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A geometric theory for 2-D systems including notions of stabilisability

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Abstract

In this paper we consider the problem of internally and externally stabilising controlled invariant and output-nulling subspaces for two-dimensional (2-D) Fornasini–Marchesini models, via static feedback. A numerically tractable procedure for computing a stabilising feedback matrix is developed via linear matrix inequality techniques. This is subsequently applied to solve, for the first time, various 2-D disturbance decoupling problems subject to a closed-loop stability constraint.

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References

  • Basile G., Marro G. (1969). Controlled and conditioned invariant subspaces in linear system theory. Journal of Optimization Theory and Applications 3(5): 306–315

    Article  MATH  MathSciNet  Google Scholar 

  • Basile G., Marro G. (1982). Self-bounded controlled invariant subspaces: A straightforward approach to constrained controllability. Journal of Optimization Theory and Applications 38(1): 71–81

    Article  MATH  MathSciNet  Google Scholar 

  • Basile G., Marro G. (1992). Controlled and conditioned invariants in linear system theory. Prentice Hall, Englewood Cliffs

    MATH  Google Scholar 

  • Bliman P.A. (2002). Lyapunov equation for the stability of 2-D systems. Multidimensional Systems and Signal Processing 13: 201–222

    Article  MATH  MathSciNet  Google Scholar 

  • Conte G., Perdon A. (1988). A geometric approach to the theory of 2-D systems. IEEE Transactions on Automatic Control AC-33(10): 946–950

    Article  MATH  MathSciNet  Google Scholar 

  • Conte G., Perdon A., Kaczorek T. (1991). Geometric methods in the theory of singular 2D linear systems. Kybernetika 27(3): 262–270

    MathSciNet  Google Scholar 

  • EbiharaY., Ito Y., Hagiwara T. (2006). Exact stability analysis of 2-D systems using LMIs. IEEE Transactions on Automatic Control AC-51(9): 1509–1513

    Article  Google Scholar 

  • Fornasini E., Marchesini G. (1978). Doubly-indexed dynamical systems: State-space models and structural properties. Mathematical System Theory 12: 59–72

    Article  MATH  MathSciNet  Google Scholar 

  • Fornasini E., Marchesini G. (1980). Stability analysis of 2-D systems. IEEE Transactions on Circuits and Systems CAS-27(12): 1210–1217

    Article  MathSciNet  Google Scholar 

  • Gałkowski K. (1996). The Fornasini–Marchesini and the Roesser models: Algebraic methods for recasting. IEEE Transactions on Automatic Control 41: 107–112

    Article  MATH  Google Scholar 

  • Hu G.D., Liu M. (2005). Simple criteria for stability of two-dimensional linear systems. IEEE Transactions on Signal Processing 53(12): 4720–4723

    Article  MathSciNet  Google Scholar 

  • Kar H., Sigh V. (2003). Stability of 2-D systems described by the Fornasini–Marchesini first model. IEEE Transactions on Signal Processing 51(6): 1675–1676

    Article  MathSciNet  Google Scholar 

  • Karamanciog̃lu, A., & Lewis, F. L. (1990). A geometric approach to 2-D implicit systems. In Proceedings of the 29th Conference on Decision and Control. Honolulu, Hawaii, December 1990.

  • Karamanciog̃lu A., Lewis F.L. (1992). Geometric theory for the singular Roesser model. IEEE Transactions on Automatic Control AC-37(6): 801–806

    Article  MathSciNet  Google Scholar 

  • Leibfritz F. (2001). An LMI-based algorithm for designing suboptimal static \(\mathcal H_2/\mathcal H_{\infty}\) output feedback controllers. SIAM Journal on Control and Optimization 39(6): 1711–1735

    Article  MATH  MathSciNet  Google Scholar 

  • Malabre M. (1982). The model-following problem for linear constant (A, B,C, D) quadruples. IEEE Transactions on Automatic Control AC-27(2): 458–461

    Article  MathSciNet  Google Scholar 

  • Malabre M., Kučera V. (1984). Infinite structure and exact model matching problem: A geometric approach. IEEE Transactions on Automatic Control AC-29(3): 266-268

    Article  Google Scholar 

  • Malabre M., Martínez-García J., Del-Muro-Cuéllar B. (1997). On the fixed poles for disturbance rejection. Automatica 33(6): 1209–1211

    Article  MATH  MathSciNet  Google Scholar 

  • Marro, G., Morbidi, F., & Prattichizzo, D. (2007). H 2-pseudo optimal model following: A geometric approach. In Proceedings of the 3rd IFAC Symposium on System Structure and Control. Foz do Iguaçu, Brasil, October 17–19, 2007.

  • Morse A.S. (1973). Structure and design of linear model following systems. IEEE Transactions on Automatic Control AC-18(4): 346–354

    Article  MathSciNet  Google Scholar 

  • Paraskevopoulos P.N. (1979). Exact model matching for 2D systems via state feedback. Journal of the Franklin Institute 308: 475–486

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  • Sebek M. (1983). 2-D exact model matching. IEEE Transactions on Automatic Control AC-28(2): 215–217

    Article  Google Scholar 

  • Wonham W.M., Morse A.S. (1970). Decoupling and pole assignment in linear multivariable systems: A geometric approach. SIAM Journal of Control 8(1): 1–18

    Article  MATH  MathSciNet  Google Scholar 

  • Zhou T. (2006). Stability and stability margin for a two-dimensional system. IEEE Transactions on Signal Processing 54(9): 3483–3488

    Article  Google Scholar 

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Authors and Affiliations

  1. Department of Electrical and Electronic Engineering, The University of Melbourne, Parkville, VIC, 3010, Australia

    Lorenzo Ntogramatzidis & Michael Cantoni

  2. School of Information Science and Technology, Sun Yat-Sen University, Guangzhou, 510275, P.R. China

    Ran Yang

Authors
  1. Lorenzo Ntogramatzidis
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  2. Michael Cantoni
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  3. Ran Yang
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Correspondence to Lorenzo Ntogramatzidis.

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Ntogramatzidis, L., Cantoni, M. & Yang, R. A geometric theory for 2-D systems including notions of stabilisability. Multidim Syst Sign Process 19, 449–475 (2008). https://doi.org/10.1007/s11045-007-0046-8

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  • DOI: https://doi.org/10.1007/s11045-007-0046-8

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