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On properties of gramians of continuous control systems

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Abstract

A new approach to the study of controllability and observabihty by means of analysis of the controhabihty and observabihty matrices and gramians was proposed. It rehes on the lemma of expansion of the matrix integrals proved by expanding the resolvents of the matrices A m and A and the properties of convolution of the matrix function in the complex domain. Analysis of the structural properties of gramians demonstrated that the coefficients of the characteristic equations and the Faddeev matrices in the decomposition of the resolvent of the system dynamics matrix play the leading part in their formation. An example illustrating the properties of gramians was given.

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References

  1. Kalman, R.E. and Bucy, R.S., New Results in Linear Filtering And Prediction Theory, J. Basic Eng. Trans. ASME, Ser. D, 1961, vol. 83, pp. 95–108.

    MathSciNet  Google Scholar 

  2. Kalman, R.E., Mathematical Description of Linear Systems, SIAM J. Control, 1963, vol. 1, pp. 152–192.

    MATH  MathSciNet  Google Scholar 

  3. Luenberger, D.G., Observing the State of Linear System, IEEE Trans. Mil. Electron., 1964, MIL-8, pp. 74–80.

  4. Korovin, S.K. and Fomichev, V.V., Nablyudateli sostoyaniya dlya lineinykh sistem s neopredelennost’yu (State Observers for Linear Systems with Uncertainty), Moscow: Fizmatlit, 2007.

    Google Scholar 

  5. Spravochnik po teorii avtomaticheskogo upravleniya (Handbook of the Theory of Automatic Control), Krasovskii, A.A., Ed., Moscow: Nauka, 1987.

    Google Scholar 

  6. The Control Handbook, Levine, W.S., Ed., Boca Raton: CRC Press, 1996, pp. 461–469.

    MATH  Google Scholar 

  7. Kailath, T., Linear Systems, New York: Prentice Hall, 1980.

    MATH  Google Scholar 

  8. Andreev, Yu.N., Upravlenie konechnomernymi lineinymi obektami (Control of Finite-dimensional Linear Plants), Moscow: Nauka, 1976.

    Google Scholar 

  9. Afanas’ev, V.N., Kolmanovskii, V.B., and Nosov, V.R., Matematicheskaya teoriya konstruirovaniya sistem upravleniya (Mathematical Theory of Control System Design), Moscow: Vysshaya Shkola, 1989.

    MATH  Google Scholar 

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

    MATH  Google Scholar 

  11. Zemlyakov, S.D. and Rutkovskii, V.Yu., Functional Controllability and Tunability of Coordinate-Parametric Control Systems, Autom. Remote Control, 1986, no. 2, part 1, pp. 169–176.

    Google Scholar 

  12. Emel’yanov, S.V., Binarnye sistemy avtomaticheskogo upravleniya (Binary Systems of Automatic Control), Moscow: Mir, 1987.

    Google Scholar 

  13. Fomin, V.N., Fradkov, A.L., and Yakubovich, V.A., Adaptivnoe upravlenie dinamicheskimi obektami (Adaptive System of Dynamic Plants), Moscow: Nauka, 1981.

    Google Scholar 

  14. Faddeev, D.K. and Faddeeva, V.N., Vychislitel’nye metody lineinoi algebry (Computational Methods of Linear Algebra), Moscow: Fizmatgiz, 1963.

    Google Scholar 

  15. Hanzon, B. and Peeters, R.L.M., A Faddeev Sequence Method for Solving Lyapunov and Sylvester Equations, Linear Algebra Appl, 1996, vol. 241–243, pp. 401–430.

    Article  MathSciNet  Google Scholar 

  16. Polyak, B.T. and Shcherbakov, P.S., Robastnaya ustoichivost’ i upravlenie (Robust Stability and Control), Moscow: Nauka, 2002.

    Google Scholar 

  17. Poznyak, A.S., Osnovy robastnogo upravleniya (H∞-Teoriya) (Fundamentals of Robust Control (H∞-theory)), Moscow: MFTI, 1991.

    Google Scholar 

  18. Balandin, D.V. and Kogan, M.M., Sintez zakonov upravleniya na osnove lineinykh matrichnykh ner-avenstv (Design of the Control Laws Based on Linear Matrix Inequalities), Moscow: Fizmatlit, 2007.

    Google Scholar 

  19. Yadykin, LB., H2-Optimal Tuning Algorithms for Fixed Structure Controllers, Autom. Remote Control, 2008, no. 8, pp. 1319–1332.

    Article  MathSciNet  Google Scholar 

  20. Yadykin, LB., On Adaptability Gramians, in Proc. Int. Conf. “Math. System Theory” MTS-2009, Moscow, Moscow: Inst. Sist. Anal., Ross. Akad. Nauk, 2009, pp. 127–141.

    Google Scholar 

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Authors and Affiliations

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

    I. B. Yadykin

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  1. I. B. Yadykin
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Original Russian Text © LB. Yadykin, 2010, published in Avtomatika i Telemekhanika, 2010, No. 6, pp. 39–50.

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Yadykin, I.B. On properties of gramians of continuous control systems. Autom Remote Control 71, 1011–1021 (2010). https://doi.org/10.1134/S0005117910060032

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  • DOI: https://doi.org/10.1134/S0005117910060032

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