Abstract
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.
Similar content being viewed by others
References
AgathoklisP. Bruton L.T. (1983) Practical-BIBO stability of n dimensional discrete systems. Proceedings of The Institution of Electrical Engineers, 130: 236–242
Amann N., Owens D.H., Rogers E. (1998) Predictive optimal iterative learning control. International Journal of Control, 69(2): 203–226
Bochniak J., Galkowski K., Rogers E. (2008) Multi-machine operations modelled and controlled as switched linear repetitive processes. International Journal of Control, 81(10): 1549–1567
Boland F.M., Owens D.H. (1980) Linear multipass processes—a two-dimensional systems interpretation. Proceedings of the Institution of Electrical Engineers, 127: 189–193
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.
Dai, L. (1989). Singular control systems. Springer.
de Oliveira M.C., Bernussou J., Geromel J.C. (1999) A new discrete-time robust stability condition. Systems and Control Letters, 37(4): 261–265
Fornasini E., Marchesini G. (1978) Doubly indexed dynamical systems: State-space models and structural properities. Mathematical System Theory, 12: 59–72
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): 768–778
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): 21–30
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): 1303–1307
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): 872–878
Roesser R.P. (1975) A discrete state-space model for linear image processing. IEEE Transactions on Automatic Control, AC- 20: 1–10
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.
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): 627–661
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 2 – l ∞ performance. International Journal of Multidimensional Systems and Signal Processing (in print).
Xu L., Saito O., Abe K. (1994) The design of practically stable nD feedback systems. Automatica, 30(9): 1389–1397
Xu L., Saito O., Abe K. (1997) nD control systems in a practical sense. International Journal of Applied Mathematics and Computer Science, 7(4): 907–941
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Dąbkowski, P., Gałkowski, K., Rogers, E. et al. Strong practical stability and stabilization of discrete linear repetitive processes. Multidim Syst Sign Process 20, 311–331 (2009). https://doi.org/10.1007/s11045-009-0080-9
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11045-009-0080-9