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International Symposium on Positive Systems

POSTA 2018: Positive Systems pp 189–201Cite as

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

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Part of the book series: Lecture Notes in Control and Information Sciences ((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.

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References

  1. Anderson, B.D.O., Moore, J.B.: Algebraic structure of generalized positive real matrices. SIAM J. Control 6(4), 615–624 (1968)

    Article  MathSciNet  Google Scholar 

  2. Anderson, B.D.O., Vongpanitlerd, S.: Network analysis and synthesis: a modern systems theory approach. In: Courier Corporation (1973)

    Google Scholar 

  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. Brogan, W.L.: Modern Control Theory. Pearson education india (1991)

    Google Scholar 

  5. Buscarino, A., Fortuna, L., Frasca, M.: Forward action to make a system negative imaginary. Syst. Control Lett. 94, 57–62 (2016)

    Article  MathSciNet  Google Scholar 

  6. Chen, C.T.: Linear System Theory and Design. Oxford University Press Inc, Oxford (1998)

    Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  8. Ferrante, A., Lanzon, A., Ntogramatzidis, L.: Discrete-time negative imaginary systems. Automatica 79, 1–10 (2017)

    Article  MathSciNet  Google Scholar 

  9. Lanzon, A., Chen, H.J.: Feedback stability of negative imaginary systems. IEEE Trans. Autom. Control 62(11), 5620–5633 (2017)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  11. Lanzon, A., Petersen, I.R.: Feedback control of negative-imaginary systems. IEEE Control Syst. 30(5), 54–72 (2010)

    Article  MathSciNet  Google Scholar 

  12. Liu, M., Xiong, J.: On non-proper negative imaginary systems. Syst. Control Lett. 88, 47–53 (2016)

    Article  MathSciNet  Google Scholar 

  13. Liu, M., Xiong, J.: Properties and stability analysis of discrete-time negative imaginary systems. Automatica 83, 58–64 (2017)

    Article  MathSciNet  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  21. Xiong, J., Petersen, I.R., Lanzon, A.: On lossless negative imaginary systems. Automatica 48(6), 1213–1217 (2012)

    Article  MathSciNet  Google Scholar 

  22. Xiong, J., Lam, J., Petersen, I.R.: Output feedback negative imaginary synthesis under structural constraints. Automatica 71, 222–228 (2016)

    Article  MathSciNet  Google Scholar 

  23. Xiong, J., Lanzon, A., Petersen, I.R.: Negative imaginary lemmas for descriptor systems. IEEE Trans. Autom. Control 61(2), 491–496 (2016)

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

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

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Authors and Affiliations

  1. Department of Automation, University of Science and Technology of China, Hefei, 230026, China

    Junlin Xiong & Yongge Guo

Authors
  1. Junlin Xiong
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  2. Yongge Guo
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Corresponding author

Correspondence to Junlin Xiong .

Editor information

Editors and Affiliations

  1. Department of Mechanical Engineering, University of Hong Kong, Hong Kong, Hong Kong

    James Lam

  2. Hangzhou Dianzi University, Hangzhou, China

    Yun Chen

  3. Southwest Minzu University, Chengdu, China

    Xingwen Liu

  4. Dalian University of Technology, Dalian, China

    Xudong Zhao

  5. Hangzhou Dianzi University, Hangzhou, China

    Junfeng Zhang

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Xiong, J., Guo, Y. (2019). Extension of Lossless Negative Imaginary Lemmas to Systems with Poles at the Origin. In: Lam, J., Chen, Y., Liu, X., Zhao, X., Zhang, J. (eds) Positive Systems . POSTA 2018. Lecture Notes in Control and Information Sciences, vol 480. Springer, Cham. https://doi.org/10.1007/978-3-030-04327-8_15

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