Automation and Remote Control

, Volume 79, Issue 10, pp 1903–1911 | Cite as

Stabilization of Nonlinear Fornasini–Marchesini Systems

  • J. P. EmelianovaEmail author
Large Scale Systems Control


This paper considers the 2D systems described by the Fornasini–Marchesini statespace model. Direct and converse theorems on the exponential stability of such systems are proved in terms of vector Lyapunov functions. The concepts of exponential passivity and a vector storage function are introduced for solving exponential stabilization problems. An example is given to illustrate the efficiency of the new results.


2D-systems Fornasini–Marchesini model stability Lyapunov function stabilizing control linear matrix inequality (LMI) 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Plotnikov, V.I. and Sumin, M.I., Problems of Stability of Nonlinear Goursat–Darboux Systems, Differ. Uravn., 1972, vol. 7, no. 5, pp. 845–856.MathSciNetzbMATHGoogle Scholar
  2. 2.
    Byrnes, C., Isidori, A., and Willems, J., Passivity, Feedback Equivalence and the Global Stabilization of Minimun Phase Nonlinear Systems, IEEE Trans. Automat. Control, 1991, vol. 36, pp. 1228–1240.MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Du, C. and Xie, L., Stability Analysis and Stabilization of Uncertain Two-Dimensional Discrete Systems: An LMI Approach, IEEE Trans. Circuits Syst. I: Fund. Theory Appl., 1999, vol. 46, pp. 1371–1374.CrossRefzbMATHGoogle Scholar
  4. 4.
    Emelianova, J., Pakshin, P., Gałkowski, K., and Rogers, E., Vector Lyapunov Function Based Stability of a Class of Applications Relevant 2D Nonlinear Systems, IFAC Proc. Volumes (IFACPapersOnLine), 2014, vol. 47, no. 3, pp. 8247–8252.CrossRefGoogle Scholar
  5. 5.
    Emelianova, J., Pakshin, P., Gałkowski, K., and Rogers, E., Stability of Nonlinear Discrete Repetitive Processes with Markovian Switching, Syst. Control Lett., 2015, vol. 75, pp. 108–116.MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Emelianova, J., Pakshin, P., Gałkowski, K., and Rogers, E., Stability of Nonlinear 2D Systems Described by the Continuous-Time Roesser Model, Autom. Remote Control, 2014, vol. 75, no. 5, pp. 845–858.MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Dymkov, M., Gałkowski, K., Rogers, E., Dymkou, V., and Dymkou, S., Modeling and Control of a Sorption Process Using 2D Systems Theory, Proc. 7th Int. Worskop on Multidimensional Systems (NDS’11), 2011, pp. 1–6.Google Scholar
  8. 8.
    Fornasini, E. and Marchesini, G., Doubly Indexed Dynamical Systems: State Models and Structural Properties, Math. Syst. Theory, 1978, vol. 12, pp. 59–72.MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Fradkov, A. and Hill, D., Exponential Feedback Passivity and Stabilizability of Nonlinear Systems, Automatica, 1998, vol. 34, pp. 697–703.MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Hmamed, A., Mesquine, F., Tadeo, F., Benhayoun, M., and Benzaouia, A., Stabilization of 2D Saturated Systems by State Feedback Control, in Multidim. Syst. Signal Proces., 2010, vol. 21, no. 3, pp. 277–292.MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Hladowski, L., Gałkowski, K., Cai, Z., Rogers, E., Freeman, C.T., and Lewin, P.L., Experimentally Supported 2D Systems Based Iterative Learning Control Law Design for Error Convergence and Performance, Control Eng. Practice, 2010, vol. 18(4), pp. 339–348.CrossRefGoogle Scholar
  12. 12.
    Khalil, H., Nonlinear Systems, New Jersey: Prentice Hall, 2002, 3rd ed.zbMATHGoogle Scholar
  13. 13.
    Kurek, J., Stability of Nonlinear Time-Varying Digital 2-D Fornasini–Marchesini System, in Multidim. Syst. Signal Proces., 2014, vol. 25, no. 1, pp. 235–244.MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Kurek, J.E. and Zaremba, M.B., Iterative Learning Control Synthesis Based on 2D System Theory, IEEE Trans. Automat. Control, 1993, vol. 38, pp. 121–125.MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Liu, D., Lyapunov Stability of Two-Dimensional Digital Filters with Overflow Nonlinearities, IEEE Trans. Circuits Syst. I: Fund. Theory Appl., 1998, vol. 45, pp. 574–577.MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Pandolfi, L., Exponential Stability of 2D Systems, Syst. Control Lett., 1984, vol. 4, pp. 381–385.CrossRefzbMATHGoogle Scholar
  17. 17.
    Pakshin, P., Gałkowski, K., and Roger, E., Stability and Stabilization of Systems Modeled by 2D Nonlinear Stochastic Roesser Models, Proc. 7th Int. Workshop on Multidimensional (nD) Systems, 2011, pp. 1–6.Google Scholar
  18. 18.
    Paszke, W., Rogers, E., Gałkowski, K., and Cai, Z., Robust Finite Frequency Range Iterative Learning Control Design and Experimental Verification, Control Eng. Pract., 2013, vol. 21, pp. 1310–1320.CrossRefGoogle Scholar
  19. 19.
    Roesser, R.P., A Discrete State-Space Model for Linear Image Processing, IEEE Trans. Automat. Control, 1975, vol. AC-20(1), pp. 1–10.MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Rogers, E., Gałkowski, K., and Owens, D.H., Control Systems Theory and Applications for Linear Repetitive Processes, Lecture Notes in Control and Information Sciences, Berlin: Springer-Verlag, 2007, vol.349.Google Scholar
  21. 21.
    Sammons, P.M., Bristow, D.A., and Landers, R.G., Iterative Learning Control of Bead Morphology in Laser Metal Deposition Processes, Proc. Am. Control Conf., 2013, pp. 5962–5967.Google Scholar
  22. 22.
    Sammons, P.M., Bristow, D.A., and Landers, R.G., Height Dependent Laser Metal Deposition Process Modeling, J. Manufact. Sci. Eng., 2013, vol. 135, no. 5, pp. 1–7.CrossRefGoogle Scholar
  23. 23.
    Willems, J., Dissipative Dynamical Systems Part I: General Theory, Arch. Rational Mech. Analysis, 1972, vol. 45, pp. 325–351.Google Scholar
  24. 24.
    Yamada, M., Xu, L., and Saito, O., 2D Model-Following Servo System, in Multidim. Syst. Signal Proces., 1999, vol. 10, no. 1, pp. 71–91.MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Yeganefar, N., Yeganefar, N., Ghamgui, M., and Moulay, E., Lyapunov Theory for 2D Nonlinear Roesser Models: Application to Asymptotic and Exponential Stability, IEEE Trans. Automat. Control, 2013, vol. 58, pp. 1299–1304.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  1. 1.Arzamas Polytechnic Institute of Alekseev Nizhny Novgorod State Technical UniversityArzamasRussia

Personalised recommendations