Abstract
In this paper, we formalize two related but different notions for state-space realization of multidimensional (\(n\hbox {D}\)) single-input–single-output discrete systems in \(n\hbox {D}\) Roesser model, namely the “absolutely minimal realization” and the “minimal realization”. We then focus our study mainly on first-degree 2D and 3D causal systems. A necessary and sufficient condition for absolutely minimal realizations is given for first-degree 2D systems. It is then shown that first-degree 2D systems that do not admit absolutely minimal realizations always admit minimal realizations of order 3. A Gröbner basis approach is also proposed which leads to a sufficient condition for the absolutely minimal realizations of some higher-degree 2D systems. We then present a symbolic method that gives simple necessary conditions for the existence of absolutely minimal realizations for first-degree 3D systems. A two-step approach to absolutely minimal realizations for first-degree 3D systems is then presented, followed by techniques for minimal realizations of first-degree 3D systems. Throughout the paper, several non-trivial examples are illustrated with the aim of helping the reader to apply the realization methods proposed in this paper.
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References
Antoniou, G. E. (1993). 2-D discrete time lossless bounded real functions: Minimal state space realization. Electronics Letters, 29, 2004–2009.
Antoniou, G. E. (2001). 2-D lattice discrete filters: Minimal delay and state space realization. IEEE Signal Processing Letters, 8, 23–25.
Antoniou, G. E., Varoufakis, S. J., & Parakevopoulos, P. N. (1986). State space realization of 2-D systems via continued fraction expansion. IEEE Transactions on Circuits and Systems, 33, 926–930.
Antoniou, G. E., Varoufakis, S. J., & Parakevopoulos, P. N. (1988). Minimal state space realization of factorable 2-D systems. Transactions on Circuits and Systems, 35, 1055–1058.
Ball, J. A., Groenewald, G., & Malakorn, T. (2005). Structured noncommutative multidimensional linear systems. SIAM Journal on Control and Optimization, 44(4), 1474–1528.
Beck, C., & D’Andrea, R. (1997). Minimality, controllability and observability for uncertain systems. In Proceedings of the American control conference (pp. 3130–3135), Albuquerque, NM, June 1997.
Beck, C., & D’Andrea, R. (2004). Noncommuting multidimensional realization theory: Minimality, reachability, and observability. IEEE Transactions on Automatic Control, 49(10), 1815–1822.
Belcastro, C. M. (1994). Uncertainty modeling of real parameter variations for robust control applications. Ph.D. thesis. USA: University of Drexel, December 1994.
Bose, N. K. (1976). An algorithm for G.C.F. extraction from two multivariable polynomials. Proceedings of IEEE, 64, 185–186.
Bose, N. K. (1982). Applied multidimensional systems theory. New York: Van Nostrand Reinhold.
Bose, N. K. (2003). Multidimensional systems theory and applications. New York: Springer.
Buchberger, B. (1985). Gröbner bases: An algorithmic method in polynomial ideal theory. In N. K. Bose (Ed.), Multidimensional systems theory. Reidel: Dordrecht.
Cockburn, J. C. (2000). Multidimensional realizations of systems with parametric uncertainty. In Proceedings of MTNS, Perpignan, France, Session si20a, June 2000.
D’Andrea, R., & Khatri, S. (1997). Kalman decomposition of linear fractional transformation representations and minimality. In Proceedings of the American control conference (pp. 3557–3561), Albuquerque, NM, June 1997.
Decker, W., Greuel, G. M., Pfister, G., & Schönemann, H. (2012). Singular 3-1-6—A computer algebra system for polynomial computations. http://www.singular.uni-kl.de.
Dymkov, M., Gaishun, I., Rogers, E., Galkowski, K., & Owens, D. H. (2004). Control theory for a class of 2D continuous-discrete linear systems. International Journal of Control, 77(9), 847–860.
Eising, R. (1978). Realization and stabilization of 2-D systems. IEEE Transactions on Automatic Control, 23(5), 793–799.
El-Kasri, B., Hmamed, A., Tissir, E. H., & Tadeo, F. (2013). Robust \(H_{\infty }\) filtering for uncertain two-dimensional continuous systems with time-varying delays. Multidimensional Systems and Signal Processing, 24(4), 685–706.
Fan, H., Cheng, H., & Xu, L. (2009). A constructive approach to minimal realization problem of 2D systems. Journal of Control Theory and Applications, 7(3), 335–343.
Galkowski, K. (2001a). State-space realizations of linear 2-D systems with extensions to the general \(n\)-\(D\) (\(n>2\)) case. In LNCIS, 2001. Springer
Galkowski, K. (2001b). Minimal state-space realization for a class of linear, discrete, \(n\text{ D }\), SISO systems. International Journal of Control, 74(13), 1279–1294.
Galkowski, K. (2005). Minimal state-space realization for a class of \(n\text{ D }\) systems. Operation Theory: Advances and Applications, 160, 179–194.
Ganapathy, V., Reddy, D. R., & Reddy, P. S. (1983). Minimal delay realization of first order 2-D all-pass digital filters. IEEE Transactions on Acoustics Speech and Signal Processing, 31, 1577–1579.
Givone, D. D., & Roesser, R. P. (1973). Minimization of multidimensional linear iterative circuits. IEEE Transactions on Computers, 22, 673–678.
Gu, G., Aravena, J. L., & Zhuo, K. (1991). On minimal realization of 2-D systems. IEEE Transactions on Circuits and Systems, 38(10), 1228–1233.
Guiver, J. P., & Bose, N. K. (1985). Causal and weakly causal 2-D filters with applications in stabilizations. In N. K. Bose (Ed.), Multidimensional systems theory: Progress, directions and open problems (pp. 52–100). Dordrecht: Reidel.
Jury, E. (1978). Stability of multidimensional scalar and matrix polynomials. Proceedings of the IEEE, 66, 1018–1047.
Kailath, T. (1980). Linear systems. Englewood Cliffs: Prentice-Hall Inc.
Kanellakis, A. J., & Theodorou, N. J. (1989). Canonical and minimal state-space realization of two-dimensional transfer functions having separable numerator or denominator. International Journal of Systems Science, 20, 1213–1219.
Kung, S. Y. (1977). New results in 2-D systems theory. Part II: 2-D state space models-realization and the notions of controllability, observability and minimality. Proceedings of IEEE, 65, 945–961.
Lin, Z. (1998). Feedback stabilizability of MIMO \(n\)-D linear systems. Multidimensional Systems and Signal Processing, 9, 149–172.
Lin, Z. (1999). Notes on \(n\text{ D }\) polynomial matrix factorizations. Multidimensional Systems and Signal Processing, 10, 379–393.
Lin, Z., Xu, L., & Anazawa, Y. (2007). Revisiting the absolutely minimal realization for two-dimensional digital filters. In Proceedings of IEEE ISCAS 2007 (pp. 597–600), New Orleans, USA.
Lin, Z., Xu, L., & Bose, N. K. (2008). A tutorial on Gröbner bases with applications in signals and systems. IEEE Transactions on Circuits and Systems I: Regular Papers, 55, 445–461.
Lu, W. S., & Antoniou, A. (1992). Two-dimensional digital filters. New York: Marcel Dekker Inc.
Magni, J. F. (2006). User manual of the linear fractional representation toolbox, version 2. Technical report, France (revised February 2006). http://www.feng.pucrs.br/~gacs/new/disciplinas/ppgee/crobusto/referencias/lfr/lfrt_manual_v20.pdf.
Maplesoft\(^{\textregistered }\), Maple 15. Released April 13, 2011.
Paraskevopoulos, P. N., & Antoniou, G. E. (1993). Minimal realization of 2-D systems via a cyclic model. International Journal of Electronics, 74, 491–511.
Reddy, M. S., Roy, S. C. D., & Hazra, S. N. (1986). Realization of first order two dimensional all pass digital filters. IEEE Transactions on Acoustics Speech and Signal Processing, 34, 1011–1013.
Varoufakis, S. J., Antoniou, G. E., & Parakevopoulos, P. N. (1987). On the minimal state space realization of all-pole and all-zero 2-D systems. Transactions on Circuits and Systems, 34, 289–292.
Xu, L., & Yan, S. (2010). A new elementary operation approach to multidimensional realization and LFR uncertainty modeling: The SISO case. Multidimensional Systems and Signal Processing, 21(4), 343–372.
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, 19, 323–359.
Xu, L., Fan, H., Lin, Z., & Xiao, Y. (2011). Coefficient-dependent direct-construction approach to realization of multidimensional systems in Roesser model. Multidimensional Systems and Signal Processing, 22(1–3), 97–129.
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 I, 59(3), 638–651.
Youla, D., & Gnavi, G. (1979). Notes on \(n\)-dimensional system theory. IEEE Transactions on Circuits and Systems, 26, 105–111.
Żak, S. H., Lee, E. B., & Lu, W.-S. (1986). Realization of 2-D filters and time delay systems. IEEE Transactions on Circuits and Systems, 33(12), 1241–1244.
Acknowledgments
We would like to thank the anonymous reviewers for constructive comments, which helped improve the quality and presentation of this paper. We wish to acknowledge the funding support for this project from Nanyang Technological University under the Undergraduate Research Experience on Campus (URECA) program.
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Appendix: Proof of the sufficiency of Theorem 3
Appendix: Proof of the sufficiency of Theorem 3
In this appendix, we show that a solutions for \(A\) can always be obtained under the assumption \(\Delta _{3D}\geqslant 0\). Noting that
we can conclude, as in the 2D case, that the following different cases due to the values of \(d_1d_2-d_4,\, d_1d_3-d_5\) and \(d_2d_3-d_6\) should be considered.
\(d_1d_2-d_4\ne 0\) | \(d_1d_3-d_5 \ne 0\) | \(d_2d_3-d_6 \ne 0\) | Case 1 |
\(d_2d_3-d_6 = 0\) | Case 2 | ||
\(d_1d_3-d_5=0\) | \(d_2d_3-d_6 \ne 0\) | Case 3 | |
\(d_2d_3-d_6 = 0\) | Case 4 | ||
\(d_1d_2-d_4=0\) | \(d_1d_3-d_5 \ne 0\) | \(d_2d_3-d_6 \ne 0\) | Case 5 |
\(d_2d_3-d_6 = 0\) | Case 6 | ||
\(d_1d_3-d_5 = 0\) | \(d_2d_3-d_6 \ne 0\) | Case 7 | |
\(d_2d_3-d_6 = 0\) | Case 8 |
We will give explicitly the solution processes for Cases 1 and 2, and only state the final results for other cases as they can be done in the similar way.
Case 1 (\(d_1d_2-d_4\ne 0,\,d_1d_3-d_5\ne 0,\,d_2d_3-d_6\ne 0\)).
Let \(F = \{f_1,f_2,f_3,f_4\}\). We can obtain the reduced Gröbner basis of the ideal generated by \(F\) with lexicographical order \(a_{12}\prec a_{13}\prec a_{21}\prec a_{23}\prec a_{31}\prec a_{32}\), say \(G_{F}:=\{{g_1,\ldots ,g_4}\}\) where:
Instead of solving \(f_i=0, i=1,2,3,4\) directly, we can find solutions for \(a_{12},a_{13},a_{21},a_{23},\) \(a_{31}\) and \(a_{32}\) by solving
Assume that \(\Delta _{3D}\geqslant 0\). Since \(\Delta _{3D} \ne S_{3D}^2\) in this case, we have that \(S_{3D} \pm \sqrt{\Delta _{3D}}\ne 0\) and the solution to (45) can be found by, e.g., Maple (2011) as
where \(u\) and \(v\) are any nonzero real numbers. That is, the solution for \(A\) is given as
Case 2 (\(d_1d_2-d_4 \ne 0,\,d_1d_3-d_5 \ne 0,\,d_2d_3-d_6=0\)).
Substituting \(d_6=d_2d_3\) into \(f_i, i=1,2,3,4\), we have that
Let \(\tilde{F} = \{\tilde{f}_1, \tilde{f}_2,\tilde{f}_3,\tilde{f}_4\}\). Then, the reduced Gröbner basis \(G_{\tilde{F}} =\{\tilde{g}_1, \tilde{g}_2, \tilde{g}_3, \tilde{g}_4\}\) of the ideal generated by \(\tilde{F}\) can be obtained by, e.g., Maple (2011) with
We can now obtain the following solution for \(A\) by solving \(\tilde{g}_i=0, i=1,2,3,4\).
where \(u,\, v\) are any nonzero real numbers and \(S_{3D}=d_2d_5+d_3d_4-d_7-d_1d_2d_3\) here.
In the similar way shown above, we can find the solutions for the other cases as follows.
Case 3 (\(d_1d_2-d_4\ne 0,\,d_1d_3-d_5=0,\,d_2d_3-d_6 \ne 0\)).
where \(u, t\) are any nonzero real numbers and \(S_{3D}=d_1d_6+d_3d_4-d_7-d_1d_2d_3\) here.
Case 4 (\(d_1d_2-d_4\ne 0,\,d_1d_3-d_5=0,\,d_2d_3-d_6=0\)), then \(S_{3D}=d_3d_4-d_7\).
For any real numbers \(t,\, u\) and \(v \ne 0\), we have that
-
when \(S_{3D}=d_3d_4-d_7=0\),
$$\begin{aligned} A=\begin{bmatrix} -d_1&\quad u&\quad 0\\ \dfrac{d_1d_2-d_4}{u}&\quad -d_2&\quad 0\\ v&\quad t&\quad -d_3 \end{bmatrix} \quad \text {or} \quad A=\begin{bmatrix} -d_1&\quad u&\quad v\\ \dfrac{d_1d_2-d_4}{u}&\quad -d_2&\quad t\\ 0&\quad 0&\quad -d_3 \end{bmatrix}; \end{aligned}$$ -
when \(S_{3D}=d_3d_4-d_7\ne 0\),
$$\begin{aligned}&A=\left[ \begin{array}{c@{\quad }c@{\quad }c} -d_1&{}u&{}0\\ \dfrac{d_1d_2-d_4}{u}&{}-d_2&{}\dfrac{d_3d_4-d_7}{uv}\\ v &{}0 &{}-d_3 \end{array}\right] \quad \text {or} \quad A=\left[ \begin{array}{c@{\quad }c@{\quad }c} -d_1 &{}\dfrac{d_1d_2-d_4}{u}&{}v\\ u &{}-d_2 &{}0\\ 0&{}\dfrac{d_3d_4-d_7}{uv}&{}-d_3 \end{array} \right] . \end{aligned}$$
Case 5 (\(d_1d_2-d_4=0,\,d_1d_3-d_5 \ne 0,\,d_2d_3-d_6 \ne 0\)).
For any nonzero real numbers \(v\) and \(t\), we have that
with \(S_{3D}=d_1d_6+d_2d_5-d_7-d_1d_2d_3\) here.
Case 6 (\(d_1d_2-d_4=0,\,d_1d_3-d_5\ne 0,\,d_2d_3-d_6=0\)), then \(S_{3D}=d_2d_5-d_7\).
For any real numbers \(u,\, t\) and \(v\ne 0\), we have that
-
when \(S_{3D}=d_2d_5-d_7=0\),
$$\begin{aligned} A=\begin{bmatrix} -d_1&\quad u&\quad v\\0&\quad -d_2&\quad 0\\ \dfrac{d_1d_3-d_5}{v}&\quad t&\quad -d_3 \end{bmatrix} \quad \text {or} \quad A=\begin{bmatrix} -d_1&\quad 0&\quad v\\u&\quad -d_2&\quad t\\ \dfrac{d_1d_3-d_5}{v}&\quad 0&\quad -d_3 \end{bmatrix}; \end{aligned}$$ -
when \(S_{3D}=d_2d_5-d_7\ne 0\),
$$\begin{aligned} A=\left[ \begin{array}{c@{\quad }c@{\quad }c} -d_1&{}0&{}v\\ \dfrac{d_2d_5-d_7}{vt}&{}-d_2&{}0\\ \dfrac{d_1d_3-d_5}{v}&{}t&{}-d_3 \end{array}\right] \quad \text {or} \quad A=\left[ \begin{array}{c@{\quad }c@{\quad }c} -d_1&{}0&{}v\\ u&{}-d_2&{}0\\ \dfrac{d_1d_3-d_5}{v}&{}\dfrac{d_2d_5-d_7}{uv}&{}-d_3 \end{array} \right] . \end{aligned}$$
Case 7 (\(d_1d_2-d_4=0,\,d_1d_3-d_5=0,\,d_2d_3-d_6\ne 0\)), then \(S_{3D}=d_1d_6-d_7\).
For any real numbers \(t,\, u\) and \(v\ne 0\), we have that
-
when \(S_{3D}=d_1d_6-d_7=0\),
$$\begin{aligned} A=\begin{bmatrix} -d_1&\quad 0&\quad 0\\u&\quad -d_2&\quad t\\v&\quad \dfrac{d_2d_3-d_6}{t}&\quad -d_3 \end{bmatrix} \quad \text {or} \quad A=\begin{bmatrix} -d_1&\quad u&\quad v\\0&\quad -d_2&\quad t\\0&\quad \dfrac{d_2d_3-d_6}{t}&\quad -d_3 \end{bmatrix}; \end{aligned}$$ -
when \(S_{3D}=d_1d_6-d_7\ne 0\),
$$\begin{aligned} A=\left[ \begin{array}{c@{\quad }c@{\quad }c} -d_1&{}\dfrac{d_1d_6-d_7}{vt}&{}0\\ 0&{}-d_2&{}t \\ v &{} \dfrac{d_2d_3-d_6}{t}&{}-d_3 \end{array}\right] \quad \text {or} \quad A=\left[ \begin{array}{c@{\quad }c@{\quad }c} -d_1&{}0&{}\dfrac{d_1d_6-d_7}{ut}\\ u &{} -d_2 &{} \dfrac{d_2d_3-d_6}{t}\\ 0&{}t&{}-d_3 \end{array}\right] . \end{aligned}$$
Case 8 (\(d_1d_2-d_4=0,\,d_1d_3-d_5=0,\,d_2d_3-d_6=0\)).
-
When \(S_{3D}=d_1d_2d_3-d_7=0\), the denominator is separable:
$$\begin{aligned} d(z_1,z_2,z_3)=(d_1z_1+1)(d_2z_2+1)(d_3z_3+1). \end{aligned}$$Then, we have the following results for any real numbers \(u,\ v,\ t\) with \(A=A_i,\,i \in \{1,\ldots , 6\}\).
$$\begin{aligned}&A_1=\left[ \begin{array}{c@{\quad }c@{\quad }c} -d_1&{}0&{}0\\ u&{}-d_2&{}t\\ v&{}0&{}-d_3 \end{array}\right] ,\quad A_2=\left[ \begin{array}{c@{\quad }c@{\quad }c} -d_1&{}0&{}0\\ u&{}-d_2&{}0\\ v&{}t&{}-d_3 \end{array}\right] ,\\&A_3=\left[ \begin{array}{c@{\quad }c@{\quad }c} -d_1&{}0&{}v\\ u&{}-d_2&{}t\\ 0&{}0&{}-d_3 \end{array}\right] ,\quad A_4=\left[ \begin{array}{c@{\quad }c@{\quad }c} -d_1&{}u&{}v\\ 0&{}-d_2&{}t\\ 0&{}0&{}-d_3 \end{array}\right] ,\\&A_5=\left[ \begin{array}{c@{\quad }c@{\quad }c} -d_1&{}u&{}v\\ 0&{}-d_2&{}0\\ 0&{}t&{}-d_3 \end{array}\right] ,\quad A_6=\left[ \begin{array}{c@{\quad }c@{\quad }c} -d_1&{}u&{}0\\ 0&{}-d_2&{}0\\ v&{}t&{}-d_3 \end{array}\right] . \end{aligned}$$ -
When \(S_{3D}=d_1d_2d_3-d_7\ne 0\), for any nonzero real numbers \(u, v\), we have that
$$\begin{aligned} A=\left[ \begin{array}{c@{\quad }c@{\quad }c} -d_1&{}0&{}v\\ u&{}-d_2&{}0\\ 0&{}\dfrac{d_1d_2d_3-d_7}{uv}&{}-d_3 \end{array}\right] \quad \text {or} \quad A=\left[ \begin{array}{c@{\quad }c@{\quad }c} -d_1&{}u&{}0\\ 0&{}-d_2&{}\dfrac{d_1d_2d_3-d_7}{uv}\\ v&{}0&{}-d_3 \end{array}\right] . \end{aligned}$$
We have now finished the proof of the sufficiency of Theorem 3. \(\square \)
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Doan, M.L., Nguyen, T.T., Lin, Z. et al. Notes on minimal realizations of multidimensional systems. Multidim Syst Sign Process 26, 519–553 (2015). https://doi.org/10.1007/s11045-014-0297-0
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DOI: https://doi.org/10.1007/s11045-014-0297-0