RELATIVE ORDER OF SAMPLED LINEAR TIME-INVARIANT SYSTEMS

Bondarko V.A. Relative order of sampled linear time-invariant systems. Scientific and Technical Journal of Information Technologies, Mechanics and Optics, 2019, vol. 19, no. 2,  pp. 339–346 (in Russian). doi: 10.17586/2226-1494-2019-19-2-339-346

The paper considers the rank behavior of the matrix leading coefficient of transfer function numerator of a sampled multidimensional linear time-invariant system when the sampling period tends to zero. Zero order hold sampling method is used. It is shown that the rank of the coefficient under consideration for all sufficiently small values of the sampling period is maximal if the natural condition of nonsingularity of the initial continuous prototype system is satisfied. In particular, if the dimensions of the system input and output coincide, then sampled system with a nonsingular leading coefficient is generated with any sufficiently small sampling period. This feature plays an important role in solving many problems of control theory. For example, the classical criterion of decouplability of linear systems requires the nonsingularity of the system interactor, and for systems with a nonsingular leading coefficient, this condition is satisfied automatically. In the following,while the synthesis of minimax regulators for general form discrete systems, the artificial reduction of the system to the form with the maximal rank of the leading coefficient is one of the first steps in an optimal regulator design.  The results of this work have proved that this step is superfluous if a discrete system under control is a result of sampling of a nonsingular system, as is often the case. The results of the work are illustrated by the example of an induction motor discrete model.

1. Krylov V.I., Bobkov V.V., Monastyrnyi P.I. Computational Methods. Vol. 2. Moscow, Nauka Publ., 1977, 400 p.
   (in Russian)
2. Rabiner, L.R., Gold B. Theory and Application of Digital Signal Processing. Englewood Cliffs,Prentice Hall, 1975, 762 p.
3. Stotent D.P., Harrison A.J.L. Generation of discrete and continuous time transfer function coefficients. International Journal of Control,1994,vol. 59,no. 5, pp. 1159–1172. doi: 10.1080/00207179408923125
4. Åström K.J., Hagander P., Sternby J. Zeros of sampled systems. Automatica, 1984, vol. 20, no. 1, pp. 21–38. doi: 10.1016/0005-1098(84)90062-1
5. Bondarko V.A. Discretization of continuous linear dynamic systems – analysis of the methods. Systems and Control Letters, 1984, vol. 5, pp. 97–101. doi: 10.1016/0167-6911(84)90016-1
6. Weller S.R., Moran W., Ninness B., Pollington A.D. Sampling zeros and the Euler–Frobenius polynomials. IEEE Transactions on Automatic Control, 2001, vol. 46, no. 2, pp. 340–343. doi: 10.1109/cdc.1997.657672
7. 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, 1993, vol. 58, no. 6,
   pp. 1325–1346. doi: 10.1080/00207179308923057
8. Błachuta M.J. On zeros of pulse transfer functions. IEEE Transactions on Automatic Control, 1999, vol. 44, no. 6, pp. 1229–1234. doi: 10.1109/9.769380
9. Weller S.R. Limiting zeros of decouplable MIMO systems. IEEE Transactions on Automatic Control, 1999, vol. 44, no. 1, pp. 129–134. doi: 10.1109/9.739097
10. Bondarko V.A. Limiting zeros of sampled systems with time delay. Automation and Remote Control,2015,vol. 76, no. 8, pp. 1327–1346.
11. Barabanov A.E., Granichin O.N. Optimal controller for linear plant with bounded noise. Automation and Remote Control, 1984, no. 5, pp. 39–46. (in Russian)
12. Barabanov A.E. Synthesis of Minimax Regulators. St. Petersburg, SPbSU Publ., 1996, 222 p. (in Russian)
13. Isidori A. Nonlinear Control Systems. 2^nd ed. Berlin, Heidelberg: Springer-Verlag, 1995, 549 p. doi: 10.1007/978-1-84628-615-5
14. Bondarko V.A. Adaptive vector control of an induction motor on the basis of the method of recurrent objective inequalities. Automation and Remote Control, 2010, vol. 71, no. 9,
    pp. 1849–1863. doi: 10.1134/S0005117910090080
15. Falb P.L., Wolovich W.A. Decoupling in the design and synthesis of multivariable control systems. IEEE Transactions on Automatic Control, 1967, vol. AC-12, no. 6, pp. 651–659. doi: 10.1109/tac.1967.1098737
16. Bondarko V.A. Discretization of infinite-dimensional linear dynamical systems. Differential Equations, 1996, vol. 32, no. 10, pp. 1309–1318.

This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License