998)
[16] Yang, Z.Y., & Chan, C.W. (2006). Fault detection for nonlinear systems with multiple periodic inputs. In Proc. int. contr. conf. ICC; Yang, Z.Y., & Chan, C.W. (2006). Fault detection for nonlinear systems with multiple periodic inputs. In Proc. int. contr. conf. ICC
[17] Yue, R.; Billings, S. A.; Lang, Z. Q., An investigation into the characteristics of non-linear frequency response functions, International Journal of Control, 78, 1031-1044 (2005), 1130-1149 · Zbl 1121.93030
[18] Zhou, G.; Giannakis, G. B., Harmonics in multiplicative and additive noise: performance analysis of cyclic estimators, IEEE Transactions on Signal Processing, 43, 1445-1460 (1995)
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