## AB05SD

### Closed-loop system for an output feedback control law

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

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.

Specification

          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,*)

Arguments

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.

Input/Output Parameters

      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.

Workspace

      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 ).

Error Indicator

      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.

Method

      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.

Numerical Aspects

      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.

Further Comments

      None

Example

Program Text

      None

Program Data

      None

Program Results

      None

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