Purpose
  To compute the matrices of the positive feedback controller
           | Ak | Bk |
       K = |----|----|
           | Ck | Dk |
  for the shaped plant
           | A | B |
       G = |---|---|
           | C | D |
  in the McFarlane/Glover Loop Shaping Design Procedure.
Specification
      SUBROUTINE SB10ID( N, M, NP, A, LDA, B, LDB, C, LDC, D, LDD,
     $                   FACTOR, NK, AK, LDAK, BK, LDBK, CK, LDCK,
     $                   DK, LDDK, RCOND, IWORK, DWORK, LDWORK, BWORK,
     $                   INFO )
C     .. Scalar Arguments ..
      INTEGER            INFO, LDA, LDAK, LDB, LDBK, LDC, LDCK, LDD,
     $                   LDDK, LDWORK, M, N, NK, NP
      DOUBLE PRECISION   FACTOR
C     .. Array Arguments ..
      INTEGER            IWORK( * )
      LOGICAL            BWORK( * )
      DOUBLE PRECISION   A( LDA, * ), AK( LDAK, * ), B( LDB, * ),
     $                   BK( LDBK, * ), C( LDC, * ), CK( LDCK, * ),
     $                   D( LDD, * ), DK( LDDK, * ), DWORK( * ),
     $                   RCOND( 2 )
Arguments
Input/Output Parameters
  N       (input) INTEGER
          The order of the plant.  N >= 0.
  M       (input) INTEGER
          The column size of the matrix B.  M >= 0.
  NP      (input) INTEGER
          The row size of the matrix C.  NP >= 0.
  A       (input) DOUBLE PRECISION array, dimension (LDA,N)
          The leading N-by-N part of this array must contain the
          system state matrix A of the shaped plant.
  LDA     INTEGER
          The leading dimension of the 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
          system input matrix B of the shaped plant.
  LDB     INTEGER
          The leading dimension of the array B.  LDB >= max(1,N).
  C       (input) DOUBLE PRECISION array, dimension (LDC,N)
          The leading NP-by-N part of this array must contain the
          system output matrix C of the shaped plant.
  LDC     INTEGER
          The leading dimension of the array C.  LDC >= max(1,NP).
  D       (input) DOUBLE PRECISION array, dimension (LDD,M)
          The leading NP-by-M part of this array must contain the
          system matrix D of the shaped plant.
  LDD     INTEGER
          The leading dimension of the array D.  LDD >= max(1,NP).
  FACTOR  (input) DOUBLE PRECISION
          = 1 implies that an optimal controller is required;
          > 1 implies that a suboptimal controller is required,
              achieving a performance FACTOR less than optimal.
          FACTOR >= 1.
  NK      (output) INTEGER
          The order of the positive feedback controller.  NK <= N.
  AK      (output) DOUBLE PRECISION array, dimension (LDAK,N)
          The leading NK-by-NK part of this array contains the
          controller state matrix Ak.
  LDAK    INTEGER
          The leading dimension of the array AK.  LDAK >= max(1,N).
  BK      (output) DOUBLE PRECISION array, dimension (LDBK,NP)
          The leading NK-by-NP part of this array contains the
          controller input matrix Bk.
  LDBK    INTEGER
          The leading dimension of the array BK.  LDBK >= max(1,N).
  CK      (output) DOUBLE PRECISION array, dimension (LDCK,N)
          The leading M-by-NK part of this array contains the
          controller output matrix Ck.
  LDCK    INTEGER
          The leading dimension of the array CK.  LDCK >= max(1,M).
  DK      (output) DOUBLE PRECISION array, dimension (LDDK,NP)
          The leading M-by-NP part of this array contains the
          controller matrix Dk.
  LDDK    INTEGER
          The leading dimension of the array DK.  LDDK >= max(1,M).
  RCOND   (output) DOUBLE PRECISION array, dimension (2)
          RCOND(1) contains an estimate of the reciprocal condition
                   number of the X-Riccati equation;
          RCOND(2) contains an estimate of the reciprocal condition
                   number of the Z-Riccati equation.
Workspace
  IWORK   INTEGER array, dimension max(2*N,N*N,M,NP)
  DWORK   DOUBLE PRECISION array, dimension (LDWORK)
          On exit, if INFO = 0, DWORK(1) contains the optimal value
          of LDWORK.
  LDWORK  INTEGER
          The dimension of the array DWORK.
          LDWORK >= 4*N*N + M*M + NP*NP + 2*M*N + N*NP + 4*N +
                    max( 6*N*N + 5 + max(1,4*N*N+8*N), N*NP + 2*N ).
          For good performance, LDWORK must generally be larger.
          An upper bound of LDWORK in the above formula is
          LDWORK >= 10*N*N + M*M + NP*NP + 2*M*N + 2*N*NP + 4*N +
                    5 + max(1,4*N*N+8*N).
  BWORK   LOGICAL array, dimension (2*N)
Error Indicator
  INFO    INTEGER
          = 0:  successful exit;
          < 0:  if INFO = -i, the i-th argument had an illegal
                value;
          = 1:  the X-Riccati equation is not solved successfully;
          = 2:  the Z-Riccati equation is not solved successfully;
          = 3:  the iteration to compute eigenvalues or singular
                values failed to converge;
          = 4:  the matrix Ip - D*Dk is singular;
          = 5:  the matrix Im - Dk*D is singular;
          = 6:  the closed-loop system is unstable.
Method
The routine implements the formulas given in [1].References
  [1] McFarlane, D. and Glover, K.
      A loop shaping design procedure using H_infinity synthesis.
      IEEE Trans. Automat. Control, vol. AC-37, no. 6, pp. 759-769,
      1992.
Numerical Aspects
The accuracy of the results depends on the conditioning of the two Riccati equations solved in the controller design (see the output parameter RCOND).Further Comments
NoneExample
Program Text
*     SB10ID EXAMPLE PROGRAM TEXT
*     Copyright (c) 2002-2010 NICONET e.V.
*
*     .. Parameters ..
      INTEGER          NIN, NOUT
      PARAMETER        ( NIN = 5, NOUT = 6 )
      INTEGER          NMAX, MMAX, PMAX
      PARAMETER        ( NMAX = 10, MMAX = 10, PMAX = 10 )
      INTEGER          LDA, LDAK, LDB, LDBK, LDC, LDCK, LDD, LDDK
      PARAMETER        ( LDA  = NMAX, LDAK = NMAX, LDB  = NMAX,
     $                   LDBK = NMAX, LDC  = PMAX, LDCK = MMAX,
     $                   LDD  = PMAX, LDDK = MMAX )
      INTEGER          LIWORK
      PARAMETER        ( LIWORK = MAX( 2*NMAX, NMAX*NMAX, MMAX, PMAX ) )
      INTEGER          LDWORK
      PARAMETER        ( LDWORK = 4*NMAX*NMAX + MMAX*MMAX + PMAX*PMAX +
     $                            2*MMAX*NMAX + NMAX*PMAX + 4*NMAX +
     $                            MAX( 10*NMAX*NMAX + 8*NMAX + 5,
     $                                    NMAX*PMAX + 2*NMAX ) )
*     .. Local Scalars ..
      DOUBLE PRECISION FACTOR
      INTEGER          I, INFO, J, M, N, NK, NP
*     .. Local Arrays ..
      LOGICAL          BWORK(2*NMAX)
      INTEGER          IWORK(LIWORK)
      DOUBLE PRECISION A(LDA,NMAX), AK(LDA,NMAX), B(LDB,MMAX),
     $                 BK(LDBK,PMAX), C(LDC,NMAX), CK(LDCK,NMAX),
     $                 D(LDD,MMAX), DK(LDDK,PMAX), DWORK(LDWORK),
     $                 RCOND( 2 )
*     .. External Subroutines ..
      EXTERNAL         SB10ID
*     .. 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, NP
      IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
         WRITE ( NOUT, FMT = 99990 ) N
      ELSE IF ( M.LT.0 .OR. M.GT.MMAX ) THEN
         WRITE ( NOUT, FMT = 99989 ) M
      ELSE IF ( NP.LT.0 .OR. NP.GT.PMAX ) THEN
         WRITE ( NOUT, FMT = 99988 ) NP
      ELSE
         READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
         READ ( NIN, FMT = * ) ( ( B(I,J), J = 1,M ), I = 1,N )
         READ ( NIN, FMT = * ) ( ( C(I,J), J = 1,N ), I = 1,NP )
         READ ( NIN, FMT = * ) ( ( D(I,J), J = 1,M ), I = 1,NP )
         READ ( NIN, FMT = * ) FACTOR
         CALL SB10ID( N, M, NP, A, LDA, B, LDB, C, LDC, D, LDD,
     $                FACTOR, NK, AK, LDAK, BK, LDBK, CK, LDCK,
     $                DK, LDDK, RCOND, IWORK, DWORK, LDWORK,
     $                BWORK, INFO )
         IF ( INFO.EQ.0 ) THEN
            WRITE ( NOUT, FMT = 99997 )
            DO 10 I = 1, NK
               WRITE ( NOUT, FMT = 99992 ) ( AK(I,J), J = 1,NK )
   10       CONTINUE
            WRITE ( NOUT, FMT = 99996 )
            DO 20 I = 1, NK
               WRITE ( NOUT, FMT = 99992 ) ( BK(I,J), J = 1,NP )
   20       CONTINUE
            WRITE ( NOUT, FMT = 99995 )
            DO 30 I = 1, M
               WRITE ( NOUT, FMT = 99992 ) ( CK(I,J), J = 1,NK )
   30       CONTINUE
            WRITE ( NOUT, FMT = 99994 )
            DO 40 I = 1, M
               WRITE ( NOUT, FMT = 99992 ) ( DK(I,J), J = 1,NP )
   40       CONTINUE
            WRITE( NOUT, FMT = 99993 )
            WRITE( NOUT, FMT = 99991 ) ( RCOND(I), I = 1, 2 )
         ELSE
            WRITE( NOUT, FMT = 99998 ) INFO
         END IF
      END IF
      STOP
*
99999 FORMAT (' SB10ID EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (/' INFO on exit from SB10ID =',I2)
99997 FORMAT (/' The controller state matrix AK is'/)
99996 FORMAT (/' The controller input matrix BK is'/)
99995 FORMAT (/' The controller output matrix CK is'/)
99994 FORMAT (/' The controller matrix DK is'/)
99993 FORMAT (/' The estimated condition numbers are'/)
99992 FORMAT (10(1X,F9.4))
99991 FORMAT ( 2(1X,D12.5))
99990 FORMAT (/' N is out of range.',/' N = ',I5)
99989 FORMAT (/' M is out of range.',/' M = ',I5)
99988 FORMAT (/' NP is out of range.',/' NP = ',I5)
      END
Program Data
SB10ID EXAMPLE PROGRAM DATA 6 2 3 -1.0 0.0 4.0 5.0 -3.0 -2.0 -2.0 4.0 -7.0 -2.0 0.0 3.0 -6.0 9.0 -5.0 0.0 2.0 -1.0 -8.0 4.0 7.0 -1.0 -3.0 0.0 2.0 5.0 8.0 -9.0 1.0 -4.0 3.0 -5.0 8.0 0.0 2.0 -6.0 -3.0 -4.0 2.0 0.0 -5.0 -7.0 4.0 -6.0 -3.0 9.0 1.0 -2.0 1.0 -1.0 2.0 -4.0 0.0 -3.0 -3.0 0.0 5.0 -1.0 1.0 1.0 -7.0 5.0 0.0 -8.0 2.0 -2.0 1.0 -2.0 0.0 4.0 5.0 -3.0 1.0Program Results
 SB10ID EXAMPLE PROGRAM RESULTS
 The controller state matrix AK is
  -39.0671    9.9293   22.2322  -27.4113   43.8655
   -6.6117    3.0006   11.0878  -11.4130   15.4269
   33.6805   -6.6934  -23.9953   14.1438  -33.4358
  -32.3191    9.7316   25.4033  -24.0473   42.0517
  -44.1655   18.7767   34.8873  -42.4369   50.8437
 The controller input matrix BK is
  -10.2905  -16.5382  -10.9782
   -4.3598   -8.7525   -5.1447
    6.5962    1.8975    6.2316
   -9.8770  -14.7041  -11.8778
   -9.6726  -22.7309  -18.2692
 The controller output matrix CK is
   -0.6647   -0.0599   -1.0376    0.5619    1.7297
   -8.4202    3.9573    7.3094   -7.6283   10.6768
 The controller matrix DK is
    0.8466    0.4979   -0.6993
   -1.2226   -4.8689   -4.5056
 The estimated condition numbers are
  0.13861D-01  0.90541D-02
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