Skip to main content
SpringerLink
Log in

Realization of multidimensional systems in Fornasini-Marchesini state-space model

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

Abstract

This paper presents a constructive procedure that can generate a Fornasini-Marchesini (local) state-space model realization for a given n-D system with lower order than the existing procedure given by Alpay and Dubi. It is clarified that the method of Alpay and Dubi usually generates a realization with unnecessarily high order even for a simple rational transfer function, and, by taking into account of the structural characteristic of the given transfer function or transfer matrix, it is possible to construct a realization with much lower order. Nontrivial examples are given to illustrate the basic idea as well as the effectiveness of the proposed approach.

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

  • Alpay D., Dubi C. (2003) A realization theorem for rational function of several complex variables. Systems & Control Letters 49: 225–229

    MathSciNet  MATH  Google Scholar 

  • Bisiacco M., Fornasini E., Marchesini G. (1985) On some connections between BIBO and internal stability of two-dimensional filters. IEEE Transactions on Circuits and Systems 32: 948–953

    Article  MathSciNet  MATH  Google Scholar 

  • Bisiacco M., Fornasini E., Marchesini G. (1989) Dynamic regulation of 2-D systems: a state-space approach. Linear Algebra and Its Applications 122/123/124: 195–218

    Article  MathSciNet  Google Scholar 

  • Bose, N. K. (eds) (2003) Multidimensional systems theory and applications. Kluwer, Dordrecht, The Netherlands

    MATH  Google Scholar 

  • Dewasurendra, D. A., & Bauer, P. H. (2008). A novel approach to grid sensor networks. In 15th IEEE international conference on electronics, circuits and systems (pp. 1191–1194).

  • Dubi C. (2008) Triangular realization of rational functions of N complex variables. Multidimensional Systems and Signal Processing 19(1): 123–129

    Article  MathSciNet  MATH  Google Scholar 

  • Dubi C. (2009) Triangular representation of Fornasini-Marchesini systems. Multidimensional Systems and Signal Processing 20(3): 217–234

    Article  MathSciNet  MATH  Google Scholar 

  • Eising R. (1978) Realization and stabilization of 2-D systems. IEEE Transactions on Automatic Control 23(5): 793–799

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  • Fornasini E., Marchesini G. (1977) Computation of reachable and observable realization of spatial filters. International Journal of Control 25(4): 621–635

    Article  MathSciNet  MATH  Google Scholar 

  • Fornasini E., Marchesini G. (1978) Doubly-indexed dynamical systems: State-space models and structural properties. Mathematical System Theory 12(1): 59–72

    Article  MathSciNet  MATH  Google Scholar 

  • Galkowski K. (2001) State-space realizations of linear 2-D systems with extensions to the general nD (n > 2) case. Springer-Verlag, LNCIS

    Google Scholar 

  • Guiver J. P., Bose N. K. (2003) Causal and weak causal 2-D filter with applications in stabilization, Chap. 2.. In: Bose N. K. (eds) Multidimensional systems theory and applications. Kluwer Academic Publishers, Dordrecht

    Google Scholar 

  • Hinamoto T. (1993) 2-D Lyapunov equation and filter design based on Fornasini-Marchesini second model. IEEE Transactions on Circuits and Systems: I 40: 102–110

    Article  Google Scholar 

  • Kaczorek T. (1985) Two-dimensional linear systems. Springer-Verlag, Berlin

    MATH  Google Scholar 

  • Kailath T. (1980) Linear systems. Prentice-Hall Inc, USA

    MATH  Google Scholar 

  • Kung, S. Y., Levy, B. C., Morf M., & Kailath, T. (1977). New results in 2-D systems theory, Part II: 2-D state-space models—realization and the notions of controllability, observability, and minimality. In Proceedinge of the IEEE (Vol. 65, pp. 945–961)

  • Lim J. S. (1990) Two-dimensional signal and image processing. Prentice Hall, Englewood Cliffs, NJ

    Google Scholar 

  • Lin Z. (1998) Feedback stabilization of MIMO n-D linear systems. Multidimensional Systems and Signal Processing 9(2): 149–172

    Article  MathSciNet  MATH  Google Scholar 

  • Lu W. S., Antoniou A. (1992) Two-dimensional digital filters. Marcel Dekker Inc, New York

    MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  • Schröder H., Blume H. (2000) One-and multidimensional signal processing: Algorithms and applications in image processing. Wiley, New York

    Google Scholar 

  • Tzafestas S. G. (1986) Multidimensional systems: Techniques and applications. Marcel Dekker, New York

    MATH  Google Scholar 

  • Xu L., Yan S. (2010) A new elementary operation approach to multidimensional realization and LFR uncertainty modeling: The SISO case. Multidimensional Systems and Signal Processing, 21(4): 343–372

    Article  MathSciNet  MATH  Google Scholar 

  • Xu L., Wu L., Wu Q., Lin Z., Xiao Y. (2005) On realization of 2D discrete systems by Fornasini-Marchesini model. International Journal of Control, Automation, and Systems 3(4): 631–639

    Google Scholar 

  • Xu L., Wu Q., Lin Z., Xiao Y. (2007) A new constructive procedure for 2-D coprime realization in Fornasini-Marchesini model. IEEE Trans Circuits Syst I: Regular Papers 54(9): 2061–2069

    Article  MathSciNet  Google Scholar 

  • Xu L., Fan H., Lin Z., Bose N. K. (2008) A direct-construction approach to multidimensional realization and LFR uncertainty modeling. Multidimensional Systems and Signal Processing 19(3–4): 323–359

    Article  MathSciNet  MATH  Google Scholar 

  • Zak S. H., Lee E. B., Lu W. S. (1986) Realizations of 2-D filters and time delay systems. IEEE Transactions on Circuits and Systems CAS-33: 1241–1244

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Electronics and Information Systems, Akita Prefectural University, Akita, 015-0055, Japan

    Hua Cheng, Tatsuya Saito, Shin-ya Matsushita & Li Xu

Authors
  1. Hua Cheng
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Tatsuya Saito
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Shin-ya Matsushita
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Li Xu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Li Xu.

Additional information

This work was partially supported by JSPS.KAKENHI 22560451.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Cheng, H., Saito, T., Matsushita, Sy. et al. Realization of multidimensional systems in Fornasini-Marchesini state-space model. Multidim Syst Sign Process 22, 319–333 (2011). https://doi.org/10.1007/s11045-010-0136-x

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11045-010-0136-x

Keywords

Search

Navigation