Structural stability, asymptotic stability and exponential stability for linear multidimensional systems: the good, the bad and the ugly

References

• 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), 1629–1652.
   Web of Science ®Google Scholar
• Bliman, P.-.A (2002). Lyapunov equation for the stability of 2-D systems. Multidimensional Systems and Signal Processing, 13 (2), 201–222.
   Web of Science ®Google Scholar
• Bochniak, J. , & Galkowski, K. (2005). LMI-based analysis for continuous-discrete linear shift invariant nD-systems. Journal of Circuits, Systems and Computers, 14 (2), 307–332.
   Web of Science ®Google Scholar
• Bose, N. K. (2010). Multidimensional systems theory and applications . Dordrecht, The Netherlands: Kluwer Academic Publishers.
   Google Scholar
• Chesi, G. , & Middleton, R. H. (2014). Necessary and sufficient LMI conditions for stability and performance analysis of 2D mixed continuous-discrete-time systems. IEEE Transaction on Automatic Control, 59 (4), 997–1007.
   Web of Science ®Google Scholar
• Chesi, G. , & Middleton, R. H. (2016). Robust stability and performance analysis of 2D mixed continuous–discrete-time systems with uncertainty. Automatica, 67 , 233–243.
   Web of Science ®Google Scholar
• Dymkov, M. , Galkowski, K. , Rogers, E. , Dymkou, V. , & Dymkou, S. (2011). Modelling and control of a sorption process using 2D systems. In Proceedings of the nds’11, 7th international workshop on multidimensional systems , Poitiers, France: IEEE.
   Google Scholar
• Ebihara, Y. , Ito, Y. , & Hagiwara, T. (2006). Exact stability analysis of 2-D systems using LMIs. IEEE Transactions on Automatic Control, 51 (9), 1509–1513.
   Web of Science ®Google Scholar
• Emelianova, J. , Pakshin, P. , Gałkowski, K. , & Rogers, E. (2013). Stabilization and H ∞ control of 2D systems. In Proceedings of the nDS’13, 8th international workshop on multidimensional systems , Erlangen, Germany: IEEE.
   Google Scholar
• Emelianova, J. P. , Pakshin, P. V. , Gałkowski, K. , & Rogers, E. (2014). Stability of nonlinear 2D systems described by the continuous-time Roesser model. Automation and Remote Control, 75 (5), 845–858.
   Web of Science ®Google Scholar
• Emelianova, J. , Pakshin, P. , Gałkowski, K. , & Rogers, E. (2014). Vector Lyapunov function based stability of a class of applications relevant 2D, . In Proceedings of the 19th IFAC world congress (Vol. 47, pp. 8247–8252). Cape Town, South Africa.
   Google Scholar
• Emelianova, J. , Pakshin, P. , Gałkowski, K. , & Rogers, E. (2015). Stability of nonlinear discrete repetitive processes with Markovian switching. Systems & Control Letters, 75 , 108–116.
   Web of Science ®Google Scholar
• Fornasini, E. , & Marchesini, G. (1976). State-space realization theory of two-dimensional filters. IEEE Transactions on Automatic Control, 21 (4), 484–492.
   Web of Science ®Google Scholar
• Fornasini, E. , & Marchesini, G. (1978). Doubly-indexed dynamical systems: State-space models and structural properties. Mathematical Systems Theory, 12 , 59–72.
   Google Scholar
• Galkowski, K. , Emelianov, M. A. , Pakshin, P. V. , & Rogers, E. (2016). Vector Lyapunov functions for stability and stabilization of differential repetitive processes. Journal of Computer and Systems Sciences International, 55 (4), 503–514.
   Web of Science ®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.
   Web of Science ®Google Scholar
• Kaczorek, T. (1985). Two-dimensional linear systems . Lecture notes in control and information science , Vol. 68. New York, NY: Springer-Verlag.
   Google Scholar
• Knorn, S. , National University if Ireland Maynooth (2012). A two-dimensional systems stability analysis of vehicle platoons ( Doctoral dissertation). Hamilton Institute, National University of Ireland Maynooth.
   Google Scholar
• Knorn, S. , & Middleton, R. H. (2013). Stability of two-dimensional linear systems with singularities on the stability boundary using LMIs. IEEE Transactions on Automatic Control, 58 (10), 2579–2590.
   Web of Science ®Google Scholar
• Knorn, S. , & Middleton, R. H. (2016). Asymptotic and exponential stability of nonlinear two-dimensional continuous–discrete Roesser models. Systems & Control Letters, 93 , 35–42.
   Web of Science ®Google Scholar
• Kurek, J. E. (2014). Stability of nonlinear time-varying digital 2D Fornasini-Marchesini system. Multidimensional Systems and Signal Processing, 25 (1), 235–244.
   Web of Science ®Google Scholar
• Li, L. , Xu, L. , & Lin, Z. (2013). Stability and stabilization of linear multidimensional discrete systems in the frequency domain. International Journal of Control, 86 (11), 1969–1989.
   Web of Science ®Google Scholar
• Liu, D. , & Michel, A. N. (1994). Stability analysis of state-space realizations for two-dimensional filters with overflow nonlinearities. IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, 41 , 127–137.
   Google Scholar
• Napp, D. , Rapisarda, P. , & Rocha, P. (2011). Time-relevant stability of 2D systems. Automatica, 47 (11), 2373–2382.
   Web of Science ®Google Scholar
• Oberst, U. , & Scheicher, M. (2014). The asymptotic stability of stable and time-autonomous discrete multidimensional behaviors. Mathematics of Control, Signals, and Systems, 26 (2), 215–258.
   Web of Science ®Google Scholar
• Pakshin, P. , Emelianova, J. , Gałkowski, K. , & Rogers, E. (2015). Stabilization of nonlinear 2D Fornasini-Marchesini and Roesser systems. In 2015 IEEE 9th international workshop on multidimensional systems (nDS) (pp. 1–6). Vila Real, Portugal: IEEE.
   Google Scholar
• Pandolfi, L. (1984). Exponential stability of 2-D systems. Systems and Control Letters, 4 , 381–385.
   Web of Science ®Google Scholar
• Roesser, R. P. (1975). Discrete state-space model for linear image processing. IEEE Transactions on Automatic Control, 20 (1), 1–10.
   Web of Science ®Google Scholar
• Rogers, E. , Galkowski, K. , & Owens, D. H. (2007). Control systems theory and applications for linear repetitive processes . Lecture notes in control and information sciences . Vol. 349.
   Google Scholar
• Rogers, E. , Galkowski, K. , Paszke, W. , Moore, K. L. , Bauer, P. H. , Hladowski, L. , & Dabkowski, P. (2015). Multidimensional control systems: Case studies in design and evaluation. Multidimensional Systems and Signal Processing, 26 (4), 895–939.
   Web of Science ®Google Scholar
• Scheicher, M. , & Oberst, U. (2008). Multidimensional BIBO stability and Jury's conjecture. Mathematics of Control, Signals, and Systems, 20 (1), 81–109.
   Web of Science ®Google Scholar
• Valcher, M. E. (2000). Characteristic cones and stability properties of two-dimensional autonomous behaviors. IEEE Transaction on Circuits and Systems I: Fundamental Theory and Applications, 47 (3), 290–302.
   Google Scholar
• Yeganefar, N. , Yeganefar, N. , Bachelier, O. , & Moulay, E. (2013). Exponential stability for 2D systems: The linear case. In Proceedings of the nDS’13, 8th international workshop on multidimensional systems (pp. 117–120).Erlangen, Germany: IEEE.
   Google Scholar
• Yeganefar, N. , Yeganefar, N. , Ghamgui, M. , & Moulay, E. (2013). Lyapunov theory for 2D nonlinear Roesser models: Application to asymptotic and exponential stability. IEEE Transactions on Automatic Control, 58 (5), 1299–1304.
   Web of Science ®Google Scholar
• Zerz, E. (1998). Topics in multidimensional linear systems theory . Lecture notes in control and information science. Vol. 256 London, England: Springer-Verlag.
   Google Scholar