Improvement of the asymptotic properties of zero dynamics for sampled-data systems in the case of a time delay. (English) Zbl 1406.93201

Summary: It is well known that the existence of unstable zero dynamics is recognized as a major barrier in many control systems, and deeply limits the achievable control performance. When a continuous-time system with relative degree greater than or equal to three is discretized using a zero-order hold (ZOH), at least one of the zero dynamics of the resulting sampled-data model is obviously unstable for sufficiently small sampling periods, irrespective of whether they involve time delay or not. Thus, attention is here focused on continuous-time systems with time delay and relative degree two. This paper analyzes the asymptotic behavior of zero dynamics for the sampled-data models corresponding to the continuous-time systems mentioned above, and further gives an approximate expression of the zero dynamics in the form of a power series expansion up to the third order term of sampling period. Meanwhile, the stability of the zero dynamics is discussed for sufficiently small sampling periods and a new stability condition is also derived. The ideas presented here generalize well-known results from the delay-free control system to time-delay case.


93C57 Sampled-data control/observation systems
93D05 Lyapunov and other classical stabilities (Lagrange, Poisson, \(L^p, l^p\), etc.) in control theory
93C55 Discrete-time control/observation systems


Full Text: DOI


[1] Åström, K. J.; Hagander, P.; Sternby, J., Zeros of sampled systems, Automatica, 20, 1, 31-38 (1984) · Zbl 0542.93047
[2] Zeng, C.; Liang, S.; Li, H.; Su, Y., Current development and future challenges for zero dynamics of discrete-time systems, Control Theory & Applications, 30, 10, 1213-1230 (2013) · Zbl 1299.93178
[3] Hagiwara, T.; Yuasa, T.; Araki, M., Stability of the limiting zeros of sampled-data systems with zero- and first-order holds, International Journal of Control, 58, 6, 1325-1346 (1993) · Zbl 0787.93067
[4] Hagiwara, T., Analytic study on the intrinsic zeros of sampled-data systems, IEEE Transactions on Automatic Control, 41, 2, 261-263 (1996) · Zbl 0850.93481
[5] Blachut, M. J., On zeros of pulse transfer functions, IEEE Transactions on Automatic Control, 44, 6, 1229-1234 (1999) · Zbl 0955.93028
[6] Weller, S. R., Limiting zeros of decouplable MIMO systems, IEEE Transactions on Automatic Control, 44, 1, 129-134 (1999) · Zbl 1056.93558
[7] Ishitobi, M., A stability condition of zeros of sampled multivariable systems, IEEE Transactions on Automatic Control, 45, 2, 295-299 (2000) · Zbl 0971.93049
[8] Isidori, A., The zero dynamics of a nonlinear system: from the origin to the latest progresses of a long successful story, European Journal of Control, 19, 5, 369-378 (2013) · Zbl 1293.93666
[9] Liang, S.; Ishitobi, M., Properties of zeros of discretised system using multirate input and hold, IEE Proceedings: Control Theory and Applications, 151, 2, 180-184 (2004)
[10] Liang, S.; Ishitobi, M.; Zhu, Q., Improvement of stability of zeros in discrete-time multivariable systems using fractional-order hold, International Journal of Control, 76, 17, 1699-1711 (2003) · Zbl 1047.93033
[11] Yuz, J. I.; Goodwin, G. C., On sampled-data models for nonlinear systems, IEEE Transactions on Automatic Control, 50, 10, 1477-1489 (2005) · Zbl 1365.93283
[12] Liang, S.; Xian, X.; Ishitobi, M.; Xie, K., Stability of zeros of discrete-time multivariable systems with gshf, International Journal of Innovative Computing, Information and Control, 6, 7, 2917-2926 (2010)
[13] Liang, S.; Ishitobi, M.; Shi, W. R.; Xian, X. D., On stability of the limiting zeros of discrete-time MIMO systems, Acta Automatica Sinica, 33, 4, 439-442 (2007)
[14] Ugalde, U.; Bárcena, R.; Basterretxea, K., Generalized sampled-data hold functions with asymptotic zero-order hold behavior and polynomic reconstruction, Automatica, 48, 6, 1171-1176 (2012) · Zbl 1244.93096
[15] Ishitobi, M.; Koga, T.; Nishi, M.; Kunimatsu, S., Asymptotic properties of zeros of sampled-data systems, Proceedings of the 49th IEEE Conference on Decision and Control (CDC ’10)
[16] Hara, S., Properties of zeros in digital control systems with computational time delay, International Journal of Control, 49, 2, 493-511 (1989) · Zbl 0679.93034
[17] Ishitobi, M., Stable zeros of a discrete system obtained by sampling a continuous-time plant with a time delay, International Journal of Control, 59, 4, 1053-1062 (1994) · Zbl 0813.93051
[18] Liang, S.; Ishitobi, M., The stability properties of the zeros of sampled models for time delay systems in fractional order hold case, Dynamics of Continuous, Discrete & Impulsive Systems B: Applications & Algorithms, 11, 3, 299-312 (2004) · Zbl 1061.93062
[19] Astrom, K. J.; Wittenmark, B., Computer Controlled Systems: Theory and Design (1990), Englewood Cliffs, NJ, USA: Prentice-Hall, Englewood Cliffs, NJ, USA
[20] Franklin, G. F.; Powell, J. D.; Workman, M. L., Digital Control of Dynamic Systems (1998), New York, NY, USA: Addison-Wesley, New York, NY, USA
[21] Vaccaro, R. J., Digital Control (1995), New York, NY, USA: McGraw-Hill, New York, NY, USA
[22] Khalil, H., Nonlinear Systems (2002), Upper Saddle River, NJ, USA: Prentice Hall, Upper Saddle River, NJ, USA
[23] Isidori, A., Nonlinear Control Systems. Nonlinear Control Systems, Berlin, Germany (1995), Springer · Zbl 0878.93001
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.