International Applied Mechanics

, Volume 50, Issue 3, pp 321–334 | Cite as

Algorithms for Solving a Unilateral Quadratic Matrix Equation and the Model Updating Problem

  • V. B. Larin

The Schur and doubling methods, which are usually used to solve the algebraic Riccati equation, are generalized to the case of unilateral quadratic matrix equations. The efficiency of the algorithms proposed to solve the unilateral quadratic matrix equation is demonstrated by way of examples. The algorithms are compared with well-known ones. It is shown that the solutions of the unilateral quadratic matrix equation can be used to update model parameters.


unilateral quadratic matrix equation linear matrix inequality model updating 


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  1. 1.
    O. M. Golubentsev and O. M. Drogovoz, “Criteria of aperiodic stability of motion,” Prikl. Mekh., 8, No. 4, 17–23 (1962).Google Scholar
  2. 2.
    M. G. Krein, “An introduction to the geometry of indefinite J-spaces and the theory of operators in these spaces,” in: Proc. 2nd Summer Math. School [in Russian], Issue I, Naukova Dumka, Kyiv (1965), pp. 15–92.Google Scholar
  3. 3.
    V. B. Larin, “Selecting the parameters of the optimal damper,” in: Proc. 1st Resp. Conf. of Young Mathematicians of Ukraine [in Russian], Inst. Mat. AN USSR, Kyiv (1965), pp. 395–405.Google Scholar
  4. 4.
    V. B. Larin, Statistical Vibration-Isolation Problems [in Russian], Naukova Dumka, Kyiv (1974).Google Scholar
  5. 5.
    V. B. Larin, “Inverting the problem of analytical design of controllers,” J. Autom. Inform. Sci., 36, No. 2, 11–18 (2004).CrossRefGoogle Scholar
  6. 6.
    V. B. Larin, “Algorithm of determination of stabilizing and antistabilizing solutions of the discrete algebraic Riccati equation,” J. Autom. Inform. Sci., 38, No. 11, 1–13 (2006).CrossRefGoogle Scholar
  7. 7.
    F. A. Aliev, B. A. Bordyug, and V. B. Larin, “Comments on a stability-enhancing scaling procedure for Schur–Riccati solvers,” Systems & Control Letters, 14, 453 (1990).CrossRefMathSciNetGoogle Scholar
  8. 8.
    F. A. Aliev and V. B. Larin, Optimization of Linear Control Systems: Analytical Methods and Computational Algorithms, Gordon and Breach Science, Amsterdam (1998).MATHGoogle Scholar
  9. 9.
    E. I. Bespalova and G. P. Urusova, “Vibrations of statically loaded shells of revolution of positive or negative Gaussian curvature,” Int. Appl. Mech., 46, No. 3, 279–286 (2010).ADSCrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    D. A. Bini, B. Meini, and F. Poloni, “Transforming algebraic Riccati equations into unilateral quadratic matrix equations,” Numer. Math., 116, 553–578 (2010).CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    S. Boyd, L. E. Ghaoui, E. Feron, and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory, SIAM, Philadelphia (1994).CrossRefMATHGoogle Scholar
  12. 12.
    V. P. Barsegyai and L. A. Movsisyan, “Optimal control of the vibration of elastic systems described by the wave equation,” Int. Appl. Mech., 48, No. 2, 234–240 (2012).ADSCrossRefGoogle Scholar
  13. 13.
    TK-W Chu, H-Y Fan, W-W Lin, and C. S. Wang, “Structure-preserving algorithms for periodic discrete-time algebraic Riccati equation,” Int. J. Control, 77, No. 8, 767–788 (2004).CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    B. N. Datta and V. Sokolov, “Quadratic inverse eigenvalue problems, active vibration control and model updating,” Appl. Comp. Math., 8, No. 2, 170–191 (2009).MATHMathSciNetGoogle Scholar
  15. 15.
    P. Gahinet, A. Nemirovski, A. J. Laub, and M. Chilali, LMI Control Toolbox Users Guide, The MathWorks Inc. (1995).Google Scholar
  16. 16.
    V. I. Gulyaev, P. Z. Lugovoi, and Yu. A. Zaets, “Shielding of elastic nonstationary waves by interfaces,” Int. Appl. Mech., 48, No. 4, 414–422 (2012).ADSCrossRefMathSciNetGoogle Scholar
  17. 17.
    I. F. Kirichok, “Forced resonant vibrations and self-heating of a flexible circular plate with piezoactuators,” Int. Appl. Mech., 48, No. 5, 583–591 (2012).ADSCrossRefGoogle Scholar
  18. 18.
    V. B. Larin, “Optimization of periodic systems with singular weight matrix which defines the quadratic form of control actions,” J. Automat. Inform Sci., 31, No. 6, 27–38 (1999).Google Scholar
  19. 19.
    V. B. Larin, “LMI approach to the inverse problem of optimal control,” System Science, 26, No. 3, 61–68 (2001).MathSciNetGoogle Scholar
  20. 20.
    V. B. Larin, “Some optimization problems for vibriprotective systems,” Int. Appl. Mech., 37, No. 4, 456–483 (2001).ADSCrossRefGoogle Scholar
  21. 21.
    V. B. Larin, “Determination both as stabilizing and antistabilizing solutions of the discrete-time algebraic Riccati equation,” Int. J. Appl. Math. Mech., 3, No. 1, 42–60 (2007).Google Scholar
  22. 22.
    V. B. Larin, “Solution of matrix equations in problems of mechanics and control,” Int. Appl. Mech., 45, No. 8, 847–872 (2009).ADSCrossRefMathSciNetGoogle Scholar
  23. 23.
    A. J. Laub, “A Schur method for solving algebraic Riccati equations,” IEEE Trans. Automat. Contr., 24, 913–921 (1979).CrossRefMATHMathSciNetGoogle Scholar
  24. 24.
    Y. Yuan, “An iterative updating method for damped gyroscopic systems,” Int. J. Comput. Math. Sci., 4:2, 63–71 (2010).Google Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.S. P. Timoshenko Institute of MechanicsNational Academy of Sciences of UkraineKyivUkraine

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