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 Discrete-Time Loop Shaping Design Procedure.
Specification
      SUBROUTINE SB10ZD( N, M, NP, A, LDA, B, LDB, C, LDC, D, LDD,
     $                   FACTOR, AK, LDAK, BK, LDBK, CK, LDCK, DK,
     $                   LDDK, RCOND, TOL, IWORK, DWORK, LDWORK, BWORK,
     $                   INFO )
C     .. Scalar Arguments ..
      INTEGER            INFO, LDA, LDAK, LDB, LDBK, LDC, LDCK, LDD,
     $                   LDDK, LDWORK, M, N, NP
      DOUBLE PRECISION   FACTOR, TOL
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( 6 )
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 input/output 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
               (not recommended);
          > 1  implies that a suboptimal controller is required
               achieving a performance FACTOR less than optimal.
          FACTOR >= 1.
  AK      (output) DOUBLE PRECISION array, dimension (LDAK,N)
          The leading N-by-N 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 N-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-N 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 (6)
          RCOND(1) contains an estimate of the reciprocal condition
                   number of the linear system of equations from
                   which the solution of the P-Riccati equation is
                   obtained;
          RCOND(2) contains an estimate of the reciprocal condition
                   number of the linear system of equations from
                   which the solution of the Q-Riccati equation is
                   obtained;
          RCOND(3) contains an estimate of the reciprocal condition
                   number of the matrix (gamma^2-1)*In - P*Q;
          RCOND(4) contains an estimate of the reciprocal condition
                   number of the matrix Rx + Bx'*X*Bx;
          RCOND(5) contains an estimate of the reciprocal condition
                                               ^
                   number of the matrix Ip + D*Dk;
          RCOND(6) contains an estimate of the reciprocal condition
                                             ^
                   number of the matrix Im + Dk*D.
Tolerances
  TOL     DOUBLE PRECISION
          Tolerance used for checking the nonsingularity of the
          matrices to be inverted. If TOL <= 0, then a default value
          equal to sqrt(EPS) is used, where EPS is the relative
          machine precision.  TOL < 1.
Workspace
  IWORK   INTEGER array, dimension 2*max(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 >= 16*N*N + 5*M*M + 7*NP*NP + 6*M*N + 7*M*NP +
                     7*N*NP + 6*N + 2*(M + NP) +
                     max(14*N+23,16*N,2*M-1,2*NP-1).
          For good performance, LDWORK must generally be larger.
  BWORK   LOGICAL array, dimension (2*N)
Error Indicator
  INFO    (output) INTEGER
          =  0:  successful exit;
          <  0:  if INFO = -i, the i-th argument had an illegal
                 value;
          =  1:  the P-Riccati equation is not solved successfully;
          =  2:  the Q-Riccati equation is not solved successfully;
          =  3:  the iteration to compute eigenvalues or singular
                 values failed to converge;
          =  4:  the matrix (gamma^2-1)*In - P*Q is singular;
          =  5:  the matrix Rx + Bx'*X*Bx is singular;
                                   ^
          =  6:  the matrix Ip + D*Dk is singular;
                                 ^
          =  7:  the matrix Im + Dk*D is singular;
          =  8:  the matrix Ip - D*Dk is singular;
          =  9:  the matrix Im - Dk*D is singular;
          = 10:  the closed-loop system is unstable.
Method
The routine implements the formulas given in [1].References
  [1] Gu, D.-W., Petkov, P.H., and Konstantinov, M.M.
      On discrete H-infinity loop shaping design procedure routines.
      Technical Report 00-6, Dept. of Engineering, Univ. of
      Leicester, UK, 2000.
Numerical Aspects
The accuracy of the results depends on the conditioning of the two Riccati equations solved in the controller design. For better conditioning it is advised to take FACTOR > 1.Further Comments
NoneExample
Program Text
*     SB10ZD EXAMPLE PROGRAM TEXT
*     Copyright (c) 2002-2010 NICONET e.V.
*
*     .. Parameters ..
      INTEGER          NIN, NOUT
      PARAMETER        ( NIN = 5, NOUT = 6 )
      INTEGER          MMAX, NMAX, PMAX
      PARAMETER        ( MMAX = 10, NMAX = 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 = 2*MAX( NMAX, MMAX + PMAX ) )
      INTEGER          LDWORK
      PARAMETER        ( LDWORK = 16*NMAX*NMAX + 5*MMAX*MMAX +
     $                            7*PMAX*PMAX + 6*MMAX*NMAX +
     $                            7*MMAX*PMAX + 7*NMAX*PMAX + 6*NMAX +
     $                            2*( MMAX + PMAX ) +
     $                            MAX( 14*NMAX + 23, 16*NMAX,
     $                                  2*MMAX - 1, 2*PMAX - 1 ) )
*     .. Local Scalars ..
      DOUBLE PRECISION FACTOR, TOL
      INTEGER          I, INFO, J, M, N, NP
*     .. Local Arrays ..
      LOGICAL          BWORK(2*NMAX)
      INTEGER          IWORK(LIWORK)
      DOUBLE PRECISION A(LDA,NMAX),   AK(LDAK,NMAX), B(LDB,MMAX),
     $                 BK(LDBK,PMAX), C(LDC,NMAX),   CK(LDCK,NMAX),
     $                 D(LDD,MMAX),   DK(LDDK,PMAX), DWORK(LDWORK),
     $                 RCOND( 6 )
*     .. External Subroutines ..
      EXTERNAL         SB10ZD
*     .. 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, TOL
         CALL SB10ZD( N, M, NP, A, LDA, B, LDB, C, LDC, D, LDD, FACTOR,
     $                AK, LDAK, BK, LDBK, CK, LDCK, DK, LDDK, RCOND,
     $                TOL, IWORK, DWORK, LDWORK, BWORK, INFO )
         IF ( INFO.EQ.0 ) THEN
            WRITE ( NOUT, FMT = 99997 )
            DO 10 I = 1, N
               WRITE ( NOUT, FMT = 99992 ) ( AK(I,J), J = 1,N )
   10       CONTINUE
            WRITE ( NOUT, FMT = 99996 )
            DO 20 I = 1, N
               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,N )
   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 = 1,6 )
         ELSE
            WRITE( NOUT, FMT = 99998 ) INFO
         END IF
      END IF
      STOP
*
99999 FORMAT (' SB10ZD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (/' INFO on exit from SB10ZD =',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,F8.4))
99991 FORMAT ( 5(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
SB10LD EXAMPLE PROGRAM DATA 6 2 3 0.2 0.0 3.0 0.0 -0.3 -0.1 -3.0 0.2 -0.4 -0.3 0.0 0.0 -0.1 0.1 -1.0 0.0 0.0 -3.0 1.0 0.0 0.0 -1.0 -1.0 0.0 0.0 0.3 0.6 2.0 0.1 -0.4 0.2 -4.0 0.0 0.0 0.2 -2.0 -1.0 -2.0 1.0 3.0 -3.0 -4.0 1.0 -2.0 0.0 1.0 1.0 5.0 1.0 -1.0 2.0 -2.0 0.0 -3.0 -3.0 0.0 1.0 -1.0 1.0 -1.0 2.0 4.0 -3.0 0.0 5.0 1.0 10.0 -6.0 -7.0 8.0 2.0 -4.0 1.1 0.0Program Results
SB10ZD EXAMPLE PROGRAM RESULTS The controller state matrix AK is 1.0128 0.5101 -0.1546 1.1300 3.3759 0.4911 -2.1257 -1.4517 -0.4486 0.3493 -1.5506 -1.4296 -1.0930 -0.6026 -0.1344 0.2253 -1.5625 -0.6762 0.3207 0.1698 0.2376 -1.1781 -0.8705 0.2896 0.5017 0.9006 0.0668 2.3613 0.2049 0.3703 1.0787 0.6703 0.2783 -0.7213 0.4918 0.7435 The controller input matrix BK is 0.4132 0.3112 -0.8077 0.2140 0.4253 0.1811 -0.0710 0.0807 0.3558 -0.0121 -0.2019 0.0249 0.1047 0.1399 -0.0457 -0.2542 -0.3472 0.0523 The controller output matrix CK is -0.0372 -0.0456 -0.0040 0.0962 -0.2059 -0.0571 0.1999 0.2994 0.1335 -0.0251 -0.3108 0.2048 The controller matrix DK is 0.0629 -0.0022 0.0363 -0.0228 0.0195 0.0600 The estimated condition numbers are 0.27949D-03 0.66679D-03 0.45677D-01 0.23433D-07 0.68495D-01 0.76854D-01
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