## AB05RD

### Closed-loop system for a mixed output and state 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 mixed output and
      state feedback control law

           u = alpha*F*y + beta*K*x + G*v
           z = H*y.

Specification

          SUBROUTINE AB05RD( FBTYPE, JOBD, N, M, P, MV, PZ, ALPHA, BETA, A,
         $                   LDA, B, LDB, C, LDC, D, LDD, F, LDF, K, LDK,
         $                   G, LDG, H, LDH, RCOND, BC, LDBC, CC, LDCC,
         $                   DC, LDDC, IWORK, DWORK, LDWORK, INFO )
    C     .. Scalar Arguments ..
          CHARACTER         FBTYPE, JOBD
          INTEGER           INFO, LDA, LDB, LDBC, LDC, LDCC, LDD, LDDC,
         $                  LDF, LDG, LDH, LDK, LDWORK, M, MV, N, P, PZ
          DOUBLE PRECISION  ALPHA, BETA, RCOND
    C     .. Array Arguments ..
          INTEGER           IWORK(*)
          DOUBLE PRECISION  A(LDA,*), B(LDB,*), BC(LDBC,*), C(LDC,*),
         $                  CC(LDCC,*), D(LDD,*), DC(LDDC,*), DWORK(*),
         $                  F(LDF,*), G(LDG,*), H(LDH,*), K(LDK,*)

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 dimension of state vector x, 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 dimension of input vector u, i.e. the number of
              columns of matrices B and D, and the number of rows of F.
              M >= 0.

      P       (input) INTEGER
              The dimension of output vector y, 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'.

      MV      (input) INTEGER
              The dimension of the new input vector v, i.e. the number
              of columns of matrix G.  MV >= 0.

      PZ      (input) INTEGER.
              The dimension of the new output vector z, i.e. the number
              of rows of matrix H.  PZ >= 0.

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

      BETA    (input) DOUBLE PRECISION.
              The coefficient beta in the state 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 intermediary input matrix B1 (see METHOD).

      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 intermediary output matrix C1+BETA*D1*K (see METHOD).

      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, if JOBD = 'D', the leading P-by-M part of this
              array must contain the system direct input/output
              transmission matrix D.
              On exit, the leading P-by-M part of this array contains
              the intermediary direct input/output transmission matrix
              D1 (see METHOD).
              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.

      K       (input) DOUBLE PRECISION array, dimension (LDK,N)
              The leading M-by-N part of this array must contain the
              state feedback matrix K.
              The array K is not referenced if BETA = 0.

      LDK     INTEGER
              The leading dimension of the array K.
              LDK >= MAX(1,M) if BETA <> 0.
              LDK >= 1 if BETA = 0.

      G       (input) DOUBLE PRECISION array, dimension (LDG,MV)
              The leading M-by-MV part of this array must contain the
              system input scaling matrix G.

      LDG     INTEGER
              The leading dimension of the array G.  LDG >= MAX(1,M).

      H       (input) DOUBLE PRECISION array, dimension (LDH,P)
              The leading PZ-by-P part of this array must contain the
              system output scaling matrix H.

      LDH     INTEGER
              The leading dimension of the array H.  LDH >= MAX(1,PZ).

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

      BC      (output) DOUBLE PRECISION array, dimension (LDBC,MV)
              The leading N-by-MV part of this array contains the input
              matrix Bc of the closed-loop system.

      LDBC    INTEGER
              The leading dimension of array BC.  LDBC >= MAX(1,N).

      CC      (output) DOUBLE PRECISION array, dimension (LDCC,N)
              The leading PZ-by-N part of this array contains the
              system output matrix Cc of the closed-loop system.

      LDCC    INTEGER
              The leading dimension of array CC.
              LDCC >= MAX(1,PZ) if N > 0.
              LDCC >= 1 if N = 0.

      DC      (output) DOUBLE PRECISION array, dimension (LDDC,MV)
              If JOBD = 'D', the leading PZ-by-MV part of this array
              contains the direct input/output transmission matrix Dc
              of the closed-loop system.
              The array DC is not referenced if JOBD = 'Z'.

      LDDC    INTEGER
              The leading dimension of array DC.
              LDDC >= MAX(1,PZ) if JOBD = 'D'.
              LDDC >= 1 if JOBD = 'Z'.

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*MV, 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 = A1 + beta*B1*K,      Bc = B1*G,
      Cc = H*(C1 + beta*D1*K),  Dc = H*D1*G,

      where

      A1 = A + alpha*B*F*E*C,   B1 = B + alpha*B*F*E*D,
      C1 = E*C,                 D1 = E*D,

      with 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

    *     AB05RD EXAMPLE PROGRAM TEXT
    *
    *     .. Parameters ..
          INTEGER          NIN, NOUT
          PARAMETER        ( NIN = 5, NOUT = 6 )
          INTEGER          NMAX, MMAX, MVMAX, PMAX, PZMAX
          PARAMETER        ( NMAX = 20, MMAX = 20, MVMAX = 20,
         $                   PMAX = 20, PZMAX = 20 )
          INTEGER          LDA, LDB, LDBC, LDC, LDCC, LDD, LDDC, LDF, LDG,
         $                 LDH, LDK, LDWORK, LIWORK
          PARAMETER        ( LDA = NMAX, LDB = NMAX, LDBC = NMAX,
         $                   LDC = PMAX, LDCC = PZMAX,
         $                   LDD = PMAX, LDDC = PZMAX, LDF = MMAX,
         $                   LDG = MMAX, LDH  = PZMAX, LDK = MMAX,
         $                   LDWORK = MAX( MMAX, PMAX*MVMAX,
         $                   PMAX*PMAX + 4*PMAX ), LIWORK = 2*PMAX )
    *     .. Parameters ..
          DOUBLE PRECISION ZERO
          PARAMETER        ( ZERO = 0.0D0 )
    *     .. Local Scalars ..
          LOGICAL          LJOBD, OUTPF
          CHARACTER*1      FBTYPE, JOBD
          INTEGER          I, INFO, J, M, MV, N, P, PZ
          DOUBLE PRECISION ALPHA, BETA, RCOND
    *     .. Local Arrays ..
          INTEGER          IWORK(LIWORK)
          DOUBLE PRECISION A(LDA,NMAX), B(LDB,MMAX), BC(LDBC,MVMAX),
         $                 C(LDC,NMAX), CC(LDCC,NMAX),
         $                 D(LDD,MMAX), DC(LDDC,MVMAX),  DWORK(LDWORK),
         $                 F(LDF,PMAX), G(LDG,MVMAX), H(LDH,PMAX),
         $                 K(LDK,NMAX)
    *     .. External functions ..
          LOGICAL          LSAME
          EXTERNAL         LSAME
    *     .. External Subroutines ..
          EXTERNAL         AB05RD
    *     .. Intrinsic Functions ..
          INTRINSIC        MAX
    *     .. Executable Statements ..
    *
          WRITE ( NOUT, FMT = 99999 )
    *     Skip the heading in the data file and read the data.
          READ ( NIN, FMT = '()' )
          READ ( NIN, FMT = * ) N, M, P, MV, PZ, ALPHA, BETA, FBTYPE, JOBD
          OUTPF = LSAME( FBTYPE, 'O' )
          LJOBD = LSAME( JOBD, 'D' )
          IF ( N.LE.0 .OR. N.GT.NMAX ) THEN
             WRITE ( NOUT, FMT = 99992 ) N
          ELSE
             READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
             IF ( M.LE.0 .OR. M.GT.MMAX ) THEN
                WRITE ( NOUT, FMT = 99991 ) M
             ELSE
                READ ( NIN, FMT = * ) ( ( B(I,J), I = 1,N ), J = 1,M )
                IF ( BETA.NE.ZERO )
         $         READ ( NIN, FMT = * ) ( ( K(I,J), J = 1,N ), I = 1,M )
                IF ( P.LE.0 .OR. P.GT.PMAX ) THEN
                   WRITE ( NOUT, FMT = 99990 ) P
                ELSE
                   READ ( NIN, FMT = * ) ( ( C(I,J), J = 1,N ), I = 1,P )
                   IF ( LJOBD )
         $            READ ( NIN, FMT = * ) ( ( D(I,J), J = 1,M ), I = 1,P )
                   IF ( OUTPF.AND.ALPHA.NE.ZERO )
         $            READ ( NIN, FMT = * ) ( ( F(I,J), J = 1,P ), I = 1,M )
                   IF ( MV.LE.0 .OR. MV.GT.MVMAX ) THEN
                      WRITE ( NOUT, FMT = 99989 ) MV
                   ELSE
                      READ ( NIN, FMT = * )
         $                 ( ( G(I,J), J = 1,MV ), I = 1,M )
                      IF ( PZ.LE.0 .OR. PZ.GT.PZMAX ) THEN
                         WRITE ( NOUT, FMT = 99988 ) PZ
                      ELSE
                         READ ( NIN, FMT = * )
         $                    ( ( H(I,J), J = 1,P ), I = 1,PZ )
    *                       Find the state-space model (A,B,C,D).
                         CALL AB05RD( FBTYPE, JOBD, N, M, P, MV, PZ, ALPHA,
         $                            BETA, A, LDA, B, LDB, C, LDC, D, LDD,
         $                            F, LDF, K, LDK, G, LDG, H, LDH, RCOND,
         $                            BC, LDBC, CC, LDCC, DC, LDDC, IWORK,
         $                            DWORK, LDWORK, INFO )
    *
                         WRITE ( NOUT, FMT = 99987 ) RCOND
                         IF ( INFO.NE.0 ) THEN
                            WRITE ( NOUT, FMT = 99998 ) INFO
                         ELSE
                            WRITE ( NOUT, FMT = 99997 )
                            DO 20 I = 1, N
                               WRITE ( NOUT, FMT = 99996 )
         $                           ( A(I,J), J = 1,N )
       20                   CONTINUE
                            WRITE ( NOUT, FMT = 99995 )
                            DO 40 I = 1, N
                               WRITE ( NOUT, FMT = 99996 )
         $                           ( BC(I,J), J = 1,MV )
       40                   CONTINUE
                            WRITE ( NOUT, FMT = 99994 )
                            DO 60 I = 1, PZ
                               WRITE ( NOUT, FMT = 99996 )
         $                           ( CC(I,J), J = 1,N )
       60                   CONTINUE
                            IF ( LJOBD ) THEN
                               WRITE ( NOUT, FMT = 99993 )
                               DO 80 I = 1, PZ
                                  WRITE ( NOUT, FMT = 99996 )
         $                              ( DC(I,J), J = 1,MV )
       80                      CONTINUE
                            END IF
                         END IF
                      END IF
                   END IF
                END IF
             END IF
          END IF
          STOP
    *
    99999 FORMAT (' AB05RD EXAMPLE PROGRAM RESULTS',/1X)
    99998 FORMAT (' INFO on exit from AB05RD = ',I2)
    99997 FORMAT (' The state transition matrix of the closed-loop system is
         $')
    99996 FORMAT (20(1X,F8.4))
    99995 FORMAT (/' The input/state matrix of the closed-loop system is ')
    99994 FORMAT (/' The state/output matrix of the closed-loop system is ')
    99993 FORMAT (/' The input/output matrix of the closed-loop system is ')
    99992 FORMAT (/' N is out of range.',/' N = ',I5)
    99991 FORMAT (/' M is out of range.',/' M = ',I5)
    99990 FORMAT (/' P is out of range.',/' P = ',I5)
    99989 FORMAT (/' MV is out of range.',/' MV = ',I5)
    99988 FORMAT (/' PZ is out of range.',/' PZ = ',I5)
    99987 FORMAT ( ' The reciprocal condition number of the matrix ',
         $         ' I - alpha*D*F is',F8.4,/1X)
          END

Program Data

     AB05RD EXAMPLE PROGRAM DATA
       3     2     2     2     2   1.0   1.0    O     D
       1.0   0.0  -1.0
       0.0  -1.0   1.0
       1.0   1.0   2.0
       1.0   1.0   0.0
       2.0   0.0   1.0
       2.0   1.0   0.0
       1.0   0.0   1.0
       3.0  -2.0   1.0
       0.0   1.0   0.0
       1.0   0.0
       0.0   1.0
       1.0   2.0
       3.0   4.0
       1.0   1.0
       0.0   1.0
       4.0   3.0
       2.0   1.0

Program Results

     AB05RD EXAMPLE PROGRAM RESULTS

     The reciprocal condition number of the matrix  I - alpha*D*F is  0.2000

     The state transition matrix of the closed-loop system is
      -4.8333   0.1667  -2.8333
      -0.8333   0.1667   0.1667
      -1.5000   0.5000   1.5000

     The input/state matrix of the closed-loop system is
      -0.5000  -0.8333
       0.5000   0.1667
      -0.5000  -0.5000

     The state/output matrix of the closed-loop system is
       1.1667  -1.8333  -0.8333
       1.8333  -1.1667  -0.1667

     The input/output matrix of the closed-loop system is
       0.5000  -0.8333
       0.5000  -0.1667

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