Abstract
Linear repetitive processes are a distinct class of 2D systems of both systems theoretic and applications interest. They are distinct from other classes of such systems by the fact that information propagation in one of the two separate directions only occurs over a finite duration. This, in turn, means that existing 2D systems theory either cannot be applied at all or only in substantially modified form. Hence a distinct systems theory must be developed for them with onward translation (where appropriate) into reliable routinely applicable analysis and design tools. This paper contributes substantial news results to this general task in the areas of controllability and observability for the sub-class of so-called discrete linear repetitive processes which arise in key applications areas and, in particular, iterative learning control.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
K.J. Smyth, Computer Aided Analysis for Linear Repetitive Processes, PhD Thesis, University of Strathclyde, UK, 1992.
N. Amann, D.H. Owens, and E. Rogers, “Predictive Optimal Iterative Learning Control,” International Journal of Control, vol. 69, no. 2, 1998, pp. 203–226.
P.D. Roberts, “Numerical Investigations of a Stability Theorem Arising from 2-Dimensional Analysis of an Iterative Optimal Control Algorithm,” Multidimensional Systems and Signal Processing, vol. 11, no. 1/2, 2000, pp. 109–124.
J.B. Edwards, “Stability Problems in the Control of Multipass Processes,” Proceedings of The Institution of Electrical Engineers, vol. 121, no. 11, 1974, pp. 1425–1431.
D.H. Owens, “Stability of Multipass Processes,” Proceedings of the Institution of Electrical Engineers, vol. 124, no. 11, 1977, pp. 1079–1082.
E. Rogers and D. H. Owens, “Stability Analysis for Linear Repetitive Processes,” Springer-Verlag Lecture Notes in Control and Information Sciences Series, vol. 175, 1992.
R.P. Roesser, “A Discrete State Space Model for Linear Image Processing,” IEEE Transactions on Automatic Control, vol. AC-20: 1, 1975, pp. 1–10.
E. Fornasini and G. Marchesini, “Doubly-Indexed Dynamical Systems: State Space Models and Structural Properties,” Mathematical Systems Theory, vol. 12, 1978, pp. 59–72.
K. Galkowski, E. Rogers, and D.H. Owens, “Matrix Rank Based Conditions for Reachability/Controllability of Discrete Linear Repetitive Processes,” Linear Algebra and its Applications, vols. 275-276, 1998, pp. 201–224.
K. Galkowski, E. Rogers, and D.H. Owens, “New 2D Models and a Transition Matrix for Discrete Linear Repetitive Processes,” International Journal of Control, vol. 72, no. 15, 1999, pp. 1365–1380.
T. Kaczorek, Two-Dimensional Linear Systems, Berlin: Springer-Verlag, 1985.
T. Kailath, Linear Systems, Englewood Cliffs: Prentice-Hall, 1980.
M.P. Dymkov, I. Gaishun, K. Galkowski, E. Rogers, and D.H. Owens, Controllability of discrete linear repetitive processes-a Volterra operator approach. Proceedings of The International Symposium on the Mathematical Theory of Networks and Systems (MTNS 2000), CD Rom, 2000.
M.P. Dymkov, I.V. Gaishun, K. Galkowski, E. Rogers, and D.H. Owens, “AVolterra Operator Approach to the Stability Analysis of a Class of 2D Linear Systems',” Proceedings of 2001 European Control Conference (ECC'01), 2001, pp. 1410–1415.
E.D. Sontag, Mathematical Control Theory, Texts in Applied Mathematics 6, Berlin: Springer-Verlag, 1990.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Dymkov, M., Gaishun, I., Rogers, E. et al. z - Transform and Volterra-Operator Based Approaches to Controllability and Observability Analysis for Discrete Linear Repetitive Processes. Multidimensional Systems and Signal Processing 14, 365–395 (2003). https://doi.org/10.1023/A:1023586803743
Issue Date:
DOI: https://doi.org/10.1023/A:1023586803743