Journal of Applied Mathematics

  • J. Appl. Math.
  • Volume 2013, Special Issue (2013), Article ID 150590, 19 pages.

Novel Observer-Based Suboptimal Digital Tracker for a Class of Time-Delay Singular Systems

Nien-Tsu Hu, Ter-Feng Wu, Sendren Sheng-Dun Xu, and Hsu-Chih Huang

Full-text: Open access

Abstract

This paper presents a novel suboptimal digital tracker for a class of time-delay singular systems. First, some existing techniques are utilized to obtain an equivalent regular time-delay system, which has a direct transmission term from input to output. The equivalent regular time-delay system is important as it enables the optimal control theory to be conveniently combined with the digital redesign approach. The linear quadratic performance index, specified in the continuous-time domain, can be discretized into an equivalent decoupled discrete-time performance index using the newly developed extended delay-free model. Additionally, although the extended delay-free model is large, its advantage is the elimination of all delay terms (which included a new extended state vector), simplifying the proposed approach. As a result, the proposed approach can be applied to a class of time-delay singular systems. An illustrative example demonstrates the effectiveness of the proposed design methodology.

Article information

Source
J. Appl. Math., Volume 2013, Special Issue (2013), Article ID 150590, 19 pages.

Dates
First available in Project Euclid: 14 March 2014

Permanent link to this document
https://projecteuclid.org/euclid.jam/1394806101

Digital Object Identifier
doi:10.1155/2013/150590

Mathematical Reviews number (MathSciNet)
MR3138981

Zentralblatt MATH identifier
06950534

Citation

Hu, Nien-Tsu; Wu, Ter-Feng; Xu, Sendren Sheng-Dun; Huang, Hsu-Chih. Novel Observer-Based Suboptimal Digital Tracker for a Class of Time-Delay Singular Systems. J. Appl. Math. 2013, Special Issue (2013), Article ID 150590, 19 pages. doi:10.1155/2013/150590. https://projecteuclid.org/euclid.jam/1394806101


Export citation

References

  • J. Lin and Z. Gao, “Exponential admissibility and ${H}_{\infty }$ control of switched singular time-delay systems: an average dwell time approach,” Journal of Applied Mathematics, vol. 2012, Article ID 482792, 28 pages, 2012.
  • P. Wang, C. Han, and B. Ding, “Stability of discrete-time networked control systems and its extension for robust ${H}_{\infty }$ control,” International Journal of Systems Science, vol. 44, no. 2, pp. 275–288, 2013.
  • L. Xie, T. Liu, G. Lu, J. Liu, and S. T. C. Wong, “Stochastic robust stability analysis for Markovian jump discrete-time delayed neural networks with multiplicative nonlinear perturbations,” in Advances in Neural Networks, vol. 3971 of Lecture Notes in Computer Science, pp. 172–178, Springer, 2006.
  • X. Zhang, S. Li, and H. Li, “Structure and BIBO stability of a three-dimensional fuzzy two-term control system,” Mathematics and Computers in Simulation, vol. 80, no. 10, pp. 1985–2004, 2010.
  • J. S.-H. Tsai, C.-T. Wang, and L. S. Shieh, “Model conversion and digital redesign of singular systems,” Journal of the Franklin Institute, vol. 330, no. 6, pp. 1063–1086, 1993.
  • B. G. Mertzios, M. A. Christodoulou, B. L. Syrmos, and F. L. Lewis, “Direct controllability and observability time domain conditions of singular systems,” IEEE Transactions on Automatic Control, vol. 33, no. 8, pp. 788–791, 1988.
  • C.-J. Wang and H.-E. Liao, “Impulse observability and impulse controllability of linear time-varying singular systems,” Automatica, vol. 37, no. 11, pp. 1867–1872, 2001.
  • J. Yu, G. Sun, and H. R. Karimi, “Fault-reconstruction-based cascaded sliding mode observers for descriptor linear systems,” Mathematical Problems in Engineering, vol. 2012, Article ID 623426, 20 pages, 2012.
  • J. Wu, H. R. Karimi, and P. Shi, “Network-based ${H}_{\infty }$ output feedback control for uncertain stochastic systems,” Information Sciences, vol. 232, pp. 397–410, 2013.
  • H. R. Karimi, “Observer-based mixed \emphH$_{2}$/${H}_{\infty }$ control design of linear systems with time-varying delays: an LMI approach,” International Journal of Control, Automation and Systems, vol. 6, no. 1, pp. 1–14, 2008.
  • M. Chadi and H. R. Karimi, “Robust observer design for unknown inputs Takagi-Sugeno models,” IEEE Transactions on Fuzzy Systems, vol. 21, no. 1, pp. 158–164, 2013.
  • S. Zhou and W. Zhang, “Discrete-time indefinite stochastic LQ control via SDP and LMI methods,” Journal of Applied Mathematics, vol. 2012, Article ID 638762, 14 pages, 2012.
  • T. Zou, “Offset-free strategy by double-layered linear model predictive control,” Journal of Applied Mathematics, vol. 2012, Article ID 808327, 14 pages, 2012.
  • H. Zhang, J. Cao, and W. Jiang, “Controllability criteria for linear fractional differential systems with state delay and impulses,” Journal of Applied Mathematics, vol. 2013, Article ID 146010, 9 pages, 2013.
  • C.-J. Wang and J.-S. Chiou, “A stability condition with delay-dependence for a class of switched large-scale time-delay systems,” Journal of Applied Mathematics, vol. 2013, Article ID 360170, 7 pages, 2013.
  • M. S. Mahmoud, Robust Control and Filtering for Time-Delay Systems, vol. 5 of Control Engineering (New York), Marcel Dekker, New York, NY, USA, 2000.
  • T. Chen and B. A. Francis, Optimal Sample-Data Control Systems, Springer, New York, NY, USA, 1995.
  • D. Li and Y. Xi, “Constrained feedback robust model predictive control for polytopic uncertain systems with time delays,” International Journal of Systems Science, vol. 42, no. 10, pp. 1651–1660, 2011.
  • L. S. Shieh, J. S. H. Tsai, and S. R. Lian, “Determining continuous-time state equations from discrete-time state equations via the principal qth root method,” IEEE Transactions on Automatic Control, vol. 31, no. 5, pp. 454–457, 1986.
  • F. R. Ganmacher, The Theory of Matrices II, Chelsea, New York, NY, USA, 1974.
  • R. Nikoukhah, A. S. Willsky, and B. C. Levy, “Boundary-value descriptor systems: well-posedness, reachability and observability,” International Journal of Control, vol. 46, no. 5, pp. 1715–1737, 1987.
  • S. L. Campbell, Singular Systems of Differential Equations II, Pitman, New York, NY, USA, 1982.
  • L. S. Shieh, Y. T. Tsay, and C. T. Wang, “Matrix sector functions and their applications to systems theory,” IEE Proceedings D: Control Theory and Applications, vol. 131, no. 5, pp. 171–181, 1984.
  • J. S. H. Tsai, L. S. Shieh, and R. E. Yates, “Fast and stable algorithms for computing the principal nth root of a complex matrix and the matrix sector function,” Computers and Mathematics with Applications, vol. 15, no. 11, pp. 903–913, 1988.
  • W.-M. Wang, S.-M. Guo, and L.-S. Shieh, “Discretization of cascaded continuous-time controllers for state and input delayed systems,” International Journal of Systems Science, vol. 31, no. 3, pp. 287–296, 2000.
  • S.-M. Guo, W. Wang, and L.-S. Shieh, “Discretisation of two degree-of-freedom controller and system with state, input and output delays,” IEE Proceedings-Control Theory and Applications, vol. 147, no. 1, pp. 87–96, 2000.
  • S.-M. Guo, L.-S. Shieh, G. Chen, and C.-F. Lin, “Effective chaotic orbit tracker: a prediction-based digital redesign approach,” IEEE Transactions on Circuits and Systems I, vol. 47, no. 11, pp. 1557–1570, 2000.
  • M. Hou, P. Zítek, and R. J. Patton, “An observer design for linear time-delay systems,” IEEE Transactions on Automatic Control, vol. 47, no. 1, pp. 121–125, 2002.
  • F. L. Lewis, Optimal Control, John Wiley & Sons, New York, NY, USA, 1986.
  • K. Ogata, Discrete-Time Control Systems, Prentice-Hall, Englewood Cliffs, NJ, USA, 1987.
  • K. Watanabe and T. Ouchi, “An observer of systems with delays in state variables,” International Journal of Control, vol. 41, no. 1, pp. 217–229, 1985.
  • J. Leyva-Ramos and A. E. Pearson, “Asymptotic modal observer for linear autonomous time lag systems,” IEEE Transactions on Automatic Control, vol. 40, no. 7, pp. 1291–1294, 1995.
  • J. S.-H. Tsai, N.-T. Hu, P.-C. Yang, S.-M. Guo, and L.-S. Shieh, “Modeling of decentralized linear observer and tracker for a class of unknown interconnected large-scale sampled-data nonlinear systems with closed-loop decoupling property,” Computers and Mathematics with Applications, vol. 60, no. 3, pp. 541–562, 2010.
  • N. T. Hu, J. S. H. Tsai, S. M. Guo, L. S. Shieh, and Y. Chen, “Low-order multi-rate linear time-invariant decentralized trackers using the new observer-based sub-optimal method for unknown sampled-data nonlinear time-delay system with closed-loop decoupling,” Optimal Control Applications and Methods, vol. 32, no. 4, pp. 433–475, 2011. \endinput