Abstract
It is well known that the presence of unstable zeros limits the control performance, which can be achieved, and some control schemes cannot be directly applied. This manuscript investigates a sufficient condition for stabilization of zeros in the discrete-time multirate sampled systems. It is shown that the discretization zeros, especially sampling zeros, can be arbitrarily placed inside the unit circle in the case of relative degree being greater than or equal to three and multirate input and hold, such as generalized sampled-data hold function (GSHF), for the linear continuous-time systems when the sampling period T tends to zero. Moreover, the authors further consider the multivariable case, which has a similar asymptotic behavior. Finally, the simulation proves the validity of the method, and we also propose a hold design that places the sampling zeros asymptotically to the origin for fast sampling rates.
Similar content being viewed by others
References
Astrom, K.J., Hagander, P., and Sternby, J., Zeros of sampled systems, Automatica, 1984, vol. 20, pp. 31–38.
Zeng, C., Liang, S., Li, H., and Su, Y., Current development and future challenges for zero dynamics of discrete- time systems, Control Theory Appl., 2013, vol. 30, no. 10, pp. 1213–1230.
Hagiwara, T. and Araki, M., Properties of limiting zeros of sampled systems, Trans. Inst. Electr. Eng. Jpn., 1990, vol. 110-C, no. 4, pp. 235–244.
Hagiwara, T., Analytic study on the intrinsic zeros of sampled-data system, IEEE Trans. Autom. Control, 1996, vol. 41, pp. 263–783.
Hagiwara, T., Yuasa, T., and Araki, M., Stability of the limiting zeros of sampled-data systems with zero- and first-order holds, Int. J. Control, 1993, vol. 58, pp. 1325–1346.
Weller, S.R., Moran, W., Ninness, B., and Pollington, A.D., Sampling zeros and the Euler-Frobenius polynomials, IEEE Trans. Autom. Control, 2001, vol. 46, pp. 340–343.
Blachuta, M.J., On zeros of pulse transfer functions, IEEE Trans. Autom. Control, 1999, vol. 44, pp. 1229–1234.
Zeng, C., Liang, S., and Li, H., The properties of zero dynamics for nonlinear discretized systems with time delay via Taylor method, Nonlinear Dyn., 2015, vol. 79, no. 1, pp. 11–25.
Weller, S.R., Limiting zeros of decouplable MIMO systems, IEEE Trans. Autom. Control, 1999, vol. 44, pp. 292–300.
Ishitobi, M., Nishi, M., and Kunimatsu, S., Asymptotic properties and stability criteria of zeros of sampled-data models for decouplable MIMO systems, IEEE Trans. Autom. Control, 2013, vol. 58, no. 11, pp. 2985–2990.
Zeng, C., Liang, S., Zhong, J., and Su, Y., Improvement of the asymptotic properties of zero dynamics for sampled- data systems in the case of a time delay, Abstr. Appl. Anal., 2014, vol. 2014, pp. 1–12.
Passino, K.M. and Antsaklis, P.J., Inverse stable sampled low-pass systems, Int. J. Control, 1988, vol. 47, pp. 1905–1913.
Ishitobi, M., Stability of zeros of sampled system with fractional order hold, IEEE Proc. Control Theory Appl., 1996, vol. 143, pp. 296–300.
Barcena, R., de la Sen, M., and Sagastabeitia, I., Improving the stability properties of the zeros of sampled systems with fractional order hold, IEEE Proc. Control Theory Appl., 2000, vol. 147, no. 4, pp. 456–464.
Barcena, R., de la Sen, M., Sagastabeitia, I., and Collantes, J.M., Discrete control for a computer hard disk by using a fractional order hold device, IEEE Proc. Control Theory Appl., 2001, vol. 148, no. 2, pp. 117–124.
Liang, S., Ishitobi, M., and Zhu, Q., Improvement of stability of zeros in discrete-time multivariable systems using fractional-order hold, Int. J. Control, 2003, vol. 76, pp. 1699–1711.
Zeng, C., Liang, S., Zhang, Y., Zhong, J., and Su, Y., Improving the stability of discretization zeros with the Taylor method using a generalization of the fractional-order hold, Int. J. Appl. Math. Comput. Sci., 2014, vol. 24, no. 4, pp. 745–757.
Zeng, C., Liang, S., and Su, Y., Improving the asymptotic properties of discrete system zeros in fractional-order hold case, J. Appl. Math., 2013, vol. 2013, pp. 1–17.
Zeng, C. and Liang, S., Zeros dynamics of sampled-data models for nonlinear multivariable systems in fractional- order hold caseI, Appl. Math. Comput., 2014, vol. 246, no. 11, pp. 88–102.
B’arcena, R. and de la Sen, M., On the sufficiently small sampling period for the convenient turning of fractional order hold circuits, IEEE Proc. Control Theory Appl., 2003, vol. 150, no. 2, pp. 183–188.
Chan, J.T., On the stabilization of discrete system zeros, Int. J. Control, 1998, vol. 69, no. 6, pp. 789–796.
Kabamba, P.T., Control of linear systems using generalized sampled data hold functions, IEEE Trans. Autom. Control, 1987, vol. AC-32, no. 7, pp. 772–783.
Chan, J.T., Stabilization of discrete system zeros: A improved design, Int. J. Control, 2002, vol. 75, no. 10, pp. 759–765.
Liang, S. and Ishitobi, M., Properties of zeros of discretised system using multirate input and hold, IEEE Proc. Control Theory Appl., 2004, vol. 151, pp. 180–184.
Yuz, J.I., Goodwin, G.C., and Garnier, H., Generalized hold functions for fast sampling rates, 43rd IEEE Conference on Decision and Control (CDC’2004), Atlantis, Bahamas, 2004, pp. 761–765.
Ugalde, U., B`arcena, R., and Basterretxea, K., Generalized sampled-data hold functions with asymptotic zeroorder hold behavior and polynomic reconstruction, Automatica, 2012, vol. 48, no. 6, pp. 1171–1176.
Liang, S., Xian, X., Ishitobi, M., and Xie, K., Stability of zeros of discrete time multivariable systems with GSHF, Int. J. Innovative Comput. Inf. Control, 2010, vol. 6, pp. 2917–2926.
Author information
Authors and Affiliations
Corresponding author
Additional information
The article is published in the original.
About this article
Cite this article
Zeng, C., Su, Y. & Liang, S. A sufficient and necessary condition for stabilization of zeros in discrete-time multirate sampled systems. Aut. Control Comp. Sci. 51, 42–49 (2017). https://doi.org/10.3103/S0146411617010084
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.3103/S0146411617010084