Improving the stability of discretization zeros with the Taylor method using a generalization of the fractional-order hold

Zeng, C. ; Liang, S. ; Zhang, Y. ; Zhong, J. ; Su, Y.

International Journal of Applied Mathematics and Computer Science, Tome 24 (2014) no. 4, pp. 745-757.

Résumé

Remarkable improvements in the stability properties of discrete system zeros may be achieved by using a new design of the fractional-order hold (FROH) circuit. This paper first analyzes asymptotic behaviors of the limiting zeros, as the sampling period T tends to zero, of the sampled-data models on the basis of the normal form representation for continuous-time systems with a new hold proposed. Further, we also give the approximate expression of limiting zeros of the resulting sampled-data system as power series with respect to a sampling period up to the third order term when the relative degree of the continuous-time system is equal to three, and the corresponding stability of the discretization zeros is discussed for fast sampling rates. Of particular interest are the stability conditions of sampling zeros in the case of a new FROH even though the relative degree of a continuous-time system is greater than two, whereas the conventional FROH fails to do so. An insightful interpretation of the obtained sampled-data model can be made in terms of minimal intersample ripple by design, where multirate sampled systems have a poor intersample behavior. Our results provide a more accurate approximation for asymptotic zeros, and certain known results on asymptotic behavior of limiting zeros are shown to be particular cases of the ideas presented here.

Détail

Keywords: stability, discretization zeros, Taylor method, signal reconstruction, sampled data model
Mots-clés : stabilność, dyskretyzacja, metoda Taylora, rekonstrukcja sygnału, model danych

Bibliographie

[1] Åström, K.J., Hagander, P. and Sternby, J. (1984). Zeros of sampled systems, Automatica 20(1): 31–38.

[2] Bàrcena, R., de la Sen, M. and Sagastabeitia, I. (2000). Improving the stability properties of the zeros of sampled systems with fractional order hold, IEE Proceedings: Control Theory and Applications 147(4): 456–464.

[3] Bàrcena, R., de la Sen, M., Sagastabeitia, I. and Collantes, J.M. (2001). Discrete control for a computer hard disk by using a fractional order hold device, IEE Proceedings: Control Theory and Applications 148(2): 117–124.

[4] Błachuta, M.J. (1998). On zeros of pulse transfer functions of systems with first-order hold, Proceedings of the 37th IEEE Conference on Decision and Control, Tampa, FL, USA, Vol. 1, pp. 307–312.

[5] Błachuta, M.J. (1999). On zeros of pulse transfer functions, IEEE Transactions on Automatic Control 44(6): 1229–1234.

[6] Błachuta, M.J. (2001). On fast sampling zeros of systems with fractional-order hold, Proceedings of the 2001 American Control Conference, Arlington, VA, USA, Vol. 4, pp. 3229–3230.

[7] Chan, J.T. (1998). On the stabilization of discrete system zeros, International Journal of Control 69(6): 789–796.

[8] Chan, J.T. (2002). Stabilization of discrete system zeros: An improved design, International Journal of Control 75(10): 759–765.

[9] Feuer, A. and Goodwin, G. (1996). Sampling in Digital Signal Processing and Control, Birkhauser, Boston, MA.

[10] Filatov, N.M., Keuchel, U. and Unbehauen, H. (1996). Dual control for an unstable mechanical plant, IEEE Control Systems Magazine 16(4): 31–37.

[11] Hagiwara, T. (1996). Analytic study on the intrinsic zeros of sampled-data system, IEEE Transactions on Automatic Control 41(2): 261–263.

[12] Hagiwara, T., Yuasa, T. and Araki, M. (1992). Limiting properties of the zeros of sampled-data systems with zero and first-order holds, Proceeding of the 31st Conference on Decision and Control, Tucson, AZ, USA, pp. 1949–1954.

[13] Hagiwara, T., Yuasa, T. and Araki, M. (1993). Stability of the limiting zeros of sampled-data systems with zero and first-order holds, International Journal of Control 58(6): 1325–1346.

[14] Hayakawa, Y., Hosoe, S. and Ito, M. (1983). On the limiting zeros of sampled multivariable systems, Systems and Control Letters 2(5): 292–300.

[15] Ishitobi, M. (1996). Stability of zeros of sampled system with fractional order hold, IEE Proceedings: Control Theory and Applications 143(2): 296–300.

[16] Ishitobi, M. (2000). A stability condition of zeros of sampled multivariable systems, IEEE Transactions on Automatic Control AC-45(2): 295–299.

[17] Ishitobi, M., Nishi, M. and Kunimatsu, S. (2013). Asymptotic properties and stability criteria of zeros of sampled-data models for decouplable MIMO systems, IEEE Transactions on Automatic Control 58(11): 2985–2990.

[18] Isidori, A. (1995). Nonlinear Control Systems: An Introduction, Springer Verlag, New York, NY.

[19] Kabamba, P.T. (1987). Control of linear systems using generalized sampled-data hold functions, IEEE Transactions on Automatic Control AC-32(7): 772–783.

[20] Kaczorek, T. (1987). Stability of periodically switched linear systems and the switching frequency, International Journal of System Science 18(4): 697–726.

[21] Kaczorek, T. (2010). Decoupling zeros of positive discrete-time linear systems, Circuits and Systems 1: 41–48.

[22] Kaczorek, T. (2013). Application of the Drazin inverse to the analysis of descriptor fractional discrete-time linear systems with regular pencils, International Journal of Applied Mathematics and Computer Science 23(1): 29–33, DOI: 10.2478/amcs-2013-0003.

[23] Karampetakis, N.P. and Karamichalis, R. (2014). Discretization of singular systems and error estimation, International Journal of Applied Mathematics and Computer Science 24(1): 65–73, DOI: 10.2478/amcs-2014-0005.

[24] Khalil, H. (2002). Nonlinear Systems, Prentice-Hall, London.

[25] Liang, S. and Ishitobi, M. (2004a). Properties of zeros of discretised system using multirate input and hold, IEE Proceedings: Control Theory and Applications 151(2): 180–184.

[26] Liang, S. and Ishitobi, M. (2004b). The stability properties of the zeros of sampled models for time delay systems in fractional order hold case, Dynamics of Continuous, Discrete and Impulsive Systems, B: Applications and Algorithms 11(3): 299–312.

[27] Liang, S., Ishitobi, M., Shi,W. and Xian, X. (2007). On stability of the limiting zeros of discrete-time MIMO systems, ACTA Automatica SINICA 33(4): 439–441, (in Chinese).

[28] Liang, S., Ishitobi, M. and Zhu, Q. (2003). Improvement of stability of zeros in discrete-time multivariable systems using fractional-order hold, International Journal of Control 76(17): 1699–1711.

[29] Liang, S., Xian, X., Ishitobi, M. and Xie, K. (2010). Stability of zeros of discrete-time multivariable systems with GSHF, International Journal of Innovative Computing, Information and Control 6(7): 2917–2926.

[30] Middleton, R. and Freudenberg, J. (1995). Non-pathological sampling for generalized sampled-data hold functions, Automatica 31(2): 315–319.

[31] Ostalczyk, P. (2012). Equivalent descriptions of a discrete-time fractional-order linear system and its stability domains, International Journal of Applied Mathematics and Computer Science 22(3): 533–538, DOI: 10.2478/v10006-012-0040-7.

[32] Passino, K.M. and Antsaklis, P.J. (1988). Inverse stable sampled low-pass systems, International Journal of Control 47(6): 1905–1913.

[33] Ruzbehani, M. (2010). A new tracking controller for discrete-time SISO non minimum phase systems, Asian Journal of Control 12(1): 89–95.

[34] Tokarzewski, J. (2009). Zeros of Linear Systems, Springer, Berlin.

[35] Ugalde, U., Bàrcena, R. and Basterretxea, K. (2012). Generalized sampled-data hold functions with asymptotic zero-order hold behavior and polynomic reconstruction, Automatica 48(6): 1171–1176.

[36] Weller, S.R. (1999). Limiting zeros of decouplable MIMO systems, IEEE Transactions on Automatic Control 44(1): 292–300.

[37] Weller, S.R., Moran, W., Ninness, B. and Pollington, A.D. (2001). Sampling zeros and the Euler–Frobenius polynomials, IEEE Transactions on Automatic Control 46(2): 340–343.

[38] Yuz, J.I., Goodwin, G.C. and Garnier, H. (2004). Generalized hold functions for fast sampling rates, 43rd IEEE Conference on Decision and Control (CDC’2004), Atlantis, The Bahamas, Vol. 46, pp. 761–765.

[39] Zeng, C., Liang, S., Li, H. and Su, Y. (2013). Current development and future challenges for zero dynamics of discrete-time systems, Control Theory Applications 30(10): 1213–1230, (in Chinese).

[40] Zhang, Y., Kostyukova, O. and Chong, K.T. (2011). A new time-discretization for delay multiple-input nonlinear systems using the Taylor method and first order hold, Discrete Applied Mathematics 159(9): 924–938.