TB01KX

Additive spectral decomposition of the transfer-function matrix of a standard system

Purpose

  To compute an additive spectral decomposition of the transfer-
  function matrix of the system (A,B,C) by reducing the system
  state-matrix A to a block-diagonal form. It is assumed that A is
  in a real Schur form, and the leading diagonal block of order NDIM
  has eigenvalues distinct from those of the trailing diagonal
  block. The system matrices are transformed as
     A <-- V*A*U, B <--V*B and C <-- C*U, where V = inv(U),
  preserving the spectra of the two diagonal blocks.

      SUBROUTINE TB01KX( N, M, P, NDIM, A, LDA, B, LDB, C, LDC, U, LDU,
     $                   V, LDV, INFO )
C     .. Scalar Arguments ..
      INTEGER          INFO, LDA, LDB, LDC, LDU, LDV, M, N, NDIM, P
C     .. Array Arguments ..
      DOUBLE PRECISION A(LDA,*), B(LDB,*), C(LDC,*), U(LDU,*), V(LDV,*)

Input/Output Parameters

  N       (input) INTEGER
          The order of the state-space representation, i.e., the
          order of the matrix A.  N >= 0.

  M       (input) INTEGER
          The number of system inputs, or of columns of B.  M >= 0.

  P       (input) INTEGER
          The number of system outputs, or of rows of C.  P >= 0.

  NDIM    (input) INTEGER
          The dimension of the leading diagonal block of A having
          eigenvalues distinct from those of the trailing diagonal
          block.  0 <= NDIM <= N.

  A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
          On entry, the leading N-by-N part of this array must
          contain the state dynamics matrix A in real Schur form.
          On exit, the leading N-by-N part of this array contains a
          block diagonal matrix inv(U) * A * U with two diagonal
          blocks in real Schur form, with the elements below the
          first subdiagonal set to zero. The leading block has
          dimension NDIM-by-NDIM.

  LDA     INTEGER
          The leading dimension of the array A.  LDA >= MAX(1,N).

  B       (input/output) DOUBLE PRECISION array, dimension (LDB,M)
          On entry, the leading N-by-M part of this array must
          contain the input matrix B.
          On exit, the leading N-by-M part of this array contains
          the transformed input matrix inv(U) * B.

  LDB     INTEGER
          The leading dimension of the array B.  LDB >= MAX(1,N).

  C       (input/output) DOUBLE PRECISION array, dimension (LDC,N)
          On entry, the leading P-by-N part of this array must
          contain the output matrix C.
          On exit, the leading P-by-N part of this array contains
          the transformed output matrix C * U.

  LDC     INTEGER
          The leading dimension of the array C.  LDC >= MAX(1,P).

  U       (input/output) DOUBLE PRECISION array, dimension (LDU,N)
          On entry, the leading N-by-N part of this array must
          contain an initial transformation matrix U.
          On exit, the leading N-by-N part of this array contains
          the transformation matrix used to reduce A to the block-
          diagonal form. The first NDIM columns of U span the
          invariant subspace of A corresponding to the eigenvalues
          of its leading diagonal block. The last N-NDIM columns of
          U span the reducing subspace of A corresponding to the
          eigenvalues of the trailing diagonal block of A.

  LDU     INTEGER
          The leading dimension of the array U.  LDU >= max(1,N).

  V       (output) DOUBLE PRECISION array, dimension (LDV,N)
          The leading N-by-N part of this array contains the
          inverse of the transformation matrix U used to reduce A
          to the block-diagonal form.

  LDV     INTEGER
          The leading dimension of the array V.  LDV >= max(1,N).

  INFO    INTEGER
          = 0: successful exit;
          < 0: if INFO = -i, the i-th argument had an illegal value;
          = 1: the separation of the two diagonal blocks failed
               because of very close eigenvalues.

  A similarity transformation U is determined that reduces the given
  system state-matrix A to a block-diagonal form (with two diagonal
  blocks), so that the eigenvalues of the leading diagonal block of
  the resulting A are preserved. The determined transformation is
  applied to the system (A,B,C) as
     A <-- inv(U)*A*U, B <-- inv(U)*B and C <-- C*U.

  [1] Safonov, M.G., Jonckheere, E.A., Verma, M., Limebeer, D.J.N.
      Synthesis of positive real multivariable feedback systems.
      Int. J. Control, pp. 817-842, 1987.

                                3
  The algorithm requires about N /2 + NDIM*(N-NDIM)*(2*N+M+P)
  floating point operations.

  None

Program Text

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