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A novel elementary operation approach with Jordan transformation to order reduction for Roesser state-space model

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Abstract

This paper proposes a novel elementary operation approach to order reduction for the Roesser state-space model of multidimensional (n-D) systems by introducing a new kind of transformation, i.e., the Jordan transformation, which guarantees the establishment of an objective matrix with more general structure than the existing one. Then two basic order reduction techniques are developed which can overcome the difficulty encountered by the existing methods and reveal, for the first time, the fact that the order reduction is still possible even when the column (or row) blocks in the related n-D polynomial matrix are full rank. Furthermore, based on the Jordan transformation, an equivalence relationship between two Roesser models after using the elementary operations among the different blocks will be clarified. Although these operations do not directly lower the total order of the model, the partial orders can be changed so that it may nevertheless yield a possibility for further order reduction. It turns out that this new approach includes our previous elementary operation order reduction approach just as a special case. Examples are given to illustrate the details as well as the effectiveness of the proposed approach.

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References

  • Beck, C., & D’Andrea, R. (1998). Computational study and comparisons of lft reducibility methods. In Proceedings of American Control Conference (Vol. 2, pp. 1013–1017).

  • 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., & 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 

  • Chen, X., & Wen, J. T. (1994). Model reduction of multidimensional positive real systems. In Proceedings of IEEE Conference on Decision and Control (Vol. 4, pp. 3758–3763).

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

  • Doan, M. L., Nguyen, T. T., Lin, Z., & Xu, L. (2015). Notes on minimal realizations of multidimensional systems. Multidimensional Systems and Signal Processing, 26(2), 519–553.

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  • Horn, R. A., & Johnson, C. R. (2012). Matrix analysis. Cambridge: Cambridge University Press.

    Book  Google Scholar 

  • Ishchenko, A., Myrzik, J., & Kling, W. (2007). Dynamic equivalencing of distribution networks with dispersed generation using Hankel norm approximation. IET Generation, Transmission and Distribution, 1(5), 818–825.

    Article  Google Scholar 

  • Kung, S. Y., Levy, B. C., Morf, M., & Kailath, T. (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 

  • Kurek, J. E. (1985). Basic properties of q-dimensional linear digital systems. International Journal of Control, 42(1), 119–128.

    Article  MathSciNet  MATH  Google Scholar 

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

  • Lastman, G., & Sinha, N. (1985). A comparison of the balanced matrix method and the aggregation method of model reduction. IEEE Transactions on Automatic Control, 30(3), 301–304.

    Article  MathSciNet  MATH  Google Scholar 

  • Li, L., & Petersen, I. R. (2007). A Gramian-based approach to model reduction for uncertain systems. In Proceedings of the IEEE Conference on Decision and Control (pp. 4373–4378).

  • Lin, Z. (1998). Feedback stabilizability of MIMO \(n-\)D linear systems. Multidimensional Systems and Signal Processing, 9(2), 149–172.

    Article  MathSciNet  MATH  Google Scholar 

  • Lin, Z., Xu, L., & Anazawa, Y. (2007). Revisiting the absolutely minimal realization for two-dimensional digital filters. In Proceedings of the IEEE ISCAS 2007 (pp. 597–600). New Orleans, USA.

  • Magni, J. (2006). User manual of the linear fractional representation toolbox version 2.0. Technical Report, France, October 2005 (revised Feb. 2006), http://w3.onera.fr/smac/.

  • Sandberg, H. (2010). An extension to balanced truncation with application to structured model reduction. IEEE Transactions on Automatic Control, 55(4), 1038–1043.

    Article  MathSciNet  Google Scholar 

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

  • Toth, R., Abbas, H., & Werner, H. (2012). On the state-space realization of LPV input–output models: Practical approaches. IEEE Transactions on Control Systems Technology, 20(1), 139–153.

    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., 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(1–3), 97–129.

    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, 59(3), 638–651.

    Article  MathSciNet  Google Scholar 

  • Yan, S., Xu, L., & Xiao, Y. (2012). Order reduction for Roesser state-space model based on elementary operations. In Proceedings of the 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 

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

    MATH  Google Scholar 

  • Zhang, C. J., & Jiang, J. M. (2012). Jordan normal form of a matrix by elementary transformation method. Journal of Hechi University. 32(5), 51–53.

  • Zhou, K., Doyle, J., Glover, K., et al. (1996). Robust and optimal control. Upper Saddle River, NJ: Prentice Hall.

    MATH  Google Scholar 

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Authors and Affiliations

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

    Shi Yan, Dongdong Zhao & Qiaoqiao Li

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

    Li Xu

  3. Department of Automation, Shanghai Jiao Tong University, Shanghai, 200240, China

    Yunze Cai

Authors
  1. Shi Yan
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  2. Dongdong Zhao
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  3. Li Xu
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  4. Yunze Cai
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  5. Qiaoqiao Li
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Corresponding author

Correspondence to Li Xu.

Additional information

This work was partly supported by the National Natural Science Foundation of China (Nos. 61104122, 61374160), the Fundamental Research Funds for the Central Universities (lzujbky-2016), and the Japan Society for the Promotion of Science (JSPS.KAKENHI15K06072).

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Yan, S., Zhao, D., Xu, L. et al. A novel elementary operation approach with Jordan transformation to order reduction for Roesser state-space model. Multidim Syst Sign Process 28, 1417–1442 (2017). https://doi.org/10.1007/s11045-016-0418-z

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  • DOI: https://doi.org/10.1007/s11045-016-0418-z

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