## AB08MZ

### Normal rank of the transfer-function matrix of a state space model (complex case)

[Specification] [Arguments] [Method] [References] [Comments] [Example]

Purpose

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

Specification

          SUBROUTINE AB08MZ( EQUIL, N, M, P, A, LDA, B, LDB, C, LDC, D, LDD,
         $                   RANK, TOL, IWORK, DWORK, ZWORK, LZWORK, INFO )
    C     .. Scalar Arguments ..
          CHARACTER         EQUIL
          INTEGER           INFO, LDA, LDB, LDC, LDD, LZWORK, M, N, P, RANK
          DOUBLE PRECISION  TOL
    C     .. Array Arguments ..
          INTEGER           IWORK(*)
          COMPLEX*16        A(LDA,*), B(LDB,*), C(LDC,*), D(LDD,*), ZWORK(*)
          DOUBLE PRECISION  DWORK(*)

Arguments

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.

Input/Output Parameters

      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) COMPLEX*16 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) COMPLEX*16 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) COMPLEX*16 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) COMPLEX*16 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.

Tolerances

      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).

Workspace

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

      DWORK   DOUBLE PRECISION array, dimension (2*MAX(M,P))

      ZWORK   COMPLEX*16 array, dimension (LZWORK)
              On exit, if INFO = 0, ZWORK(1) returns the optimal value
              of LZWORK.

      LZWORK  INTEGER
              The length of the array ZWORK.
              LZWORK >= (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 LZWORK should be larger.

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

Error Indicator

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

Method

      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.

References

      [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.

Numerical Aspects

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

Further Comments

      None

Example

Program Text

      None

Program Data

      None

Program Results

      None

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