On characteristic cones of discrete nD autonomous systems: theory and an algorithm

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Abstract

In this paper, we provide a complete answer to the question of characteristic cones for discrete autonomous nD systems, with arbitrary \(n\geqslant 2\), described by linear partial difference equations with real constant coefficients. A characteristic cone is a special subset (having the structure of a cone) of the domain (here \(\mathbb {Z}^n\)) such that the knowledge of the trajectories on this set uniquely determines them over the whole domain. Despite its importance in numerous system-theoretic issues, the question of characteristic sets for multidimensional systems has not been answered in its full generality except for Valcher’s seminal work for the special case of 2D systems (Valcher in IEEE Trans Circuits and Syst Part I Fundam Theory Appl 47(3):290–302, 2000). This apparent lack of progress is perhaps due to inapplicability of a crucial intermediate result by Valcher to cases with \(n\geqslant 3\). We illustrate this inapplicability of the above-mentioned result in Sect. 3 with the help of an example. We then provide an answer to this open problem of characterizing characteristic cones for discrete nD autonomous systems with general n; we prove an algebraic condition that is necessary and sufficient for a given cone to be a characteristic cone for a given system of linear partial difference equations with real constant coefficients. In the second part of the paper, we convert this necessary and sufficient condition to another equivalent algebraic condition, which is more suited from algorithmic perspective. Using this result, we provide an algorithm, based on Gröbner bases, that is implementable using standard computer algebra packages, for testing whether a given cone is a characteristic cone for a given discrete autonomous nD system.

Keywords

Characteristic cones Discrete nD systems Autonomous systems Algebraic methods Affine semigroups 

Notes

Acknowledgements

This work has been supported in parts by DST-INSPIRE Faculty Grant, the Department of Science and Technology (DST), Govt. of India (Grant Code: IFA14-ENG-99); and the Industrial Research and Consultancy Centre (IRCC) IIT Bombay (Project ID: 15IRCCSG012).

References

  1. Adams, W. W., & Loustaunau, P. (2012). An introduction to Gröbner Bases (Vol. 3). New Delhi: American Mathematical Society.MATHGoogle Scholar
  2. Atiyah, M., & MacDonald, I. (1969). Introduction to commutative algebra. Britain: Addison-Wesley Publishing Company.MATHGoogle Scholar
  3. Avelli, D. N., Rapisarda, P., & Rocha, P. (2011a). Lyapunov stability of \(2\)D finite-dimensional behaviours. International Journal of Control, 84(4), 737–745.MathSciNetCrossRefMATHGoogle Scholar
  4. Avelli, D. N., Rapisarda, P., & Rocha, P. (2011b). Time-relevant \(2\)D behaviors. Automatica, 47(11), 2373–2382.MathSciNetCrossRefMATHGoogle Scholar
  5. Avelli, D. N., Rapisarda, P., & Rocha, P. (2012). Lyapunov functions for time-relevant \(2\)D systems, with application to first-orthant stable systems. Automatica, 48(9), 1998–2006.MathSciNetCrossRefMATHGoogle Scholar
  6. Björk, J. E. (1979). Rings of differential operators. New York: North Holland Publishing Company.Google Scholar
  7. Cox, D., Little, J., & O’Shea, D. (2007). Ideals, varieties and algorithms. New York: Springer.CrossRefMATHGoogle Scholar
  8. Fornasini, E., Rocha, P., & Zampieri, S. (1993). State space realizations of 2-D finite-dimensional behaviours. SIAM Journal of Control and Optimization, 31(6), 1502–1517.MathSciNetCrossRefMATHGoogle Scholar
  9. Fornasini, E., & Valcher, M. E. (1997). nD polynomial matrices with applications to multidimensional signal analysis. Multidimensional Systems and Signal Processing, 8, 387–407.MathSciNetCrossRefMATHGoogle Scholar
  10. Kreuzer, M., & Robbiano, L. (2000). Computational Commutative Algebra 1. Berlin: Springer.CrossRefMATHGoogle Scholar
  11. Limaye, B. V. (1996). Functional analysis. New Delhi: New Age International (P) Ltd., Publishers.MATHGoogle Scholar
  12. Miller, E., & Sturmfels, B. (2004). Combinatorial commutative algebra. New York: Springer.MATHGoogle Scholar
  13. Mukherjee, M., & Pal, D. (2016). On characteristic cones of scalar autonomous \(n\)D systems, with general \(n\). In 22nd International symposium on mathematical theory of networks and systems, Minneapolis, USA (pp. 839–845).Google Scholar
  14. Mukherjee, M., & Pal, D. (2017). Algorithms for verification of characteristic sets of discrete autonomous \(n\)D systems with \(n\geqslant 2\). IFAC PapersOnline, 50(1), 1840–1846.CrossRefGoogle Scholar
  15. Oberst, U. (1990). Multidimensional constant linear systems. Acta Applicandae Mathematicae, 20, 1–175.MathSciNetCrossRefMATHGoogle Scholar
  16. Pal, D. (2015). Every discrete 2D autonomous system admits a finite union of parallel lines as a characteristic set. Multidimensional Systems and Signal Processing,.  https://doi.org/10.1007/s11045-015-0330-y.MATHGoogle Scholar
  17. Pal, D., & Pillai, H. K. (2011). Lyapunov stability of \(n\)-D strongly autonomous systems. International Journal of Control, 84(11), 1759–1768.MathSciNetCrossRefMATHGoogle Scholar
  18. Pal, D., & Pillai, H. K. (2014). On restrictions of \(n\)-d systems to 1-d subspaces. Multidimensional Systems and Signal Processing, 25, 115–144.MathSciNetCrossRefMATHGoogle Scholar
  19. Pal, D., & Pillai, H. K. (2016). Multidimensional behaviors: The state-space paradigm. Systems and Control Letters, 95, 27–34.MathSciNetCrossRefMATHGoogle Scholar
  20. Pauer, F., & Unterkircher, A. (1999). Gröbner bases for ideals in Laurent polynomial rings and their application to systems of difference equations. Applicable Algebra in Engineering, Communication and Computing, 9, 271–291.MathSciNetCrossRefMATHGoogle Scholar
  21. Pillai, H. K., & Shankar, S. (1998). A behavioral approach to control of distributed systems. SIAM Journal on Control and Optimization, 37(2), 388–408.MathSciNetCrossRefMATHGoogle Scholar
  22. Pommaret, J. F., & Quadrat, A. (1999). Algebraic analysis of linear multidimensional control systems. IMA Journal of Mathematical Control & Information, 16, 275–297.MathSciNetCrossRefMATHGoogle Scholar
  23. Rocha, P., & Willems, J. C. (1989). State for 2-D systems. Linear Algebra and Its Applications, 122–124, 1003–1038.CrossRefMATHGoogle Scholar
  24. Rocha, P., & Willems, J. C. (2006). Markov properties for systems described by PDEs and first-order representations. Systems and Control Letters, 55, 538–542.MathSciNetCrossRefMATHGoogle Scholar
  25. Rogers, E., Galkowski, K., Paszke, W., Moore, K. L., Bauer, P. H., Hladowski, L., et al. (2015). Multidimensional control systems: case studies in design and evaluation. Multidimensional Systems and Signal Processing, 26, 895–939.MathSciNetCrossRefMATHGoogle Scholar
  26. Valcher, M. E. (2000). Characteristic cones and stability properties of two-dimensional autonomous behaviors. IEEE Transactions On Circuits and Systems Part I Fundamental Theory and Applications, 47(3), 290–302.CrossRefGoogle Scholar
  27. Willems, J. C. (1989). Models for dynamics. Dynamics Reported, 2, 171–269.CrossRefGoogle Scholar
  28. Willems, J. C. (1991). Paradigms and puzzles in theory of dynamical systems. IEEE Transactions On Automatic Control, 36(6), 259–294.MathSciNetCrossRefMATHGoogle Scholar
  29. Wood, J. (2000). Modules and behaviours in \(n\)D systems theory. Multidimensional Systems and Signal Processing, 11, 11–48.MathSciNetCrossRefMATHGoogle Scholar
  30. Wood, J., Rogers, E., & Owens, D. H. (1999). Controllable and autonomous \(n\)D linear systems. Multidimensional Systems and Signal Processing, 10, 33–69.MathSciNetCrossRefMATHGoogle Scholar
  31. Wood, J., Sule, V. R., & Rogers, E. (2005). Causal and stable input/output structures on multidimensional behaviors. SIAM Journal on Control and Optimization, 43(4), 1493–1520.MathSciNetCrossRefMATHGoogle Scholar
  32. Youla, D. C., & Gnavi, G. (1979). Notes on n-dimensional system theory. IEEE Transactions on Circuits and Systems, Cas–26(2), 105–111.MathSciNetCrossRefMATHGoogle Scholar
  33. Zerz, E., & Oberst, U. (1993). The canonical Cauchy problem for linear systems of partial difference equations with constant coefficients over the complete \(r\)-dimensional integral lattice \(\mathbb{Z}^{r}\). Acta Applicandae Mathematicae, 31, 249–273.MathSciNetCrossRefMATHGoogle Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.EE DepartmentIIT BombayPowai, MumbaiIndia

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