## SB10DD

### H-infinity (sub)optimal controller for a discrete-time system

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

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

      None

Example

Program Text

    *     SB10DD 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, 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   0.00000001

Program 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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