Stability and limit oscillations of a control event-based sampling criterion. (English) Zbl 1244.93131

de La Sen, M.; Soto, J. C.; Ibeas, A.

J. Appl. Math. 2012, Article ID 684292, 25 p. (2012).

Summary: This paper investigates the presence of limit oscillations in an adaptive sampling system. The basic sampling criterion operates in the sense that each next sampling occurs when the absolute difference of the signal amplitude with respect to its currently sampled signal equalizes a prescribed threshold amplitude. The sampling criterion is extended involving a prescribed set of amplitudes. The limit oscillations might be interpreted through the equivalence of the adaptive sampling and hold device with a nonlinear one consisting of a relay with multiple hysteresis whose parameterization is, in general, dependent on the initial conditions of the dynamic system. The performed study is performed on the time domain.

MSC:

93D05	Lyapunov and other classical stabilities (Lagrange, Poisson, \(L^p, l^p\), etc.) in control theory
34C25	Periodic solutions to ordinary differential equations
94A20	Sampling theory in information and communication theory

References:

[1]	J. C. Soto and M. De La Sen, “On the derivation and analysis of a non-linear model for describing a class of adaptive sampling laws,” International Journal of Control, vol. 42, no. 6, pp. 1347-1368, 1985. · Zbl 0614.93037
[2]	M. de la Sen, “A method for improving the adaptation transient using adaptive sampling,” International Journal of Control, vol. 40, no. 4, pp. 639-665, 1984. · Zbl 0546.93048
[3]	M. de la Sen, “Adaptive sampling for improving the adaptation transients in hybrid adaptive control,” International Journal of Control, vol. 41, no. 5, pp. 1189-1205, 1985. · Zbl 0565.93043
[4]	M. De La Sen, “A new modelling for aperiodic sampling systems,” International Journal of Systems Science, vol. 15, no. 3, pp. 315-328, 1984. · Zbl 0542.93048
[5]	G. Ge and W. Zhiwen, “Stability of control systems with time-varying sampling,” in Proceedings of the 6th World Congress on Intelligent Control and Automation (WCICA ’06), pp. 651-656, June 2006.
[6]	S. Luo, “Error estimation for non-uniform sampling in shift invariant space,” Applicable Analysis, vol. 86, no. 4, pp. 483-496, 2007. · Zbl 1135.94003
[7]	W. R. Dieter, S. Datta, and W. K. Kai, “Power reduction by varying sampling rate,” in Proceedings of the International Symposium on Low Power Electronics and Design, pp. 227-232, August 2005.
[8]	G. M. Andrew, “Control and guidance systems with automatic aperiodic sampling,” Journal of Spacecraft and Rockets, vol. 12, no. 1, p. 59, 1975.
[9]	M. de la Sen and A. Gallego, “Adaptive control for nonperiodic sampling using bilinear models,” International Journal of Systems Science, vol. 22, no. 8, pp. 1403-1418, 1991. · Zbl 0737.93037
[10]	M. J. Chambers, “Testing for unit roots with flow data and varying sampling frequency,” Journal of Econometrics, vol. 144, no. 2, pp. 524-525, 2008. · Zbl 1418.62292
[11]	T. Shibata, T. Bando, and S. Ishii, “Visual tracking achieved by adaptive sampling from hierarchical and parallel predictions,” in Proceedings of the 14th International Conference on Neural Information Processing (ICONIP ’07), vol. 4984 of Neural Information Processing, Part I Book Series: Lecture Notes in Computer Science, pp. 604-613, Kitakyushu, Japan, 2008. · Zbl 05298178
[12]	M. H. F. Zarandi, A. Alaeddini, and I. B. Turksen, “A hybrid fuzzy adaptive sampling-run rules for Shewhart control charts,” Information Sciences, vol. 178, no. 4, pp. 1152-1170, 2008. · Zbl 05248533
[13]	M. De la Sen, “On the properties of reachability, observability, controllability, and constructibility of discrete-time positive time-invariant linear systems with aperiodic choice of the sampling instants,” Discrete Dynamics in Nature and Society, vol. 2007, Article ID 84913, 23 pages, 2007. · Zbl 1179.93126
[14]	M. de la Sen, “Application of the nonperiodic sampling to the identifiability and model matching problems in dynamic systems,” International Journal of Systems Science, vol. 14, no. 4, pp. 367-383, 1983. · Zbl 0511.93023
[15]	M. Delasen, J. M. Sandoval, and Yu. Kryukov, “An adaptive sampling approach to the problem of learning and control in manipulator systems,” Revista de Informatica y Automatica, vol. 22, no. 4, pp. 31-44, 1989.
[16]	M. de la Sen, “Non-periodic and adaptive sampling. A tutorial review,” Lithuanian Academy of Sciences. Informatica, vol. 7, no. 2, pp. 175-228, 1996. · Zbl 0910.93055
[17]	M. De la Sen, J. J. Miñambres, A. J. Garrido, A. Almansa, and J. C. Soto, “Basic theoretical results for expert systems. Application to the supervision of adaptation transients in planar robots,” Artificial Intelligence, vol. 152, no. 2, pp. 173-211, 2004. · Zbl 1082.68831
[18]	J. C. Soto and M. De La Sen, “Nonlinear oscillations in nonperiodic sampling systems,” Electronics Letters, vol. 20, no. 20, pp. 816-818, 1984.
[19]	C. Kadilar, Y. Unyazici, and H. Cingi, “Ratio estimator for the population mean using ranked set sampling,” Statistical Papers, vol. 50, no. 2, pp. 301-309, 2009. · Zbl 1309.62023
[20]	C. Kadilar and H. Cingi, “Improvement in variance estimation in simple random sampling,” Communications in Statistics. Theory and Methods, vol. 36, no. 9-12, pp. 2075-2081, 2007. · Zbl 1124.62001
[21]	C. Kadilar and H. Cingi, “Ratio estimators for the population variance in simple and stratified random sampling,” Applied Mathematics and Computation, vol. 173, no. 2, pp. 1047-1059, 2006. · Zbl 1086.62005
[22]	C. Kadilar and H. Cingi, “Improvement in estimating the population mean in simple random sampling,” Applied Mathematics Letters, vol. 19, no. 1, pp. 75-79, 2006. · Zbl 1125.62002
[23]	M. Miskowicz, “Asymptotic effectiveness of the event-based sampling according to the integral criterion,” Sensors, vol. 7, no. 1, pp. 16-37, 2007.
[24]	M. Miskowicz, “Send-on-delta concept: an event-based data reporting strategy,” Sensors, vol. 6, no. 1, pp. 49-63, 2006.
[25]	E. Oral and C. Kadilar, “Improved ratio estimators via modified maximum likelihood,” Pakistan Journal of Statistics, vol. 27, no. 3, pp. 269-282, 2011. · Zbl 1243.62011
[26]	A. J. Garrido, M. De la Sen, and R. Bárcena, “Semi-heuristically obtained discrete models for LTI Systems under real sampling with choice of the hold device,” in Proceedings of the American Control Conference, vol. 1-6, pp. 71-76, June 2003.
[27]	M. De la Sen, R. Bárcena, and A. J. Garrido, “On the intrinsic limiting zeros as the sampling period tends to zero,” IEEE Transactions on Circuits and Systems. I. Regular Papers, vol. 48, no. 7, pp. 898-900, 2001. · Zbl 1009.93051
[28]	M. Mellado, S. Dormido, and J. M. Guillen, “Introduction to control of fixed interval systems,” Anales de Fisica, vol. 66, no. 1-2, p. 33, 1970.
[29]	J. Sánchez, M. A. Guarnes, and S. Dormido, “On the application of different event-based sampling strategies to the control of a zsimple industrial process,” Sensors, vol. 9, no. 9, pp. 6795-6818, 2009.
[30]	S. Dormido, M. Mellado, J. Ruiz, and J. M. Guillen, “Sistemas de muestreo adaptivo mediante un criterio de diferencia de amplitudes constante,” Revista de Informatica y Automatica, vol. 16, pp. 13-17, 1973.
[31]	S. Dormido and M. Mellado, “Determinacion de ciclos limite en sistemas de muestreo adaptivo,” Revista de Informatica y Automatica, vol. 26, no. 4, pp. 21-33, 1975.
[32]	M. B. Paz, M. de la Sen, S. Dormido, and M. Mellado, “Compensation of discrete Systems to variations in their parameters by changing sampling period,” Electronics Letters, vol. 18, no. 10, pp. 404-406, 1982.
[33]	K. J. Åström and B. Bernhardsson, “Systems with Lebesgue sampling,” in Directions in Mathematical Systems Theory and Optimization, A. Rantzer and C. I. Byrnes, Eds., vol. 286 of Lecture Notes in Control and Information Sciences, pp. 1-13, Springer, Berlin, Germany, 2003. · Zbl 1047.93031
[34]	E. Kofman and J. H. Braslavsky, “Level crossing sampling in feedback stabilization under data-rate constraints,” in Proceedings of the 45th IEEE Conference on Decision and Control (CDC ’06), vol. 1-14, pp. 4423-4428, December 2006.
[35]	M. De la Sen, “Sufficiency-type stability and stabilization criteria for linear time-invariant systems with constant point delays,” Acta Applicandae Mathematicae, vol. 83, no. 3, pp. 235-256, 2004. · Zbl 1067.34078
[36]	D. P. Atherton, Nonlinear Control Engineering, Van Nostrand Reinhold, London, UK, 1975. · Zbl 0322.49019