Unified Modeling for Singularly Perturbed Systems by Delta Operators: Pole Assignment Case

  • Kyungtae Lee
  • Kyu-Hong Shim
  • M. Edwin Sawan
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3397)


A unified modeling method by using the δ-operators for the singularly perturbed systems is introduced. The unified model unifies the continuous and the discrete models. When compared with the discrete model, the unified model has an improved finite word-length characteristics and its δ-operator is handled conveniently like that of the continuous system. In additions, the singular perturbation method, a model approximation technique, is introduced. A pole placement example is used to show such advantages of the proposed methods. It is shown that the error of the reduced model in its eigenvalues is less than the order of ε (singular perturbation parameter). It is shown that the error in the unified model is less than that of the discrete model.


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  1. 1.
    Middleton, R.H., Goodwin, G.C.: Improved finite word length characteristics in digital control using delta operators. IEEE Trans. on Automatic Control 31, 1015–1021 (1986)zbMATHCrossRefGoogle Scholar
  2. 2.
    Janecki, D.: Model reference adaptive control using delta operator. IEEE Trans. on Automatic Control 33, 771–775 (1988)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Salgado, M., Middleton, R.H., Goodwin, G.C.: Connection between continuous and discrete Riccati equations with application to Kalman filtering. IEE Proceedings Pt. D 135, 28–34 (1988)zbMATHMathSciNetGoogle Scholar
  4. 4.
    Middleton, R.H., Goodwin, G.C.: Digital Control and Estimation: A Unified Approach. Prentice-Hall, Englewood Cliffs (1990)zbMATHGoogle Scholar
  5. 5.
    Li, G., Gevers, M.: Round-off noise minimization using delta-operator realizations. IEEE Trans. on Signal Processing 41, 629–637 (1993)zbMATHCrossRefGoogle Scholar
  6. 6.
    Li, G., Gevers, M.: Comparative study of finite word-length effects in shift and delta operator parameterizations. IEEE Trans. on Automatic Control 38, 803–807 (1993)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Shim, K.H., Sawan, M.E.: Linear quadratic regulator design for singularly perturbed systems by unified approach using delta operators. International Journals of Systems Science 32, 1119–1125 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Shim, K.H., Sawan, M.E.: Near-optimal state feedback design for singularly perturbed systems by unified approach. Int.l J. of Systems Science 33, 197–212 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Chang, K.W.: Diagonalization method for a vector boundary problem of singular perturbation type. J. of Mathematical Analysis and Application 48, 652–665 (1974)zbMATHCrossRefGoogle Scholar
  10. 10.
    Chow, J., Kokotovic, P.V.: Eigenvalue placement in two-time-scale systems. IFAC Symposium on Large Scale Systems, 321–326 (1976)Google Scholar
  11. 11.
    Kokotovic, P.V.: A Riccati equation for block diagonalization of ill-conditioned systems. IEEE Trans. on Automatic Control 20, 812–814 (1975)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Kokotovic, P.V., Khalil, H., O’ Reilly, J.: Singular perturbation methods in control analysis and design. Academic Press, Orlando (1986)zbMATHGoogle Scholar
  13. 13.
    Naidu, D.S.: Singular Perturbation Methodology in Control Systems. Peter Peregrinus, London, United Kingdom (1988)zbMATHGoogle Scholar
  14. 14.
    Naidu, D.S., Rao, A.K.: Singular perturbation analysis of the closed-loop discrete optimal control system. Optimal Control Application & Methods 5, 19–37 (1984)zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Naidu, D.S., Price, D.B.: Time-scale synthesis of a closed-loop discrete optimal Control system. J. of Guidance 10, 417–421 (1987)zbMATHCrossRefGoogle Scholar
  16. 16.
    Naidu, D.S., Price, D.B.: Singular perturbations and time scales in the design of digital flight control systems. NASA Technical Paper 2844 (1988)Google Scholar
  17. 17.
    Mahmoud, M.S.: Order reduction and control of discrete systems. IEE PROC. 129 Pt.D, 129–135 (1982)Google Scholar
  18. 18.
    Mahmoud, M.S., Chen, Y.: Design of feedback controllers by two-time-stage methods. Appl. Math. Modelling 7, 163–168 (1983)zbMATHCrossRefGoogle Scholar
  19. 19.
    Mahmoud, M.S., Singh, M.G.: On the use of reduced-order models in output feedback design of discrete systems. Automatica 21, 485–489 (1985)zbMATHCrossRefGoogle Scholar
  20. 20.
    Mahmoud, M.S., Chen, Y., Singh, M.G.: Discrete two-time-scale systems. Int.l J. of Systems Science 17, 1187–1207 (1986)zbMATHCrossRefGoogle Scholar
  21. 21.
    Tran, M.T., Sawan, M.E.: Reduced order discrete-time models, Int.l J. Systems Science 14, 745–752 (1983)zbMATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Tran, M.T., Sawan, M.E.: Decentralized control for two time-scale systems. Int. J. Systems Science 15, 1295–1300 (1984)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Kyungtae Lee
    • 1
  • Kyu-Hong Shim
    • 2
  • M. Edwin Sawan
    • 3
  1. 1.Aerospace Engineering DepartmentSejong UniversitySeoulKorea
  2. 2.Sejong-Lockheed Martin Aerospace Research CenterSejong UniversitySeoulKorea
  3. 3.Electrical Engineering DepartmentWichita State UniversityWichitaUSA

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