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3-D modelling of rectangular circuits as the particular class of spatially interconnected systems on the plane

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Abstract

This paper is devoted to modelling the spatially interconnected system, particularly the regular circuit displayed in the plane as a 3-D system with two spatial indeterminates and one temporal indeterminate. First, the singular and nonsingular 3-D models of the Fornasini–Marchesini type are derived. Next, these models are transformed into the equivalent Roesser models. This work is a basis for further control applications of spatially interconnected systems on the plane.

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References

  • Ahn, C. K., Shi, P., & Basin, M. V. (2015). Two-dimensional dissipative control and filtering for roesser model. IEEE Transactions on Automatic Control, 60(7), 1745–1759.

    MathSciNet  MATH  Google Scholar 

  • Antsaklis, P . J., & Michel, A . N. (2007). A linear systems primer (Vol. 1). Boston: Birkhäuser.

    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.

    MathSciNet  MATH  Google Scholar 

  • Bose, N., Buchberger, B., & Guiver, J. (2003). Multidimensional systems theory and applications. Dordrecht: Kluwer.

    Google Scholar 

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

    MATH  Google Scholar 

  • Boudellioua, M. S. (2005). Equivalence to Smith form over a multivariate polynomial ring. In The fourth internatinal workshop on multidimensional systems (pp. 259–262).

  • Boudellioua, M. S., Galkowski, K., & Rogers, E. (2017). Equivalent 2-D nonsingular Roesser models for discrete linear repetitive processes. International Journal of Control, 1–9. https://doi.org/10.1080/00207179.2017.1414307

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

    Google Scholar 

  • Cockburn, J. C. (2000). Multidimensional realizations of systems with parametric uncertainty. In Proceedings of MTNS (p. 6).

  • Cunha, D. P. (2013). Reduced order state-space models for 2-D systems. Master’s thesis, Dept. Elect. Comput. Eng., Univ. Porto, Portugal.

  • Ding, D.-W., Wang, H., & Li, X. (2015). \({H}_{-}/{H}_{\infty }\) fault detection observer design for two-dimensional roesser systems. Systems & Control Letters, 82, 115–120.

    MathSciNet  MATH  Google Scholar 

  • Dorf, R . C., & Bishop, R . H. (2011). Modern control systems. London: Pearson.

    MATH  Google Scholar 

  • Du, C., Xie, L., & Zhang, C. (2001). \({H}_\infty \) control and robust stabilization of two-dimensional systems in Roesser models. Automatica, 37(2), 205–211.

    MathSciNet  MATH  Google Scholar 

  • Feng, Z., & Xu, L. (2018). Optimization of coordinate transformation matrix for \({H}_\infty \) static-output-feedback control of 2-D discrete systems in FM second model. Multidimensional Systems and Signal Processing, 29(4), 1727–1737.

    MathSciNet  Google Scholar 

  • Feng, Z.-Y., Wu, Q., & Xu, L. (2012). \({H}_{\infty }\) control of linear multidimensional discrete systems. Multidimensional Systems and Signal Processing, 23(3), 381–411.

    MathSciNet  MATH  Google Scholar 

  • Fornasini, E., & Marchesini, G. (1976). State space realization theory of two-dimensional filters. IEEE Transactions on Automatic Control, 21(4), 484–492.

    MathSciNet  MATH  Google Scholar 

  • Friedland, B. (1986). Control systems design: An introduction to state-space methods. New York: McGraw-Hill.

    MATH  Google Scholar 

  • Galkowski, K. (2001). State space realization of linear 2-D systems with extensions to the general nD (\(n>2\)) case. London: Springer.

  • Kaczorek, T. (1985). Two-dimensional linear systems. London: Springer.

    MATH  Google Scholar 

  • Kaczorek, T. (1988). Singular models of 2-D systems. In Proceedings of the 12th World Congress on Scientific Computation, Paris (France).

  • Kailath, T. (1980). Linear systems. Upper Saddle River: Prentice-Hall.

    MATH  Google Scholar 

  • Lin, Z., & Bruton, L. (1989). BIBO stability of inverse 2-D digital filters in the presence of nonessential singularities of the second kind. IEEE Transactions on Circuits and Systems, 36(2), 244–254.

    MathSciNet  MATH  Google Scholar 

  • Lin, Z., Lam, J., Galkowski, K., & Xu, S. (2001). A constructive approach to stabilizability and stabilization of a class of nD systems. Multidimensional Systems and Signal Processing, 12(3–4), 329–343.

    MathSciNet  MATH  Google Scholar 

  • Lu, W.-S. (1995). On robust stability of 2-D discrete systems. IEEE Transactions on Automatic Control, 40(3), 502–506.

    MathSciNet  MATH  Google Scholar 

  • Magni, J.-F. (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.

  • Paszke, W., Dabkowski, P., Rogers, E., & Gałkowski, K. (2015). New results on strong practical stability and stabilization of discrete linear repetitive processes. Systems & Control Letters, 77, 22–29.

    MathSciNet  MATH  Google Scholar 

  • Roesser, R. (1975). A discrete state-space model for linear image processing. IEEE Transactions on Automatic Control, 20(1), 1–10.

    MathSciNet  MATH  Google Scholar 

  • Rogers, E., Galkowski, K., & Owens, D. H. (2007). Control Systems theory and applications for linear repetitive processes. Control and information sciences. Berlin, Heidelberg: Springer.

    MATH  Google Scholar 

  • Sulikowski, B. (2018). Robust \({H}_2\) control of ladder circuits modeled as a subclass of \(2\)D systems. In Proceedings of the 17th annual European control conference (ECC2018) (pp. 888–893). Limassol, Cyprus.

  • Sulikowski, B., Galkowski, K., & Kummert, A. (2015). Proportional plus integral control of ladder circuits modeled in the form of two-dimensional (2D) systems. Multidimesional Systems and Signal Processing, 26(1), 267–290.

    MathSciNet  MATH  Google Scholar 

  • Wang, K., Chen, M. Z., & Chen, G. (2017). Realization of a transfer function as a passive two-port RC ladder network with a specified gain. International Journal of Circuit Theory and Applications, 45(11), 1467–1481.

    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.

    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, 59(3), 638–651.

    MathSciNet  Google Scholar 

  • Yan, S., Xu, L., Zhang, Y., Cai, Y., & Zhao, D. (2018). Order evaluation to new elementary operation approach for MIMO multidimensional systems. International Journal of Control, (just-accepted):1–20.

  • 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.

    Google Scholar 

  • Yan, S., Zhao, D., Xu, L., Cai, Y., & Li, Q. (2017). A novel elementary operation approach with Jordan transformation to order reduction for Roesser state-space model. Multidimensional Systems and Signal Processing, 28(4), 1417–1442.

    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.

    MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  • Zhao, D., Yan, S., Matsushita, S., & Xu, L. (2017). Order reduction for Roesser state-space model based on a certain system of equations. In 2017 10th international workshop on multidimensional (nD) systems (nDS) (pp. 1–5).

  • Zhao, D., Yan, S., & Xu, L. (2018). Eigenvalue trim approach to exact order reduction for Roesser state-space model of multidimensional systems. Multidimensional Systems and Signal Processing, 29(4), 1905–1934.

    MathSciNet  Google Scholar 

  • Zhou, K., Doyle, J . C., & Glover, K. (1996). Robust and optimal control (Vol. 40). New Jersey: Prentice hall.

    MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Intelligent Mechatronics, Akita Prefectural University, Akita, 015-0055, Japan

    Dongdong Zhao & Li Xu

  2. Institute of Control and Computation Engineering, University of Zielona Gora, 65-516, Zielona Gora, Poland

    Krzysztof Galkowski & Bartlomiej Sulikowski

Authors
  1. Dongdong Zhao
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  2. Krzysztof Galkowski
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  3. Bartlomiej Sulikowski
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  4. Li Xu
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Correspondence to Dongdong Zhao.

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This work is partially supported by National Science Centre in Poland, Grant No. 2015/17/B/ST7/03703, the Japan Society for the Promotion of Science (JSPS.KAKENHI15K06072).

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Zhao, D., Galkowski, K., Sulikowski, B. et al. 3-D modelling of rectangular circuits as the particular class of spatially interconnected systems on the plane. Multidim Syst Sign Process 30, 1583–1608 (2019). https://doi.org/10.1007/s11045-018-0619-8

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  • DOI: https://doi.org/10.1007/s11045-018-0619-8

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