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Eigenvalue trim approach to exact order reduction for roesser state-space model of multidimensional systems

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Abstract

In this paper, a new notion of eigenvalue trim or co-trim for n-D Roesser (state-space) model is first introduced, which reveals the internal connection between the eigenvalues of the system matrix and the reducibility of the considered Roesser model. Then, new reducibility conditions and the corresponding order reduction algorithms based on eigenvalue trim or co-trim are proposed for exact order reduction of a given n-D Roesser model, and it will be shown that this eigenvalue trim approach can be applied even to those systems for which the existing approaches cannot do any further order reduction. Furthermore, a new transformation for n-D Roesser models, by swapping certain rows and columns and interchanging certain entries that belong to different blocks corresponding to different variables, will be established, which can transform an n-D Roesser model whose order cannot be reduced any more by the proposed approach to another equivalent Roesser model with the same order so that this transformed Roesser model can still be reduced further. Examples are given to illustrate the details as well as the effectiveness of the proposed approach.

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References

  • Andersson, L., Rantzer, A., & Beck, C. (1999). Model comparison and simplification. International Journal of Robust and Nonlinear Control, 9(3), 157–181.

    Article  MathSciNet  MATH  Google Scholar 

  • Anton, H. (2010). Elementary linear algebra. London: Wiley.

    Google Scholar 

  • Antoniou, G., Paraskevopoulos, P., & Varoufakis, S. (1988). Minimal state-space realization of factorable 2-D transfer functions. IEEE Transactions on Circuits and Systems, 35(8), 1055–1058.

    Article  MathSciNet  Google Scholar 

  • Bachelier, O., Cluzeau, T., David, R., & Yeganefar, N. (2016). Structural stabilization of linear 2D discrete systems using equivalence transformations. Multidimensional Systems and Signal Processing, 28(4), 1–24.

    MathSciNet  MATH  Google Scholar 

  • 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  MATH  Google Scholar 

  • Beck, C., Doyle, J., & Glover, K. (1996). Model reduction of multidimensional and uncertain systems. IEEE Transactions on Automatic Control, 41(10), 1466–1477.

    Article  MathSciNet  MATH  Google Scholar 

  • Beck, C. L., & Doyle, J. (1999). A necessary and sufficient minimality condition for uncertain systems. IEEE Transactions on Automatic Control, 44(10), 1802–1813.

    Article  MathSciNet  MATH  Google Scholar 

  • Bose, N., Guiver, J., Kamen, E., Valenzuela, H., & Buchberger, B. (1985). Multidimensional systems theory. Dordrecht: Reidel.

    Book  Google Scholar 

  • Bose, N. K. (1982). Applied multidimensional systems theory. Berlin: Springer.

    MATH  Google Scholar 

  • Boudellioua, M., Galkowski, K., & Rogers, E. (2017). On the connection between discrete linear repetitive processes and 2-D discrete linear systems. Multidimensional Systems and Signal Processing, 28(1), 341–351.

    Article  MathSciNet  MATH  Google Scholar 

  • Chen, C.-T. (1995). Linear system theory and design. Oxford: Oxford University Press, Inc.

    Google Scholar 

  • Cockburn, J. C., & Morton, B. G. (1997). Linear fractional representations of uncertain systems. Automatica, 33(7), 1263–1271.

    Article  MathSciNet  MATH  Google Scholar 

  • Cunha, D. P. (2013). Reduced order state-space models for 2-D systems. Master’s thesis. Porto, Portugal: Department of Electrical and Computer Engineering university.

  • D’Andrea, R. & Khatri, S. (1997). Kalman decomposition of linear fractional transformation representations and minimality. In Proceeding of American control conference (Vol. 6, pp. 3557–3561). IEEE.

  • 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. (1997). Elementary operation approach to state-space realizations of 2-D systems. IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, 44(2), 120–129.

    Article  MathSciNet  MATH  Google Scholar 

  • Ghamgui, M., Yeganefar, N., Bachelier, O., & Mehdi, D. (2013). Robust stability of hybrid roesser models against parametric uncertainty: A general approach. Multidimensional Systems and Signal Processing, 24(4), 667–684.

    Article  MathSciNet  MATH  Google Scholar 

  • Ghous, I., & Xiang, Z. (2016). Robust state feedback H\(_{\infty }\) control for uncertain 2-D continuous state delayed systems in the roesser model. Multidimensional Systems and Signal Processing, 27(2), 297–319.

    Article  MathSciNet  MATH  Google Scholar 

  • Gilbert, J., & Gilbert, L. (2014). Linear algebra and matrix theory. Cambridge: Academic Press.

    MATH  Google Scholar 

  • Givone, D. D., & Roesser, R. P. (1973). Minimization of multidimensional linear iterative circuits. IEEE Transactions on Computers, C–22(7), 673–678.

    Article  MathSciNet  MATH  Google Scholar 

  • Hecker, S., & Varga, A. (2006). Symbolic manipulation techniques for low order LFT-based parametric uncertainty modelling. International Journal of Control, 79(11), 1485–1494.

    Article  MathSciNet  MATH  Google Scholar 

  • Hecker, S., Varga, A., & Magni, J. F. (2005). Enhanced lfr-toolbox for matlab. Aerospace Science and Technology, 9(2), 173–180.

    Article  MATH  Google Scholar 

  • Kaczorek, T. (2008). Realization problem for positive 2D hybrid systems. COMPEL-The International Journal for Computation and Mathematics in Electrical and Electronic Engineering, 27(3), 613–623.

    Article  MathSciNet  MATH  Google Scholar 

  • Kailath, T. (1980). Linear systems (Vol. 156). Englewood Cliffs, NJ: Prentice-Hall.

    MATH  Google Scholar 

  • Kaltofen, E., Krishnamoorthy, M. S., & Saunders, B. D. (1990). Parallel algorithms for matrix normal forms. Linear Algebra and its Applications, 136, 189–208.

    Article  MathSciNet  MATH  Google Scholar 

  • Kaltofen, E. L., & Storjohann, A. (2015). Complexity of computational problems in exact linear algebra. Berlin: Springer.

    Google Scholar 

  • Kung, S. Y., Levy, B., Morf, M., Kailath, T., Kung, S.-Y., Levy, B. C., et al. (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 the IEEE, 65(6), 945–961.

    Article  Google Scholar 

  • Lambrechts, P., Terlouw, J., Bennani, S., & Steinbuch, M. (1993). Parametric uncertainty modeling using LFTs. In Proceeding of American control conference (pp. 267–272).

  • Li, X., Lam, J., & Cheung, K. C. (2016). Generalized H\(_{\infty }\) model reduction for stable two-dimensional discrete systems. Multidimensional Systems and Signal Processing, 27(2), 359–382.

    Article  MathSciNet  MATH  Google Scholar 

  • Lu, W.-S. (1992). Two-dimensional digital filters (Vol. 80). Boca Raton: CRC Press.

    MATH  Google Scholar 

  • Lunze, J., & Lamnabhi-Lagarrigue, F. (2009). Handbook of hybrid systems control: Theory, tools, applications. Cambridge: Cambridge University Press.

    Book  MATH  Google Scholar 

  • Magni, J. (2006). User manual of the linear fractional representation toolbox version 2.0. Systems Control and Flight Dynamics Department, http://w3.onera.fr/smac/lfrt. Accessed 17 Mar 2014.

  • Manrique, T. & Patino, D. (2010). Mathematical modelling on hybrid dynamical systems: An application to the bouncing ball and a two-tanks system. In Andescon (pp. 1–8). IEEE.

  • Nguyen, T. L., Xu, L., Lin, Z., & Tay, D. B. (2017). On minimal realizations of first-degree 3D systems with separable denominators. Multidimensional Systems and Signal Processing, 28(1), 305–314.

    Article  MathSciNet  MATH  Google Scholar 

  • Sugie, T. & Kawanishi, M. (1995). \(\mu \) analysis/synthesis based on exact expression of physical parameter variations. In Proceedings of ECC (pp. 159–164).

  • Van Hien, L., & Trinh, H. (2017). Observers design for 2-D positive time-delay Roesser systems. IEEE Transactions on Circuits and Systems II: Express Briefs, 99, 1–5. https://doi.org/10.1109/TCSII.2017.2723425.

  • Varga, A. & Looye, G. (1999). Symbolic and numerical software tools for lft-based low order uncertainty modeling. In Proceedings of the 1999 IEEE international symposium on computer aided control system design (pp. 1–6). IEEE.

  • Xu, L., Fan, H., Lin, Z., & Bose, N. (2008). A direct-construction approach to multidimensional realization and LFR uncertainty modeling. Multidimensional Systems and Signal Processing, 19(3–4), 323–359.

    Article  MathSciNet  MATH  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), 97–129.

    Article  MathSciNet  MATH  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  MathSciNet  MATH  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: Regular Papers, 59(3), 638–651.

    Article  MathSciNet  Google Scholar 

  • Yan, S., Xu, L., & Anazawa, Y. (2007). A two-stage approach to the establishment of state-space formulation of 2-D frequency transformation. IEEE Signal Processing Letters, 14(12), 960–963.

    Article  Google Scholar 

  • Yan, S., Xu, L., & Xiao, Y. (2012). Order reduction for Roesser state-space model based on elementary operations. In IEEE International symposium on circuits and systems (ISCAS) (pp. 1468–1471).

  • Yan, S., Xu, L., Zhao, Q., & Tian, Y. (2014). Elementary operation approach to order reduction for Roesser state-space model of multidimensional systems. IEEE Transactions on Circuits and Systems I: Regular Papers, 61(3), 789–802.

    Article  Google Scholar 

  • Yan, S., Zhao, D., Xu, L., Cai, Y., & Li, Q. (2016). A novel elementary operation approach with Jordan transformation to order reduction for Roesser state-space model. Multidimensional Systems and Signal Processing, 28(4), 1–26.

    MathSciNet  MATH  Google Scholar 

  • Zerz, E. (1999). LFT representations of parametrized polynomial systems. IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, 46(3), 410–416.

    Article  MathSciNet  MATH  Google Scholar 

  • Zerz, E. (2000). Topics in multidimensional linear systems theory. London: Springer.

    MATH  Google Scholar 

  • Zhou, K., Doyle, J . C., Glover, K., et al. (1996). Robust and optimal control (Vol. 40). Englewood Cliffs, NJ: Prentice Hall.

    MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Electronics and Information Systems, Akita Prefectural University, Akita, 015-0055, Japan

    Dongdong Zhao & Li Xu

  2. School of Information Science and Engineering, Lanzhou University, Lanzhou, 730000, China

    Shi Yan

Authors
  1. Dongdong Zhao
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  2. Shi Yan
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  3. Li Xu
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Correspondence to Li Xu.

Additional information

This work was partly supported by the Japan Society for the Promotion of Science (JSPS.KAKENHI15K06072), the National Natural Science Foundation of China (No. 61104122) and the Fundamental Research Funds for the Central Universities (lzujbky-2016-136).

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Zhao, D., Yan, S. & Xu, L. Eigenvalue trim approach to exact order reduction for roesser state-space model of multidimensional systems. Multidim Syst Sign Process 29, 1905–1934 (2018). https://doi.org/10.1007/s11045-017-0536-2

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