Skip to main content

Introduction to Two-Dimensional Systems

  • Chapter
  • First Online:
Two-Dimensional Systems

Part of the book series: Studies in Systems, Decision and Control ((SSDC,volume 28))

Abstract

This chapter presents the backgrounds on two-dimensional (2-D) systems: first, 2-D representations are discussed, then definitions of 2-D stability are provided and complemented with some previous results that will be useful in the rest of the book; then some techniques used for dealing with actuator saturation are discussed, as used in Chaps. 2 and 4.

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

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. E.I. Jury, Stability of multidimensional scalar and matrix polynomials. Proc. IEEE 66, 1018–1047 (1978)

    Article  Google Scholar 

  2. M. Piekarski, Algebraic characterization of matrices whose multivariable characteristic polynomial is Hermitian, in Proceedings of the International Symposium on the Operator Theory of Networks and Systems, Lubbock, Texas, 17–19 August, pp. 121–126 (1977)

    Google Scholar 

  3. J.L. Shanks, S. Treitel, J.H. Justice, Stability and synthesis of two-dimensional recursive filters. IEEE Trans. Audio Electroacoust. 20(2), 115–128 (1979)

    Article  Google Scholar 

  4. J.H. Justice, J.L. Shanks, Stability criterion for N-dimensional digital filters. IEEE Trans. Autom. Control 18(3), 284–286 (1973)

    Article  MATH  MathSciNet  Google Scholar 

  5. N.K. Bose, Applied Multidimensional Systems Theory (Van Nostrand Reinhold, New York, 1982)

    MATH  Google Scholar 

  6. R.N. Bracewell, Two Dimensional Imaging (Prentice Hall Inc., Englewood Cliffs, 1995)

    MATH  Google Scholar 

  7. W.S. Lu, A. Antoniou, Two Dimensional Digital Filters, Electrical Engineering and Electronics Series, vol. 80 (Marcel Dekker, New York, 1992)

    Google Scholar 

  8. T. Kaczorek, Two Dimensional Linear Systems (Springer, Berlin, 1985)

    MATH  Google Scholar 

  9. J.R. Cui, G.D. Hu, Q. Zhu, Stability and robust stability of 2-D discrete stochastic systems. Discret. Dyn Nat. Soc., Article ID 545361, 11 pp. (2011)

    Google Scholar 

  10. E. Fornasini, G. Marchesini, State-space realization theory of two-dimensional filters. IEEE Trans. Autom. Control 21(4), 484–492 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  11. E. Fornasini, G. Marchesini, Doubly-indexed dynamical systems: state-space models and structural properties. Math. Syst. Theory 12(1), 59–72 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  12. R. Eising, Realization and stabilization of 2D systems. IEEE Trans. Autom. Control 23(5), 793–799 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  13. H. Xu, Y. Zou, \(H_{\infty }\) control for 2-D singular delayed systems. Int. J. Syst. Sci. 42(4), 609–619 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  14. B.O. Anderson, P. Agathoklis, E.I. Jury, M. Mansour, Stability and the matrix Lyapunov equation for discrete 2-dimensional systems. IEEE Trans. Circuits Syst. CAS-33(3), 261–266 (1986)

    Google Scholar 

  15. K. Galkowski, LMI based stability analysis for 2-D continuous systems, in International Conference on Electronics Circuits and Systems, vol. 3, Dubrovnik, Croatia, 15–18 September, pp. 923–926 (2002)

    Google Scholar 

  16. H.D. Tuan, P. Apkarian, T.Q. Nguyen, T. Narikiys, Robust mixed \(H_{2}/H_{\infty }\) filtering of 2-D systems. IEEE Trans. Signal Process 50(7), 1759–1771 (2002)

    Article  Google Scholar 

  17. T. Ooba, On stability analysis of 2-D systems based on 2-D Lyapunov matrix inequalities. IEEE Trans. Circuits Syst. I 47(8), 1263–1265 (2000)

    Article  MathSciNet  Google Scholar 

  18. Y. Zou, S.L. Campbell, The jump behavior and stability analysis for 2-D singular systems. Multidimens. Syst. Signal Process. 11(3), 339–358 (2000)

    Article  MathSciNet  Google Scholar 

  19. C. Cai, W. Wang, Y. Zou, A note on the internal stability for 2-D acceptable linear singular discrete systems. Multidimens. Syst. Signal Process. 15(2), 197–204 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  20. T. Hu, Z. Lin, The equivalence of several set invariance conditions under saturations, in Proceedings of the 41st IEEE Conference on Decision and Control, 10–13 December, Las Vegas, USA (2002)

    Google Scholar 

  21. K. Galkowski, E. Rogers, W. Paszke, D.H. Owens, Linear repetitive process control theory applied to physical example. Int. J. Appl. Math. Comput. Sci. 13(1), 87–99 (2003)

    MATH  MathSciNet  Google Scholar 

  22. W. Paszke, K. Galkowski, E. Rogers, D.H. Owens, \(H\infty \) control of differential linear repetitive processes. IEEE Trans. Circuits Syst. II: Analog Digit. Signal Process. 53(1), 39–44 (2006)

    Article  Google Scholar 

  23. A. Berman, R.J. Plemmon, Nonnegative matrices in the mathematical sciences. SIAM Class. Appl. Math. 9 (1994)

    Google Scholar 

  24. R. Horn, C. Johnson, Topics in Matrix Analysis (Cambridge University Press, Cambridge, 1991)

    Book  MATH  Google Scholar 

  25. R.E. Skelton, T. Iwasaki, K. Grigoriadis, A Unified Algebraic Approach to Linear Control Design (Taylor-Francis, Bristol, 1998)

    Google Scholar 

  26. S. Boyd, L. EI Ghaoui, E. Feron, V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory, SIAM Studies in Applied Mathematics (SIAM, Philadelphia, 1994)

    Book  MATH  Google Scholar 

  27. P. Gahinet, P. Apkarian, A linear matrix inequality approach to \(H_{\infty }\) control. Int. J. Robust Nonlinear Control 4(4), 421–448 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  28. F. Delmotte, T.M. Guerra, M. Ksantini, Continuous Takagi–Sugeno’s models: reduction of the number of LMI conditions in various fuzzy control design technics. IEEE Trans. Fuzzy Syst. 15(3), 426–438 (2007)

    Article  Google Scholar 

  29. J. Qiu, G. Feng, J. Yang, A new design of delay-dependent robust \(H_{\infty }\) filtering for continuous-time polytopic systems with time-varying delay. Int. J. Robust Nonlinear Control 20(3), 346–365 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  30. S. Xu, J. Lam, Z. Lin, K. Galkowski, Positive real control for uncertain two-dimensional systems. IEEE Trans. Circuits Syst. I 49(11), 1659–1666 (2002)

    Article  MathSciNet  Google Scholar 

  31. K. Gu, An integral inequality in the stability problem of time-delay systems, in The 39th IEEE Conference on Decision Control, Sydney, Australia, 12–15 December, pp. 2805–2810 (2000)

    Google Scholar 

  32. P.A. Bliman, R.C.L.F. Oliveira, V.F. Montagner, P.L.D. Peres, Existence of homogeneous polynomial solutions for parameter-dependent linear matrix inequalities with parameters in the simplex, in Proceedings of the 45th IEEE Conference on Decision and Control, San Diego, CA, USA, pp. 1486–1491, 13–15 December 2006

    Google Scholar 

  33. A. Hmamed, F. Mesquine, M. Benhayoun, A. Benzaouia, F. Tadeo, Stabilization of 2-D saturated systems by state feedback control. Multidimens. Syst. Signal Process. 21(3), 277–292 (2010)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Abdellah Benzaouia .

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Benzaouia, A., Hmamed, A., Tadeo, F. (2016). Introduction to Two-Dimensional Systems. In: Two-Dimensional Systems. Studies in Systems, Decision and Control, vol 28. Springer, Cham. https://doi.org/10.1007/978-3-319-20116-0_1

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-20116-0_1

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-20115-3

  • Online ISBN: 978-3-319-20116-0

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics