Combining Symbolic tools with interval analysis. An application to solve robust control problems.

DOI: 10.4236/ajcm.2014.43015   PDF   HTML   XML   3,103 Downloads   4,100 Views  


Complex systems are often subjected to uncertainties that make its model difficult, if not impossible to obtain. A quantitative model may be inadequate to represent the behavior of systems which require an explicit representation of imprecision and uncertainty. Assuming that the uncertainties are structured, these models can be handled with interval models in which the values of the parameters are allowed to vary within numeric intervals. Robust control uses such mathematical models to explicitly have uncertainty into account. Solving robust control problems, like finding the robust stability or designing a robust controller, involves hard symbolic and numeric computation. When interval models are used, it also involves interval computation. The main advantage using interval analysis is that it provides guaranteed solutions, but as drawback its use requires the interaction with multiple kinds of data. We present a methodology and a framework that combines symbolic and numeric computation with interval analysis to solve robust control problems.

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Ferrer-Mallorquí, I. and Vehí, J. (2014) Combining Symbolic tools with interval analysis. An application to solve robust control problems.. American Journal of Computational Mathematics, 4, 183-196. doi: 10.4236/ajcm.2014.43015.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] The MathWorks Inc. (1999) The Polynomial Toolbox for Matlab, v. 2.0. Technical Report.
[2] Sakabe, K., Yanami, H., Anai, H. and Hara, S. (2004) A Matlab Toolbox for Robustcontrol Synthesis by Symbolic Computation. SICE Annual Conference in Sapporo, Hokkaido Institute of Tecnology, Japan, August 2004, 1968-1973.
[3] Sienel, W., Bunte, T. and Ackermann, J. (1996) Paradise: Parametric Robust Analysisand Design Interactive Software Environment: A Matlab-Based Robust Control Toolbox. Proceedings of the 1996 IEEE International Symposium on Computer-Aided Control System Design, Dearborn, 15-18 September 1996, 380-385.
[4] Nataraj, P.S.V. and Sardar, G. (2000) Computation of QFT Bounds for Robust Sensitivity and Gain-Phase Margins Specifications. Journal of Dynamic Systems Measurement and Control, 122, 528-834.
[5] Ackermann, J. (1993) Robust Control: Systems with Uncertain Physical Parameters. Communications and Control Engineering Series. Springer-Verlag, Berlin.
[6] Hyodo, N., Hong, M., Yanami, H. and Anai, H., et al. (2006) Development of a Matlab Toolbox for Parametric Robust Control: New Algorithms and Functions. SICE-ICASE International Joint Conference 2006, Busan, 18-21 October 2006, 2856-2861,
[7] Ackermann, J. (2000) PARADISE—Parametric Robustness Analysis and Design Interactive Software Environment. Technical Report, Institute of Robotics and Mechatronics.
[8] Moore, R.E. (1979) Methods and Applications of Interval Analysis. In: SIAM Studies in Applied Mathematics, SIAM, Philadelphia.
[9] Vehí, J., Herrero, P., Sainz, M.A. and Jaulin, L. (2004) Quantified Set Inversion with Applications to Control. IEEE International Symposium on ComputerAided Control Systems Design, Taipei, 2-4 September 2004, 179-183.
[10] Ackermann, J. (2002) Robust Control. The Parameter Space Approach. Springer, Berlin.
[11] Frazer, R. and Duncan, W. (1929) On the Criteria for Stability of Small Motions. Proceedings of the Royal Society A, 124, 642-654.
[12] Munro, N. (1999) Symbolic Methods in Control System Analysis and Design. IEEE Control Engineering Series, 56, 393 p.
[13] Chapellat, H., Battacharyya, S.P and Keel, L.H. (1995) Robust Control: The Parametric Approach. In: Information and System Science Series, N.J. Prentice Hall, Englewood Cliffs.
[14] Garloff, J. and Graft, B. (1999) Robust Schur Stability of Polynomials with Polynomial Parameter Dependency. Multidimensional Systems and Signal Processing, 10, 189-199.
[15] Jaulin, L. and Walter, E. (1996) Guaranteed Tuning, with Application to Robust Control and Motion Planning. Automatica, 32, 1217-1221.
[16] Stoorvogel, A. (1992) The Control Problem. A State Space Approach. Prentice Hall International Series in Systems and Control Engineering, London.
[17] Salapaka, M.V., Dahleh, M. and Voulgaris, P. (1997) Mixed Objective Control Synthesis: Optimal l1/h2 Control. SIAM: SIAM Journal on Control and Optimization (SICON), 35, 1672-1689.
[18] Zhou, K., Doyle, J.C. and Glover, K. (1996) Robust and Optimal Control. Prentice Hall Inc., Upper Saddle River.
[19] Gagnon, E., Pomerleau, A. and Desbiens, A. (1999) Mu-Synthesis of Robust Decentralized PI Controllers. IEEE Proceedings Control Theory Applications, 146, 289-294.
[20] Vehí, J. (1998) Analysis and Design of Robust Controllers by Means of Modal Intervals. Ph.D. Thesis, Universitat de Girona, Spain. (in Catalan)
[21] SIGLA/X. (1999) Modal Intervals. In: J. Vehí and M. Sainz, Eds., Application of Interval Analysis to Systems and Control, Girona, Spain, 157-221.
[22] Fiorio, G., Malan, S., Milanese, M. and Taragna, M. (1993) Robust Performance Design of Fixed Structure Controllers for Systems with Uncertain Parameters. Proceedings of the 32nd Conference on Decision and Control, San Antonio, 15-17 December 1993, 3029-3031.
[23] Malan, S., Milanese, M. and Taragna, M. (1997) Robust Analysis and Design of Control Systems Using Interval Arithmetic. Automatica, 33, 1363-1372.

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