On Hyper–Regularity and Unimodularity of Ore Polynomial Matrices

Fritzsche, Klemens; Rӧbenack, Klaus

doi:10.2478/amcs-2018-0045

Abstract

We investigate Ore polynomial matrices, i. e., matrices with polynomial entries in d/dt whose coefficients are meromorphic functions in t and as such constitute a non-commutative ring. In particular, we study the properties of hyper-regularity and unimodularity of such matrices and derive conditions which make it possible to efficiently check for these characteristics. In addition, this approach enables computation of hyper-regular left and right and unimodular inverses.

References

Antritter, F., Cazaurang, F., Lévine, J. and Middeke, J. (2014). On the computation of π-flat outputs for linear time-varying differential-delay systems, Systems & Control Letters 71: 14-22.10.1016/j.sysconle.2014.07.002
Search in Google Scholar Back to article
Antritter, F. and Middeke, J. (2011). A toolbox for the analysis of linear systems with delays, Proceedings of CDC-ECC, Orlando, FL, USA, pp. 1950-1955.10.1109/CDC.2011.6160636
Search in Google Scholar Back to article
Beckermann, B., Cheng, H. and Labahn, G. (2006). Fraction-free row reduction of matrices of Ore polynomials, Journal of Symbolic Computation 41(5): 513-543.10.1016/j.jsc.2005.10.002
Search in Google Scholar Back to article
Ben-Israel, A. and Greville, T.N. (2003). Generalized Inverses: Theory and Applications, Springer, New York, NY.
Search in Google Scholar Back to article
Bose, N.K. and Mitra, S.K. (1978). Generalized inverse of polynomial matrices, IEEE Transactions on Automatic Control 23(3): 491-493.10.1109/TAC.1978.1101751
Search in Google Scholar Back to article
Boullion, T.L. and Odell, P.L. (1971). Generalized Inverse Matrices, Wiley, New York, NY.
Search in Google Scholar Back to article
Campbell, S.L. and Meyer, C.D. (2008). Generalized Inverses of Linear Transformations, SIAM, London.10.1137/1.9780898719048
Search in Google Scholar Back to article
Cluzeau, T. and Quadrat, A. (2013). Isomorphisms and Serre’s reduction of linear systems, Proceedings of the 8th International Workshop on Multidimensional Systems, Erlangen, Germany, pp. 11-16.
Search in Google Scholar Back to article
Cohn, P. (1985). Free Rings and Their Relations, Academic Press, London.
Search in Google Scholar Back to article
Davies, P., Cheng, H. and Labahn, G. (2008). Computing Popov form of general Ore polynomial matrices, Proceedings of the Conference on Milestones in Computer Algebra (MICA), Stonehaven Bay, Trinidad and Tobago, pp. 149-156.
Search in Google Scholar Back to article
Fabianska, A. and Quadrat, A. (2007). Applications of the Quillen-Suslin theorem to multidimensional systems theory, in H. Park and G. Regensburger (Eds.), Grӧbner Bases in Control Theory and Signal Processing, De Gruyter, Berlin/New York, NY, pp. 23-106.10.1515/9783110909746.23
Search in Google Scholar Back to article
Franke, M. and Rӧbenack, K. (2013). On the computation of flat outputs for nonlinear control systems, Proceedings of the European Control Conference (ECC), Z¨urich, Switzerland, pp. 167-172.10.23919/ECC.2013.6669771
Search in Google Scholar Back to article
Fritzsche, K. (2018). Toolbox for checking hyper regularity and unimodularity of polynomial matrices in the differential operator d/dt, https://github.com/klim-/hypore.10.2478/amcs-2018-0045
Search in Google Scholar Back to article
Fritzsche, K., Knoll, C., Franke, M. and R¨obenack, K. (2016). Unimodular completion and direct flat representation in the context of differential flatness, Proceedings in Applied Mathematics and Mechanics 16(1): 807-808.10.1002/pamm.201610392
Search in Google Scholar Back to article
Gupta, R.N., Khurana, A., Khurana, D. and Lam, T.Y. (2009). Rings over which the transpose of every invertible matrix is invertible, Journal of Algebra 322(5): 1627-1636.10.1016/j.jalgebra.2009.05.029
Search in Google Scholar Back to article
Jacobson, N. (1953). Lectures in Abstract Algebra II: Linear Algebra, Springer, New York, NY.10.1007/978-1-4684-7053-6
Search in Google Scholar Back to article
Knoll, C. (2016). Regelungstheoretische Analyse- und Entwurfsans ¨atze f¨ur unteraktuierte mechanische Systeme, PhD thesis, TU Dresden, Dresden.
Search in Google Scholar Back to article
Knoll, C. and Fritzsche, K. (2017). Symbtools: A toolbox for symbolic calculations in nonlinear control theory, DOI: 10.5281/zenodo.275073.10.5281/zenodo.275073
Open DOI Search in Google Scholar Back to article
Kondratieva, M.V., Mikhalev, A.V. and Pankratiev, E.V. (1982). On Jacobi’s bound for systems of differential polynomials, Algebra, Moscow University Press, Moscow, pp. 79-85.
Search in Google Scholar Back to article
Lam, T. (1978). Serre’s Conjecture, Springer, Berlin/Heidelberg.
Search in Google Scholar Back to article
Lévine, J. (2011). On necessary and sufficient conditions for differential flatness, Applicable Algebra in Engineering, Communication and Computing 22(1): 47-90.10.1007/s00200-010-0137-x
Search in Google Scholar Back to article
Logar, A. and Sturmfels, B. (1992). Algorithms for the Quillen-Suslin theorem, Journal of Algebra 145(1): 231-239.10.1016/0021-8693(92)90189-S
Search in Google Scholar Back to article
Meurer, A., Smith, C., Paprocki, M., ˇ Cert´ık, O., Kirpichev, S., Rocklin, M., Kumar, A., Ivanov, S., Moore, J., Singh, S., Rathnayake, T., Vig, S., Granger, B., Muller, R., Bonazzi, F., Gupta, H., Vats, S., Johansson, F., Pedregosa, F., Curry, M., Terrel, A., Rouˇcka, v., Saboo, A., Fernando, I., Kulal, S., Cimrman, R. and Scopatz, A. (2017). SymPY: Symbolic computing in Python, PeerJ Computer Science 3: e103, DOI: 10.7717/peerj-cs.103.10.7717/peerj-cs.103
Open DOI Search in Google Scholar Back to article
Middeke, J. (2011). A Computational View on Normal Forms of Matrices of Ore Polynomials, PhD thesis, Johannes Kepler Universit¨at Linz, Linz.
Search in Google Scholar Back to article
Newman, M. (1972). Integral Matrices, Academic Press, New York, NY/London.
Search in Google Scholar Back to article
Ollivier, F. (1990). Standard bases of differential ideals, Proceedings of the 8th International Conference on Applied Algebra, Algebraic Algorithms and Error-Correcting10.1007/3-540-54195-0_60
Search in Google Scholar Back to article
Codes, Tokyo, Japan, pp. 304-321. Ollivier, F. and Brahim, S. (2007). La borne de Jacobi pour une diffi´et´e d´efinie par un syst`eme quasi r´egulier (Jacobi’s bound for a diffiety defined by a quasi-regular system), Comptes rendus Math´ematiques 345(3): 139-144.10.1016/j.crma.2007.06.010
Search in Google Scholar Back to article
Ritt, J.F. (1935). Jacobi’s problem on the order of a system of differential equations, Annals of Mathematics 36(2): 303-312.10.2307/1968572
Search in Google Scholar Back to article
Rӧbenack, K. and Reinschke, K. (2011). On generalized inverses of singular matrix pencils, International Journal of Applied Mathematics and Computer Science 21(1): 161-172, DOI: 10.2478/v10006-011-0012-3.10.2478/v10006-011-0012-3
Open DOI Search in Google Scholar Back to article
Verhoeven, G.G. (2016). Symbolic Software Tools for Flatness of Linear Systems with Delays and Nonlinear Systems, PhD thesis, Universit¨at der Bundeswehr M¨unchen, Munich.
Search in Google Scholar Back to article
Youla, D. and Pickel, P. (1984). The Quillen-Suslin theorem and the structure of n-dimensional elementary polynomial matrices, IEEE Transactions on Circuits and Systems 31(6): 513-518.10.1109/TCS.1984.1085545
Search in Google Scholar Back to article
Zhou, W. and Labahn, G. (2014). Unimodular completion of polynomial matrices, Proceedings of the 41st International Symposium on Symbolic and Algebraic Computation (ISSAC), Kobe, Japan, pp. 414-420.10.1145/2608628.2608640
Search in Google Scholar Back to article

On Hyper–Regularity and Unimodularity of Ore Polynomial Matrices

Abstract

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