Multidimensional Systems and Signal Processing

, Volume 20, Issue 4, pp 311–331 | Cite as

Strong practical stability and stabilization of discrete linear repetitive processes

  • Paweł Dąbkowski
  • Krzysztof Gałkowski
  • Eric Rogers
  • Anton Kummert


This paper considers two-dimensional (2D) discrete linear systems recursive over the upper right quadrant described by well known state-space models. Included are discrete linear repetitive processes that evolve over subset of this quadrant. A stability theory exists for these processes based on a bounded-input bounded-output approach and there has also been work on the design of stabilizing control laws, elements of which have led to the assertion that this stability theory is too strong in many cases of applications interest. This paper develops so-called strong practical stability as an alternative in such cases. The analysis includes computationally efficient tests that lead directly to the design of stabilizing control laws, including the case when there is uncertainty associated with the process model. The results are illustrated by application to a linear model approximation of the dynamics of a metal rolling process.


2D information propagation Recursive updating Stability Stabilization 


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  1. AgathoklisP. Bruton L.T. (1983) Practical-BIBO stability of n dimensional discrete systems. Proceedings of The Institution of Electrical Engineers, 130: 236–242Google Scholar
  2. Amann N., Owens D.H., Rogers E. (1998) Predictive optimal iterative learning control. International Journal of Control, 69(2): 203–226MATHCrossRefMathSciNetGoogle Scholar
  3. Bochniak J., Galkowski K., Rogers E. (2008) Multi-machine operations modelled and controlled as switched linear repetitive processes. International Journal of Control, 81(10): 1549–1567MATHCrossRefMathSciNetGoogle Scholar
  4. Boland F.M., Owens D.H. (1980) Linear multipass processes—a two-dimensional systems interpretation. Proceedings of the Institution of Electrical Engineers, 127: 189–193MathSciNetGoogle Scholar
  5. Chaabane, M., Bachelier, O., & Mehdi, D. (2007). Admissibility and state feedback stabilization of discrete singular systems: An LMI Approach, LAII-ESIP Technical Report 20070208DM, Poitiers, France.Google Scholar
  6. Dai, L. (1989). Singular control systems. Springer.Google Scholar
  7. de Oliveira M.C., Bernussou J., Geromel J.C. (1999) A new discrete-time robust stability condition. Systems and Control Letters, 37(4): 261–265MATHCrossRefMathSciNetGoogle Scholar
  8. Fornasini E., Marchesini G. (1978) Doubly indexed dynamical systems: State-space models and structural properities. Mathematical System Theory, 12: 59–72MATHCrossRefMathSciNetGoogle Scholar
  9. Galkowski K., Rogers E., Xu S., Lam J., Owens D.H. (2002) LMIs—a fundamental tool in analysis and controller design for discrete linear repetitive processes. IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, 49(6): 768–778CrossRefMathSciNetGoogle Scholar
  10. Peaucelle D., Arzelier D., Bachelier O., Bernussou J. (2000) A new robust D-stability condition for real convex polytopic uncertainty. Systems and Control Letters, 40(1): 21–30MATHCrossRefMathSciNetGoogle Scholar
  11. Ratcliffe J.D., Lewin P.L., Rogers E., Hatonen J.J., Owens D.H. (2006) Norm-optimal iterative learning control applied to gantry robots for automation applications. IEEE Transactions on Robotics, 22(6): 1303–1307CrossRefGoogle Scholar
  12. Roberts P.D. (2002) Two-dimensional analysis of an iterative nonlinear optimal control algorithm. IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, 49(6): 872–878CrossRefMathSciNetGoogle Scholar
  13. Roesser R.P. (1975) A discrete state-space model for linear image processing. IEEE Transactions on Automatic Control, AC- 20: 1–10MATHCrossRefMathSciNetGoogle Scholar
  14. Rogers, E., Galkowski, K., & Owens, D. H. (2007). Control systems theory and applications for linear repetitive processes. Lecture Notes in Control and Information Sciences Series (Vol. 349). Springer.Google Scholar
  15. Wood J., Oberst U., Rogers E., Owens H.H. (2000) A behavioural approach to the pole structure of one-dimensional and multidimensional linear systems. SIAM Journal on Control and Optimization, 38(2): 627–661MATHCrossRefMathSciNetGoogle Scholar
  16. Wu, L., Lam, J., Paszke, W., Galkowski, K., & Rogers, E. (2009). Control and filtering for discrete linear repetitive processes with \({\mathcal{H}_\infty}\) and l 2l performance. International Journal of Multidimensional Systems and Signal Processing (in print).Google Scholar
  17. Xu L., Saito O., Abe K. (1994) The design of practically stable nD feedback systems. Automatica, 30(9): 1389–1397MATHCrossRefMathSciNetGoogle Scholar
  18. Xu L., Saito O., Abe K. (1997) nD control systems in a practical sense. International Journal of Applied Mathematics and Computer Science, 7(4): 907–941MATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  • Paweł Dąbkowski
    • 1
  • Krzysztof Gałkowski
    • 1
  • Eric Rogers
    • 2
  • Anton Kummert
    • 3
  1. 1.Institute of PhysicsNicolaus Copernicus University in TorunTorunPoland
  2. 2.School of Electronics and Computer ScienceUniversity of SouthamptonSouthamptonUK
  3. 3.Faculty of Electrical, Information and Media EngineeringUniversity of WuppertalWuppertalGermany

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