References
- Attasi, S . (1976). Modelling and recursive estimation for double indexed sequences. In R. K. Mehra & D. G. Lainiotis (Eds.) System identification: Advances and case studies, edited by R. K. Mehra and D. G. Lainiotis (pp. 289–348). New York, NY: Academic Press.
- Auzinger, W. , & Stetter, H. J . (1988). An elimination algorithm for the computation of all zeros of a system of multivariate polynomial equations. In R. P. Agarwal , Y. M. Chow & S. J. Wilson (Eds.) Proceedings of the International Conference on Numerical Mathematics , (pp. 11–30). Basel: Birkhäuser.
- Ball, J. A. , Boquet, G. M. , & Vinnikov, V. (2012). A behavioral interpretation of Livšic systems. Multidimensional Systems and Signal Processing , 23 (1), 17–48.
- Ball, J. A. , & Vinnikov, V. (2003). Overdetermined multidimensional systems: State space and frequency domain methods. In J. Rosenthal & D. S. Gilliam (Eds.), Mathematical systems theory in biology, communications, computation, and finance (pp. 63–119). New York, NY: Springer.
- Batselier, K. , Dreesen, P. , & De Moor, B. (2014a). The canonical decomposition of Cn d and numerical Gröbner and border bases. SIAM Journal on Matrix Analysis and Applications , 35 (4), 1242–1264.
- Batselier, K. , Dreesen, P. , & De Moor, B. (2014b). On the null spaces of the Macaulay matrix. Linear Algebra and its Applications , 460 (1), 259–289.
- Batselier, K. , & Wong, N. (2016). Computing the state difference equations for discrete overdetermined linear systems. Automatica , 64 , 254–261.
- Bleylevens, I. , Peeters, R. , & Hanzon, B. (2007). Efficiency improvement in an nD-systems approach to polynomial optimization. Journal of Symbolic Computation , 42 (1–2), 30–53.
- Bose, N. K . (2016). Applied multidimensional systems theory (2nd ed.). Springer International Publishing.
- Bose, N. K. (2007). Two decades (1985–2005) of Gröbner bases in multidimensional systems. In H. A. Park & G. Regensburger (Eds.), Gröbner bases in control theory and signal processing (Vol. 3, pp. 1–22). Berlin: Walter de Gruyter.
- Bose, N. K. , Buchberger, B. , & Guiver, J. P . (2003). Multidimensional systems theory and applications . New York: Springer.
- Buchberger, B. (2001). Gröbner bases and systems theory. Multidimensional Systems and Signal Processing , 12 , 223–251.
- Cox, D. A. , Little, J. B. , & O’Shea, D . (2005). Using algebraic geometry (2nd ed.). New York, NY: Springer-Verlag.
- Cox, D. A. , Little, J. B. , & O’Shea, D . (2007). Ideals, varieties and algorithms (3rd ed.). New York, NY: Springer-Verlag.
- Dayton, B. H. , Li, T.-Y. , & Zeng, Z. (2011). Multiple zeros of nonlinear systems. Mathematics of Computation , 80 , 2143–2168.
- Dreesen, P. (2013). Back to the roots – polynomial system solving using linear algebra ( Unpublished doctoral dissertation). KU Leuven, Leuven, Belgium: Faculty of Engineering Science.
- Fornasini, E. , & Marchesini, G. (1976). State-space realization theory of two-dimensional filters. IEEE Transactions on Automatic Control , 21 , 484–492.
- Fornasini, E. , Rocha, P. , & Zampieri, S. (1993). State space realization of 2-D finite-dimensional behaviors. SIAM Journal on Control and Optimization , 31 (6), 1502–1517.
- Gałkowski, K . (2001). State-space realizations of linear 2-D systems with extensions to the general nD (n > 2) case . London: Springer.
- Gantmacher, F . (1960). The theory of matrices (Vol. 2). New York, NY: Chelsea Publishing Company.
- Gerdin, M. (2004). Computation of a canonical form for linear differential-algebraic equations. Proceeding of the Reglermöte 2004 , Göteborg.
- Giusti, M. , & Schost, E . (1999). Solving some overdetermined polynomial systems. In K. O. Geddes , B. Salvy & S. S. Doley (Eds.) Proceeding of the 1999 International Symposium on Symbolic and Algebraic Computation (ISSAC 1999) , (pp. 1–8). New York NY: ACM.
- Hanzon, B. , & Hazewinkel, M. (Eds.) . (2006a). Constructive algebra and systems theory . Amsterdam, Netherlands: Royal Netherlands Academy of Arts and Sciences.
- Hanzon, B. , & Hazewinkel, M. (2006b). An introduction to constructive algebra and systems theory. In B. Hanzon & M. Hazewinkel (Eds.), Constructive algebra and systems theory (pp. 2–7). Amsterdam, Netherlands: Royal Netherlands Academy of Arts and Sciences.
- Ho, B. L. , & Kalman, R. E. (1966). Effective construction of linear state-variable models from input/output functions. Regelungstechnik , 14 (12), 545–548.
- Jónsson, G. F. , & Vavasis, S. A. (2004). Accurate solution of polynomial equations using Macaulay resultant matrices. Mathematics of Computation , 74 (249), 221–262.
- Kaczorek, T. (1988). The singular general model of 2D systems and its solution. IEEE Transactions on Automatic Control , 33 , 1060–1061.
- Kailath, T . (1980). Linear systems . New Jersey: Prentice-Hall International.
- Kurek, J. E. (1985). Basic properties of q-dimensional linear digital systems. International Journal of Control , 42 , 119–128.
- Lazard, D. (1983). Groebner bases, Gaussian elimination and resolution of systems of algebraic equations. In J. van Hulzen (Ed.), Computer algebra (Vol. 162, pp. 146–156). Berlin, Heidelberg: Springer.
- Livšic, M. S. (1983). Cayley-Hamilton theorem, vector bundles and divisors of commuting operators. Integral Equations Operator Theory , 6 , 250–273.
- Livšic, M. S. , Kravitsky, N. , Markus, A. S. , & Vinnikov, V . (1995). Theory of commuting nonselfadjoint operators (Vol. 332). Dordrecht, NL: Kluwer Academic Publisher Group.
- Luenberger, D. G. (1978). Time-invariant descriptor systems. Automatica , 14 , 473–480.
- Macaulay, F. S . (1916). The algebraic theory of modular systems . Cambridge: Cambridge University Press.
- Möller, H. M. , & Stetter, H. J. (1995). Multivariate polynomial equations with multiple zeros solved by matrix eigenproblems. Numerische Mathematik , 70 , 311–329.
- Moonen, M. , De Moor, B. , Ramos, J. , & Tan, S. (1992). A subspace identification algorithm for descriptor systems. Systems & Control Letters , 19 , 47–52.
- Mourrain, B. (1998). Computing the isolated roots by matrix methods. Journal of Symbolic Computation , 26 , 715–738.
- Oberst, U. (1990). Multidimensional constant linear systems. Acta Applicandae Mathematicae , 20 (1), 1–175.
- Ramos, J. A. , & Mercère, G. (2016). Subspace algorithms for identifying separable-in-denominator 2D systems with deterministic-stochastic inputs. International Journal of Control , 89 , 2584–2610.
- Rocha, P. , & Willems, J. C. (2006). Markov properties for systems described by PDEs and first-order representations. Systems & Control Letters , 55 , 538–542.
- Roesser, R. (1975). A discrete state-space model for linear image processing. IEEE Transactions on Automatic Control , 20 , 1–10.
- Rogers, E. , Gałkowski, K. , Paszke, W. , Moore, K. L. , Bauer, P. H. , Hladowski, L. , & Dabkowski, P. (2015). Multidimensional control systems: Case studies in design and evaluation. Multidimensional Systems and Signal Processing , 26 (4), 895–939.
- Shafarevich, I. R . (2013). Basic algebraic geometry I — Varieties in projective space (3rd ed.). Berlin: Springer.
- Shaul, L. , & Vinnikov, V . (2009). State feedback for overdetermined 2D systems: Pole placement for bundle maps over an algebraic curve. In International Workshop on Multidimensional (nD) Systems (nDS 2009) (pp. 1–2). IEEE.
- Stetter, H. J . (2004). Numerical polynomial algebra . Philadelphia: SIAM.
- Van Overschee, P. , & De Moor, B . (1996). Subspace identification for linear systems: Theory, implementation, applications . Dordrecht, Netherlands: Kluwer Academic Publishers.
- Willems, J. C. (1986a). From time series to linear system – Part I. Finite dimensional linear time invariant systems. Automatica , 22 (5), 561–580.
- Willems, J. C. (1986b). From time series to linear system – Part II. Exact modeling. Automatica , 22 (6), 675–694.
- Willems, J. C. (1987). From time series to linear system – Part III. Approximate modeling. Automatica , 23 (1), 87–115.
- Xu, L. , Fan, H. , Lin, Z. , & Bose, N. K. (2008). A direct-construction approach to multidimensional realization and LFR uncertainty modeling. Multidimensional Systems and Signal Processing , 22 (1–3), 323–359.
- Xu, L. , Yan, S. , Lin, Z. , & Matsushita, S. (2012). A new elementary operation approach to multidimensional realization and LFR uncertainty modeling: The MIMO case. IEEE Transactions on Circuits and Systems , 59 (3), 638–651.
- Zerz, E . (2000). Topics in multidimensional linear systems theory . London: Springer.
- Zerz, E. (2008). The discrete multidimensional MPUM. Multidimensional Systems and Signal Processing , 19 , 307–321.