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Synthesis of digital H ∞-controllers for multidimensional systems with given accuracy with the mean squared criterion

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Abstract

We consider linear multidimensional objects subject to power bounded polyharmonic perturbances and measurement noise that contains an arbitrary number of harmonics with unknown amplitudes and frequencies. For such objects, we propose a synthesis method for digital state controllers and controllers with respect to measurable output. The problem of guaranteeing a desired accuracy is formulated as the problem of guaranteeing a given mean squared radius of the stabilized state [1]; our solution of this problem is based on the choice of weight matrices in the minimal quadratic functional of a discrete H ∞-optimization problem. We give a synthesis algorithm for a digital controller in the LMI Control Toolbox package and a numerical example for an interconnected electric drive.

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References

  1. Chestnov, V.N., Synthesis of Multidimensional Systems of Prescribed Accuracy by the Mean-Square Criterion, Autom. Remote Control, 1998, vol. 59, no. 12, part 2, pp. 1786–1792.

    MathSciNet  MATH  Google Scholar 

  2. Kwakernaak, H. and Sivan, R., Linear Optimal Control Systems, New York: Wiley Interscience, 1972. Translated under the title Lineinye optimal’nye sistemy upravleniya, Moscow: Mir, 1977.

    MATH  Google Scholar 

  3. Doyle, J.C., Glover, K., Khargonekar, P.P., and Francis, B.A., State-Space Solution to Standard H 2 and H ∞ Control Problem, IEEE Trans. Automat. Control, 1989, vol. 34, no. 8, pp. 831–846.

    Article  MathSciNet  MATH  Google Scholar 

  4. Spravochnik po teorii avtomaticheskogo upravleniya (Automated Control Theory Reference), Krasovskii, A.A., Ed., Moscow: Nauka, 1987.

    Google Scholar 

  5. Petrov, Yu.P., Guaranteeing Control in Linear Systems, Izv. Akad. Nauk SSSR, Tekh. Kibern., 1989, no. 3, pp. 105–109.

  6. Aleksandrov, A.G. and Chestnov, V.N., Synthesis of Multivariable Systems of Prescribed Accuracy. II. Use of Procedures of H ∞-Optimization, Autom. Remote Control, 1998, vol. 59, no. 8, part 2, pp. 1153–1164.

    MathSciNet  MATH  Google Scholar 

  7. Chestnov, V.N., Design of Digital State H ∞-Controllers of the MIMO Systems with Prescribed Precision, Autom. Remote Control, 2005, vol. 66, no. 8, pp. 1233–1238.

    Article  MathSciNet  MATH  Google Scholar 

  8. Iglesias, P.A. and Glover, K., State-Space Discrete-Time H ∞ Control Theory, Eur. Control Conf., Grenoble, 1991, vol. 2, pp. 1730–1735.

    Google Scholar 

  9. Yaesh, I. and Shaked, U., A Transfer Function Approach to the Problem of Discrete-Time Systems: H ∞-Optimal Linear Control and Filtering, IEEE Trans. Automat. Control, 1991, vol. 36, no. 11, pp. 1264–1271.

    Article  MathSciNet  MATH  Google Scholar 

  10. Iglesias, P.A. and Glover, K., State-Space Approach to Discrete-Time H ∞ Control, Int. J. Control, 1991, vol. 54, no. 5, pp. 1031–1073.

    Article  MathSciNet  MATH  Google Scholar 

  11. Tsypkin, Ya.Z., Teoriya impul’snykh sistem (Impulse Systems Theory), Moscow: Fizmatgiz, 1958.

    Google Scholar 

  12. Grimble, M.J., Robust Industrial Control Systems: Optimal Design Approach for Polynomial Systems, New York: Wiley, 2006.

    Google Scholar 

  13. Basar, T. and Bernhard, P., H ∞-Optimal Control and Related Minimax Design Problems: A Dynamic Game Approach, Boston: Birkhauser, 1991.

    Google Scholar 

  14. Agafonov, P.A. and Chestnov, V.N., H ∞-Control for Guaranteed Simultaneous Input and Output Stability Margins for a Multivariate System, Autom. Remote Control, 2004, vol. 65, no. 9, pp. 1452–1460.

    Article  MathSciNet  MATH  Google Scholar 

  15. Agafonov, P.A. and Chestnov, V.N., Controllers of a Given Radius of Stability Margin: Their Design by the H ∞-approach with Regard for External Disturbances, Autom. Remote Control, 2004, vol. 65, no. 10, pp. 1611–1617.

    Article  MathSciNet  MATH  Google Scholar 

  16. Barabanov, A.E. and Pervozvanskii, A.A., Optimization by Frequency-Uniform Criteria (H-Theory), Autom. Remote Control, 1992, vol. 53, no. 9, part 1, pp. 1301–1327.

    MathSciNet  Google Scholar 

  17. Chestnov, V.N., Synthesizing H ∞-Controllers for Multidimensional Systems with Given Accuracy and Degree of Stability, Autom. Remote Control, 2011, no. 10, pp. 2161–2175.

  18. Boyd, S., Chaoui, L.E., Feron, E., and Balakrishnan, V., Linear Matrix Inequality in System and Control Theory, Philadelphia: SIAM, 1994.

    Google Scholar 

  19. Gahinet, P., Nemirovski, A., and Laub, A., LMI Control Toolbox. User’s Guide, Natick: MathWorks, 1995.

    Google Scholar 

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Authors and Affiliations

  1. Moscow State Institute of Steel and Alloys, Moscow, Russia

    Zh. V. Zatsepilova

  2. Trapeznikov Institute of Control Sciences, Russian Academy of Sciences, Moscow, Russia

    V. N. Chestnov

Authors
  1. Zh. V. Zatsepilova
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  2. V. N. Chestnov
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Original Russian Text © Zh.V. Zatsepilova, V.N. Chestnov, 2011, published in Avtomatika i Telemekhanika, 2011, No. 10, pp. 39–51.

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Zatsepilova, Z.V., Chestnov, V.N. Synthesis of digital H ∞-controllers for multidimensional systems with given accuracy with the mean squared criterion. Autom Remote Control 72, 2041–2052 (2011). https://doi.org/10.1134/S0005117911100055

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