Skip to main content
Log in

Notes on minimal realizations of multidimensional systems

  • Published:
Multidimensional Systems and Signal Processing Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Antoniou, G. E. (1993). 2-D discrete time lossless bounded real functions: Minimal state space realization. Electronics Letters, 29, 2004–2009.

    Article  Google Scholar 

  • Antoniou, G. E. (2001). 2-D lattice discrete filters: Minimal delay and state space realization. IEEE Signal Processing Letters, 8, 23–25.

    Article  MathSciNet  Google Scholar 

  • 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.

    Article  MATH  Google Scholar 

  • 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.

    Article  Google Scholar 

  • Ball, J. A., Groenewald, G., & Malakorn, T. (2005). Structured noncommutative multidimensional linear systems. SIAM Journal on Control and Optimization, 44(4), 1474–1528.

    Article  MATH  MathSciNet  Google Scholar 

  • 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.

    Article  MathSciNet  Google Scholar 

  • 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.

    Article  Google Scholar 

  • Bose, N. K. (1982). Applied multidimensional systems theory. New York: Van Nostrand Reinhold.

    MATH  Google Scholar 

  • Bose, N. K. (2003). Multidimensional systems theory and applications. New York: Springer.

    MATH  Google Scholar 

  • Buchberger, B. (1985). Gröbner bases: An algorithmic method in polynomial ideal theory. In N. K. Bose (Ed.), Multidimensional systems theory. Reidel: Dordrecht.

    Google Scholar 

  • 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.

    Article  MATH  MathSciNet  Google Scholar 

  • Eising, R. (1978). Realization and stabilization of 2-D systems. IEEE Transactions on Automatic Control, 23(5), 793–799.

    Article  MATH  MathSciNet  Google Scholar 

  • 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.

    Article  MATH  MathSciNet  Google Scholar 

  • 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.

    Article  MathSciNet  Google Scholar 

  • 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.

    Article  MATH  MathSciNet  Google Scholar 

  • Galkowski, K. (2005). Minimal state-space realization for a class of \(n\text{ D }\) systems. Operation Theory: Advances and Applications, 160, 179–194.

    MathSciNet  Google Scholar 

  • 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.

    Article  Google Scholar 

  • Givone, D. D., & Roesser, R. P. (1973). Minimization of multidimensional linear iterative circuits. IEEE Transactions on Computers, 22, 673–678.

    Article  MATH  MathSciNet  Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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.

    Chapter  Google Scholar 

  • Jury, E. (1978). Stability of multidimensional scalar and matrix polynomials. Proceedings of the IEEE, 66, 1018–1047.

    Article  Google Scholar 

  • Kailath, T. (1980). Linear systems. Englewood Cliffs: Prentice-Hall Inc.

    MATH  Google Scholar 

  • 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.

    Article  MATH  MathSciNet  Google Scholar 

  • 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.

    Article  Google Scholar 

  • Lin, Z. (1998). Feedback stabilizability of MIMO \(n\)-D linear systems. Multidimensional Systems and Signal Processing, 9, 149–172.

    Article  MATH  MathSciNet  Google Scholar 

  • Lin, Z. (1999). Notes on \(n\text{ D }\) polynomial matrix factorizations. Multidimensional Systems and Signal Processing, 10, 379–393.

    Article  MATH  MathSciNet  Google Scholar 

  • 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.

    Article  MathSciNet  Google Scholar 

  • Lu, W. S., & Antoniou, A. (1992). Two-dimensional digital filters. New York: Marcel Dekker Inc.

    MATH  Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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.

    Article  MATH  MathSciNet  Google Scholar 

  • 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.

    Article  MATH  MathSciNet  Google Scholar 

  • 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.

    Article  MATH  MathSciNet  Google Scholar 

  • 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.

    Article  MathSciNet  Google Scholar 

  • Youla, D., & Gnavi, G. (1979). Notes on \(n\)-dimensional system theory. IEEE Transactions on Circuits and Systems, 26, 105–111.

    Article  MATH  MathSciNet  Google Scholar 

  • 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.

    Article  Google Scholar 

Download references

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.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Li Xu.

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

$$\begin{aligned} \Delta _{3D}=S_{3D}^2-4(d_1d_2-d_4)(d_1d_3-d_5)(d_2d_3-d_6), \end{aligned}$$

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:

$$\begin{aligned}&g_1=a_{12}a_{21}-d_1d_2+d_4 \\&g_2=(d_2d_3-d_6)a_{13}^2a_{21}^2-S_{3D}a_{13}a_{21}a_{23}+(d_1d_2-d_4)(d_1d_3-d_5)a_{23}^2\\&g_3=a_{13}a_{31}-d_1d_3+d_5 \\&g_4=-S_{3D}a_{12}a_{31}+a_{12}^2a_{23}a_{31}^2+(d_1d_2-d_4)(d_1d_3-d_5)a_{32}. \end{aligned}$$

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

$$\begin{aligned} g_i=0,\quad i=1,2,3,4. \end{aligned}$$
(45)

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

$$\begin{aligned} a_{12}&= u, \quad a_{13}=v, \quad a_{21}=\dfrac{d_1d_2-d_4}{u}, \quad a_{23}=\dfrac{2(d_1d_2-d_4)(d_2d_3-d_6)v}{(S_{3D}\pm \sqrt{\Delta _{3D}})u},\\ a_{31}&=\dfrac{d_1d_3-d_5}{v}, \quad a_{32} =\dfrac{(S_{3D}\pm \sqrt{\Delta _{3D}})u}{2(d_1d_2-d_4)v}, \end{aligned}$$

where \(u\) and \(v\) are any nonzero real numbers. That is, the solution for \(A\) is given as

$$\begin{aligned} A=\left[ \begin{array}{c@{\quad }c@{\quad }c} -d_1 &{} u &{} v\\ \dfrac{d_1d_2-d_4}{u} &{} -d_2 &{} \dfrac{2(d_1d_2-d_4)(d_2d_3-d_6)v}{(S_{3D} \pm \sqrt{\Delta _{3D}})u} \\ \dfrac{d_1d_3-d_5}{v} &{} \dfrac{(S_{3D}\pm \sqrt{\Delta _{3D}})u}{2(d_1d_2-d_4)v} &{} -d_3 \end{array} \right] . \end{aligned}$$

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

$$\begin{aligned} \tilde{f}_1&= a_{12}a_{21}-d_1d_2+d_4,\\ \tilde{f}_2&= a_{13}a_{31}-d_1d_3+d_5,\\ \tilde{f}_3&= a_{23}a_{32},\\ \tilde{f}_4&= a_{12}a_{23}a_{31}+a_{13}a_{32}a_{21}+d_1d_2d_3-d_2d_5-d_3d_4+d_7. \end{aligned}$$

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

$$\begin{aligned} \tilde{g}_1&= a_{12}a_{21}-d_1d_2+d_4, \end{aligned}$$
(46a)
$$\begin{aligned} \tilde{g}_2&= -S_{3D}a_{13}a_{21}a_{23}+(d_1d_2-d_4)(d_1d_3-d_5)a_{23}^2, \end{aligned}$$
(46b)
$$\begin{aligned} \tilde{g}_3&= a_{13}a_{31}-d_1d_3+d_5, \end{aligned}$$
(46c)
$$\begin{aligned} \tilde{g}_4&= -S_{3D}a_{12}a_{31}+a_{12}^2a_{23}a_{31}^2+(d_1d_2-d_4)(d_1d_3-d_5)a_{32} . \end{aligned}$$
(46d)

We can now obtain the following solution for \(A\) by solving \(\tilde{g}_i=0, i=1,2,3,4\).

$$\begin{aligned} A=\left[ \begin{array}{c@{\quad }c@{\quad }c} -d_1&{}u&{}v \\ \dfrac{d_1d_2-d_4}{u}&{} -d_2&{}0 \\ \dfrac{d_1d_3-d_5}{v} &{} \dfrac{S_{3D}u}{(d_1d_2-d_4)v}&{}-d_3 \end{array} \right] \quad \text {or} \quad A\!=\!\left[ \begin{array}{c@{\quad }c@{\quad }c} -d_1&{}u&{}v \\ \dfrac{d_1d_2-d_4}{u} &{} -d_2 &{} \dfrac{S_{3D}v}{(d_1d_3-d_5)u}\\ \dfrac{d_1d_3-d_5}{v}&{} 0 &{} -d_3 \end{array}\right] , \end{aligned}$$

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 (\(d_1d_2-d_4\ne 0,\,d_1d_3-d_5=0,\,d_2d_3-d_6 \ne 0\)).

$$\begin{aligned} A=\left[ \begin{array}{c@{\quad }c@{\quad }c} -d_1&{}u&{}0 \\ \dfrac{d_1d_2-d_4}{u}&{}-d_2&{}\dfrac{d_2d_3-d_6}{t}\\ \dfrac{S_{3D}t}{(d_2d_3-d_6)u}&{}t&{}-d_3 \end{array} \right] \quad \text {or} \quad A=\left[ \begin{array}{c@{\quad }c@{\quad }c} -d_1&{}u&{}\dfrac{S_{3D}u}{(d_1d_2-d_4)t}\\ \dfrac{d_1d_2-d_4}{u}&{}-d_2&{}\dfrac{d_2d_3-d_6}{t}\\ 0&{}t&{}-d_3 \end{array} \right] \end{aligned}$$

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

$$\begin{aligned} A=\left[ \begin{array}{c@{\quad }c@{\quad }c} -d_1&{}0&{}v\\ \dfrac{S_{3D}t}{(d_2d_3-d_6)v}&{}-d_2&{}t\\ \dfrac{d_1d_3-d_5}{v}&{}\dfrac{d_2d_3-d_6}{t}&{}-d_3 \end{array} \right] \quad \text {or} \quad A=\left[ \begin{array}{c@{\quad }c@{\quad }c} -d_1&{}\dfrac{S_{3D}v}{(d_1d_3-d_5)t}&{}v\\ 0&{}-d_2&{}t\\ \dfrac{d_1d_3-d_5}{v} &{} \dfrac{d_2d_3-d_6}{t}&{}-d_3 \end{array} \right] \end{aligned}$$

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. \(\square \)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

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

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11045-014-0297-0

Keywords

Navigation