Algorithm 747: a Fortran subroutine to solve the eigenvalue assignment problem for multiinput systems using state feedback

ABSTRACT

The implementation of an algorithm for the computation of a state feedback for multiinput linear systems, resulting in a closed-loop matrix with a specified self-conjugate set of eigenvalues, is presented. The computation uses only real arithmetic, assigning complex conjugate eigenvalues in one double step. The implementation uses level-1 BLAS routines where possible. A brief description of the algorithm is also given.

REFERENCES

• 1: J. W. Demmel, On condition numbers and the distance to the nearest III-posted problem, Numerische Mathematik, v.51 n.3, p.251-289, July 1987 [doi>10.1007/BF01400115]
• 2: GcLUB, G. H. AND VAN LOAN, C. F. 1989. Matrix Computations. 2nd ed. John Hopkins University Press, Baltimore, Md.
• 3: KAUTSKY, J., NICHOLS, N. K., AND VAN DOOREN, P. 1985. Robust pole assignment in linear state feedback. Int. J. Control 41, 5 (May), 1129-1155.
• 4: C. L. Lawson , R. J. Hanson , D. R. Kincaid , F. T. Krogh, Basic Linear Algebra Subprograms for Fortran Usage, ACM Transactions on Mathematical Software (TOMS), v.5 n.3, p.308-323, Sept. 1979 [doi>10.1145/355841.355847]
• 5: LUENBERGER, D.G. 1979. Introduction to Dynamic Systems. Wfiey, New York.
• 6: MIMINIS, G. S. AND PAIGE, C.C. 1982. An algorithm for pole assignment of time mvamant multiqnput hnear systems. In Proceedings of the 21st IEEE Conference on Decision and Control (Orlando, Fla., Dec.). Vol. 1. IEEE, New York, 62-67.
• 7: George S. Miminis , Chris C. Paige, A direct algorithm for pole assignment of time-invariant multi-input linear systems using state feedback, Automatica (Journal of IFAC), v.24 n.3, p.343-356, May, 1988 [doi>10.1016/0005-1098(88)90075-1]
• 8: Mm~NIS, G. S. AND PAIGE, C. C. 1994. A QR-like approach for the eigenvalue assignment problem In Procee&ngs of the 2nd Hellenic European Conference on Mathematics and Informat~cs ~Athens, Greece, Sept. 22-24).
• 9: George Miminis , Michael Reid, Algorithm 718; a FORTRAN subroutine to solve the eigenvalue allocation problem for single-input systems, ACM Transactions on Mathematical Software (TOMS), v.19 n.2, p.224-232, June 1993 [doi>10.1145/152613.152618]
• 10: PAINE, C. C. 1981. Properties of numerical algorithms related to computing controllabihty. IEEE Trans. Autom. Control AC-26, i (Feb.), 130-138.
• 11: PATEL, R. V. AND MISRA, P. 1984. Numerical algorithms for eigenvalue assignment by state feedback. Proc. IEEE 72, 12 (Dec.), 1755-1764.
• 12: PETKOV, P., CHRISTOV, N, AND KONSTANTINOV, M. 1986. A computational algorithm for pole assignment of linear multi-input systems. IEEE Trans. Autom Control AC-31, 11 (Nov.), 1044-1047.
• 13: VAN DOOREN, P. 1981. The generahzed eigenstructure problem in linear system theory IEEE Trans. Aurora. Control AC-26, i (Feb.), 111-129
• 14: VARGA, A. 1981. A Schur method for pole assignment IEEE Trans. Aurora. ControlAC-26, 2 (Apr.), 517-519.
• 15: WONHAM, W.M. 1967. On pole assignment in multi-input controllable linear systems. IEEE Trans. Autom Control AC-12, 6 (Dec.), 660 665.

INDEX TERMS

Primary Classification:
  G. Mathematics of Computing
  G.1 NUMERICAL ANALYSIS
      G.1.3 Numerical Linear Algebra
          Subjects: Eigenvalues and eigenvectors (direct and iterative methods)

Additional Classification:
  F. Theory of Computation
  F.2 ANALYSIS OF ALGORITHMS AND PROBLEM COMPLEXITY
      F.2.1 Numerical Algorithms and Problems
          Subjects: Computations on matrices

  G. Mathematics of Computing
  G.1 NUMERICAL ANALYSIS
      G.1.0 General
          Subjects: Numerical algorithms
      G.1.3 Numerical Linear Algebra
          Subjects: Linear systems (direct and iterative methods)

  J. Computer Applications
  J.2 PHYSICAL SCIENCES AND ENGINEERING
      Subjects: Aerospace; Engineering

General Terms:
Algorithms

Keywords:
deflation, double steps, eigenvalue assignment, numerical efficiency, pole assignment