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Admissible Control of Linear Singular Delta Operator Systems

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Abstract

In this paper, we study the problem of state feedback admissible control for the linear singular delta operator systems that result from linear singular continuous systems. By introducing the concept of delta operator, for a given linear singular continuous system, we establish its corresponding delta operator model and this discrete model converges to its continuous counterpart as the sampling period decreases. Sufficient conditions for desirable controllers in terms of matrix inequalities and linear matrix inequalities are given, and the explicit expressions of the controllers are derived. Some examples as well as numerical simulations are provided to demonstrate the effectiveness of the proposed approaches.

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References

  1. O. Barambones, P. Alkorta, J. Durana, Sliding mode position control for real-time control of induction motors. Int. J. Innov. Comput. I. 9(7), 2741–2754 (2013)

    Google Scholar 

  2. Y. Cao, Z. Lin, A descriptor system approach to robust satbility analysis and controller synthesis. IEEE Trans. Autom. Control 49, 2081–2084 (2004)

    Article  MathSciNet  Google Scholar 

  3. L. Dai, Singular Control Systems (Springer, Berlin, 1989)

    Book  MATH  Google Scholar 

  4. X. Dong, Controllability analysis of linear singular delta operator systems, in 12th International Conference on Control, Automation, Robotics and Vision (2012), pp. 1199–1204.

  5. X. Dong, Admissibility analysis of linear singular systems via a delta operator method. Int. J. Syst. Sci. (2013). doi:10.1080/00207721.2013.769646

  6. X. Dong, Q. Mao, W. Tian, D. Wang, Observability analysis of linear singular delta operator systems, in 10th IEEE International Conference on Control and Automation (2013), pp. 10–15.

  7. X. Dong, W. Tian, Q. Mao, D. Wang, Robust admissibility analysis and synthesis of uncertain singular systems via delta operator approach, in 10th IEEE International Conference on Control and Automation (2013), pp. 1059–1064.

  8. E. Fridman, U. Shaked, A descriptor system approach to \(H_\infty \) control of linear time-delay systems. IEEE Trans. Autom. Control 47, 253–270 (2002)

    Article  MathSciNet  Google Scholar 

  9. G. Goodwin, R. Lozano-Leal, D. Mayne, R. Middleton, Rapproachement between continuous and discrete model reference adaptive control. Automatica 22(2), 199–207 (1986)

    Article  MATH  Google Scholar 

  10. H.J. Lee, J.B. Park, Y.H. Joo, Further refinement on LMI-based digital redesign: delta-operator approach. IEEE Trans. Circuits-II 53(6), 447–473 (2006)

    Google Scholar 

  11. G. Li, M. Gevers, Comparative study of finite word length effects in shift and delta operator parameterizations. IEEE Trans. Autom. Control 38(5), 803–807 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  12. H. Li, B. Wu, G. Li, C. Yang, Basic Theory of Delta Operator Control and its Robust Control (National Defence Industry Press, Beijing, 2005)

    Google Scholar 

  13. R. Lu, H. Su, A. Xue, J. Chu, Robust Control Theory of Singular Systems (Science Press, Beijing, 2008)

    Google Scholar 

  14. Q. Mao, X. Dong, W. Tian, Admissibility conditions for linear singular delta operator systems: analysis and synthesis, in 10th World Congress on Intelligent Control and Automation (2012), pp. 1870–1875.

  15. R. Middleton, G. Goodwin, Improved finite word length characteristics in digital control using delta operators. IEEE Trans. Autom. Control 31(11), 1015–1021 (1986)

    Article  MATH  Google Scholar 

  16. C.P. Neuman, Transformations between delta and forward shift operator transfer function models. IEEE Trans. Syst. Man Cybern. 23(1), 295–296 (1993)

    Article  Google Scholar 

  17. C.P. Neuman, Properties of the delta operator model of dynamic physical systems. IEEE Trans. Syst. Man Cybern. 23(1), 296–301 (1993)

    Article  MATH  Google Scholar 

  18. J. Qiu, H. Yang, P. Shi, Y. Xia, Robust \(H_\infty \) control for a class of discrete-time Markovian jump systems with time-varying delays based on delta operator. Circuits Syst. Signal Process 27, 627–643 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  19. X. Su, P. Shi, L. Wu, Y. Song, A novel control design on discrete-time Takagi–Sugeno fuzzy systems with time-varying delays. IEEE Trans. Fuzzy Syst. 21(4), 655–671 (2013)

    Article  Google Scholar 

  20. P. Suchomski, Numerically robust delta-domain solutions to discrete-time Lyapunov equations. Syst. Control Lett. 47(4), 319–326 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  21. V. Veselý, D. Rosinová, T. Quang, Networked output feedback robust predictive controller design. Int. J. Innov. Comput. I. 9(10), 3941–3953 (2013)

    Google Scholar 

  22. J. Wu, G. Li, R. Istepanian, J. Chu, Shift and delta operator realisation for digital controllers with finite word length consideration. IEE Proc. Control Theory Appl. 147(6), 664–672 (2000)

    Article  Google Scholar 

  23. L. Wu, X. Su, P. Shi, J. Qiu, A new approach to stability analysis and stabilization of discrete-time T–S fuzzy time-varying delay systems. IEEE Trans. Syst. Man Cybern. 41(1), 273–286 (2011)

    Article  Google Scholar 

  24. L. Wu, W. Zheng, H. Gao, Dissipativity-based sliding mode control of switched stochastic systems. IEEE Trans. Autom. Control 58(3), 785–793 (2013)

    Article  Google Scholar 

  25. Z. Wu, P. Shi, H. Su, J. Chu, \(l_2-l_\infty \) filter design for discrete-time singular Markovian jump systems with time-varying delays. Inf. Sci. 181(24), 5534–5547 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  26. Z. Wu, P. Shi, H. Su, J. Chu, Delay-dependent stability analysis for discrete-time singular Markovian jump systems with time-varying delay. Int. J. Syst. Sci. 43(11), 2095–2106 (2012)

    Article  MathSciNet  Google Scholar 

  27. Z. Wu, H. Su, J. Chu, Delay-dependent \(H_\infty \) filtering for singular Markovian jump time-delay systems. Signal Process 90(6), 1815–1824 (2010)

    Article  MATH  Google Scholar 

  28. Z. Wu, H. Su, P. Shi, J. Chu, Analysis and Synthesis of Singular Systems with Time-Delays (Springer, Berlin, 2013)

    Book  MATH  Google Scholar 

  29. S. Xu, J. Lam, Robust stability and stabilization of discrete singular systems: an equivalent characterization. IEEE Trans. Autom. Control 49(4), 568–574 (2004)

    Article  MathSciNet  Google Scholar 

  30. S. Xu, J. Lam, New positive realness conditions for uncertain discrete descriptor systems: analysis and synthesis. IEEE Trans. Circuits-I 51(9), 1897–1905 (2004)

    Article  MathSciNet  Google Scholar 

  31. S. Xu, C. Yang, Stabilization of discrete-time singular systems: a matrix inequalities approach. Automatica 35, 1613–1617 (1999)

    Article  MATH  Google Scholar 

  32. S. Xu, C. Yang, Y. Niu, L. Lam, Robust stabilization for uncertain discrete singular systems. Automatica 37, 769–774 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  33. R. Yang, P. Shi, G.-P. Liu, H. Gao, Network-based feedback control for systems with mixed delays based on quantization and dropout compensation. Automatica 47(12), 2805–2809 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  34. H. Yang, P. Shi, J. Zhang, J. Qiu, Robust \(H_\infty \) control for a class of discrete time fuzzy systems via delta operator approach. Inf. Sci. 184, 230–245 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  35. H. Yang, Y. Xia, Low frequency positive real control for delta operator systems. Automatica 48(8), 1791–1795 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  36. H. Yang, Y. Xia, P. Shi, Observer-based sliding mode control for a class of discrete systems via delta operator approach. J. Franklin Inst. 347(5), 1199–1213 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  37. H. Yang, Y. Xia, P. Shi, M. Fu, A novel delta operator Kalman filter design and convergence analysis. IEEE Trans. Circuits-I 58(10), 2458–2468 (2011)

    Article  MathSciNet  Google Scholar 

  38. H. Yang, Y. Xia, P. Shi, M. Fu, Stability of Markovian jump systems over networks via delta operator approach. Circuits Syst. Signal Process 31, 107–125 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  39. H. Yang, Y. Xia, P. Shi, M. Fu, Stability analysis for high frequency networked control systems. IEEE Trans. Autom. Control 57(10), 2694–2700 (2012)

    Article  MathSciNet  Google Scholar 

  40. H. Yang, Y. Xia, P. Shi, L. Zhao, Analysis and Synthesis of Delta Operator Systems (Springer, Berlin, 2012)

    Book  Google Scholar 

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Acknowledgments

The authors would like to thank the anonymous reviewers for their valuable and helpful comments and suggestions which helped improving the quality of the paper greatly. This publication was made possible by NPRP grant #[4-451-2-168] from the Qatar National Research Fund (a member of Qatar Foundation). The statements made herein are solely the responsibility of the authors. This research was also supported by the National Natural Science Foundation of China under Grant 61104001 and the International Cooperation Program for Excellent Lecturers of 2012 by Shandong Provincial Education Department, China and in part by NSF 1021203 of United States.

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Authors and Affiliations

  1. College of Automation Engineering, Qingdao University, Qingdao, 266071, People’s Republic of China

    Xin-zhuang Dong

  2. Department of Mathematics, Southern Illinois University, Carbondale, IL, 62901, USA

    Xin-zhuang Dong & Mingqing Xiao

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  1. Xin-zhuang Dong
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  2. Mingqing Xiao
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Correspondence to Xin-zhuang Dong.

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Dong, Xz., Xiao, M. Admissible Control of Linear Singular Delta Operator Systems. Circuits Syst Signal Process 33, 2043–2064 (2014). https://doi.org/10.1007/s00034-013-9732-y

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  • DOI: https://doi.org/10.1007/s00034-013-9732-y

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