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Extension of Lossless Negative Imaginary Lemmas to Systems with Poles at the Origin

  • Junlin Xiong
  • Yongge Guo
Chapter
Part of the Lecture Notes in Control and Information Sciences book series (LNCIS, volume 480)

Abstract

This paper is concerned with extending lossless negative imaginary lemmas to the dynamical system with poles at the origin. Firstly, two versions of lossless negative imaginary lemma are established in terms of a set of linear matrix equations. They can be considered as extensions of the previous results. Secondly, a new type of lossless negative imaginary lemma is derived based on Kalman canonical decomposition of system state-space realization. The second type of lossless negative imaginary lemma does not require solving linear matrix equations, which is more computationally efficient. Finally, the validity of the developed lemmas is illustrated by a numerical example.

Keywords

Linear systems Lossless negative imaginary systems Kalman decomposition 

Notes

Acknowledgements

The work in this paper was financially supported by National Natural Science Foundation of China (No.61374026, No. 61773357).

References

  1. 1.
    Anderson, B.D.O., Moore, J.B.: Algebraic structure of generalized positive real matrices. SIAM J. Control 6(4), 615–624 (1968)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Anderson, B.D.O., Vongpanitlerd, S.: Network analysis and synthesis: a modern systems theory approach. In: Courier Corporation (1973)Google Scholar
  3. 3.
    Bhikkaji, B., Moheimani, S.O.R.: Fast scanning using piezoelectric tube nanopositioners: a negative imaginary approach. In: IEEE/ASME International Conference on Advanced Intelligent Mechatronics, pp. 274–279 (2009)Google Scholar
  4. 4.
    Brogan, W.L.: Modern Control Theory. Pearson education india (1991)Google Scholar
  5. 5.
    Buscarino, A., Fortuna, L., Frasca, M.: Forward action to make a system negative imaginary. Syst. Control Lett. 94, 57–62 (2016)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Chen, C.T.: Linear System Theory and Design. Oxford University Press Inc, Oxford (1998)Google Scholar
  7. 7.
    Chu, D., Tan, R.C.E.: Algebraic characterizations for positive realness of descriptor systems. SIAM J. Matrix Analy. Appl. 30(1), 197–222 (2008)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Ferrante, A., Lanzon, A., Ntogramatzidis, L.: Discrete-time negative imaginary systems. Automatica 79, 1–10 (2017)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Lanzon, A., Chen, H.J.: Feedback stability of negative imaginary systems. IEEE Trans. Autom. Control 62(11), 5620–5633 (2017)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Lanzon, A., Petersen, I.R.: Stability robustness of a feedback interconnection of systems with negative imaginary frequency response. IEEE Trans. Autom. Control 53(4), 1042–1046 (2008)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Lanzon, A., Petersen, I.R.: Feedback control of negative-imaginary systems. IEEE Control Syst. 30(5), 54–72 (2010)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Liu, M., Xiong, J.: On non-proper negative imaginary systems. Syst. Control Lett. 88, 47–53 (2016)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Liu, M., Xiong, J.: Properties and stability analysis of discrete-time negative imaginary systems. Automatica 83, 58–64 (2017)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Mabrok, M.A., Kallapur, A.G., Petersen, I.R.: Spectral conditions for negative imaginary systems with applications to nanopositioning. IEEE/ASME Trans. Mechatron. 19(3), 895–903 (2014)CrossRefGoogle Scholar
  15. 15.
    Mabrok, M.A., Kallapur, A.G., Petersen, I.R.: Generalizing negative imaginary systems theory to include free body dynamics: control of highly resonant structures with free body motion. IEEE Trans. Autom. Control 59(10), 2692–2707 (2014)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Mabrok, M., Kallapur, A.G., Petersen, I.R.: A generalized negative imaginary lemma and Riccati-based static state-feedback negative imaginary synthesis. Syst. Control Lett. 77, 63–68 (2015)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Mahmood, I.A., Moheimani, S.O.R., Bhikkaji, B.: A new scanning method for fast atomic force microscopy. IEEE Trans. Nanotechnol. 10(2), 203–216 (2011)CrossRefGoogle Scholar
  18. 18.
    Rao, S., Rapisarda, P.: An algebraic approach to the realization of lossless negative imaginary behaviors. SIAM J. Control Optim. 50(3), 1700–1720 (2012)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Song, Z., Lanzon, A., Patra, S.: Towards controller synthesis for systems with negative imaginary frequency response. IEEE Trans. Autom. Control 55(6), 1506–1511 (2010)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Xiong, J., Petersen, I.R., Lanzon, A.: A negative imaginary lemma and the stability of interconnections of linear negative imaginary systems. IEEE Trans. Autom. Control 55(10), 2342–2347 (2010)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Xiong, J., Petersen, I.R., Lanzon, A.: On lossless negative imaginary systems. Automatica 48(6), 1213–1217 (2012)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Xiong, J., Lam, J., Petersen, I.R.: Output feedback negative imaginary synthesis under structural constraints. Automatica 71, 222–228 (2016)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Xiong, J., Lanzon, A., Petersen, I.R.: Negative imaginary lemmas for descriptor systems. IEEE Trans. Autom. Control 61(2), 491–496 (2016)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of AutomationUniversity of Science and Technology of ChinaHefeiChina

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