The paper addresses the problem of the transformation of nonlinear discrete-time systems, described by implicit higherorder difference equations, into the strong row-reduced form. The motivating example illustrates the phenomenon that sometimes equations in the row-reduced form may contain higher-order shifts of output variables than the corresponding row degrees. This means that, in general, linear transformations of equations are not enough for transforming equations into the strong row-reduced form. Therefore, in this paper we study the possibility of using local nonlinear transformations to reduce the order of a system. A constructive (up to the solution of a system of partial differential equations) step-by-step algorithm is provided. It is followed by several illustrative examples.
1. Beckermann, B., Cheng, H., and Labahn, G. Fraction-free row reduction of matrices of Ore polynomials. J. Symb. Comput., 2006, 41, 513–543.
2. Davies, P., Cheng, H., and Labahn, G. Computing Popov form of general Ore polynomial matrices. In Milestones in Computer Algebra, Stonehaven Bay, Trinidad and Tobago. 2008, 149–156.
3. Kotta, Ü. Towards solution of the state-space realization problem of a set of multi-input multi-output nonlinear difference equations. In European Control Conference. Karlsruhe, Germany, August–September 1999, 1–6.
4. Kotta, Ü. and Tõnso, M. Realization of discrete-time nonlinear input-output equations: Polynomial approach. Automatica, 2012, 48, 255–262.
5. Kotta, Ü., Bartosiewicz, Z., Nõmm, S., and Pawłuszewicz, E. Linear input-output equivalence and row reducedness of discrete-time nonlinear systems. IEEE Trans. Autom. Contr., 2011, 56, 1421–1426.
6. Middeke, J. A Computational View on Normal Forms of Matrices of Ore Polynomials. PhD thesis, Research Institute for Symbolic Computation, Johannes Kepler University, Linz, Austria, 2011.
7. Popov, V. M. Some properties of the control systems with irreducible matrix – Transfer functions. In Seminar on Differential Equations and Dynamical Systems (Yorke, J. A., ed.), Lecture Notes Math., 1970, 144, 169–180.
8. Van der Schaft, A. J. Correction. On realization of nonlinear systems described by higher-order differential equations. Math. Syst. Theory, 1987, 20, 305–306.
9. Van der Schaft, A. J. On realization of nonlinear systems described by higher-order differential equations. Math. Syst. Theory, 1987, 19, 239–275.
10. Van der Schaft, A. J. Transformations of nonlinear systems under external equivalence. In New Trends in Nonlinear Control Theory (Descusse, Y., Fliess, M., Isidori, A., and Leborgne, D., eds), Lecture Notes Control Inform. Sci., 1989, 122, 33–43.