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A singular perturbations approach to reduced-order modeling and decoupling for large scale linear systems

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Part of the Lecture Notes in Mathematics book series (LNM,volume 968)

Keywords

  • Riccati Equation
  • Singular Perturbation
  • Fundamental Matrix
  • Order Reduction
  • Algebraic Riccati Equation

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References

  1. L. Anderson, "Decoupling of two-time-scale linear systems," Proceedings, 1978 Joint Automatic Control Conference, vol. 4, 153–164.

    Google Scholar 

  2. L. R. Anderson and W. L. Hallauer, Jr., "A method of order reduction for structural dynamics," Proceedings, 21st Structures, Structural Dynamics, and Materials Conference, 1980.

    Google Scholar 

  3. R. Bellman, Introduction to Matrix Analysis, second edition, McGraw-Hill, New York, 1970.

    MATH  Google Scholar 

  4. W. A. Coppel, Dichotomies in Stability Theory, Lecture Notes in Math. 629, Springer-Verlag, Berlin, 1978.

    MATH  Google Scholar 

  5. G. Dahlquist, "A numerical method for some ordinary differential equations with large Lipschitz constants," Information Processing 68, A. J. H. Morell, editor, North-Holland, Amsterdam, 1969, 183–186.

    Google Scholar 

  6. R. L. deHoff and W. E. Hall, Jr., "Optimal control of turbine engines," J. Dynamic Systems, Measurement, and Control 101 (1979), 117–126.

    CrossRef  Google Scholar 

  7. F. de Hoog and R. Weiss, "The numerical solution of boundary value problems with an essential singularity," SIAM J. Numerical Analysis 16 (1979), 637–669.

    CrossRef  MathSciNet  MATH  Google Scholar 

  8. B. Etkin, Dynamics of Atmospheric Flight, Wiley, New York, 1972.

    Google Scholar 

  9. C. A. Harvey and R. E. Pope, "Synthesis techniques for insensitive aircraft control systems," Proceedings, 1976 IEEE Decision and Control Conference, 990–1001.

    Google Scholar 

  10. P. V. Kokotovic, J. B. Cruz, Jr., J. V. Medanic, and W. R. Perkins, editors, Multimodeling and Control of Large Scale Systems, Report DC-28, Coordinated Science Laboratory, University of Illinois, Urbana, 1979.

    Google Scholar 

  11. P. V. Kokotovic, P. Sannuti, and R. E. O'Malley, Jr., "Singular perturbations and order reduction in control theory—an overview," Automatica 12 (1976), 123–132.

    CrossRef  MathSciNet  MATH  Google Scholar 

  12. H.-O. Kreiss, "Difference methods for stiff ordinary differential equations," SIAM J. Numerical Analysis 15 (1978), 21–58.

    CrossRef  MathSciNet  MATH  Google Scholar 

  13. H.-O. Kreiss, "Problems with different time scales for ordinary differential equations," SIAM J. Numerical Analysis 16 (1979), 980–998.

    CrossRef  MathSciNet  MATH  Google Scholar 

  14. M. Lentini and H. B. Keller, "Boundary value problems on semiinfinite intervals and their numerical solution," SIAM J. Numerical Analysis, 17, 577 (1980).

    CrossRef  MathSciNet  MATH  Google Scholar 

  15. J. Medanic, "Geometric properties and invariant manifolds of the Riccati equation," Technical Report, Coordinated Science Laboratory, University of Illinois-Urbana, 1979.

    Google Scholar 

  16. W. L. Miranker and G. Wahba, "An averaging method for the stiff highly oscillatory problem," Math. Computation 30 (1976), 383–399.

    CrossRef  MathSciNet  MATH  Google Scholar 

  17. C. Moler and C. Van Loan, "Nineteen dubious ways to compute the exponential of a matrix," SIAm Review 20 (1978), 801–836.

    CrossRef  MathSciNet  MATH  Google Scholar 

  18. L. Oden, "An experimental and theoretical analysis of the SAPS method for stiff ordinary differential equations," technical report, Department of Information Processing, Royal Institute of Technology, Stockholm, 1971.

    Google Scholar 

  19. R. E. O'Malley, Jr., Introduction to Singular Perturbations, Academic Press, New York, 1974.

    MATH  Google Scholar 

  20. R. E. O'Malley, Jr., "Singular perturbations and optimal control," Lecture Notes in Math. 680 (1978), Springer-Verlag, Berlin, 170–218.

    MATH  Google Scholar 

  21. R. E. O'Malley, Jr. and L. R. Anderson, "Decoupling and order reduction for linear time-varying two-time-scale systems,” Optim. Contr. 3, 133 (1982).

    CrossRef  MathSciNet  MATH  Google Scholar 

  22. R. E. O'Malley, Jr. and J. E. Flaherty, "Analytical and numerical methods for nonlinear singular singularly perturbed initial value problems," SIAM J. Applied Math. 38 (1980), 225–248.

    CrossRef  MathSciNet  MATH  Google Scholar 

  23. L. R. Petzold, "An efficient numerical method for highly oscillatory ordinary differential equations," technical report 78–933, Department of Computer Science, University of Illinois, Urbana, 1978.

    Google Scholar 

  24. M. K. Sain, "The theme problem," Proceedings, International Forum on Alternatives for Multivariable Control, 1977, 1–12.

    Google Scholar 

  25. M. R. Scott and W. A. Watts, "Computational solution of linear two-point boundary value problems via orthogonormalization," SIAM J. Numerical Analysis 14 (1977), 40–70.

    CrossRef  MathSciNet  MATH  Google Scholar 

  26. G. W. Stewart, "Methods of simultaneous iteration for calculating eigenvectors of matrices," Topics in Numerical Analysis II, J. J. H. Miller, editor, Academic Press, London, 1975, 185–196.

    CrossRef  Google Scholar 

  27. D. Teneketzis and N. R. Sandell, Jr., "Linear regulator design for stochastic systems by a multiple time-scales method," IEEE Trans. Automatic Control 22 (1977), 615–621.

    CrossRef  MathSciNet  MATH  Google Scholar 

  28. P. Van Dooren, "Updating the QZ-algorithm for the computation of deflating subspaces," internal report, Department of Computer Science, Stanford University, 1980.

    Google Scholar 

  29. A. B. Vasil'eva and V. F. Butuzov, Asymptotic Expansions of Solutions of Singularly Perturbed Equations, Nauka, Moscow, 1973.

    MATH  Google Scholar 

  30. W. R. Wasow, Asymptotic Expansions for Ordinary Differential Equations, Wiley, New York, 1965.

    MATH  Google Scholar 

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© 1982 Springer-Verlag

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O'Malley, R.E. (1982). A singular perturbations approach to reduced-order modeling and decoupling for large scale linear systems. In: Hinze, J. (eds) Numerical Integration of Differential Equations and Large Linear Systems. Lecture Notes in Mathematics, vol 968. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0064892

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  • DOI: https://doi.org/10.1007/BFb0064892

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