Nondifferentiable design optimization involving the eigenvalues of control system matrices

  • Nedyalko I. Krushev
III Optimal Control III.1 Control Problems
Part of the Lecture Notes in Control and Information Sciences book series (LNCIS, volume 180)


The designer of linear dynamic systems is always concerned with their eigenvalues because of such important issues as stability, reaction speed, robustness etc. Mathematical programming has proved to be a powerful instrument for control systems design. Requirements upon eigenvalues can also be formulated as part of the design optimization problem. This paper presents a nondifferentiable approach for solving such problems. It considers the general case when the system matrices are non-symmetric. The approach is based on the numerical calculation of the Jordan canonical form and on the generalized gradients of F.Clarke. An algorithm was developed and implemented. Finally, some control systems design examples illustrate the features of the considered problems and approaches.


Descent Direction Control System Design Design Optimization Problem Elementary Divisor Multiple Eigenvalue 
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  1. 1.
    Antsiferov, E.G. (1978). Some Optimization Problems Connected to the Synthesis of Stable Linear Dynamic Systems (in russian), USSR Prikladnaja matematika — Novosibirsk: Nauka, pp. 5–35.Google Scholar
  2. 2.
    Antsiferov, E.G., L.T. Ashtepkov and V.P. Bulatov (1990). Optimization Methods and their Applications. Part I: Mathematical Programming (in russian), Nauka, Novosibirsk, pp.87–111.zbMATHGoogle Scholar
  3. 3.
    Clarke, F. (1983) Optimization and Nonsmooth Analysis, Wiley, New York.zbMATHGoogle Scholar
  4. 4.
    Demjanov, V.F. and V.N. Malozemov (1974). Introduction to Minimax, Wiley, New York.Google Scholar
  5. 5.
    Dutta, S.R.K. and M. Vidyasagar (1977). New Algorithms for Constrained Minimax Optimization, Math.Programming, 13, pp.140–155.zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Kagstrom, B. and A. Ruhe (1980). An Algorithm for Numerical Computation of the Jordan Normal Form of a Complex Matrix, ACM Trans.Math.Software 6(3), pp.398–419.CrossRefMathSciNetGoogle Scholar
  7. 7.
    Kiwiel, K.C. (1985). Methods of Descent for Nondifferentiable Optimization, LN in Mathematics 1133, Springer-Verlag, New York.zbMATHGoogle Scholar
  8. 8.
    Lancaster, P. (1969). Theory of Matrices, Academic Press, New YorkzbMATHGoogle Scholar
  9. 9.
    Owens, D.H. (1978). Feedback and Multivariable Systems, P.Peregrinus Ltd., Stevenage.zbMATHGoogle Scholar
  10. 10.
    Parlett, B.N. (1980). The Symmetric Eigenvalue Problem, Prentice-Hall, Englewood Cliffs, N.J.zbMATHGoogle Scholar
  11. 11.
    Polak, E. and Y. Wardi (1982). Nondifferentiable Optimization Algorithm for Designing Control Systems Having Singular Value Inequalities, Automatica, 18(3), pp.267–283.zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Polak, E., K. Mayne and D.M. Stimler (1984). Control System Design Via Semi-infinite Optimization: A Review, Trans.of IEEE, 72(12), pp.1777–1794.Google Scholar
  13. 13.
    Rockafellar, R.T. (1970). Convex Analysis, Princeton Univ.Press, Princeton, N.J.zbMATHGoogle Scholar
  14. 14.
    Venkov, G.I. and N.I. Krushev (1991). Nondifferentiable Design Optimization Problems with Application to Control Engineering, to appear in Journal Optimization, Ilmenau, Germany.Google Scholar
  15. 15.
    Wilkinson, J.H. (1965). The Algebraic Eigenvalue Problem, Clarendon Press, Oxford.zbMATHGoogle Scholar
  16. 16.
    Wilkinson, J.H. and C. Reinsch (Eds.) (1971). Handbook for Automatic Computation II: Linear Algebra, Springer-Verlag, New York.zbMATHGoogle Scholar

Copyright information

© International Federation for Information Processing 1992

Authors and Affiliations

  • Nedyalko I. Krushev
    • 1
  1. 1.Institute of Applied Mathematics and Computer ScienceTechnical University of SofiaSofia

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