Abstract
This paper is concerned with the problem of delay-dependent H ∞ control for two-dimensional (2-D) discrete state delay systems described by the second Fornasini and Marchesini (FM) state-space model. Based on a summation inequality, a sufficient condition to have a delay-dependent H ∞ noise attenuation for this 2-D system is given in terms of linear matrix inequalities (LMIs). A delay-dependent optimal state feedback H ∞ controller is obtained by solving an LMI optimization problem. Finally, a simulation example of thermal processes is given to illustrate the effectiveness of the proposed result.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bracewell, R. N. (1995). Two-dimensional Imaging. Prentice-Hall, Englewood Cliffs, NJ: Prentice-Hall Signal Processing Series.
Bisiacco M. (1995) New results in 2D optimal control theory. Multidimensional Systems and Signal Processing 6: 189–222
Du, C., & Xie, L. (2002). H ∞ control and filtering of two-dimensional systems, Lecture Notes in Control and Information Sciences (Vol. 278). Berlin: Springer.
Dymkov M., Gaishun I., Galkowski K., Rogers E., Owens D.H. (2002) Exponential stability of discrete linear repetitive processes. International Journal of Control 75(12): 861–869
Fridman E., Sharked U. (2003) Delay-dependent stability and H ∞ control: Constant and time-varying delays. International Journal of Control 76(1): 48–60
Fornasini E., Marchesini G. (1976) State-space realization theory of two-dimensional filters. IEEE Transactions on Automatic Control 21(4): 484–491
Hinamoto T. (1997) Stability of 2-D discrete systems described by the Fornasini–Marchesini second model. IEEE Transactions on Circuits Systems I: Fundamental Theory and Applications 44(3): 254–257
Kaczorek T. (1985) Two-dimensional linear systems, Lecture Notes in Control and Information Sciences (Vol. 68). Springer, Berlin
Lee J., Kim S., Kwon W. (1994) Memoreless H ∞ controllers for delayed systems. IEEE Transactions on Automatic Control 39(1): 159–162
Lee Y.S., Moon Y.S., Kwon W.H., Park P.G. (2004) Delay-dependent robust H ∞ control for uncertain systems with a state-delay. Automatica 40(1): 65–72
Li X.D., Ho J.K.L., Chow T.W.S. (2005) Iterative learning control for linear time-variant discrete systems based on 2-D system theory. IEE Proceeding Control Theory and Applications 152(1): 13–18
Lu W.S., Antoniou A. (1992) Two-dimensional digital filters, electrical engineering and electronics (Vol. 80). Marcel Dekker, New York
Niculescu S.I. (1998) H ∞ memoreless control with α-stability constraint for time-delay systems: An LMI approach. IEEE Transactions on Automatic Control 43(5): 739–743
Niculescu S.I. (2001) Delay effects on stability: A robust control approach, Lecture Notes in Control and Information Sciences (Vol. 269). Springer, London
Owens D.H., Amann N., Rogers E., French M. (2000) Analysis of linear iterative learning control schemes—a 2D systems/repetitive processes approach. Multidimensional Systems and Singnal Processing 11: 125–177
Paszke W., Lam J., Galkowski K., Xu S., Lin Z. (2004) Robust stability and stabilisation of 2D discrete state-delayed systems. Systems & Control Letters 51(3-4): 277–291
Roesser R.P. (1975) A discrete state-space model for linear image processing. IEEE Transactions on Automatic Control 20(1): 1–10
Sebek, M. (1993). H ∞ problem of 2-D systems. European Control Conference’93 (pp. 1476–1479).
Sulikowski B., Galkowski K., Rogers E., Owens D.H. (2004) Output feedback control of discrete linear repetitive processes. Automatica 40(12): 2167–2173
Wu M., He Y., She J.-H. (2006) Delay-dependent stabilization for systems with multiple unknown time-varying delays. International Journal of Control, Automation, and Systems 4(6): 662–668
Xu H., Zou Y., Lu. J., Xu S. (2005) Robust H ∞ control for a class of uncertain nonlinear two-dimensional systems with state delay. Journal of the Franklin Institute 342(7): 877–891
Xu J.M., Yu L. (2006) H ∞ control of 2-D discrete state delay systems. International Journal of Control, Automation, and Systems 4(4): 516–523
Xu S., Lam J. (2005) Improved delay-dependent stability criteria for time-delay systems. IEEE Transactions on Automatic Control 50(3): 384–387
Xu S., Lam J., Zou Y. (2004) Improved conditions on delay-dependent robust stability and stabilization of uncertain discrete time-delay systems. Asian Journal of Control 7(3): 348–352
Xu S., LamJ. Zou Y., Lin Z., Paszke W. (2005) Robust H ∞ filtering for uncertain 2-D continuous systems. IEEE Transactions on Signal Processing 53(5): 1731–1738
Zhang X.M., Wu M., She J.H., He Y. (2005) Delay-dependent stabilization of linear systems with time-varying state and input delays. Automatica 41(8): 1405–1412
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Xu, J., Yu, L. Delay-dependent H ∞ control for 2-D discrete state delay systems in the second FM model. Multidim Syst Sign Process 20, 333–349 (2009). https://doi.org/10.1007/s11045-008-0074-z
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11045-008-0074-z