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

Article

First Online:

Received:

- 73 Downloads
- 6 Citations

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.

## Keywords

unilateral quadratic matrix equation linear matrix inequality model updating## Preview

Unable to display preview. Download preview PDF.

## References

- 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.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.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.V. B. Larin,
*Statistical Vibration-Isolation Problems*[in Russian], Naukova Dumka, Kyiv (1974).Google Scholar - 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.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.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.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.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.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.S. Boyd, L. E. Ghaoui, E. Feron, and V. Balakrishnan,
*Linear Matrix Inequalities in System and Control Theory*, SIAM, Philadelphia (1994).CrossRefMATHGoogle Scholar - 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.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.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.P. Gahinet, A. Nemirovski, A. J. Laub, and M. Chilali,
*LMI Control Toolbox Users Guide*, The MathWorks Inc. (1995).Google Scholar - 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.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.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.V. B. Larin, “LMI approach to the inverse problem of optimal control,”
*System Science*,**26**, No. 3, 61–68 (2001).MathSciNetGoogle Scholar - 20.V. B. Larin, “Some optimization problems for vibriprotective systems,”
*Int. Appl. Mech.*,**37**, No. 4, 456–483 (2001).ADSCrossRefGoogle Scholar - 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.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.A. J. Laub, “A Schur method for solving algebraic Riccati equations,”
*IEEE Trans. Automat. Contr.*,**24**, 913–921 (1979).CrossRefMATHMathSciNetGoogle Scholar - 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