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A Survey on the Stability of 2-D Discrete Systems Described by Fornasini-Marchesini Second Model ()

A key issue of practical importance in the two-dimensional (2-D) discrete system is stability analysis. Linear state-space models describing 2-D discrete systems have been proposed by several researchers. A popular model, called Forna- sini-Marchesini (FM) second model was proposed by Fornasini and Marchesini in 1978. The aim of this paper is to present a survey of the existing literature on the stability of FM second model.

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M. Tiwari and A. Dhawan, "A Survey on the Stability of 2-D Discrete Systems Described by Fornasini-Marchesini Second Model,"

*Circuits and Systems*, Vol. 3 No. 1, 2012, pp. 17-22. doi: 10.4236/cs.2012.31003.Conflicts of Interest

The authors declare no conflicts of interest.

[1] | N. K. Bose, “Applied Multidimensional System Theory,” Van Nostrand Reinhold, New York, 1982. |

[2] | T. Kaczorek, “Two-Dimensional Linear Systems,” Springer-Verlag, Berlin, 1985. |

[3] | W.-S. Lu and A. Antoniou, “Two-Dimensional Digital Filters,” Marcel Dekker, Electrical Engineering and Electronics, Vol. 80, New York, 1992. |

[4] | R. N. Bracewell, “Two-Dimensional Imaging,” Prentice-Hall Signal Processing Series, Prentice-Hall, Englewood Cliffs, 1995. |

[5] | R. P. Roesser, “A Discrete State-Space Model for Linear Image Processing,” IEEE Transactions on Automatic Control, Vol. 20, No. 1, 1975, pp. 1-10. doi:10.1109/TAC.1975.1100844 |

[6] | E. Fornasini and G. Marchesini, “State-Space Realization Theory of Two Dimensional Filters,” IEEE Transactions on Automatic Control, Vol. 21, No. 4, 1976, pp. 484-492. doi:10.1109/TAC.1976.1101305 |

[7] | E. Fornasini and G. Marchesini, “Doubly Indexed Dynamical Systems: State-Space Models and Structural Properties,” Mathematical Systems Theory, Vol. 12, No. 1, 1978, pp. 59-72. doi:10.1007/BF01776566 |

[8] | R. Gnanasekaran, “A Note on the New 1-D and 2-D Stability Theorems for Discrete Systems,” IEEE Transactions on Acoustics, Speech, & Signal Processing, Vol. 29, No. 6, 1981, pp. 1211-1212. doi:10.1109/TASSP.1981.1163695 |

[9] | T. Bose and D. A. Trautman, “Two’s Complement Quantization in Two-Dimensional State-Space Digital Filters,” IEEE Transactions on Signal Processing, Vol. 40, No. 10, 1992, pp. 2589-2592. doi:10.1109/78.157299 |

[10] | P. Agathoklis, E. I. Jury and Mohamed Mansour, “Algebraic Necessary and Sufficient Conditions for the Stability of 2-D Discrete Systems,” IEEE Transactions on Circuits and Systems II: Analog and Digital Signal Processing, Vol. 40, No. 10, 1993, pp. 251-258. |

[11] | T. Bose, “Stability of 2-D State-Space System with Overflow and Quantization,” IEEE Transactions on Circuits and Systems II, Vol. 42, No. 6, 1995, pp. 432-434. doi:10.1109/82.392319 |

[12] | Y. Su and A. Bhaya, “On the Bose-Trautman Condition for Stability of Two-Dimensional Linear Systems,” IEEE Transactions on Signal Processing, Vol. 46, No. 7, 1998, pp. 2069-2070. doi:10.1109/78.700987 |

[13] | T. Fernando and H. Trinh, “Lower Bounds for Stability Margin of Two-Dimensional Discrete Systems Using the MacLaurine Series,” Computer and Electrical Engineering, Vol. 25, No. 2, 1999, pp. 95-109. doi:10.1016/S0045-7906(98)00036-6 |

[14] | R. Thamvichai and T. Bose, “Stability of 2-D Periodically Shift Variant Filters,” IEEE Transactions on Circuits and Systems II, Vol. 49, No. 1, 2002, pp. 61-64. doi:10.1109/82.996060 |

[15] | Yuval Bistritz, “Testing Stability of 2-D Discrete Systems by a Set of Real 1-D Stability Tests,” IEEE Transactions on Circuits and Systems I, Vol. 51, N0. 7, 2004, pp. 1312-1320. |

[16] | G. -D. Hu and M. Liu, “Simple Criteria for Stability of two-Dimensional Linear Systems,” IEEE Transactions on Signal Processing, Vol. 53, No. 12, 2005, pp. 4720-4723. doi:10.1109/TSP.2005.859265 |

[17] | T. Kaczorek, “New Stability Tests of Positive Standard and Fractional Linear Systems,” Circuits and Systems, Vol. 2, No. 4, 2011, pp. 261-268. doi:10.4236/cs.2011.24036 |

[18] | T. Liu, “Stability Analysis of Linear 2-D Systems,” Signal Processing, Vol. 88, No. 8, 2008, pp. 2078-2084. doi:10.1016/j.sigpro.2008.02.007 |

[19] | E. Fornasini and G. Marchesini, “Stability Analysis of 2-D Systems,” IEEE Transactions on Circuits and Systems, Vol. 27, No. 12, 1980, pp. 1210-1217. doi:10.1109/TCS.1980.1084769 |

[20] | W.-S. Lu and E. B. Lee, “Stability Analysis for Two-Dimensional Systems via a Lyapunov Approach,” IEEE Transactions on Circuits and Systems, Vol. 32, No. 1, 1985, pp. 61-68. doi:10.1109/TCS.1985.1085639 |

[21] | P. Agathoklis, “The Lyapunov Equation for n-Dimensional Discrete Systems,” IEEE Transactions on Circuits and Systems, Vol. 35, No. 4, 1988, pp. 448-451. doi:10.1109/31.1762 |

[22] | P. Agathoklis, E. I. Jury and M. Mansour, “The Discrete-Time Strictly Bounded-Real Lemma and the Computation of Positive Definite Solutions to the 2-D Lyapunov Equation,” IEEE Transactions on Circuits and Systems, Vol. 36, No. 6, 1989, pp. 830-837. doi:10.1109/31.90402 |

[23] | D. Liu and A. N. Michel, “Stability Analysis of State-Space Realizations for Two-Dimensional Filters with Overflow Nonlinearities,” IEEE Transactions on Circuits and Systems I, Vol. 41, No. 2, 1994, pp. 127-137. doi:10.1109/81.269049 |

[24] | C. Xiao, D. J. Hill and P. Agathoklis, “Stability and the Lyapunov Equation for n-Dimensional Digital Systems,” IEEE Transactions on Circuits and Systems I, Vol. 44, No. 7, 1997, pp. 614-621. doi:10.1109/81.596942 |

[25] | H. Kar and V. Singh, “Stability of 2-D Systems Described by Fornasini-Marchesini First Model,” IEEE Transactions on Signal Processing, Vol. 51, No. 6, 2003, pp. 1675-1676. doi:10.1109/TSP.2003.811237 |

[26] | H. Kar and V. Singh, “Stability Analysis of 2-D Digital Filters with Saturation Arithmetic: An LMI Approach,” IEEE Transactions on Signal Processing, Vol. 53, No. 6, 2005, pp. 2267-2271. doi:10.1109/TSP.2005.847857 |

[27] | W.-S. Lu, “Some New Results on Stability Robustness of 2-D Digital Filters,” Multidimensional Systems and Signal Processing, Vol. 5, No. 4, 1994, pp. 345-361. doi:10.1007/BF00989278 |

[28] | W.-S. Lu, “On Robust Stability of 2-D Discrete Systems,” IEEE Transactions on Automatic Control, Vol. 40, No. 3, 1995, pp. 502-506. doi:10.1109/9.376069 |

[29] | L. Xie, “LMI Approach to Output Feedback Stabilization of 2-D Discrete Systems,” International Journal of Control, Vol. 72, No. 2, 1999, pp. 97-106. doi:10.1080/002071799221262 |

[30] | C. Du and L. Xie, “Stability Analysis and Stabilization of Uncertain Two-Dimensional Discrete Systems: An LMI Approach,” IEEE Transactions on Circuits and Systems I, Vol. 46, No. 11, 1999, pp. 1371-1374. doi:10.1109/81.802835 |

[31] | K. Galkowski, J. Lam, S. Xu and Z. Lin, “LMI Approach to State-Feedback Stabilization of Multidimensional Systems,” International Journal of Control, Vol. 76, No. 14, 2003, pp. 1428-1436. doi:10.1080/00207170310001599113 |

[32] | Z. Wang and X. Liu, “Robust Stability of Two-Dimensional Uncertain Discrete Systems,” IEEE Signal Processing Letters, Vol. 10, No. 5, 2003, pp. 133-136. doi:10.1109/LSP.2003.810754 |

[33] | H. Kar and V. Singh, “Corrections to Robust Stability of Two-Dimensional Uncertain Discrete Systems,” IEEE Signal Processing Letters, Vol. 10, No. 8, 2003, p. 250. doi:10.1109/LSP.2003.816071 |

[34] | X. Guan, C. Long and G. Duan, “Robust Optimal Guaranteed Cost Control for 2D Discrete Systems,” IEE Proceedings—Control Theory & Applications, Vol. 148, No. 5, 2001, pp. 355-361. doi:10.1049/ip-cta:20010596 |

[35] | A. Dhawan and H. Kar, “Comment on Robust Optimal Guaranteed Cost Control for 2-D Discrete Systems,” IEE Proceedings—Control Theory & Applications, Vol. 1, No. 4, 2007, pp. 1188-1190. doi:10.1049/iet-cta:20060327 |

[36] | Dhawan and H. Kar, “LMI-Based Criterion for the Robust Guaranteed Cost Control of 2-D Systems Described by the Fornasini-Marchesini Second Model,” Signal Processing, Vol. 87, No. 3, 2007, pp. 479-488. doi:10.1016/j.sigpro.2006.06.002 |

[37] | Dhawan and H. Kar, “Optimal Guaranteed Cost Control of 2-D Discrete Uncertain Systems: An LMI Approach,” Signal Processing, Vol. 87, No. 12, 2007, pp. 3075-3085. doi:10.1016/j.sigpro.2007.06.001 |

[38] | Dhawan and H. Kar, “An LMI Approach to Robust Optimal Guaranteed Cost Control of 2-D Discrete Systems Described by the Roesser Model,” Signal Processing, Vol. 90, No. 9, 2010, pp. 2648-2654. doi:10.1016/j.sigpro.2010.03.008 |

[39] | Dhawan and H. Kar, “An Improved LMI-Based Criterion for the Design of Optimal Guaranteed Cost Controller for 2-D Discrete Uncertain Systems,” Signal Processing, Vol. 91, No. 4, 2011, pp. 1032-1035. doi:10.1016/j.sigpro.2010.07.014 |

[40] | W. A. Porter and J. L. Aravena, “State Estimation in Discrete M-D Systems,” IEEE Transactions on Automatic Control, Vol. 31, No. 3, 1986, pp. 280-283. doi:10.1109/TAC.1986.1104249 |

[41] | Du, L. Xie and Y. C. Soh, “ Filtering of 2-D Discrete Systems,” IEEE Transactions on Signal Processing, Vol. 48, No. 6, 2000, pp. 1760-1768. doi:10.1109/78.845933 |

[42] | Du and L. Xie, “ Control and Filtering of Two Dimensional Systems,” Springer-Verlag, Berlin, 2002. |

[43] | M. Sebek, “ Problem of 2-D Systems,” Proceeding of the European Control Conference, Groningen, Netherlands, 28 June-1 July 1993, pp. 1476-1479. |

[44] | H. D. Tuan, P. Apkarian, T. Q. Nguyen and T. Narikiyo, “Robust Mixed Filtering of 2-D Systems,” IEEE Transactions on Signal Processing, Vol. 50, No. 7, 2002, pp. 1759-1771. |

[45] | L. Xie, C. Du, C. Zhang and Y. C. Soh, “ Deconvolution Filtering of 2-D Digital Systems,” IEEE Transactions on Signal Processing Vol. 50, No. 9, 2002, pp. 2319-2331. doi:10.1109/TSP.2002.800401 |

[46] | Du, L. Xie and C. Zhang, “ Control and Robust Stabilization of Two Dimensional Systems in Roesser Models,” Automatica, Vol. 37, No. 2, 2001, pp. 205-211. doi:10.1016/S0005-1098(00)00155-2 |

[47] | E. de Souza, L. Xie and D. F. Coutinho, “Robust Filtering for 2-D Discrete-Time Linear Systems with Convex-Bounded Parameter Uncertainty,” Automatica, Vol. 46, No. 4, 2010, pp. 673-681. doi:10.1016/j.automatica.2010.01.017 |

[48] | S. Boyd, L. El Ghaoui, E. Feron and V. Balakrishnan, “Linear Matrix Inequalities in System and Control Theory,” SIAM, Philadelphia, 1994. doi:10.1137/1.9781611970777 |

[49] | Dumitrescu, “LMI Stability Tests for the Fornasini-Marchesini Model,” IEEE Transactions on Signal Processing, Vol. 56, No 8, 2008, pp. 4091-4095. doi:10.1109/TSP.2008.921768 |

[50] | P. Gahinet, A. Nemirovski, A. J. Laub and M. Chilali, “LMI Control Toolbox—For Use with Matlab,” The MATH Works Inc., Natick, 1995. |

[51] | T. Hinamoto, “2-D Lyapunov Equation and Filter Design Based on the Fornasini-Marchesini Second Model,” IEEE Transactions on Circuits and Systems I, Vol. 40, No. 2, 1993, pp. 102-110. doi:10.1109/81.219824 |

[52] | A. Kanellakis, “New Stability Results for 2-D Discrete Systems Based on the Fornasini-Marchesini State Space Model,” IEE Proceedings Circuits, Devices & Systems, Vol. 141, No. 5, 1994, pp. 427-432. doi:10.1049/ip-cds:19941368 |

[53] | W.-S. Lu, “On a Lyapunov Approach to Stability Analysis of 2-D Digital Filters,” IEEE Transactions on Circuits and Systems I, Vol. 41, No. 10, 1994, pp. 665-669. doi:10.1109/81.329727 |

[54] | T. Hinamoto, “Stability of 2-D Discrete Systems Described by Fornasini-Marchesini Second Model,” IEEE Transactions on Circuits and Systems I, Vol. 44, No. 3, 1997, pp. 254-257. doi:10.1109/81.557373 |

[55] | T. Ooba, “On Stability Analysis of 2-D Systems Based on 2-D Lyapunov Matrix Inequalities,” IEEE Transactions on Circuits and Systems I, Vol. 47, No. 8, 2000, pp. 1263-1265. doi:10.1109/81.873883 |

[56] | T. Ooba, “On Stability Robustness of 2-D Systems Described by the Fornasini-Marchesini Model,” Multidimensional Systems and Signal Processing, Vol. 12, No. 1, 2001, pp. 81-88. doi:10.1023/A:1008420911857 |

[57] | P. Fu, J. Chen and S.-I. Niculescu, “Generalized Eigenvalue-Based Stability Tests for 2-D Linear Systems: Necessary and Sufficient Conditions,” Automatica, Vol. 42, No. 9, 2006, pp. 1569-1576. doi:10.1016/j.automatica.2006.04.015 |

[58] | T. Kaczorek, “LMI Approach to Stability of 2D Positive Systems,” Multidimensional Systems and Signal Processing, Vol. 20, No. 1, 2009, pp. 39-54. doi:10.1007/s11045-008-0050-7 |

[59] | W. Paszke, J. Lam, K. Galkowski, S. Xu and Z. Lin, “Robust Stability and Stabilization of 2D Discrete State-Delayed Systems,” Systems & Control Letters, Vol. 51, No. 3-4, 2004, pp. 277-291. doi:10.1016/j.sysconle.2003.09.003 |

[60] | S. Ye, W. Wang and Y. Zou, “Robust Guaranteed Cost Control for a Class of Two-Dimensional Discrete Systems with Shift-Delays,” Multidimensional Systems and Signal Processing, Vol. 20, No. 3, 2009, pp. 297-307. doi:10.1007/s11045-008-0063-2 |

[61] | M. Tiwari and A. Dhawan, “Comment on ‘Robust Guaranteed Cost Control for a Class of two-Dimensional Discrete Systems with Shift-Delays’,” Multidimensional Systems and Signal Processing, 2011. |

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