Skip to main content
SpringerLink
Log in

On the dual of a mixed H 2/l 1 optimisation problem

  • Published:
International Journal of Automation and Computing Aims and scope Submit manuscript

Abstract

The general discrete-time Single-Input Single-Output (SISO) mixed H 2/l 1 control problem is considered in this paper. It is found that the existing results of duality theory cannot be directly applied to this infinite dimension optimisation problem. By means of two finite dimension approximate problems, to which duality theory can be applied, the dual of the mixed H 2/l 1 control problem is verified to be the limit of the duals of these two approximate problems.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. B. A. Francis. A Course in H ∞ Control Theory, Springer-Verlag, Berlin, 1987.

    Book  MATH  Google Scholar 

  2. G. Zames. Feedback and optimal sensitivity: model reference transformations, multiplicative seminorms and approximate inverses. IEEE Transactions on Automatic Control, vol. 26, no. 2, pp. 301–320, 1981.

    Article  MathSciNet  Google Scholar 

  3. D. C. Youla, H. A. Jabr, J. J. Bongiorno. Modern Wiener-Hopf design of optimal controllers-part II: the multivariable case. IEEE Transactions on Automatic Control, vol. AC-21, no. 3, pp. 319–338, 1976.

    Article  MathSciNet  Google Scholar 

  4. M. Vidyasagar, Optimal rejection of persistent bounded disturbances. IEEE Transactions on Automatic Control, vol. 31, no. 6, pp. 527–534, 1986.

    Article  MathSciNet  Google Scholar 

  5. I. Kaminer, P. P. Khargonekar, M. A. Rotea. Mixed H 2/H ∞ control for discrete time systems via convex optimization. Automatica, vol. 29, no. 1, pp. 57–70, 1993.

    Article  MathSciNet  Google Scholar 

  6. M. Sznaier, J. Bu. On the properties of the solutions to mixed l 1/H ∞ control problems, In Preprints 13th IFAC Congress, San Francisco, USA, vol. G, pp. 249–254, 1996.

    Google Scholar 

  7. M. V. Salapaka, M. Dahleh, P. Voulgaris. Mixed objective control synthesis: optimal l 1/H 2 control. SIAM Journal of Control and Optimization, vol. 35, no. 5, pp. 1672–1689, 1997.

    Article  MathSciNet  Google Scholar 

  8. P. Voulgaris. Optimal H 2/l 1 control: the SISO case. In Proceedings of IEEE International Conference on Decision and Control, vol. 4, pp. 3181–3186, 1994.

    Google Scholar 

  9. P. Voulgaris. Optimal H 2/l 1 control via duality theory. IEEE Transaction on Automatic Control, vol. 40, pp. 1881–1888, 1995.

    Article  MathSciNet  Google Scholar 

  10. J. Wu, J. Chu. Mixed H 2/l 1 control for discrete time systems. In Preprints 13th IFAC Congress, San Francisco, USA, vol. G, pp. 453–457, 1996.

    Google Scholar 

  11. J. Wu, J. Chu. Approximation methods of scalar mixed H 2/l 1 problems for discrete-time systems. IEEE Transactions on Automatic Control, vol. 44, no. 10, pp. 1869–1874, 1999.

    Article  MathSciNet  Google Scholar 

  12. M. V. Salapaka, M. Khammash, M. Dahleh. Solution of MIMO H 2/l 1 problem without zero interpolation. SIAM Journal of Control and Optimization, vol. 37, no. 6, 1865–1873, 1999.

    Article  MathSciNet  Google Scholar 

  13. N. Elia, M. A. Dahleh. Control design with multiple objectives. IEEE Transactions on Automatic Control, vol. 42, no. 5, pp. 596–613, 1997.

    Article  MathSciNet  Google Scholar 

  14. M. V. Salapaka, M. Dahleh, P. G. Voulgaris. MIMO optimal control design: the interplay between the H 2 and the l 1 norms. IEEE Transactions on Automatic Control, vol. 43, no. 10, pp. 1374–1388, 1998.

    Article  MathSciNet  Google Scholar 

  15. J. Brinkhuis. Introduction to duality in optimization theory. Journal of Optimization Theory and Applications, vol. 91, pp. 523–542, 1996.

    Article  MathSciNet  Google Scholar 

  16. D. G. Luenberger. Optimization by Vector Space Methods, Wiley, New York, 1969.

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. National Key Laboratory of Industrial Control Technology Institute of Advanced Process Control, Zhejiang University, Hangzhou, 310027, PRC

    Jun Wu & Jian Chu

  2. School of Electronics and Computer Science, University of Southampton Highfield, Southampton, SO17 1BJ, UK

    Sheng Chen

Authors
  1. Jun Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jian Chu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Sheng Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Sheng Chen.

Additional information

This work is supported by the National Natural Science Foundation of China (No.60374002 and No.60421002), the 973 program of China (No.2002CB312200), and the program for New Century Excellent Talents in University (No.NCET-04-0547).

Jun Wu received the BEng and PhD degrees both in industrial automation from Huazhong University of Science and Technology, China in 1989 and 1994, respectively. Since 1994, he has been a faculty member at the Institute of Advanced Process Control, Zhejiang University, China, where he is currently a Professor. Dr Wu’s recent research work includes, finite-precision digital controller design, networked control, model reduction, and robust control and optimization. He has published over 50 papers in journals and conference proceedings, and was awarded a 1997/1998 IEE Heaviside Premium.

Sheng Chen obtained a BEng degree in control engineering from the East China Petroleum Institute, Dongying, China, in 1982, and a PhD degree in control engineering from City University, London, UK, in 1986. He joined the School of Electronics and Computer Science at the University of Southampton, UK, in September 1999. He previously held research and academic appointments at the Universities of Sheffield, Edinburgh, and Portsmouth, all in the UK. Professor Chen, holds a higher doctorate degree, DSc, from the University of Southampton. His recent research work includes adaptive nonlinear signal processing, wireless communications, modelling and identification of nonlinear systems, neural network and machine learning, finite-precision digital controller design, and evolutionary computation methods and optimization. He has published over 240 research papers. In the database of the world’s most highly cited researchers in various disciplines, compiled by the Institute for Scientific Information (ISI) of the USA, Dr Chen is on the list of highly cited researchers in the engineering category, see www.ISIHighlyCited.com.

Jian Chu received BEng and MSc degrees both in industrial automation from Zhejiang University, China, in 1982 and 1984 respectively. As a PhD candidate of the joint education programme between Zhejiang University and Kyoto University Japan, he received a PhD degree in industrial automation from Zhejiang University in 1989. Since 1989, he has been a faculty member at the Institute of Advanced Process Control, Zhejiang University, where he currently is the Institute Director, and is a Professor of the Cheung Kong Scholars Programme. He is also the Director of the National Laboratory of Industrial Control Technology, and a Deputy Director of the National Engineering Research Center for Industrial Automation. His research areas include advanced process control, robust control, and fieldbus control systems. He has published over 300 papers in journals and conference proceedings, and was awarded a 1997/1998 IEE Heaviside Premium.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Wu, J., Chu, J. & Chen, S. On the dual of a mixed H 2/l 1 optimisation problem. Int J Automat Comput 3, 91–98 (2006). https://doi.org/10.1007/s11633-006-0091-z

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11633-006-0091-z

Keywords

Search

Navigation