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Ukrainian Mathematical Journal

, Volume 70, Issue 11, pp 1777–1790 | Cite as

Evaluation of the Weighted Level of Damping of Bounded Disturbances in Descriptor Systems

  • A. G. Mazko
Article
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We establish necessary and sufficient conditions for the realization of the upper bounds in the performance criteria for linear descriptor systems characterizing the weighted level of damping of the external and initial disturbances. The verification of these conditions is reduced to the solution of matrix equations and inequalities. The main statements are formulated with an aim of their subsequent application in the problems of robust stabilization and H-optimization of descriptor control systems.

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References

  1. 1.
    B. T. Polyak and P. S. Shcherbakov, Robust Stability and Control [in Russian], Nauka, Moscow (2002).Google Scholar
  2. 2.
    B. T. Polyak, M. V. Khlebnikov, and P. S. Shcherbakov, Control over Linear Systems under External Perturbations. Technique of Linear Matrix Inequalities [in Russian], Lenand, Moscow (2014).Google Scholar
  3. 3.
    D. V. Balandin and M. M. Kogan, Synthesis of the Regularities of Control on the Basis of Linear Matrix Inequalities [in Russian], Fizmatlit, Moscow (2007).Google Scholar
  4. 4.
    G. E. Dullerud and F. G. Paganini, A Course in Robust Control Theory. A Convex Approach, Springer, Berlin (2000).zbMATHCrossRefGoogle Scholar
  5. 5.
    P. Gahinet and P. Apkarian, “A linear matrix inequality approach to H control,” Internat. J. Robust Nonlin. Control, 4, 421–448 (1994).MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    I. M. Inoue, T.Wada, M. Ikeda, and E. Uezato, “State-space H controller design for descriptor systems,” Automatica, 59, 164–170 (2015).MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    A. G. Mazko, Robust Stability and Stabilization of Dynamical Systems. Methods of Matrix and Conic Inequalities [in Russian], Proc. of the Institute of Mathematics, Ukrainian National Academy of Sciences, 102, Kiev (2016).Google Scholar
  8. 8.
    P. P. Khargonekar, K. M. Nagpal, and K. R. Poolla, “H control with transients,” SIAM J. Control Optim., 29, No. 6, 1373–1393 (1991).MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    D. V. Balandin andM. M. Kogan, “GeneralizedH -optimal control as a compromise betweenH -optimal and 𝛾-optimal controls,” Avtomat. Telemekh., No. 6, 20–38 (2010).Google Scholar
  10. 10.
    D. V. Balandin, M. M. Kogan, L. N. Krivdina, and A. A. Fedyukov, “Synthesis of a generalized H -optimal control in discrete time on finite and infinite intervals,” Avtomat. Telemekh., No. 1, 3–22 (2014).Google Scholar
  11. 11.
    R. S. Biryukov, “Generalized H -optimal filter for a continuous object on the basis of time-discrete observations,” Inform. Sist. Upravl., No. 4, (42), 89–101 (2014).Google Scholar
  12. 12.
    O. H. Mazko and S. N. Kusii, “Robust stabilization and damping of external disturbances in systems with control and observable outputs,” in: Proc. of the Institute of Mathematics, Ukrainian National Academy of Sciences [in Ukrainian], 13, No. 3 (2016), pp. 129–145.Google Scholar
  13. 13.
    A. G. Mazko and S. N. Kusii, “Stabilization with respect to output and weighted suppression of disturbances in discrete control systems,” Probl. Upravl. Inform., No. 6, 78–93 (2017).Google Scholar
  14. 14.
    S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishman, Linear Matrix Inequalities in System and Control Theory, SIAM, Philadelphia (1994).zbMATHCrossRefGoogle Scholar
  15. 15.
    S. Xu, J. Lam, and Y. Zou, “New versions of bounded real lemmas for continuous and discrete uncertain systems,” Circuits, Systems, Signal Process, 26, 829–838 (2007).MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    M. Chadli, P. Shi, Z. Feng, and J. Lam, “New bounded real lemma formulation and H control for continuous-time descriptor systems,” Asian J. Control, 20, No. 1, 1–7 (2018).MathSciNetCrossRefGoogle Scholar
  17. 17.
    F. Gao, W. Q. Liu, V. Sreeram, and K. L. Teo, “Bounded real lemma for descriptor systems and its application,” in: IFAC 14th Triennial World Congress (Beijing, China) (1999), pp. 1631–1636.Google Scholar
  18. 18.
    I. Masubushi, Y. Kamitane, A. Ohara, and N. Suda, “H control for descriptor systems: a matrix inequalities approach,” Automatica, 33, No. 4, 669–673 (1997).Google Scholar
  19. 19.
    L. Dai, Singular Control Systems, Springer, New York (1989).zbMATHCrossRefGoogle Scholar
  20. 20.
    R. Riaza, Differential-Algebraic Systems. Analytical Aspects and Circuit Applications, World Scientific, Singapore (2008).zbMATHCrossRefGoogle Scholar
  21. 21.
    A. M. Samoilenko, M. I. Shkil’, and V. P. Yakovets’, Linear Systems of Differential Equations with Degenerations [in Ukrainian], Vyshcha Shkola, Kyiv (2000).Google Scholar
  22. 22.
    A. A. Boichuk, A. A. Pokutnyi, and V. F. Chistyakov, “Application of perturbation theory to the solvability analysis of differential algebraic equations,” Comput. Math. Math. Phys., 53, No. 6, 777–788 (2013).MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    P. Gahinet, A. Nemirovski, A. J. Laub, and M. Chilali, The LMI control Toolbox. For Use with Matlab. User’s Guide, The Math- Works, Inc., Natick (1995).Google Scholar
  24. 24.
    F. R. Gantmakher, Theory of Matrices [in Russian], Nauka, Moscow (1988).zbMATHGoogle Scholar
  25. 25.
    D. J. Bender and A. J. Laub, “The linear-quadratic optimal regulator for descriptor systems,” IEEE Trans. Automat. Control, AC-32, No. 8, 672–688 (1987).MathSciNetzbMATHCrossRefGoogle Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  • A. G. Mazko
    • 1
  1. 1.Institute of MathematicsUkrainian National Academy of SciencesKyivUkraine

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