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Journal of Systems Science and Complexity

, Volume 27, Issue 3, pp 445–452 | Cite as

Performance limitations for a class of Kleinman control systems

  • Xin CaiEmail author
  • Shaoyuan Li
Article
  • 88 Downloads

Abstract

This paper provides preliminary results on performance limitations for a class of discrete time Kleinman control systems whose open loop poles lie strictly outside the unit circle. By exploiting the properties of the Kleinman controllers and using of algebraic Riccati equation (ARE), the relationship between total control energy of Kleinman control systems and the minimum energy needed to stabilize the open-loop systems is revealed. The result reflects how the horizon length of Kleinman controllers affects the performance of the closed-loop systems and quantifies how close the performance of Kleinman control systems is to the minimum energy.

Keywords

Algebraic Riccati equation Kleinman control minimum energy control performance limitations zero terminal receding horizon control 

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References

  1. [1]
    Bode H W, Network Analysis and Feedback Amplifier Design, Van Nostrand, New York, 1945.Google Scholar
  2. [2]
    Chen J and Middleton R H, Special issues on new developments and applications in performance limitation of feedback control, IEEE Trans. Automatic Control, 2003, 48(8): 1297–1393.CrossRefMathSciNetGoogle Scholar
  3. [3]
    Kwakernaak H and Sivan R, The maximally achievable accuracy of linear optimal regulators and linear optimal filter, IEEE Trans. Automatic Control, 1972, 17(1): 79–86.CrossRefzbMATHMathSciNetGoogle Scholar
  4. [4]
    Qiu L and Davison E J, Performance limitations of nonminimum phase systems in the servomechanism problem, Automatica, 1993, 29(2): 337–349.CrossRefzbMATHMathSciNetGoogle Scholar
  5. [5]
    Qiu L and Chen J, Time domain performance limitations of feedback control, Mathematical Theory of Networks and Systems, 1998, 369–372.Google Scholar
  6. [6]
    Chen J, Qiu L, and Toker O, Limitations on maximal tracking accuracy, IEEE Trans. Automatic Control, 2000, 45(2): 326–331.CrossRefzbMATHMathSciNetGoogle Scholar
  7. [7]
    Jemaa L B and Davison E J, Performance limitations in the robust servomechanism problem for discrete-time LTI systems, IEEE Trans. Automatic Control, 2003, 48(8): 1299–1311.CrossRefGoogle Scholar
  8. [8]
    Seron M M, Braslavsky J H, Kokotović P V, and Mayne D Q, Feedback limitations in nonlinear systems: From Bode integrals to cheap control, IEEE Trans. Automatic Control, 1999, 44(4): 829–833.CrossRefzbMATHGoogle Scholar
  9. [9]
    Braslavsky J H, Seron M M, and Kokotović P V, Near-optimal cheap control of nonlinear systems, Proceedings of 4th IFAC Nonlinear Control Systems Design Symposium, Enschede, The Netherlands, 1998.Google Scholar
  10. [10]
    Wu J B and Su W Z, Performance limits in output regulation of an uncertain nonlinear system under disturbances, Control Theory & Applications, 2009, 26(1): 15–22.zbMATHGoogle Scholar
  11. [11]
    Xie L L and Guo L, Fundamental limitations of discrete-time adaptive nonlinear control, IEEE Trans. Automatic Control, 1999, 44(9): 1777–1782.CrossRefzbMATHMathSciNetGoogle Scholar
  12. [12]
    Xie L L and Guo L, How much uncertainty can be dealt with by feedback?, IEEE Trans. Automatic Control, 2000, 45(12): 2203–2217.CrossRefzbMATHMathSciNetGoogle Scholar
  13. [13]
    Chen J, Hara S, and Chen G, Best tracking and regulation performance under control energy constraint, IEEE Trans. Automatic Control, 2003, 48(8): 1320–1336.CrossRefMathSciNetGoogle Scholar
  14. [14]
    Perez T, Goodwin G C, and Serón M M, Performance degradation in feedback control due to constraints, IEEE Trans. Automatic Control, 2003, 48(8): 1381–1385.CrossRefGoogle Scholar
  15. [15]
    Su W Z, Qiu L, and Chen J, An average performance limit of MIMO systems in tracking multisinusoids with partial signal information, IEEE Trans. Automatic Control, 2009, 54(8): 2001–2006.CrossRefMathSciNetGoogle Scholar
  16. [16]
    Kleinman D L, On the iterative technique for Riccati equation computations, IEEE Trans. Automatic Control, 1968, 13: 114–115.CrossRefGoogle Scholar
  17. [17]
    Kleinman D L, An easy way to stabilize a linear constant system, IEEE Trans. Automatic Control, 1970, AC-15: 692.CrossRefGoogle Scholar
  18. [18]
    Kleinman D L, Stabilizing a discrete, constant, linear system with application to iterative methods for solving the Riccati equation, IEEE Trans. Automatic Control, 1974, 19: 252–254.CrossRefzbMATHMathSciNetGoogle Scholar
  19. [19]
    Thomas Y A, Linear quadratic optimal estimation and control with receding horizon, Electronics Letters, 1975, 11: 19–21.CrossRefGoogle Scholar
  20. [20]
    Clarke D W and Mohtadi C, Properties of generalized predictive control, Automatica, 1989, 25(6): 859–875.CrossRefzbMATHMathSciNetGoogle Scholar
  21. [21]
    Ding B C and Xi Y G, Stability analysis of generalized predictive control based on Kleinman’s Controllers, Science in China Ser. F: Information Sciences, 2004, 47(4): 458–474.CrossRefzbMATHMathSciNetGoogle Scholar
  22. [22]
    Ding B C, Modern Predictive Control, CRC Press, 2010.Google Scholar

Copyright information

© Institute of Systems Science, Academy of Mathematics and Systems Science, CAS and Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Department of Automation, Shanghai Jiao Tong University, and Key Laboratory of System Control and Information ProcessingMinistry of EducationShanghaiChina

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