Purpose
  To compute the matrices of an H-infinity (sub)optimal n-state
  controller
                        | AK | BK |
                    K = |----|----|,
                        | CK | DK |
  for the discrete-time system
                | A  | B1  B2  |   | A | B |
            P = |----|---------| = |---|---|
                | C1 | D11 D12 |   | C | D |
                | C2 | D21 D22 |
  and for a given value of gamma, where B2 has as column size the
  number of control inputs (NCON) and C2 has as row size the number
  of measurements (NMEAS) being provided to the controller.
  It is assumed that
  (A1) (A,B2) is stabilizable and (C2,A) is detectable,
  (A2) D12 is full column rank and D21 is full row rank,
            j*Theta
  (A3) | A-e       *I  B2  | has full column rank for all
       |    C1         D12 |
       0 <= Theta < 2*Pi ,
            j*Theta
  (A4) | A-e       *I  B1  |  has full row rank for all
       |    C2         D21 |
       0 <= Theta < 2*Pi .
Specification
      SUBROUTINE SB10DD( N, M, NP, NCON, NMEAS, GAMMA, A, LDA, B, LDB,
     $                   C, LDC, D, LDD, AK, LDAK, BK, LDBK, CK, LDCK,
     $                   DK, LDDK, X, LDX, Z, LDZ, RCOND, TOL, IWORK,
     $                   DWORK, LDWORK, BWORK, INFO )
C     .. Scalar Arguments ..
      INTEGER            INFO, LDA, LDAK, LDB, LDBK, LDC, LDCK, LDD,
     $                   LDDK, LDWORK, LDX, LDZ, M, N, NCON, NMEAS, NP
      DOUBLE PRECISION   GAMMA, TOL
C     .. Array Arguments ..
      INTEGER            IWORK( * )
      DOUBLE PRECISION   A( LDA, * ), AK( LDAK, * ), B( LDB, * ),
     $                   BK( LDBK, * ), C( LDC, * ), CK( LDCK, * ),
     $                   D( LDD, * ), DK( LDDK, * ), DWORK( * ),
     $                   RCOND( * ), X( LDX, * ), Z( LDZ, * )
      LOGICAL            BWORK( * )
Arguments
Input/Output Parameters
  N       (input) INTEGER
          The order of the system.  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.
  NCON    (input) INTEGER
          The number of control inputs (M2).  M >= NCON >= 0,
          NP-NMEAS >= NCON.
  NMEAS   (input) INTEGER
          The number of measurements (NP2).  NP >= NMEAS >= 0,
          M-NCON >= NMEAS.
  GAMMA   (input) DOUBLE PRECISION
          The value of gamma. It is assumed that gamma is
          sufficiently large so that the controller is admissible.
          GAMMA > 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.
  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.
  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.
  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.
  LDD     INTEGER
          The leading dimension of the array D.  LDD >= max(1,NP).
  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,NMEAS)
          The leading N-by-NMEAS 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 NCON-by-N part of this array contains the
          controller output matrix CK.
  LDCK    INTEGER
          The leading dimension of the array CK.
          LDCK >= max(1,NCON).
  DK      (output) DOUBLE PRECISION array, dimension (LDDK,NMEAS)
          The leading NCON-by-NMEAS part of this array contains the
          controller input/output matrix DK.
  LDDK    INTEGER
          The leading dimension of the array DK.
          LDDK >= max(1,NCON).
  X       (output) DOUBLE PRECISION array, dimension (LDX,N)
          The leading N-by-N part of this array contains the matrix
          X, solution of the X-Riccati equation.
  LDX     INTEGER
          The leading dimension of the array X.  LDX >= max(1,N).
  Z       (output) DOUBLE PRECISION array, dimension (LDZ,N)
          The leading N-by-N part of this array contains the matrix
          Z, solution of the Z-Riccati equation.
  LDZ     INTEGER
          The leading dimension of the array Z.  LDZ >= max(1,N).
  RCOND   (output) DOUBLE PRECISION array, dimension (8)
          RCOND contains estimates of the reciprocal condition
          numbers of the matrices which are to be inverted and
          estimates of the reciprocal condition numbers of the
          Riccati equations which have to be solved during the
          computation of the controller. (See the description of
          the algorithm in [2].)
          RCOND(1) contains the reciprocal condition number of the
                   matrix R3;
          RCOND(2) contains the reciprocal condition number of the
                   matrix R1 - R2'*inv(R3)*R2;
          RCOND(3) contains the reciprocal condition number of the
                   matrix V21;
          RCOND(4) contains the reciprocal condition number of the
                   matrix St3;
          RCOND(5) contains the reciprocal condition number of the
                   matrix V12;
          RCOND(6) contains the reciprocal condition number of the
                   matrix Im2 + DKHAT*D22
          RCOND(7) contains the reciprocal condition number of the
                   X-Riccati equation;
          RCOND(8) contains the reciprocal condition number of the
                   Z-Riccati equation.
Tolerances
  TOL     DOUBLE PRECISION
          Tolerance used in neglecting the small singular values
          in rank determination. If TOL <= 0, then a default value
          equal to 1000*EPS is used, where EPS is the relative
          machine precision.
Workspace
  IWORK   INTEGER array, dimension max(2*max(M2,N),M,M2+NP2,N*N)
  DWORK   DOUBLE PRECISION array, dimension (LDWORK)
          On exit, if INFO = 0, DWORK(1) contains the optimal
          LDWORK.
  LDWORK  INTEGER
          The dimension of the array DWORK.
          LDWORK >= max(LW1,LW2,LW3,LW4), where
          LW1 = (N+NP1+1)*(N+M2) + max(3*(N+M2)+N+NP1,5*(N+M2));
          LW2 = (N+NP2)*(N+M1+1) + max(3*(N+NP2)+N+M1,5*(N+NP2));
          LW3 = 13*N*N + 2*M*M + N*(8*M+NP2) + M1*(M2+NP2) + 6*N +
                max(14*N+23,16*N,2*N+M,3*M);
          LW4 = 13*N*N + M*M + (8*N+M+M2+2*NP2)*(M2+NP2) + 6*N +
                N*(M+NP2) + max(14*N+23,16*N,2*N+M2+NP2,3*(M2+NP2));
          For good performance, LDWORK must generally be larger.
          Denoting Q = max(M1,M2,NP1,NP2), an upper bound is
          max((N+Q)*(N+Q+6),13*N*N + M*M + 2*Q*Q + N*(M+Q) +
              max(M*(M+7*N),2*Q*(8*N+M+2*Q)) + 6*N +
              max(14*N+23,16*N,2*N+max(M,2*Q),3*max(M,2*Q)).
  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;
                                   j*Theta
          = 1:  if the matrix | A-e       *I  B2  | had not full
                              |      C1       D12 |
                column rank;
                                   j*Theta
          = 2:  if the matrix | A-e       *I  B1  | had not full
                              |      C2       D21 |
                row rank;
          = 3:  if the matrix D12 had not full column rank;
          = 4:  if the matrix D21 had not full row rank;
          = 5:  if the controller is not admissible (too small value
                of gamma);
          = 6:  if the X-Riccati equation was not solved
                successfully (the controller is not admissible or
                there are numerical difficulties);
          = 7:  if the Z-Riccati equation was not solved
                successfully (the controller is not admissible or
                there are numerical difficulties);
          = 8:  if the matrix Im2 + DKHAT*D22 is singular.
          = 9:  if the singular value decomposition (SVD) algorithm
                did not converge (when computing the SVD of one of
                the matrices |A   B2 |, |A   B1 |, D12 or D21).
                             |C1  D12|  |C2  D21|
Method
The routine implements the method presented in [1].References
  [1] Green, M. and Limebeer, D.J.N.
      Linear Robust Control.
      Prentice-Hall, Englewood Cliffs, NJ, 1995.
  [2] Petkov, P.Hr., Gu, D.W., and Konstantinov, M.M.
      Fortran 77 routines for Hinf and H2 design of linear
      discrete-time control systems.
      Report 99-8, Department of Engineering, Leicester University,
      April 1999.
Numerical Aspects
With approaching the minimum value of gamma some of the matrices which are to be inverted tend to become ill-conditioned and the X- or Z-Riccati equation may also become ill-conditioned which may deteriorate the accuracy of the result. (The corresponding reciprocal condition numbers are given in the output array RCOND.)Further Comments
NoneExample
Program Text
*     SB10DD 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, LDB, LDC, LDD, LDAK, LDBK, LDCK, LDDK, LDX,
     $                 LDZ
      PARAMETER        ( LDA = NMAX, LDB = NMAX, LDC = PMAX, LDD = PMAX,
     $                   LDAK = NMAX, LDBK = NMAX, LDCK = PMAX,
     $                   LDDK = PMAX, LDX = NMAX, LDZ = NMAX )
      INTEGER          LIWORK
      PARAMETER        ( LIWORK = MAX( 2*MAX( MMAX, NMAX ),
     $                                 MMAX + PMAX, NMAX*NMAX ) )
      INTEGER          MPMX
      PARAMETER        ( MPMX = MAX( MMAX, PMAX ) )
      INTEGER          LDWORK
      PARAMETER        ( LDWORK =
     $                   MAX( ( NMAX + MPMX )*( NMAX + MPMX + 6 ),
     $                        13*NMAX*NMAX + MMAX*MMAX + 2*MPMX*MPMX +
     $                        NMAX*( MMAX + MPMX ) +
     $                        MAX( MMAX*( MMAX + 7*NMAX ),
     $                             2*MPMX*( 8*NMAX + MMAX + 2*MPMX ) )
     $                             + 6*NMAX +
     $                             MAX( 14*NMAX + 23, 16*NMAX,
     $                                  2*NMAX + MAX( MMAX, 2*MPMX ),
     $                                  3*MAX( MMAX, 2*MPMX ) ) ) )
*     .. Local Scalars ..
      DOUBLE PRECISION GAMMA, TOL
      INTEGER          I, INFO, J, M, N, NCON, NMEAS, 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), X(LDX,NMAX),
     $                 Z(LDZ,NMAX), DWORK(LDWORK), RCOND( 8 )
*     .. External Subroutines ..
      EXTERNAL         SB10DD
*     .. 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, NCON, NMEAS
      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 IF ( NCON.LT.0 .OR. NCON.GT.MMAX ) THEN
         WRITE ( NOUT, FMT = 99987 ) NCON
      ELSE IF ( NMEAS.LT.0 .OR. NMEAS.GT.PMAX ) THEN
         WRITE ( NOUT, FMT = 99986 ) NMEAS
      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 = * ) GAMMA, TOL
         CALL SB10DD( N, M, NP, NCON, NMEAS, GAMMA, A, LDA, B, LDB,
     $                C, LDC, D, LDD, AK, LDAK, BK, LDBK, CK, LDCK,
     $                DK, LDDK, X, LDX, Z, LDZ, 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,NMEAS )
   20       CONTINUE
            WRITE ( NOUT, FMT = 99995 )
            DO 30 I = 1, NCON
               WRITE ( NOUT, FMT = 99992 ) ( CK(I,J), J = 1,N )
   30       CONTINUE
            WRITE ( NOUT, FMT = 99994 )
            DO 40 I = 1, NCON
               WRITE ( NOUT, FMT = 99992 ) ( DK(I,J), J = 1,NMEAS )
   40       CONTINUE
            WRITE( NOUT, FMT = 99993 )
            WRITE( NOUT, FMT = 99991 ) ( RCOND(I), I = 1, 8 )
         ELSE
            WRITE( NOUT, FMT = 99998 ) INFO
         END IF
      END IF
      STOP
*
99999 FORMAT (' SB10DD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (/' INFO on exit from SB10DD =',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)
99987 FORMAT (/' NCON is out of range.',/' NCON = ',I5)
99986 FORMAT (/' NMEAS is out of range.',/' NMEAS = ',I5)
      END
Program Data
SB10DD EXAMPLE PROGRAM DATA 6 5 5 2 2 -0.7 0.0 0.3 0.0 -0.5 -0.1 -0.6 0.2 -0.4 -0.3 0.0 0.0 -0.5 0.7 -0.1 0.0 0.0 -0.8 -0.7 0.0 0.0 -0.5 -1.0 0.0 0.0 0.3 0.6 -0.9 0.1 -0.4 0.5 -0.8 0.0 0.0 0.2 -0.9 -1.0 -2.0 -2.0 1.0 0.0 1.0 0.0 1.0 -2.0 1.0 -3.0 -4.0 0.0 2.0 -2.0 1.0 -2.0 1.0 0.0 -1.0 0.0 1.0 -2.0 0.0 3.0 1.0 0.0 3.0 -1.0 -2.0 1.0 -1.0 2.0 -2.0 0.0 -3.0 -3.0 0.0 1.0 -1.0 1.0 0.0 0.0 2.0 0.0 -4.0 0.0 -2.0 1.0 -3.0 0.0 0.0 3.0 1.0 0.0 1.0 -2.0 1.0 0.0 -2.0 1.0 -1.0 -2.0 0.0 0.0 0.0 1.0 0.0 1.0 0.0 2.0 -1.0 -3.0 0.0 1.0 0.0 1.0 0.0 1.0 -1.0 0.0 0.0 1.0 2.0 1.0 111.294 0.00000001Program Results
SB10DD EXAMPLE PROGRAM RESULTS The controller state matrix AK is -18.0030 52.0376 26.0831 -0.4271 -40.9022 18.0857 18.8203 -57.6244 -29.0938 0.5870 45.3309 -19.8644 -26.5994 77.9693 39.0368 -1.4020 -60.1129 26.6910 -21.4163 62.1719 30.7507 -0.9201 -48.6221 21.8351 -0.8911 4.2787 2.3286 -0.2424 -3.0376 1.2169 -5.3286 16.1955 8.4824 -0.2489 -12.2348 5.1590 The controller input matrix BK is 16.9788 14.1648 -18.9215 -15.6726 25.2046 21.2848 20.1122 16.8322 1.4104 1.2040 5.3181 4.5149 The controller output matrix CK is -9.1941 27.5165 13.7364 -0.3639 -21.5983 9.6025 3.6490 -10.6194 -5.2772 0.2432 8.1108 -3.6293 The controller matrix DK is 9.0317 7.5348 -3.4006 -2.8219 The estimated condition numbers are 0.24960D+00 0.98548D+00 0.99186D+00 0.63733D-05 0.48625D+00 0.29430D-01 0.56942D-02 0.12470D-01
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