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Nest Algebras, Causality Constraints, and Multirate Robust Control

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Robust Control Theory

Part of the book series: The IMA Volumes in Mathematics and its Applications ((IMA,volume 66))

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Abstract

Nest operators and nest algebras present a natural framework for studying causality constraints in multirate control systems [8]. In this article, we first give a tutorial on this framework and then look at robust stabilization of analog plants via multirate controllers and provide an explicit solution to the problem.

The authors would like to thank the organizers of the IMA Workshop on Robust Control Theory, Professors Bruce A. Francis and Pramod P. Khargonekar, and IMA for inviting them to participate in the workshop.

Dept. of Elect. Si Comp. Engg., University of Calgary, Calgary, Alberta, CANADA T2N 1N4. Phone: 403-220-8357. Email. chent@enel.ucalgary.ca. The work of this author was supported by the Natural Sciences and Engineering Research Council of Canada.

Previously with Inst. for Math.& Its Appl., University of Minnesota, Minneapolis, MN USA 55455. Now with Dept. of Elect. & Electronic Engg., Hong Kong University of Science & Technology, Kowloon, Hong Kong. Phone: 852-358-7067. Email: eegiu@uxmail.ust.hk.

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References

  1. H. AL-RAHMANI AND G. F. FRANKLIN, A new optimal multirate control of linear periodic and time-varying systems, IEEE Trans. Automat. Control 35 (1990), pp. 406–415.

    Article  MathSciNet  MATH  Google Scholar 

  2. M. ARAKI AND K. YAMAMOTO, Multivariable multirate sampled-data systems: state-space description, transfer characteristics, and Nyquist criterion, IEEE Trans. Automat. Control 30 (1986), pp. 145–154.

    Article  MathSciNet  Google Scholar 

  3. W. ARVESON, Interpolation problems in nest algebras, J. Functional Analysis 20 (1975), pp. 208–233.

    Article  MathSciNet  MATH  Google Scholar 

  4. B. BAMIEH AND J. B. PEARSON, A general framework for linear periodic systems with application to H221E; sampled-data control, IEEE Trans. Automat. Control 37 (1992), pp. 418–435.

    Article  MathSciNet  MATH  Google Scholar 

  5. M. C. BERG, N. AMIT, AND J. POWELL, Multirate digital control system design, IEEE Trans. Automat. Control 33 (1988), pp. 1139–1150.

    Article  MATH  Google Scholar 

  6. T. CHEN AND B. A. FRANCIS, On the £2-induced norm of a sampled-data system, Systems & Control Letters 36 (1990), pp. 211–219.

    Article  MathSciNet  Google Scholar 

  7. T. CHEN AND B. A. FRANCIS, Sampled-data optimal design and robust stabilization, A.ME J. Dynamic Systems, Measurement, and Control 114 (1992), pp. 538–543.

    Article  MATH  Google Scholar 

  8. T. CHEN AND L. QIU, H oc , design of general multirate sampled-data control systems,Automatica (1994) (to appear).

    Google Scholar 

  9. P. COLANERI, R. SCATTOLINI, AND N. SCHIAVONI, Stabilization of multirate sampled-data linear systems, Automatica 26 (1990), pp. 377–380.

    Article  MATH  Google Scholar 

  10. M. A. DAHLEH, P. G. VOULGARIS, AND L. S. VALAVANI, Optimal and robust controllers for periodic and multirate systems, IEEE Trans. Automat. Control 37 (1992), pp. 90–99.

    Article  MathSciNet  MATH  Google Scholar 

  11. C. DAVIS, M. M. KAHAN, AND H. F. WEINSBERGER, Norm-preserving dilations and their applications to optimal error bounds, SIAM J. Numer. Anal. 19 (1982), pp. 445–469.

    Article  MATH  Google Scholar 

  12. K. R. DAVIDSON, Nest Algebras, Pitman Research Notes in Mathematics Series 191 Longman Scientific & Technical 1988.

    Google Scholar 

  13. A. FEINTUCH, P. P. KHARGONEKAR, AND A. TANNENBAUM, On the sensitivity minimization problem for linear time-varying periodic systems, SIAM J. Control and Optimization 24 (1986), pp. 1076–1085.

    Article  MathSciNet  MATH  Google Scholar 

  14. B. A. FRANCIS, A Course in Control Theory, Springer-Verlag, New York 1987.

    Book  MATH  Google Scholar 

  15. B. A. FRANCIS AND T. T. GEORGIOU, Stability theory for linear time-invariant plants with periodic digital controllers, IEEE Trans. Automat. Control 33 (1988), pp. 820–832.

    Article  MathSciNet  MATH  Google Scholar 

  16. T. T. GEORGIOU AND P. P. KHARGONEKAR, A constructive algorithm for sensitivity optimization of periodic systems, SIAM J. Control and Optimization 25 (1987), pp. 334–340.

    Article  MathSciNet  Google Scholar 

  17. K. GLOVER, D. J. N. LIMEBEER, J. C. DOYLE, E. M. KASENALLY, AND M. G. SAFONOV, A characterization of all solution to the four block general distance problem, SIAM J. Control and Optimization 29 (1991), pp. 283–324.

    Article  MathSciNet  MATH  Google Scholar 

  18. T. HAGIWARA AND M. ARAKI, Design of a stable feedback controller based on the multirate sampling of the plant output, IEEE Trans. Automat. Control 33 (1988), pp. 812–819.

    Article  MathSciNet  MATH  Google Scholar 

  19. S. HARA AND P. T. KABAMBA, Worst case analysis and design of sampled-data control systems, Proc. CDC (1990).

    Google Scholar 

  20. Y. HAYAKAWA, Y. YAMAMOTO, AND S. HARA, 7-too type problem for sampled-data control system—a solution via minimum energy characterization,Proc. CDC (1992) (to appear).

    Google Scholar 

  21. E. I. JURY AND F. J. MULLIN, The analysis of sampled-data control systems with a periodically time-varying sampling rate, IRE Trans. Automat. Control 4 (1959), pp. 15–21.

    Google Scholar 

  22. R. E. KALMAN AND J. E. BERTRAM, A unified approach to the theory of sampling systems, J. Franklin Inst. (267) (1959), pp. 405–436.

    Google Scholar 

  23. P. P. KHARGONEKAR, K. POOLLA, AND A. TANNENBAUM, Robust control of linear time-invariant plants using periodic compensation, IEEE Trans. Automat. Control 30 (1985), pp. 1088–1096.

    Article  MathSciNet  MATH  Google Scholar 

  24. G. M. KRANC, Input-output analysis of multirate feedback systems, IRE Trans. Automat. Control 3 (1957), pp. 21–28.

    Google Scholar 

  25. D. G. MEYER, A parametrization of stabilizing controllers for multirate sampled-data systems, IEEE Trans. Automat. Control 35 (1990), pp. 233–236.

    Article  MathSciNet  MATH  Google Scholar 

  26. D. G. MEYER, A new class of shift-varying operators, their shift-invariant equivalents, and multirate digital systems, IEEE Trans. Automat. Control 35 (1990), pp. 429–433.

    Article  MathSciNet  MATH  Google Scholar 

  27. D. G. MEYER, Cost translation and a lifting approach to the multirate LQG problem, IEEE Trans. Automat. Control 37 (1992), pp. 1411–1415.

    Article  MathSciNet  MATH  Google Scholar 

  28. R. A. MEYER AND C. S. BURRUS, A unified analysis of multirate and periodically time-varying digital filters, IEEE Trans. Circuits and Systems 22 (1975), pp. 162–168.

    Article  MathSciNet  Google Scholar 

  29. S. PARROTT, On a quotient norm and the Sz.-Nagy-Foias lifting theorem, J. Functional Analysis 30 (1978), pp. 311–328.

    Article  MathSciNet  MATH  Google Scholar 

  30. L. QIU AND T. CHEN, 7í2 and 7i ß designs of multirate sampled-data systems, Proc. ACC 1993. (Also appeared as) IMA Preprint Series (# 1062 ) 1992.

    Google Scholar 

  31. R. RAVI, P. P. KHARGONEKAR, K. D. MINTO, AND C. N. NETT, Controller parametrization for time-varying multirate plants, IEEE Trans. Automat. Control 35 (1990), pp. 1259–1262.

    Article  MathSciNet  MATH  Google Scholar 

  32. M. E. SEZER AND D. D. SILJAK, Decentralized multirate control, IEEE Trans. Automat. Control 35 (1990), pp. 60–65.

    Article  MATH  Google Scholar 

  33. W. SUN, K. M. NAGFAL, AND P. P. KHARGONEKAR, 7 00 control and filtering with sampled measurements, Proc. ACC (1991).

    Google Scholar 

  34. G. TADMOR, Optimal sampled-data control in continuous time systems, Proc. ACC (1991).

    Google Scholar 

  35. H. T. TOIVONEN, Sampled-data control of continuous-time systems with an 7 - L eo optimality criterion, Automatica 28 (1) (1992), pp. 45–54.

    Article  MathSciNet  MATH  Google Scholar 

  36. P. G. VOULGARIS, M. A. DAHLEH, AND L. S. VALAVANI, 1-(00 and î 2 optimal controllers for periodic and multi-rate systems, Automatica 32 (1994), pp. 251–264.

    Article  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. Dept. of Elect. & Comp. Engg., University of Calgary, Calgary, Alberta, Canada, T2N 1N4

    Tongwen Chen

  2. Inst. for Math. & Its Appl., University of Minnesota, Minneapolis, MN, USA, 55455

    Li Qiu

  3. Dept. of Elect. & Electronic Engg., Hong Kong University of Science & Technology, Kowloon, Hong Kong, China

    Li Qiu

Authors
  1. Tongwen Chen
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  2. Li Qiu
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Editor information

Editors and Affiliations

  1. Department of Electrical and Computer Engineering, University of Toronto, M5S 1A4, Toronto, Ontario, Canada

    Bruce A. Francis

  2. Department of Electrical Engineering and Computer Science, University of Michigan, 48109, Ann Arbor, MI, USA

    Pramod P. Khargonekar

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© 1995 Springer-Verlag New York, Inc.

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Chen, T., Qiu, L. (1995). Nest Algebras, Causality Constraints, and Multirate Robust Control. In: Francis, B.A., Khargonekar, P.P. (eds) Robust Control Theory. The IMA Volumes in Mathematics and its Applications, vol 66. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-8451-9_1

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  • DOI: https://doi.org/10.1007/978-1-4613-8451-9_1

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-1-4613-8453-3

  • Online ISBN: 978-1-4613-8451-9

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