SB10MD - SLICOT Library Routine Documentation

## SB10MD

### D-step in the D-K iteration for continuous-time case

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

Purpose

      To perform the D-step in the D-K iteration. It handles
      continuous-time case.

Specification

          SUBROUTINE SB10MD( NC, MP, LENDAT, F, ORD, MNB, NBLOCK, ITYPE,
         $                   QUTOL, A, LDA, B, LDB, C, LDC, D, LDD, OMEGA,
         $                   TOTORD, AD, LDAD, BD, LDBD, CD, LDCD, DD, LDDD,
         $                   MJU, IWORK, LIWORK, DWORK, LDWORK, ZWORK,
         $                   LZWORK, INFO )
    C     .. Scalar Arguments ..
          INTEGER           F, INFO, LDA, LDAD, LDB, LDBD, LDC, LDCD, LDD,
         $                  LDDD, LDWORK, LENDAT, LIWORK, LZWORK, MNB, MP,
         $                  NC, ORD, TOTORD
          DOUBLE PRECISION  QUTOL
    C     .. Array Arguments ..
          INTEGER           ITYPE(*), IWORK(*), NBLOCK(*)
          DOUBLE PRECISION  A(LDA, *), AD(LDAD, *), B(LDB, *), BD(LDBD, *),
         $                  C(LDC, *), CD(LDCD, *), D(LDD, *), DD(LDDD, *),
         $                  DWORK(*), MJU(*), OMEGA(*)
          COMPLEX*16        ZWORK(*)

Arguments

Input/Output Parameters

      NC      (input) INTEGER
              The order of the matrix A.  NC >= 0.

      MP      (input) INTEGER
              The order of the matrix D.  MP >= 0.

      LENDAT  (input) INTEGER
              The length of the vector OMEGA.  LENDAT >= 2.

      F       (input) INTEGER
              The number of the measurements and controls, i.e.,
              the size of the block I_f in the D-scaling system.
              F >= 0.

      ORD     (input/output) INTEGER
              The MAX order of EACH block in the fitting procedure.
              ORD <= LENDAT-1.
              On exit, if ORD < 1 then ORD = 1.

      MNB     (input) INTEGER
              The number of diagonal blocks in the block structure of
              the uncertainty, and the length of the vectors NBLOCK
              and ITYPE.  1 <= MNB <= MP.

      NBLOCK  (input) INTEGER array, dimension (MNB)
              The vector of length MNB containing the block structure
              of the uncertainty. NBLOCK(I), I = 1:MNB, is the size of
              each block.

      ITYPE   (input) INTEGER array, dimension (MNB)
              The vector of length MNB indicating the type of each
              block.
              For I = 1 : MNB,
              ITYPE(I) = 1 indicates that the corresponding block is a
              real block. IN THIS CASE ONLY MJU(JW) WILL BE ESTIMATED
              CORRECTLY, BUT NOT D(S)!
              ITYPE(I) = 2 indicates that the corresponding block is a
              complex block. THIS IS THE ONLY ALLOWED VALUE NOW!
              NBLOCK(I) must be equal to 1 if ITYPE(I) is equal to 1.

      QUTOL   (input) DOUBLE PRECISION
              The acceptable mean relative error between the D(jw) and
              the frequency responce of the estimated block
              [ADi,BDi;CDi,DDi]. When it is reached, the result is
              taken as good enough.
              A good value is QUTOL = 2.0.
              If QUTOL < 0 then only mju(jw) is being estimated,
              not D(s).

      A       (input/output) DOUBLE PRECISION array, dimension (LDA,NC)
              On entry, the leading NC-by-NC part of this array must
              contain the A matrix of the closed-loop system.
              On exit, if MP > 0, the leading NC-by-NC part of this
              array contains an upper Hessenberg matrix similar to A.

      LDA     INTEGER
              The leading dimension of the array A.  LDA >= MAX(1,NC).

      B       (input/output) DOUBLE PRECISION array, dimension (LDB,MP)
              On entry, the leading NC-by-MP part of this array must
              contain the B matrix of the closed-loop system.
              On exit, the leading NC-by-MP part of this array contains
              the transformed B matrix corresponding to the Hessenberg
              form of A.

      LDB     INTEGER
              The leading dimension of the array B.  LDB >= MAX(1,NC).

      C       (input/output) DOUBLE PRECISION array, dimension (LDC,NC)
              On entry, the leading MP-by-NC part of this array must
              contain the C matrix of the closed-loop system.
              On exit, the leading MP-by-NC part of this array contains
              the transformed C matrix corresponding to the Hessenberg
              form of A.

      LDC     INTEGER
              The leading dimension of the array C.  LDC >= MAX(1,MP).

      D       (input) DOUBLE PRECISION array, dimension (LDD,MP)
              The leading MP-by-MP part of this array must contain the
              D matrix of the closed-loop system.

      LDD     INTEGER
              The leading dimension of the array D.  LDD >= MAX(1,MP).

      OMEGA   (input) DOUBLE PRECISION array, dimension (LENDAT)
              The vector with the frequencies.

      TOTORD  (output) INTEGER
              The TOTAL order of the D-scaling system.
              TOTORD is set to zero, if QUTOL < 0.

      AD      (output) DOUBLE PRECISION array, dimension (LDAD,MP*ORD)
              The leading TOTORD-by-TOTORD part of this array contains
              the A matrix of the D-scaling system.
              Not referenced if QUTOL < 0.

      LDAD    INTEGER
              The leading dimension of the array AD.
              LDAD >= MAX(1,MP*ORD), if QUTOL >= 0;
              LDAD >= 1,             if QUTOL <  0.

      BD      (output) DOUBLE PRECISION array, dimension (LDBD,MP+F)
              The leading TOTORD-by-(MP+F) part of this array contains
              the B matrix of the D-scaling system.
              Not referenced if QUTOL < 0.

      LDBD    INTEGER
              The leading dimension of the array BD.
              LDBD >= MAX(1,MP*ORD), if QUTOL >= 0;
              LDBD >= 1,             if QUTOL <  0.

      CD      (output) DOUBLE PRECISION array, dimension (LDCD,MP*ORD)
              The leading (MP+F)-by-TOTORD part of this array contains
              the C matrix of the D-scaling system.
              Not referenced if QUTOL < 0.

      LDCD    INTEGER
              The leading dimension of the array CD.
              LDCD >= MAX(1,MP+F), if QUTOL >= 0;
              LDCD >= 1,           if QUTOL <  0.

      DD      (output) DOUBLE PRECISION array, dimension (LDDD,MP+F)
              The leading (MP+F)-by-(MP+F) part of this array contains
              the D matrix of the D-scaling system.
              Not referenced if QUTOL < 0.

      LDDD    INTEGER
              The leading dimension of the array DD.
              LDDD >= MAX(1,MP+F), if QUTOL >= 0;
              LDDD >= 1,           if QUTOL <  0.

      MJU     (output) DOUBLE PRECISION array, dimension (LENDAT)
              The vector with the upper bound of the structured
              singular value (mju) for each frequency in OMEGA.

Workspace

      IWORK   INTEGER array, dimension (LIWORK)

      LIWORK  INTEGER
              The length of the array IWORK.
              LIWORK >= MAX( NC, 4*MNB-2, MP, 2*ORD+1 ), if QUTOL >= 0;
              LIWORK >= MAX( NC, 4*MNB-2, MP ),          if QUTOL <  0.

      DWORK   DOUBLE PRECISION array, dimension (LDWORK)
              On exit, if INFO = 0, DWORK(1) returns the optimal value
              of LDWORK, DWORK(2) returns the optimal value of LZWORK,
              and DWORK(3) returns an estimate of the minimum reciprocal
              of the condition numbers (with respect to inversion) of
              the generated Hessenberg matrices.

      LDWORK  INTEGER
              The length of the array DWORK.
              LDWORK >= MAX( 3, LWM, LWD ), where
              LWM = LWA + MAX( NC + MAX( NC, MP-1 ),
                               2*MP*MP*MNB - MP*MP + 9*MNB*MNB +
                               MP*MNB + 11*MP + 33*MNB - 11 );
              LWD = LWB + MAX( 2, LW1, LW2, LW3, LW4, 2*ORD ),
                               if QUTOL >= 0;
              LWD = 0,         if QUTOL <  0;
              LWA = MP*LENDAT + 2*MNB + MP - 1;
              LWB = LENDAT*(MP + 2) + ORD*(ORD + 2) + 1;
              LW1 = 2*LENDAT + 4*HNPTS;  HNPTS = 2048;
              LW2 =   LENDAT + 6*HNPTS;  MN  = MIN( 2*LENDAT, 2*ORD+1 );
              LW3 = 2*LENDAT*(2*ORD + 1) + MAX( 2*LENDAT, 2*ORD + 1 ) +
                    MAX( MN + 6*ORD + 4, 2*MN + 1 );
              LW4 = MAX( ORD*ORD + 5*ORD, 6*ORD + 1 + MIN( 1, ORD ) ).

      ZWORK   COMPLEX*16 array, dimension (LZWORK)

      LZWORK  INTEGER
              The length of the array ZWORK.
              LZWORK >= MAX( LZM, LZD ), where
              LZM = MAX( MP*MP + NC*MP + NC*NC + 2*NC,
                         6*MP*MP*MNB + 13*MP*MP + 6*MNB + 6*MP - 3 );
              LZD = MAX( LENDAT*(2*ORD + 3), ORD*ORD + 3*ORD + 1 ),
                               if QUTOL >= 0;
              LZD = 0,         if QUTOL <  0.

Error Indicator

      INFO    (output) INTEGER
              =  0:  successful exit;
              <  0:  if INFO = -i, the i-th argument had an illegal
                     value;
              =  1:  if one or more values w in OMEGA are (close to
                     some) poles of the closed-loop system, i.e., the
                     matrix jw*I - A is (numerically) singular;
              =  2:  the block sizes must be positive integers;
              =  3:  the sum of block sizes must be equal to MP;
              =  4:  the size of a real block must be equal to 1;
              =  5:  the block type must be either 1 or 2;
              =  6:  errors in solving linear equations or in matrix
                     inversion;
              =  7:  errors in computing eigenvalues or singular values.
              = 1i:  INFO on exit from SB10YD is i. (1i means 10 + i.)

Method

      I.   First, W(jw) for the given closed-loop system is being
           estimated.
      II.  Now, AB13MD SLICOT subroutine can obtain the D(jw) scaling
           system with respect to NBLOCK and ITYPE, and colaterally,
           mju(jw).
           If QUTOL < 0 then the estimations stop and the routine exits.
      III. Now that we have D(jw), SB10YD subroutine can do block-by-
           block fit. For each block it tries with an increasing order
           of the fit, starting with 1 until the
           (mean quadratic error + max quadratic error)/2
           between the Dii(jw) and the estimated frequency responce
           of the block becomes less than or equal to the routine
           argument QUTOL, or the order becomes equal to ORD.
      IV.  Arrange the obtained blocks in the AD, BD, CD and DD
           matrices and estimate the total order of D(s), TOTORD.
      V.   Add the system I_f to the system obtained in IV.

References

      [1] Balas, G., Doyle, J., Glover, K., Packard, A. and Smith, R.
          Mu-analysis and Synthesis toolbox - User's Guide,
          The Mathworks Inc., Natick, MA, USA, 1998.

Further Comments

      None

Example

Program Text

      None

Program Data

      None

Program Results

      None

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