AB05SD

Closed-loop system for an output feedback control law

Purpose

  To construct for a given state space system (A,B,C,D) the closed-
  loop system (Ac,Bc,Cc,Dc) corresponding to the output feedback
  control law

       u = alpha*F*y + v.

      SUBROUTINE AB05SD( FBTYPE, JOBD, N, M, P, ALPHA, A, LDA, B, LDB,
     $                   C, LDC, D, LDD, F, LDF, RCOND, IWORK, DWORK,
     $                   LDWORK, INFO)
C     .. Scalar Arguments ..
      CHARACTER         FBTYPE, JOBD
      INTEGER           INFO, LDA, LDB, LDC, LDD, LDF, LDWORK, M, N, P
      DOUBLE PRECISION  ALPHA, RCOND
C     .. Array Arguments ..
      INTEGER           IWORK(*)
      DOUBLE PRECISION  A(LDA,*), B(LDB,*), C(LDC,*), D(LDD,*),
     $                  DWORK(*), F(LDF,*)

Mode Parameters

  FBTYPE  CHARACTER*1
          Specifies the type of the feedback law as follows:
          = 'I':  Unitary output feedback (F = I);
          = 'O':  General output feedback.

  JOBD    CHARACTER*1
          Specifies whether or not a non-zero matrix D appears in
          the given state space model:
          = 'D':  D is present;
          = 'Z':  D is assumed a zero matrix.

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

  M       (input) INTEGER
          The number of input variables, i.e. the number of columns
          of matrices B and D, and the number of rows of F.  M >= 0.

  P       (input) INTEGER
          The number of output variables, i.e. the number of rows of
          matrices C and D, and the number of columns of F.  P >= 0
          and P = M if FBTYPE = 'I'.

  ALPHA   (input) DOUBLE PRECISION
          The coefficient alpha in the output feedback law.

  A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
          On entry, the leading N-by-N part of this array must
          contain the system state transition matrix A.
          On exit, the leading N-by-N part of this array contains
          the state matrix Ac of the closed-loop system.

  LDA     INTEGER
          The leading dimension of 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 system input matrix B.
          On exit, the leading N-by-M part of this array contains
          the input matrix Bc of the closed-loop system.

  LDB     INTEGER
          The leading dimension of 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 system output matrix C.
          On exit, the leading P-by-N part of this array contains
          the output matrix Cc of the closed-loop system.

  LDC     INTEGER
          The leading dimension of array C.
          LDC >= MAX(1,P) if N > 0.
          LDC >= 1 if N = 0.

  D       (input/output) DOUBLE PRECISION array, dimension (LDD,M)
          On entry, the leading P-by-M part of this array must
          contain the system direct input/output transmission
          matrix D.
          On exit, if JOBD = 'D', the leading P-by-M part of this
          array contains the direct input/output transmission
          matrix Dc of the closed-loop system.
          The array D is not referenced if JOBD = 'Z'.

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

  F       (input) DOUBLE PRECISION array, dimension (LDF,P)
          If FBTYPE = 'O', the leading M-by-P part of this array
          must contain the output feedback matrix F.
          If FBTYPE = 'I', then the feedback matrix is assumed to be
          an M x M order identity matrix.
          The array F is not referenced if FBTYPE = 'I' or
          ALPHA = 0.

  LDF     INTEGER
          The leading dimension of array F.
          LDF >= MAX(1,M) if FBTYPE = 'O' and ALPHA <> 0.
          LDF >= 1 if FBTYPE = 'I' or ALPHA = 0.

  RCOND   (output) DOUBLE PRECISION
          The reciprocal condition number of the matrix
          I - alpha*D*F.

  IWORK   INTEGER array, dimension (LIWORK)
          LIWORK >= MAX(1,2*P) if JOBD = 'D'.
          LIWORK >= 1 if JOBD = 'Z'.
          IWORK is not referenced if JOBD = 'Z'.

  DWORK   DOUBLE PRECISION array, dimension (LDWORK)

  LDWORK  INTEGER
          The length of the array DWORK.
          LDWORK >= wspace, where
                    wspace = MAX( 1, M, P*P + 4*P ) if JOBD = 'D',
                    wspace = MAX( 1, M ) if JOBD = 'Z'.
          For best performance, LDWORK >= MAX( wspace, N*M, N*P ).

  INFO    INTEGER
          = 0:  successful exit;
          < 0:  if INFO = -i, the i-th argument had an illegal
                value;
          = 1:  if the matrix I - alpha*D*F is numerically singular.

  The matrices of the closed-loop system have the expressions:

  Ac = A + alpha*B*F*E*C,  Bc = B + alpha*B*F*E*D,
  Cc = E*C,                Dc = E*D,

  where E = (I - alpha*D*F)**-1.

  The accuracy of computations basically depends on the conditioning
  of the matrix I - alpha*D*F.  If RCOND is very small, it is likely
  that the computed results are inaccurate.

  None

Program Text

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