Skip to main content
SpringerLink
Log in

On the reduction of repetitive processes into singular and non-singular Roesser models

  • Published:
Multidimensional Systems and Signal Processing Aims and scope Submit manuscript

Abstract

In this paper we present new methods for the reduction of a polynomial system matrix describing a discrete linear repetitive process, to equivalent singular and non-singular 2-D state space representations. Particularly, a zero coprime system equivalence transformation resulting in a singular Roesser state space model, preserving the core algebraic structure of the original system matrix, is proposed. As a second step utilizing the singular Roesser model introduced, we further reduce the system to a non-singular, zero coprime system equivalent Roesser model. Both models are constructed by inspection or by applying elementary matrix manipulations and have significantly smaller dimensions compared to similar reductions found in the literature.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Amann, N., Owens, D. H., & Rogers, E. (1996). Iterative learning control using optimal feedback and feedforward actions. International Journal of Control, 65(2), 277–293. https://doi.org/10.1080/00207179608921697.

    Article  MathSciNet  MATH  Google Scholar 

  • Antoniou, E., & Vologiannidis, S. (2020). On the reduction of 2-D polynomial systems into first order equivalent models. Multidimensional Systems and Signal Processing, 31(1), 249–268.

    Article  MathSciNet  Google Scholar 

  • Boralevi, A., van Doornmalen, J., Draisma, J., Hochstenbach, M. E., & Plestenjak, B. (2017). Uniform determinantal representations. SIAM Journal on Applied Algebra and Geometry, 1(1), 415–441.

    Article  MathSciNet  Google Scholar 

  • Boudellioua, M. S., Galkowski, K., & Rogers, E. (2017a). Equivalent 2-D nonsingular Roesser models for discrete linear repetitive processes. International Journal of Control, 1–9.

  • Boudellioua, M. S., Galkowski, K., & Rogers, E. (2017b). On the connection between discrete linear repetitive processes and 2-D discrete linear systems. Multidimensional Systems and Signal Processing, 28(1), 341–351.

    Article  MathSciNet  Google Scholar 

  • Boudellioua, M. S., Galkowski, K., & Rogers, E. (2017c). Reduction of discrete linear repetitive processes to nonsingular Roesser models via elementary operations. IFAC-PapersOnLine, 50(1), 1865–1870.

    Article  Google Scholar 

  • Edwards, J. B. (1974). Stability problems in the control of multipass processes. Proceedings of the Institution of Electrical Engineers, 121(11), 1425–1432.

    Article  Google Scholar 

  • Fornasini, E., & Marchesini, G. (1976). State-space realization theory of two-dimensional filters. IEEE Transactions on Automatic Control, 21(4), 484–492.

    Article  MathSciNet  Google Scholar 

  • Fuhrmann, P. A. (1977). On strict system equivalence and similarity. International Journal of Control, 25(1), 5–10.

    Article  MathSciNet  Google Scholar 

  • Galkowski, K. (2001). State-space realisations of linear 2-D systems with extensions to the general nD (n\(>\)2) case. Lecture notes in control and information sciences. London: Springer.

  • Galkowski, K., Rogers, E., & Owens, D. H. (1998). Matrix rank based conditions for reachability/controllability of discrete linear repetitive processes. Linear Algebra and its Applications, 275–276, 201–224.

    Article  Google Scholar 

  • Galkowski, K., Rogers, E., & Owens, D. H. (1999). New 2D models and a transition matrix for discrete linear repetitive processes. International Journal of Control, 72(15), 1365–1380. https://doi.org/10.1080/002071799220173.

    Article  MathSciNet  MATH  Google Scholar 

  • Johnson, D. S. (1993). Coprimeness in Multidimensional System Theory and Symbolic Computation, Ph.D. Thesis. U.K: Loughborough University of Technology.

  • Johnson, D. S., Pugh, A. C., Rogers, E., Hayton, G. E., & Owens, D. H. (1996). A polynomial matrix theory for a certain class of two-dimensional linear systems. Linear Algebra and its Applications, 241–243, 669–703.

    Article  MathSciNet  Google Scholar 

  • Levy, B. C. (1981). 2-D polynomial and rational matrices and their applications for the modelling of 2-D dynamical systems. PhD thesis, Stanford University, USA.

  • Pugh, A. C., McInerney, S. J., Hou, M., & Hayton, G. E. (1996). A transformation for 2-D systems and its invariants. In Proceedings of the 35th IEEE conference on decision and control, Kobe, Japan.

  • Pugh, A. C., Mcinerney, S. J., Boudellioua, M. S., Johnson, D. S., & Hayton, G. E. (1998). A transformation for 2-D linear systems and a generalization of a theorem of Rosenbrock. International Journal of Control, 71(3), 491–503.

    Article  MathSciNet  Google Scholar 

  • Pugh, A. C., McInerney, S. J., & El-Nabrawy, E. M. O. (2005). Equivalence and reduction of 2-D systems. IEEE Transactions on Circuits and Systems II: Express Briefs, 52(5), 271–275.

    Article  Google Scholar 

  • Pugh, A. C., McInerney, S. J., & El-Nabrawy, E. M. O. (2005). Zero structures of n-D systems. International Journal of Control, 78(4), 277–285.

    Article  MathSciNet  Google Scholar 

  • Roesser, R. P. (1975). A discrete state-space model for linear image processing. IEEE Transactions Automatic Control, AC–20, 1–10.

    Article  MathSciNet  Google Scholar 

  • Rogers, E., Galkowski, K., & Owens, D. H. (2007). Control systems theory and applications for linear repetitive processes. Control and information sciences. Berlin, Heidelberg: Springer.

    MATH  Google Scholar 

  • Rosenbrock, H. H. (1970). State space and multivariable theory. London: Nelson.

    MATH  Google Scholar 

  • Smyth, K. (1992). Computer aided analysis for linear repetitive processes. Library catalog: www.semanticscholar.org.

Download references

Acknowledgements

The authors wish to thank the anonymous reviewers for their insightful and constructive comments.

Author information

Authors and Affiliations

  1. Department of Computer, Informatics and Telecommunications Engineering, International Hellenic University, Terma Magnisias, 62124, Serres, Greece

    Stavros Vologiannidis

  2. Department of Information and Electronic Engineering, International Hellenic University, 57400, Thessaloniki, Greece

    Efstathios Antoniou

Authors
  1. Stavros Vologiannidis
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Efstathios Antoniou
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Stavros Vologiannidis.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Vologiannidis, S., Antoniou, E. On the reduction of repetitive processes into singular and non-singular Roesser models. Multidim Syst Sign Process 32, 559–572 (2021). https://doi.org/10.1007/s11045-020-00750-z

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11045-020-00750-z

Keywords

Search

Navigation