Abstract
It is well known that the existence of unstable zero dynamics is recognized as a major barrier in many control systems. When the usual digital control with zero-order hold (ZOH) or fractional-order hold (FROH) input is used, unstable zero dynamics inevitably appear in the discrete-time model even though the continuous-time system with relative degree more than two is of minimum phase. This paper investigates the zero dynamics, as the sampling period tends to zero, of sampled-data models composed of a generalized sample hold function (GSHF), a continuous-time nonlinear plant and a sampler in cascade. More precisely, we show how an approximate sampled-data model can be obtained for nonlinear systems with two special GSHF cases such that sampled zero dynamics of the resulting model can be arbitrarily placed. Further, two GSHFs with appropriate parameters provide nonlinear zero dynamics as stable as possible, or with improved stability properties even when unstable, for a given continuous-time plant. It is also shown that the intersample behavior arising from the multirate input can be localised by appropriately selecting the design parameters based on the stability condition of the zero dynamics. The results presented here generalize well-known ideas from the linear to nonlinear cases.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
D. Nešic and A. Teel, Perspectives in Robust control, ch14: Sampled-data control of nonlinear systems: an overview of recent results, Springer-Verlag, 2001.
D. Nešic and A. Teel, “A framework for stabilization of nonlinear sampled-data systems based on their approximate discrete-time models,” IEEE Trans. on Automatic Control, vol. 49, no. 7, pp. 1103–1122, 2004.
J. I. Yuz and G. C. Goodwin, “On sampleddatamodels for nonlinear systems,” IEEE Trans. on Automatic Control, vol. 50, no. 10, pp. 1447–1489, 2005.
K. J. Åström, P. Hagander, and J. Sternby, “Zeros of sampled systems,” Automatica, vol. 20, no. 1, pp. 31–38, 1984.
S. Monaco and D. Normand-Cyrot, “Zero dynamics sampled nonlinear systems,” System Control Letter, vol. 11, no. 3, pp. 229–234, 1988.
M. Ishitobi and M. Nishi, “Zero dynamics of sampled-data models for nonlinear systems,” Proc. of the American Control Conference, Washington, USA, pp. 1184–1189, June 2008.
M. Ishitobi, M. Nishi, and S. Kunimatsu, “Stability of zero dynamics of sampled-data nonlinear systems,” Proc. of the 17th IFAC World Congress, Seoul, Korea, pp. 5969–5973, July 2008.
M. Nishi and M. Ishitobi, “Sampled-data models and zero dynamics for nonlinear systems,” Proc. of ICROS-SICE International Joint Conference, Fukuoka, Japan, pp. 2448-2453, August 2009.
M. Ishitobi and M. Nishi, “sampled-data models for nonlinear systems with a fractional-order hold,” Proc. of 18th IEEE International Conference on Control Applications, pp. 153–158, July 2009.
M. Nishi and M. Ishitobi, “Sampled-data models for affine nonlinear systems using a fractional order hold and their zero dynamics,” Artif Life Robotics, vol. 15, no. 4, pp. 500–503, 2010.
K. J. Åström and B. Wittenmark, Adaptive Control, Addison-Wesley, 1989.
K. J. Åström and B. Wittenmark, Computer Controlled Systems: Theory and Design, 2nd ed., Prentice-Hall, Englewood Cliffs, New Jersey, 1990.
A. Isidori, Nonlinear Control Systems: An Introduction, Springer Verlag, 1995.
H. Khalil, Nonlinear Systems, Prentice-Hall, 2002.
P. T. Kabamba, “Control of linear systems using generalized sampled-data hold functions,” IEEE Trans. on Automatic Control, vol. AC-32, no. 7, pp. 772–783, Sep. 1987.
J. T. Chan, “On the stabilization of discrete system zeros,” International Journal of Control, vol. 69, no. 6, pp. 789–796, 1998.
J. T. Chan, “Stabilization of discrete system zeros: A improved design,” International Journal of Control, vol. 75, no. 10, pp. 759–765, 2002.
S. Liang and M. Ishitobi, “Properties of zeros of discretised system using multirate input and hold,” IEE Proceedings-Control Theory and Applications, vol. 151, no. 2, pp. 180–184, 2004.
J. I. Yuz, G. C. Goodwin, and H. Garnier, “Generalized hold functions for fast sampling rates,” Proc. of 43rd IEEE Conference on Decision and Control, Atlantis, Bahamas, pp. 761–765, 2004.
A. G. Aghdam and E. J. Davison, “An optimization algorithm for decentralized digital control of continuous-time systems which accounts for intersample ripple,” Proc. of the IFAC ACC, pp. 4273–4278, 2004.
A. G. Aghdam, E. J. Davison, and R. Becerril-Arreola, “Structural modification of systems using descretization and generalized sampled-data hold functions,” Automatica, vol. 42, no. 11, pp. 1935–1941, 2006.
S. Liang, X. Xian, M. Ishitobi, and K. Xie, “stability of zeros of discrete-time multivariable systems with GSHF,” International Journal of Innovative Computing, Information and Control, vol. 6, no. 7, pp. 2917–2926, 2010.
U. Ugalde, R. Bàrcena, and K. Basterretxea, “Generalized sampled-data hold functions with asymptotic zero-order hold behavior and polynomic reconstruction,” Automatica, vol. 48, no. 6, pp. 1171–1176, 2012.
A. Feuer and G. Goodwin, Sampling in Digital Signal Processing and Control, Birkhauser, Boston, 1996.
K. L. Moore, S. P. Bhattacharyya, and M. Dahleh, “Capabilities and limitations of multirate control schemes,” Automatica, vol. 29, no. 4, pp. 941–951, 1993.
J. Butcher, The Numerical Analysis of Ordinary Differential Equations, Wiley, New York, 1987.
M. Arcak and D. Nešic, “A framework for nonlinear sampled-data observer design via approximate discrete-time models and emulation,” Automatica, vol. 40, no. 11, pp. 1931–1938, 2004.
D. Nešic and A. Teel, “Stabilization of sampleddata nonlinear systems via backstepping on their Euler approximate model,” Automatica, vol. 42, no. 10, pp. 1801–1808, 2006.
E. Gyurkovics and A. Elaiw, “Stabilization of sampled-data nonlinear systems by receding horizon control via discrete-time approximations,” Automatica, vol. 40, no. 12, pp. 2017–2028, 2004.
S. Liang, Studies on Stability of Zeros of Discretetime Control System, Ph.D. dissertation, Kumamoto University, 2004.
Author information
Authors and Affiliations
Corresponding author
Additional information
Recommended by Associate Editor Young Soo Suh under the direction of Editor Ju Hyun Park.
This work is supported by the National Basic Research Program of China (“973” Grant No. 2013CB328903), the National Natural Science Foundation of China (61403055), the Joint Funds of the Natural Science Foundation Project of Guizhou (Grant No. LH[2014]7364), the Research Project of Chongqing Science & Technology Commission (cstc2014jcyjA40005). The authors also gratefully acknowledge the helpful comments and suggestions of the reviewers, which have improved the presentation.
Cheng Zeng received his Ph.D. degree from the control theory and control engineering at the University of Chongqing, China. Since 2014, he has been working in the College of Science of Guizhou Institute of Technology, China. His research areas include sampling system, nonlinear system and adaptive control.
Shan Liang obtained a Ph.D. degree from the Department of Mechanical Systems Engineering of Kumamoto University, Japan, in 2004. He received the professional title at Chongqing University, China. Since 2000, his research interest covers nonlinear system, adaptive control and sensor network.
Rights and permissions
About this article
Cite this article
Zeng, C., Liang, S. Comparative study of discretization zero dynamics behaviors in two multirate cases. Int. J. Control Autom. Syst. 13, 831–842 (2015). https://doi.org/10.1007/s12555-014-0115-3
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12555-014-0115-3