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.
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Notes
A matrix \(P(\varvec{\xi },\varvec{\xi }^{-1})\) is called left-factor-prime if any decomposition \(P = E P_1\), where E is square, implies that E is invertible as a Laurent polynomial matrix, i.e., \(\mathrm{det~}E\) is a unit in \(\mathcal {A}\). A matrix \(R(\varvec{\xi },\varvec{\xi }^{-1})\) is said to be right-factor-prime if \(R^T(\varvec{\xi },\varvec{\xi }^{-1})\) is left-factor-prime. See Youla and Gnavi (1979) for more details.
A closed, pointed, solid, convex cone is called a proper cone; we elaborate more on this in Sect. 4.
A Hamel basis of a possibly infinite dimensional vector space \(\mathcal {V}\) over a field \(\mathbb {K}\) is a subset \(\mathcal {E}\) of \(\mathcal {V}\) that satisfies:
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1.
elements in \(\mathcal {E}\) are linearly independent over \(\mathbb {K}\), that is, no finite non-zero linear combination of elements in \(\mathcal {E}\) equals zero, and
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2.
every element of \(\mathcal {V}\) can be written as a finite linear combination of elements from \(\mathcal {E}\).
See Limaye (1996, Section 2). Note also that \(\mathcal {M}\) admits a countable Hamel basis. This justifies writing down a basis of \(\mathcal {M}\) as a set \(\mathcal {E}\) which is indexed by the natural numbers.
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1.
The ordering requires to satisfy the following properties: 1) it must be a total ordering on \(\mathbb {Z}^n_{\geqslant 0}\), 2) it must respect the additive structure of \(\mathbb {Z}^n_{\geqslant 0}\), 3) it must be a well-ordering. See (Cox et al. 2007, Chapter 2, Section 2)
It is important to note that division here refers to division in \(\mathbb {R}[\varvec{\xi }]^q\). An algorithm for this can be found in Adams and Loustaunau (2012, Algorithm 3.5.1)
The scalar multiplication property of the homomorphism obeys the following relation: for \(\alpha \in \mathbb {R}[\varvec{\xi },\varvec{\eta }]\), \(\varvec{t} \in \mathbb {R}[\varvec{\xi },\varvec{\eta }]^{1\times q}\)
Refer to Eq. (27).
Refer to Eq. (27).
An algorithm for calculating the Gröbner basis of a module can be found in Adams and Loustaunau (2012, Algorithm 3.5.2).
Note that, the symbols \(z_1,z_2\) in Valcher (2000) carry the meaning of \(\sigma _1^{-1},\sigma _2^{-1}\) of this paper, respectively.
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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).
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Mukherjee, M., Pal, D. On characteristic cones of discrete nD autonomous systems: theory and an algorithm. Multidim Syst Sign Process 30, 611–640 (2019). https://doi.org/10.1007/s11045-018-0571-7
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DOI: https://doi.org/10.1007/s11045-018-0571-7