AB08MD

Normal rank of the transfer-function matrix of a state space model

Purpose

  To compute the normal rank of the transfer-function matrix of a
  state-space model (A,B,C,D).

      SUBROUTINE AB08MD( EQUIL, N, M, P, A, LDA, B, LDB, C, LDC, D, LDD,
     $                   RANK, TOL, IWORK, DWORK, LDWORK, INFO )
C     .. Scalar Arguments ..
      CHARACTER         EQUIL
      INTEGER           INFO, LDA, LDB, LDC, LDD, LDWORK, M, N, P, RANK
      DOUBLE PRECISION  TOL
C     .. Array Arguments ..
      INTEGER           IWORK(*)
      DOUBLE PRECISION  A(LDA,*), B(LDB,*), C(LDC,*), D(LDD,*), DWORK(*)

Mode Parameters

  EQUIL   CHARACTER*1
          Specifies whether the user wishes to balance the compound
          matrix (see METHOD) as follows:
          = 'S':  Perform balancing (scaling);
          = 'N':  Do not perform balancing.

  N       (input) INTEGER
          The number of state variables, i.e., the order of the
          matrix A.  N >= 0.

  M       (input) INTEGER
          The number of system inputs.  M >= 0.

  P       (input) INTEGER
          The number of system outputs.  P >= 0.

  A       (input) DOUBLE PRECISION array, dimension (LDA,N)
          The leading N-by-N part of this array must contain the
          state dynamics matrix A of the system.

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

  B       (input) DOUBLE PRECISION array, dimension (LDB,M)
          The leading N-by-M part of this array must contain the
          input/state matrix B of the system.

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

  C       (input) DOUBLE PRECISION array, dimension (LDC,N)
          The leading P-by-N part of this array must contain the
          state/output matrix C of the system.

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

  D       (input) DOUBLE PRECISION array, dimension (LDD,M)
          The leading P-by-M part of this array must contain the
          direct transmission matrix D of the system.

  LDD     INTEGER
          The leading dimension of array D.  LDD >= MAX(1,P).

  RANK    (output) INTEGER
          The normal rank of the transfer-function matrix.

  TOL     DOUBLE PRECISION
          A tolerance used in rank decisions to determine the
          effective rank, which is defined as the order of the
          largest leading (or trailing) triangular submatrix in the
          QR (or RQ) factorization with column (or row) pivoting
          whose estimated condition number is less than 1/TOL.
          If the user sets TOL to be less than SQRT((N+P)*(N+M))*EPS
          then the tolerance is taken as SQRT((N+P)*(N+M))*EPS,
          where EPS is the machine precision (see LAPACK Library
          Routine DLAMCH).

  IWORK   INTEGER array, dimension (2*N+MAX(M,P)+1)

  DWORK   DOUBLE PRECISION array, dimension (LDWORK)
          On exit, if INFO = 0, DWORK(1) returns the optimal value
          of LDWORK.

  LDWORK  INTEGER
          The length of the array DWORK.
          LDWORK >= (N+P)*(N+M) +
                    MAX( MIN(P,M) + MAX(3*M-1,N), 1,
                         MIN(P,N) + MAX(3*P-1,N+P,N+M) )
          For optimum performance LDWORK should be larger.

          If LDWORK = -1, then a workspace query is assumed;
          the routine only calculates the optimal size of the
          DWORK array, returns this value as the first entry of
          the DWORK array, and no error message related to LDWORK
          is issued by XERBLA.

  INFO    INTEGER
          = 0:  successful exit;
          < 0:  if INFO = -i, the i-th argument had an illegal
                value.

  The routine reduces the (N+P)-by-(M+N) compound matrix (B  A)
                                                         (D  C)

  to one with the same invariant zeros and with D of full row rank.
  The normal rank of the transfer-function matrix is the rank of D.

  [1] Svaricek, F.
      Computation of the Structural Invariants of Linear
      Multivariable Systems with an Extended Version of
      the Program ZEROS.
      System & Control Letters, 6, pp. 261-266, 1985.

  [2] Emami-Naeini, A. and Van Dooren, P.
      Computation of Zeros of Linear Multivariable Systems.
      Automatica, 18, pp. 415-430, 1982.

  The algorithm is backward stable (see [2] and [1]).

  None

Program Text

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