Skip to main content

H 2 Design of Nominal and Robust Discrete Time Filters

  • Chapter
Polynomial Methods for Control Systems Design

Abstract

Polynomial methods were originally developed with control applications in mind [1, 2], but have turned out to be very useful within digital signal processing and communications. The present chapter 1 will outline a polynomial equations framework for nominal amnd robust multivariable linear filtering and, at the same time, illustrate its utility for signal processing problems in digital communications.

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

Access this chapter

eBook
USD 16.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Kučera V. Discrete Linear Control: The Polynomial Equation Approach. Wiley, Chichester, 1979

    MATH  Google Scholar 

  2. Kučera V. Analysis and Design of Discrete Linear Control Systems. Academia, Prague and Prentice Hall International, London, 1991

    Google Scholar 

  3. Kwakernaak H, Sivan R. Linear Optimal Control Systems. Wiley, New York, 1972.

    MATH  Google Scholar 

  4. Öhrn K, Ahlén A, Sternad M. A probabilistic approach to multivariable robust filtering and open-loop control. IEEE Transactions on Automatic Control,1995; 40:405–418

    Article  MATH  Google Scholar 

  5. Tidestav C, Ahlén A, Sternad M. Narrowband and broadband multiuser detection using a multivariable DFE. IE EE PIMRC’95. Toronto, 1995, pp 732–736

    Google Scholar 

  6. H. Kwakernaak. The polynomial approach to H∞–optimal regulation. In E. Mosca and L. Pandolfi, editors, H∞–Control Theory, volume 1496 of Lecture Notes in Mathematics, pages 141–221. Springer-Verlag, Heidelberg, etc., 1991.

    Google Scholar 

  7. Honig ML, Messerschmidt DG. Adaptive Filters: Structures, Algorithms and Applications. Kluwer Academic, Boston, 1984

    MATH  Google Scholar 

  8. Haykin S. Adaptive Filter Theory. Prentice-Hall, Englewood Cliffs, Third ed. 1996.

    Google Scholar 

  9. Oppenheim AV, Shafer RW. Digital Signal Processing. Prentice-Hall, Englewood Cliffs, 1975

    MATH  Google Scholar 

  10. Widrow B, Stearns SD. Adaptive Signal Processing. Prentice Hall, Englewood Cliffs, 1985

    Google Scholar 

  11. Treichler JR, Johnson Jr CR, Larimore MG. Theory and Design of Adaptive Filters. Wiley, New York, 1987

    MATH  Google Scholar 

  12. Grimble, M. J., 1987 H∞ robust controller for self-tuning control application; Part I: Controller design, Int. J. Cont., 46, 4.

    Google Scholar 

  13. Bode HW, Shannon CE. A simplified derivation of linear least square smoothing and prediction theory. Proceedings of the I.R.E. 1950; 38:417– 425

    Google Scholar 

  14. Kailath T. Lectures on Wiener and Kalman Filtering. Springer, Wien, 1981

    MATH  Google Scholar 

  15. Cadzow JA. Foundations of Digital Signal Processing and Data Analysis. Macmillan, New York, 1987

    Google Scholar 

  16. Ahlén A, Sternad M. Optimal Filtering Problems. In Hunt K (ed) Polynomial Methods in Optimal Control and Filtering, chapter 5. Control Engineering Series, Peter Peregrinus, London, 1993

    Google Scholar 

  17. Ahlén A, Sternad M. Wiener filter design using polynomial equations. IEEE Transactions on Signal Processing. 1991; 39:2387–2399

    Article  MATH  Google Scholar 

  18. Grimble MJ. Polynomial systems approach to optimal linear filtering and prediction. International Journal of Control. 1985; 41:1545–1564

    Article  MATH  MathSciNet  Google Scholar 

  19. Leondes CT, Pearson JO. A minimax filter for systems with large plant uncertainties. IEEE Transactions on Automatic Control. 1972; 17:266–268

    Article  MATH  Google Scholar 

  20. Grimble MJ. Polynomial systems approach to optimal linear filtering and prediction. International Journal of Control. 1985; 41:1545–1564

    Google Scholar 

  21. Kassam SA, Poor HV. Robust techniques for signal processing: a survey. Proceedings of the IEEE. 1985; 73:433–481

    Google Scholar 

  22. Francis B, Zames G. On H∞ optimal sensitivity theory for SISO feedback systems. IEEE Transactions on Automatic Control, 1984;29:9–16

    Article  MATH  MathSciNet  Google Scholar 

  23. Doyle J, Glover K, Khargonekar P, Francis B. State space solutions to standard H∞ and H∞ control problems. IEEE Transactions on Automatic Control, 1989;34:831–847

    Article  MATH  MathSciNet  Google Scholar 

  24. Grimble MJ, ElSayed A. Solution to the H∞ optimal linear filtering problem for discrete-time systems. IEEE Transactions on Signal Processing. 1990; 38:1092–1104

    Article  MATH  MathSciNet  Google Scholar 

  25. Shaked U, Theodor Y. A frequency domain approach to the problems of H∞–minimum error state estimation and deconvolution. IEEE Transactions on Signal Processing. 1992; 40:3001–3011.

    Article  MATH  Google Scholar 

  26. Bolzern P, Colaneri P, De Nicolao G. Optimal robust filtering for linear systems subject to time-varying parameter perturbations In: Proceedings of the IEEE 32nd Conference on Decision and Control. San Antonio, 1993, pp 1018–1023

    Google Scholar 

  27. Bolzern P, Colaneri P, De Nicolao G. Optimal robust filtering for linear systems subject to time-varying parameter perturbations In: Proceedings of the IEEE 32nd Conference on Decision and Control. San Antonio, 1993, pp 1018–1023

    Google Scholar 

  28. Xie L, Soh YC. Robust Kalman filtering for uncertain systems. Systems & Control Letters. 1994; 22:123–129

    Article  MATH  MathSciNet  Google Scholar 

  29. Xie L, Soh YC, de Souza CE. Robust Kalman filtering for uncertain discrete-time systems. IEEE Transactions on Automatic Control. 1994; 39:1310–1314

    Article  MATH  Google Scholar 

  30. Shaked U, de Souza C. Robust minimum variance filtering. IEEE Transactions on Signal Processing. 1995; 43:2474–2483

    Article  Google Scholar 

  31. Chung RC, Bélanger PR. Minimum-sensitivity filter for linear time-invariant stochastic systems with uncertain parameters. IEEE Transactions on Automatic Control. 1976 ; 21,98–100

    Article  MATH  Google Scholar 

  32. Grimble MJ. Wiener and Kalman filters for systems with random parameters. IEEE Transactions on Automatic Control. 1984; 29:552–554

    Article  Google Scholar 

  33. Grimble MJ. Wiener and Kalman filters for systems with random parameters. IEEE Transactions on Automatic Control. 1984; 29:552–554

    Article  MATH  MathSciNet  Google Scholar 

  34. Sternad M, Ahlén A. Robust filtering and feedforward control based on probabilistic descriptions of model errors. Automatica. 1993; 29:661–679

    Article  MATH  Google Scholar 

  35. Öhrn K, Ahlén A, Sternad M. A probabilistic approach to multivariable robust filtering, prediction and smoothing. In: Proceedings of the 32nd IEEE Conference on Decision and Control. San Antonio, 1993, pp 1227–1232

    Google Scholar 

  36. Sternad M, Öhrn K, Ahlén A. Robust H2 filtering for structured uncertainty: the performance of probabilistic and minimax schemes. European Control Conference. Rome, 1995, pp 87–92

    Google Scholar 

  37. Öhrn K. Design of Multivariable Cautious Wiener Filters: A Probabilistic Approach. PhD thesis, Uppsala University, 1996

    Google Scholar 

  38. Lindskog E, Sternad M, Ahlén A. Designing decision feedback equalizers to be robust with respect to channel time variations. Nordic Radio Society Symposium on interference resistant radio and radar. Uppsala, 1993

    Google Scholar 

  39. Guo L, Ljung L. The role of model validation for assessing the size of the unmodelled dynamics. In: Proceedings of the 33rd IEEE Conference of Decision and Control. Lake Buena Vista, 1994, pp 3894–3899

    Google Scholar 

  40. Gevers M. Towards a joint design of identification and control. In Trentel– man HL and Willems JC ed. Essays on Control: Perspectives in the Theory and itn Applications. Birkhauser, 1993,111–151.

    Google Scholar 

  41. Kosut RL, Goodwin GC and Polis, eds. Special issue on system identification for robust control design. IEEE Transactions on Automatic Control, 1992,37(7)

    Google Scholar 

  42. Bigi S. On the use of system identification for design purposes and parameter estimation. Licentiate Thesis, Uppsala University, 1995

    Google Scholar 

  43. Lee W. Mobile Communications Engineering. McGraw Hill, New York, 1982

    Google Scholar 

  44. Lee W. Mobile Cellular Telecommunications Systems McGraw Hill, New York, 1989

    Google Scholar 

  45. Proakis JG. Digital Communications. McGraw Hill, New York, 3rd ed. 1995

    Google Scholar 

  46. Lee EA, Messerschmitt DG. Digital Communication. Kluwer Academic Publisher, Boston, Second ed. 1994

    Google Scholar 

  47. Lindskog E. On Equalization and Beamforming for Mobile Radio Applications. Licentiate Thesis, Uppsala University, 1995

    Google Scholar 

  48. Lindskog E, Ahlén A, Sternad M. Combined spatial and temporal equalization using an adaptive antenna array and a decision feedback equalization scheme. In: Proceedings of the IEEE International Conference on Acoustics, Speech and Signal Processing. Detroit, 1995

    Google Scholar 

  49. Lindskog E, Ahlén A, Sternad M. Spatio-temporal equalization for multi-path environments in mobile radio application. In: Proceedings of the 45th IEEE Vehicular Technology Conference, Chicago, 1995, pp 399–403

    Google Scholar 

  50. Benveniste A. Design of adaptive algorithms for tracking of time-varying systems. International Journal of Adaptive Control and Signal Processing. 1987; 2:3–29

    Article  Google Scholar 

  51. Kubin G. Adaptation in rapidly time-varying environments using coefficient filters. In: Proceedings of the IEEE International Conference on Acoustics, Speech and Signal Processing, vol 3. Toronto, 1991,pp 2097–2100

    Google Scholar 

  52. Lindbom L. Simplified Kalman estimation of fading mobile radio channels: high performance at LMS computational load. In: Proceedings of the IEEE International Conference on Acoustics, Speech and Signal Processing, vol 3. Minneapolis, 1993, pp 352–355

    Google Scholar 

  53. Lindbom L. Adaptive equalizers for fading mobile radio channels. Licenciate Thesis, Uppsala University, 1992

    Google Scholar 

  54. Lindbom L. A Wiener filtering approach to the design of tracking algorithms: with applications in mobile radio communications. PhD Thesis, Uppsala University, 1995

    Google Scholar 

  55. Kailath T. Linear Systems. Prentice-Hall, Englewood Cliffs, New Jersey, 1980

    MATH  Google Scholar 

  56. Anderson BDO, Moore JB. Optimal Filtering. Prentice-Hall, Englewood Cliffs, 1979

    MATH  Google Scholar 

  57. Ahlén A, Sternad M. Optimal deconvolution based on polynomial methods. IEEE Transactions on Acoustics, Speech and Signal Processing. 1989; 37:217–226

    Article  Google Scholar 

  58. Carlsson B, Ahlén A, Sternad M, Optimal differentiation based on stochastic signal models. IEEE Transactions on Signal Processing. 1991; 39:341–353

    Article  MATH  Google Scholar 

  59. Carlsson B, Ahlén A, Sternad M, Optimal differentiation based on stochastic signal models. IEEE Transactions on Signal Processing. 1991; 39:341–353

    Article  Google Scholar 

  60. Moir TJ. A polynomial approach to optimal and adaptive filtering with application to speech enhancement. IEEE Transactions on Signal Processing. 1991; 39:1221–1224

    Article  Google Scholar 

  61. Moir TJ. A polynomial approach to optimal and adaptive filtering with application to speech enhancement. IEEE Transactions on Signal Processing. 1991; 39:1221-1224

    Google Scholar 

  62. Grimble MJ. Time-varying polynomial systems approach to multichannel optimal linear filtering. Proceedings of the American Control Conference. WA6–10:45, Boston, 1985, pp 168–174

    Google Scholar 

  63. Fitch SM, Kurz L. Recursive equalization in data transmission – A design procedure and performance evaluation. IEEE Transactions on Communication. 1975; 23:546–550

    Article  Google Scholar 

  64. Qureshi SUH. Adaptive equalization. Proceedings of the IEEE. 1985; 73:1349–1387

    Google Scholar 

  65. Nelson PA, Hamada H and Elliot SJ. Adaptive inverse filters for stereophonic sound reproduction. IEEE Transactions on Signal Processing. 1992; 40:1621-1632

    Article  MATH  Google Scholar 

  66. Mendel JM. Optimal Seismic Deconvolution. An Estimation-Based Approach. Academic Press, New York, 1983

    Google Scholar 

  67. Ahlén A, Sternad M. Filter design via inner-outer factorization: Comments on „Optimal deconvolution filter design based on orthogonal principle„. Signal Processing, 1994; 35:51–58

    Article  Google Scholar 

  68. Kučera V. Factorization of rational spectral matrices: a survey. In: Preprints of IEE ControP91. Edinburgh, 1991, pp 1074–1078

    Google Scholar 

  69. Ježek J, Kučera V. Efficient algorithm for matrix spectral factorization. Automatica. 1985; 21:663–669

    Article  MATH  Google Scholar 

  70. Demeure CJ, Mullis CT. A Newton-Raphson method for moving average spectral factorization using the Euclid algorithm. IEEE Transactions on Acoustics, Speech and Signal Processing. 1990; 38:1697–1709

    Article  MATH  MathSciNet  Google Scholar 

  71. Deng ZL. White-noise filter and smoother with application to seismic data deconvolution. In: Preprints 7th IFAC/IFORS Symp. Identification Syst. Parameter Estimation. York, 1985, pp 621–624

    Google Scholar 

  72. Demeure CJ, Mullis CT. A Newton-Raphson method for moving average spectral factorization using the Euclid algorithm. IEEE Transactions on Acoustics, Speech and Signal Processing. 1990; 38:1697–1709

    Google Scholar 

  73. Ahlén A, Sternad M. Adaptive input estimation. IFAC Symposium ACASP–89, Adaptive Systems in Control and Signal Processing. Glasgow, 1989, pp 631–636

    Google Scholar 

  74. Ahlén A, Sternad M. Adaptive deconvolution based on spectral decomposition. SPIE Annual Symposium, vol 1565. San Diego, 1991, pp 130–142

    Google Scholar 

  75. Ahlén A. Identifiability of the deconvolution problem. Automatica. 1990; 26:177–181

    Article  MATH  Google Scholar 

  76. Nikias CL, Petropulu AP. Higher Order Spectral Analysis: A Nonlinear Signal Processing Framework. Prentice Hall, Englewood Cliffs, 1993.

    Google Scholar 

  77. Tong L, Xu G, Kailath T. Fast blind equalization via antenna arrays. In: Proceedings of the IEEE International Conference on Acoustics, Speech and Signal Processing. Minneapolis, 1993, 4:272–275.

    Google Scholar 

  78. Moulines E, Duhamel P, Cardoso J-F, Mayrargue S. Subspace methods for the blind identification of multichannel FIR filters. IEEE Transactions on Signal Processing. 1995; 43(2):516–525.

    Article  Google Scholar 

  79. Bernhardsson B, Sternad M. Feedforward control is dual to deconvolution. International Journal of Control. 1993; 47:393–405

    Article  MathSciNet  Google Scholar 

  80. Sternad M, Ahlén A. The structure and design of realizable decision feedback equalizers for IIR-channels with coloured noise. IEEE Transactions on Information Theory. 1990; 36:848–858

    Article  MATH  Google Scholar 

  81. Tidestav C. Optimum diversity combining in multi-user digital mobile telephone systems. Master’s Thesis, Uppsala University, 1993

    Google Scholar 

  82. Forney Jr GD. The Viterbi algorithm. In: Proceedings of IEEE. 1973; 61:268–278

    Google Scholar 

  83. Belfiore CA, Park JH. Decision feedback equalization. Proceedings of the IEEE. 1979;67:1143–1156

    Google Scholar 

  84. Monsen P. Feedback equalization for fading dispersive channels. IEEE Transactions on Information Theory. 1971;17:56–64

    Article  MATH  Google Scholar 

  85. Salz J. Optimum mean-square decision feedback equalization. Bell System Technical Journal. 1973; 52:1341–1374

    Google Scholar 

  86. Balaban P, Salz J. Optimum diversity combining and equalization in digital data transmission with applications to cellular mobile radio - part I: theoretical considerations. IEEE Transactions on Communications 1992; 40:885–894

    Article  MATH  Google Scholar 

  87. Lo NWK, Falconer DD, Sheikh AUH. Adaptive equalizer MSE performance in the presence of multipath fading, interference and noise. In: Proceesings of the IEEE Vehicular Technology Conference, Rosemont, 1995, 409–413

    Google Scholar 

  88. Sternad M, Ahlén A, Lindskog E. Robust decision feedback equalization. In: Proceedings of the IEEE International Conference on Acoustics, Speech and Signal Processing, Minneapolis, 1993;3:555–558.

    Google Scholar 

  89. Winters J. On the capacity of radio communication systems with diversity in a Rayleigh fading environment. IEEE Journal on Selected Areas in Communication. 1987; 5(5):871–878

    Google Scholar 

  90. Falconer DD, Abdulrahman M, Lo NWK, Petersen BR, Sheikh AUH. Advances in equalization and diversity for portable wireless systems. Digital Signal Processing. 1993; 3(3):148–162

    Article  Google Scholar 

  91. Gilhousen KS, Jacobs IW, Padovani R, Viterbi A, Weaver LA, Wheatly CE. On the capacity of a cellular CDMA system. IEEE Transactions on Vehicular Technology. 1991; 40:303–312

    Article  Google Scholar 

  92. Verdú S. Optimum Multi-user Signal Detection. PhD thesis, University of Illinois, 1984.

    Google Scholar 

  93. Goodwin GC. Salgado ME. A stochastic embedding approach for quantifying uncertainty in the estimation of restricted complexity models. International Journal of Adaptive Control and Signal Processing. 1989; 3:333–356

    MATH  Google Scholar 

  94. Grimble MJ. Generalized Wiener and Kalman filters for uncertain parameters. In: Proceedings of the 21th IEEE Conference on Decision and Control. Orlando, 1982, pp 221–227

    Google Scholar 

  95. D’Appolito JA, Hutchinson CE. A minimax approach to the design of low sensitivity state estimators. Automatica. 1972; 8:599–608

    Article  MATH  MathSciNet  Google Scholar 

  96. Martin CJ, Mintz M. Robust filtering and prediction for linear systems with uncertain dynamics: a game-theoretic approach. IEEE Transactions on Automatic Control. 1983; 28:888–896

    Article  MATH  Google Scholar 

  97. Moustakides G. Kassam SA. Robust Wiener filters for random signals in correlated noise. IEEE Transactions on Information Theory. 1983; 29:614– 619

    Article  MATH  MathSciNet  Google Scholar 

  98. Poor HV. On robust Wiener filtering. IEEE Transactions on Automatic Control. 1980; 25:531–536

    Article  MATH  MathSciNet  Google Scholar 

  99. Vastola KS, Poor HV. Robust Wiener-Kolmogorov theory. IEEE Transactions on Information Theory. 1984; 30:316–327

    Article  MATH  MathSciNet  Google Scholar 

  100. Petersen IR, McFarlane DC. Robust state estimation for uncertain systems. In: Proceedings of the 30th Conference on Decision and Control. Brighton, 1991, pp 2630–2631

    Google Scholar 

  101. Haddad WF, Bernstein DS. Robust, reduced-order, nonstrictly proper state estimation via the optimal projection equations with guaranteed cost bounds. IEEE Transactions on Automatic Control. 1988; 33:591–595

    Article  MATH  Google Scholar 

  102. Haddad WF, Bernstein DS. The optimal projection equations for reduced– order discrete-time state estimation for linear systems with multiplicative noise. System & Control Letters. 1987; 8:381–388

    Article  MATH  MathSciNet  Google Scholar 

  103. Ljung L, Söderström T. Theory and Practice of Recursive Identification. MIT Press, Cambridge, 1983

    MATH  Google Scholar 

  104. Ljung L. System Identification: Theory for the User. Prentice Hall, Engle– wood Cliffs, 1987

    MATH  Google Scholar 

  105. Wahlberg B. System identification using Laguerre models. IEEE Transactions on Automatic Control. 1991; 36:551-562

    Article  MathSciNet  Google Scholar 

  106. P. Lindskog and B. Walhberg, Applications of Kautz models in system identification. In: Proceedings of the IFAC World Congress, Sydney, 1993(5) :309-312

    Google Scholar 

  107. Hakvoort R. System identification for robust process control. PhD Thesis, Delft University of Technology, The Netherlands, 1994.

    Google Scholar 

  108. Widrow B, Walach E. On the statistical efficiency of the LMS algorithm with nonstationary inputs. IEEE Transactions on Information Theory. 1984; 30:211–221

    Article  Google Scholar 

  109. Benveniste A, Métivier M, Priouret P. Adaptive Algorithms and Stochastic Approximations. Springer, Berlin, 1990

    MATH  Google Scholar 

  110. Clark AP. Adaptive Detectors for Digital Modems. Pentech Press, London, 1989

    Google Scholar 

Download references

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 1996 Springer-Verlag London Limited

About this chapter

Cite this chapter

Sternad, M., Ahlén, A. (1996). H 2 Design of Nominal and Robust Discrete Time Filters. In: Grimble, M.J., Kučera, V. (eds) Polynomial Methods for Control Systems Design. Springer, London. https://doi.org/10.1007/978-1-4471-1027-9_5

Download citation

  • DOI: https://doi.org/10.1007/978-1-4471-1027-9_5

  • Publisher Name: Springer, London

  • Print ISBN: 978-3-540-76077-1

  • Online ISBN: 978-1-4471-1027-9

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics