SB10HD - SLICOT Library Routine Documentation

## SB10HD

### H2 optimal state controller for a continuous-time system

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

Purpose

      To compute the matrices of the H2 optimal n-state controller

               | AK | BK |
           K = |----|----|
               | CK | DK |

      for the system

                    | A  | B1  B2  |   | A | B |
                P = |----|---------| = |---|---| ,
                    | C1 |  0  D12 |   | C | D |
                    | C2 | D21 D22 |

      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) The block D11 of D is zero,

      (A3) D12 is full column rank and D21 is full row rank.

Specification

          SUBROUTINE SB10HD( N, M, NP, NCON, NMEAS, A, LDA, B, LDB, C, LDC,
         $                   D, LDD, 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, NCON, NMEAS, NP
          DOUBLE PRECISION   TOL
    C     .. Array Arguments ..
          LOGICAL            BWORK( * )
          INTEGER            IWORK( * )
          DOUBLE PRECISION   A( LDA, * ), AK( LDAK, * ), B( LDB, * ),
         $                   BK( LDBK, * ), C( LDC, * ), CK( LDCK, * ),
         $                   D( LDD, * ), DK( LDDK, * ), DWORK( * ),
         $                   RCOND( 4 )

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.

      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).

      RCOND   (output) DOUBLE PRECISION array, dimension (4)
              RCOND(1) contains the reciprocal condition number of the
                       control transformation matrix;
              RCOND(2) contains the reciprocal condition number of the
                       measurement transformation matrix;
              RCOND(3) contains an estimate of the reciprocal condition
                       number of the X-Riccati equation;
              RCOND(4) contains an estimate of the reciprocal condition
                       number of the Y-Riccati equation.

Tolerances

      TOL     DOUBLE PRECISION
              Tolerance used for controlling the accuracy of the applied
              transformations for computing the normalized form in
              SLICOT Library routine SB10UD. Transformation matrices
              whose reciprocal condition numbers are less than TOL are
              not allowed. If TOL <= 0, then a default value equal to
              sqrt(EPS) is used, where EPS is the relative machine
              precision.

Workspace

      IWORK   INTEGER array, dimension (max(2*N,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 >= N*M + NP*(N+M) + M2*M2 + NP2*NP2 +
                        max(max(M2 + NP1*NP1 +
                                max(NP1*N,3*M2+NP1,5*M2),
                                NP2 + M1*M1 +
                                max(M1*N,3*NP2+M1,5*NP2),
                                N*M2,NP2*N,NP2*M2,1),
                                N*(14*N+12+M2+NP2)+5),
              where M1 = M - M2 and NP1 = NP - NP2.
              For good performance, LDWORK must generally be larger.
              Denoting Q = max(M1,M2,NP1,NP2), an upper bound is
              2*Q*(3*Q+2*N)+max(1,Q*(Q+max(N,5)+1),N*(14*N+12+2*Q)+5).

      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:  if the matrix D12 had not full column rank in
                    respect to the tolerance TOL;
              = 2:  if the matrix D21 had not full row rank in respect
                    to the tolerance TOL;
              = 3:  if the singular value decomposition (SVD) algorithm
                    did not converge (when computing the SVD of one of
                    the matrices D12 or D21).
              = 4:  if the X-Riccati equation was not solved
                    successfully;
              = 5:  if the Y-Riccati equation was not solved
                    successfully.

Method

      The routine implements the formulas given in [1], [2].

References

      [1] Zhou, K., Doyle, J.C., and Glover, K.
          Robust and Optimal Control.
          Prentice-Hall, Upper Saddle River, NJ, 1996.

      [2] Balas, G.J., Doyle, J.C., Glover, K., Packard, A., and
          Smith, R.
          mu-Analysis and Synthesis Toolbox.
          The MathWorks Inc., Natick, Mass., 1995.

Numerical Aspects

      The accuracy of the result depends on the condition numbers of the
      input and output transformations and on the condition numbers of
      the two Riccati equations, as given by the values of RCOND(1),
      RCOND(2), RCOND(3) and RCOND(4), respectively.

Further Comments

      None

Example

Program Text

    *     SB10HD EXAMPLE PROGRAM TEXT
    *
    *     .. 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
          PARAMETER        ( LDA = NMAX, LDB = NMAX, LDC = PMAX, LDD = PMAX,
         $                   LDAK = NMAX, LDBK = NMAX, LDCK = PMAX,
         $                   LDDK = PMAX )
          INTEGER          LIWORK
          PARAMETER        ( LIWORK = MAX( 2*NMAX, NMAX*NMAX ) )
          INTEGER          MPMX
          PARAMETER        ( MPMX = MAX( MMAX, PMAX ) )
          INTEGER          LDWORK
          PARAMETER        ( LDWORK = 2*MPMX*( 2*NMAX + 3*MPMX ) +
         $                   MAX( MPMX*( MPMX + MAX( NMAX, 5 ) + 1 ),
         $                   NMAX*( 14*NMAX + 12 + 2*MPMX ) + 5 ) )
    *     .. Local Scalars ..
          DOUBLE PRECISION 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,MMAX), C(LDC,NMAX), CK(LDCK,NMAX),
         $                 D(LDD,MMAX), DK(LDDK,MMAX), DWORK(LDWORK),
         $                 RCOND( 4 )
    *     .. External Subroutines ..
          EXTERNAL         SB10HD
    *     .. 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 = * ) TOL
    *        Compute the optimal H2 controller
             CALL SB10HD( N, M, NP, NCON, NMEAS, A, LDA, B, LDB,
         $                C, LDC, D, LDD, 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,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, 4 )
             ELSE
                WRITE( NOUT, FMT = 99998 ) INFO
             END IF
          END IF
          STOP
    *
    99999 FORMAT (' SB10HD EXAMPLE PROGRAM RESULTS',/1X)
    99998 FORMAT (/' INFO on exit from SB10HD =',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 (6(1X,F10.4))
    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 (/' N is out of range.',/' N = ',I5)
    99987 FORMAT (/' NCON is out of range.',/' NCON = ',I5)
    99986 FORMAT (/' NMEAS is out of range.',/' NMEAS = ',I5)
          END

Program Data

     SB10HD EXAMPLE PROGRAM DATA
       6     5     5     2     2
      -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  1.0  0.0
       2.0  0.0  1.0 -5.0  2.0
      -5.0 -7.0  0.0  7.0 -2.0
       4.0 -6.0  1.0  1.0 -2.0
      -3.0  9.0 -8.0  0.0  5.0
       1.0 -2.0  3.0 -6.0 -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
       9.0 -3.0  4.0  0.0  3.0  7.0
       0.0  1.0 -2.0  1.0 -6.0 -2.0
       0.0  0.0  0.0 -4.0 -1.0
       0.0  0.0  0.0  1.0  0.0
       0.0  0.0  0.0  0.0  1.0
       3.0  1.0  0.0  1.0 -3.0
      -2.0  0.0  1.0  7.0  1.0
       0.00000001

Program Results

     SB10HD EXAMPLE PROGRAM RESULTS

     The controller state matrix AK is

        88.0015  -145.7298   -46.2424    82.2168   -45.2996   -31.1407
        25.7489   -31.4642   -12.4198     9.4625    -3.5182     2.7056
        54.3008  -102.4013   -41.4968    50.8412   -20.1286   -26.7191
       108.1006  -198.0785   -45.4333    70.3962   -25.8591   -37.2741
      -115.8900   226.1843    47.2549   -47.8435   -12.5004    34.7474
        59.0362  -101.8471   -20.1052    36.7834   -16.1063   -26.4309

     The controller input matrix BK is

         3.7345     3.4758
        -0.3020     0.6530
         3.4735     4.0499
         4.3198     7.2755
        -3.9424   -10.5942
         2.1784     2.5048

     The controller output matrix CK is

        -2.3346     3.2556     0.7150    -0.9724     0.6962     0.4074
         7.6899    -8.4558    -2.9642     7.0365    -4.2844     0.1390

     The controller matrix DK is

         0.0000     0.0000
         0.0000     0.0000

     The estimated condition numbers are

      0.23570D+00  0.26726D+00  0.22747D-01  0.21130D-02

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