Microsystem Technologies

, Volume 19, Issue 9–10, pp 1407–1413 | Cite as

Probabilistic analytic center cutting plane method in robust \({\mathcal{H}}_2\) track following control

  • Mohammadreza Chamanbaz
  • Venkatakrishnan Venkataramanan
  • Qing-Guo Wang
Technical Paper


The present paper addresses the design of discrete time robust \({\mathcal{H}}_2\) track following dynamic output feedback controller for hard disk drives where uncertain parameters enter in a non-linear fashion into plant description. Uncertain parameters are considered as random variables with uniform distribution. The controller is designed to meet the performance specification with desired probabilistic levels (accuracy and confidence). The design is benefited from convex optimization in design parameter space and randomization in the uncertainty space. A localization method based on analytic center cutting plane algorithm is employed in order to find the probabilistic robust feasible solution. As a result of randomization, the computational complexity of the algorithm does not depend on the number of uncertain parameters and no conservatism is introduced while handling uncertain parameters. The effectiveness of the designed controller is verified through simulation as well as experiment.


Linear Matrix Inequality Uncertain Parameter Hard Disk Drive Voice Coil Motor Transfer Function Matrix 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



We would like to express our gratitude to Prof. Roberto Tempo and Dr. Fabrizio Dabbene for their constructive comments which led to improvement of the paper.


  1. Bai EW, Tempo R, Fu M (1998) Worst-case properties of the uniform distribution and randomized algorithms for robustness analysis. Math Control Signals Syst 11:183–196MathSciNetzbMATHCrossRefGoogle Scholar
  2. Boyd S, El Ghaoui L (1993) Method of centers for minimizing generalized eigenvalues. Linear Algebra Appl 188(189(0):63–111MathSciNetCrossRefGoogle Scholar
  3. Calafiore G, Campi M (2004) Uncertain convex programs: randomized solutions and confidence levels. Math Program 102(1):25–46MathSciNetCrossRefGoogle Scholar
  4. Calafiore GC, Dabbene F (2007) A probabilistic analytic center cutting plane method for feasibility of uncertain LMIs. Automatica 43(12):2022–2033MathSciNetzbMATHCrossRefGoogle Scholar
  5. Chamanbaz M, Dabbene F, Tempo R, Venkataramanan V, Wang QG (2013a) Sequential randomized algorithms for convex optimization in the presence of uncertainty. IEEE Trans Autom Control. Submitted. arXiv:1304.2222 Google Scholar
  6. Chamanbaz M, Dabbene F, Tempo R, Venkataramanan V, Wang QG (2013b) A statistical learning theory approach for uncertain linear and bilinear matrix inequalities. Automatica. Submitted. arXiv:1305.4952 Google Scholar
  7. Chen BM, Peng K, Lee TH, Venkataramanan V (2006) Hard disk drive servo systems. In: Advances in industrial control series. Springer, New YorkGoogle Scholar
  8. De Oliveira MC, Geromel JC, Bernussou J (2002) Extended H 2 and H norm characterizations and controller parameterizations for discrete-time systems. Int J Control 75(9):666–679MathSciNetzbMATHCrossRefGoogle Scholar
  9. Doyle J, Glover K, Khargonekar P, Francis B (1989) State-space solutions to standard H 2 and H control problems. IEEE Trans Autom Control 34(8):831–847MathSciNetzbMATHCrossRefGoogle Scholar
  10. Goffin JL, Vial JP (2002) Convex nondifferentiable optimization: a survey focused on the analytic center cutting plane method. Optim Methods Softw 17:805–867MathSciNetzbMATHCrossRefGoogle Scholar
  11. Hernandez D, Park SS, Horowitz R, Packard AK (1999) Dual-stage track-following servo design for hard disk drives. In: Proceedings of the 1999 American control conference, vol 6. IEEE, pp 4116–4121Google Scholar
  12. Horn RA, Johnson CR (1990) Matrix analysis. Cambridge University Press, CambridgezbMATHGoogle Scholar
  13. Kanev S, De Schutter B, Verhaegen M (2003) An ellipsoid algorithm for probabilistic robust controller design. Syst Control Lett 49(5):365–375MathSciNetzbMATHCrossRefGoogle Scholar
  14. Kanev S, Scherer C, Verhaegen M, De Schutter B (2004) Robust output-feedback controller design via local BMI optimization. Automatica 40(7):1115–1127MathSciNetzbMATHCrossRefGoogle Scholar
  15. Löfberg J (2004) YALMIP: a toolbox for modeling and optimization in MATLAB. In: Proceedings of IEEE international symposium on computer aided control systems design. Taipei, Taiwan, pp 284–289Google Scholar
  16. Markov AA (1884) On certain applications of algebraic continued fraction. PhD Thesis, St. Petersburg (in Russian)Google Scholar
  17. Mitchell JE (2003) Polynomial interior point cutting plane methods. Optim Methods Softw 18(5):507–534MathSciNetzbMATHCrossRefGoogle Scholar
  18. Myo K, Zhou W, Yu S, Hua W (2011) Direct monte carlo simulation of air bearing effects in heat-assisted magnetic recording. Microsyst Technol 17(5):903–909CrossRefGoogle Scholar
  19. Nagamune R, Huang X, Horowitz R (2010) Robust control synthesis techniques for multirate and multisensing track-following servo systems in hard disk drives. J Dyn Syst Meas Control 132(2):1896–1904Google Scholar
  20. Oishi Y (2003) Probabilistic1 design of a robust state-feedback controller based on parameter-dependent Lyapunov functions. In: Proceedings of the 42nd IEEE conference on decision and control 2003, vol 2. pp 1920–1925Google Scholar
  21. Overton ML (1988) On minimizing the maximum eigenvalue of a symmetric matrix. SIAM J matrix Anal Appl 9:256–268Google Scholar
  22. Packard A, Doyle J (1993) The complex structured singular value. Automatica 29(1):71–109MathSciNetzbMATHCrossRefGoogle Scholar
  23. Polyak BT, Tempo R (2001) Probabilistic robust design with linear quadratic regulators. Syst Control Lett 43(5):343–353MathSciNetzbMATHCrossRefGoogle Scholar
  24. Stengel RF (1980) Some effects of parameter variations on the lateral-directional stability of aircraft. J Guid Control 3(2):124–131CrossRefGoogle Scholar
  25. Tempo R, Bai EW, Dabbene F (1996) Probabilistic robustness analysis: explicit bounds for the minimum number of samples. In: Proceedings of the 35th IEEE decision and control 1996, vol 3. IEEE, New York, pp 3424–3428Google Scholar
  26. Tempo R, Calafiore G, Dabbene F (2005) Randomized algorithms for analysis and control of uncertain systems. Springer-Verlag London Limited, LondonzbMATHGoogle Scholar
  27. Tremba A, Calafiore G, Dabbene F, Gryazina E, Polyak B, Shcherbakov P, Tempo R (2008) RACT: randomized algorithms control toolbox for MATLAB. In: Proceedings of the 17th world congress of IFAC, Seoul, pp 390–395Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Mohammadreza Chamanbaz
    • 1
    • 2
  • Venkatakrishnan Venkataramanan
    • 1
  • Qing-Guo Wang
    • 2
  1. 1.Data Storage InstituteSingaporeSingapore
  2. 2.National University of SingaporeSingaporeSingapore

Personalised recommendations