## AB01OD

### Staircase form for multi-input systems using orthogonal state and input transformations

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

Purpose

      To reduce the matrices A and B using (and optionally accumulating)
      state-space and input-space transformations U and V respectively,
      such that the pair of matrices

         Ac = U' * A * U,    Bc = U' * B * V

      are in upper "staircase" form. Specifically,

              [ Acont     *    ]         [ Bcont ]
         Ac = [                ],   Bc = [       ],
              [   0    Auncont ]         [   0   ]

         and

                 [ A11 A12  . . .  A1,p-1 A1p ]         [ B1 ]
                 [ A21 A22  . . .  A2,p-1 A2p ]         [ 0  ]
                 [  0  A32  . . .  A3,p-1 A3p ]         [ 0  ]
         Acont = [  .   .   . . .    .     .  ],   Bc = [ .  ],
                 [  .   .     . .    .     .  ]         [ .  ]
                 [  .   .       .    .     .  ]         [ .  ]
                 [  0   0   . . .  Ap,p-1 App ]         [ 0  ]

      where the blocks  B1, A21, ..., Ap,p-1  have full row ranks and
      p is the controllability index of the pair.  The size of the
      block Auncont is equal to the dimension of the uncontrollable
      subspace of the pair (A, B).  The first stage of the reduction,
      the "forward" stage, accomplishes the reduction to the orthogonal
      canonical form (see SLICOT library routine AB01ND). The blocks
      B1, A21, ..., Ap,p-1 are further reduced in a second, "backward"
      stage to upper triangular form using RQ factorization. Each of
      these stages is optional.

Specification

          SUBROUTINE AB01OD( STAGES, JOBU, JOBV, N, M, A, LDA, B, LDB, U,
         $                   LDU, V, LDV, NCONT, INDCON, KSTAIR, TOL, IWORK,
         $                   DWORK, LDWORK, INFO )
    C     .. Scalar Arguments ..
          CHARACTER         JOBU, JOBV, STAGES
          INTEGER           INDCON, INFO, LDA, LDB, LDU, LDV, LDWORK, M, N,
         $                  NCONT
          DOUBLE PRECISION  TOL
    C     .. Array Arguments ..
          INTEGER           IWORK(*), KSTAIR(*)
          DOUBLE PRECISION  A(LDA,*), B(LDB,*), DWORK(*), U(LDU,*), V(LDV,*)

Arguments

Mode Parameters

      STAGES  CHARACTER*1
              Specifies the reduction stages to be performed as follows:
              = 'F':  Perform the forward stage only;
              = 'B':  Perform the backward stage only;
              = 'A':  Perform both (all) stages.

      JOBU    CHARACTER*1
              Indicates whether the user wishes to accumulate in a
              matrix U the state-space transformations as follows:
              = 'N':  Do not form U;
              = 'I':  U is internally initialized to the unit matrix (if
                      STAGES <> 'B'), or updated (if STAGES = 'B'), and
                      the orthogonal transformation matrix U is
                      returned.

      JOBV    CHARACTER*1
              Indicates whether the user wishes to accumulate in a
              matrix V the input-space transformations as follows:
              = 'N':  Do not form V;
              = 'I':  V is initialized to the unit matrix and the
                      orthogonal transformation matrix V is returned.
              JOBV is not referenced if STAGES = 'F'.

Input/Output Parameters

      N       (input) INTEGER
              The actual state dimension, i.e. the order of the
              matrix A.  N >= 0.

      M       (input) INTEGER
              The actual input dimension.  M >= 0.

      A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
              On entry, the leading N-by-N part of this array must
              contain the state transition matrix A to be transformed.
              If STAGES = 'B', A should be in the orthogonal canonical
              form, as returned by SLICOT library routine AB01ND.
              On exit, the leading N-by-N part of this array contains
              the transformed state transition matrix U' * A * U.
              The leading NCONT-by-NCONT part contains the upper block
              Hessenberg state matrix Acont in Ac, given by U' * A * U,
              of a controllable realization for the original system.
              The elements below the first block-subdiagonal are set to
              zero.  If STAGES <> 'F', the subdiagonal blocks of A are
              triangularized by RQ factorization, and the annihilated
              elements are explicitly zeroed.

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

      B       (input/output) DOUBLE PRECISION array, dimension (LDB,M)
              On entry, the leading N-by-M part of this array must
              contain the input matrix B to be transformed.
              If STAGES = 'B', B should be in the orthogonal canonical
              form, as returned by SLICOT library routine AB01ND.
              On exit with STAGES = 'F', the leading N-by-M part of
              this array contains the transformed input matrix U' * B,
              with all elements but the first block set to zero.
              On exit with STAGES <> 'F', the leading N-by-M part of
              this array contains the transformed input matrix
              U' * B * V, with all elements but the first block set to
              zero and the first block in upper triangular form.

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

      U       (input/output) DOUBLE PRECISION array, dimension (LDU,N)
              If STAGES <> 'B' or JOBU = 'N', then U need not be set
              on entry.
              If STAGES = 'B' and JOBU = 'I', then, on entry, the
              leading N-by-N part of this array must contain the
              transformation matrix U that reduced the pair to the
              orthogonal canonical form.
              On exit, if JOBU = 'I', the leading N-by-N part of this
              array contains the transformation matrix U that performed
              the specified reduction.
              If JOBU = 'N', the array U is not referenced and can be
              supplied as a dummy array (i.e. set parameter LDU = 1 and
              declare this array to be U(1,1) in the calling program).

      LDU     INTEGER
              The leading dimension of array U.
              If JOBU = 'I', LDU >= MAX(1,N);  if JOBU = 'N', LDU >= 1.

      V       (output) DOUBLE PRECISION array, dimension (LDV,M)
              If JOBV = 'I', then the leading M-by-M part of this array
              contains the transformation matrix V.
              If STAGES = 'F', or JOBV = 'N', the array V is not
              referenced and can be supplied as a dummy array (i.e. set
              parameter  LDV = 1 and declare this array to be V(1,1) in
              the calling program).

      LDV     INTEGER
              The leading dimension of array V.
              If STAGES <> 'F' and JOBV = 'I', LDV >= MAX(1,M);
              if STAGES = 'F' or JOBV = 'N', LDV >= 1.

      NCONT   (input/output) INTEGER
              The order of the controllable state-space representation.
              NCONT is input only if STAGES = 'B'.

      INDCON  (input/output) INTEGER
              The number of stairs in the staircase form (also, the
              controllability index of the controllable part of the
              system representation).
              INDCON is input only if STAGES = 'B'.

      KSTAIR  (input/output) INTEGER array, dimension (N)
              The leading INDCON elements of this array contain the
              dimensions of the stairs, or, also, the orders of the
              diagonal blocks of Acont.
              KSTAIR is input if STAGES = 'B', and output otherwise.

Tolerances

      TOL     DOUBLE PRECISION
              The tolerance to be used in rank determination when
              transforming (A, B). If the user sets TOL > 0, then
              the given value of TOL is used as a lower bound for the
              reciprocal condition number (see the description of the
              argument RCOND in the SLICOT routine MB03OD);  a
              (sub)matrix whose estimated condition number is less than
              1/TOL is considered to be of full rank.  If the user sets
              TOL <= 0, then an implicitly computed, default tolerance,
              defined by  TOLDEF = N*N*EPS,  is used instead, where EPS
              is the machine precision (see LAPACK Library routine
              DLAMCH).
              TOL is not referenced if STAGES = 'B'.

Workspace

      IWORK   INTEGER array, dimension (M)
              IWORK is not referenced if STAGES = 'B'.

      DWORK   DOUBLE PRECISION array, dimension (LDWORK)
              On exit, if INFO = 0, DWORK(1) returns the optimal value
              of LDWORK.

      LDWORK  INTEGER
              The length of the array DWORK.
              If STAGES <> 'B', LDWORK >= MAX(1, N + MAX(N,3*M));
              If STAGES =  'B', LDWORK >= MAX(1, M + MAX(N,M)).
              For optimum performance LDWORK should be larger.

Error Indicator

      INFO    INTEGER
              = 0:  successful exit;
              < 0:  if INFO = -i, the i-th argument had an illegal
                    value.

Method

      Staircase reduction of the pencil [B|sI - A] is used. Orthogonal
      transformations U and V are constructed such that

                         |B |sI-A      *  . . .  *      *       |
                         | 1|    11       .      .      .       |
                         |  |  A    sI-A    .    .      .       |
                         |  |   21      22    .  .      .       |
                         |  |        .     .     *      *       |
      [U'BV|sI - U'AU] = |0 |     0    .     .                  |
                         |  |            A     sI-A     *       |
                         |  |             p,p-1    pp           |
                         |  |                                   |
                         |0 |         0          0   sI-A       |
                         |  |                            p+1,p+1|

      where the i-th diagonal block of U'AU has dimension KSTAIR(i),
      for i = 1,...,p. The value of p is returned in INDCON. The last
      block contains the uncontrollable modes of the (A,B)-pair which
      are also the generalized eigenvalues of the above pencil.

      The complete reduction is performed in two stages. The first,
      forward stage accomplishes the reduction to the orthogonal
      canonical form. The second, backward stage consists in further
      reduction to triangular form by applying left and right orthogonal
      transformations.

References

      [1] Van Dooren, P.
          The generalized eigenvalue problem in linear system theory.
          IEEE Trans. Auto. Contr., AC-26, pp. 111-129, 1981.

      [2] Miminis, G. and Paige, C.
          An algorithm for pole assignment of time-invariant multi-input
          linear systems.
          Proc. 21st IEEE CDC, Orlando, Florida, 1, pp. 62-67, 1982.

Numerical Aspects

      The algorithm requires O((N + M) x N**2) operations and is
      backward stable (see [1]).

Further Comments

      If the system matrices A and B are badly scaled, it would be
      useful to scale them with SLICOT routine TB01ID, before calling
      the routine.

Example

Program Text

    *     AB01OD EXAMPLE PROGRAM TEXT
    *
    *     .. Parameters ..
          INTEGER          NIN, NOUT
          PARAMETER        ( NIN = 5, NOUT = 6 )
          INTEGER          NMAX, MMAX
          PARAMETER        ( NMAX = 20, MMAX = 20 )
          INTEGER          LDA, LDB, LDU, LDV
          PARAMETER        ( LDA = NMAX, LDB = NMAX, LDU = NMAX,
         $                   LDV = MMAX )
          INTEGER          LIWORK
          PARAMETER        ( LIWORK = MMAX )
          INTEGER          LDWORK
          PARAMETER        ( LDWORK = NMAX + MAX( NMAX, 3*MMAX ) )
    *     .. Local Scalars ..
          DOUBLE PRECISION TOL
          INTEGER          I, INDCON, INFO, J, M, N, NCONT
          CHARACTER*1      JOBU, JOBV, STAGES
    *     .. Local Arrays ..
          DOUBLE PRECISION A(LDA,NMAX), B(LDB,MMAX), DWORK(LDWORK),
         $                 U(LDU,NMAX), V(LDV,MMAX)
          INTEGER          IWORK(LIWORK), KSTAIR(NMAX)
    *     .. External Subroutines ..
          EXTERNAL         AB01OD
    *     .. 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, TOL, STAGES, JOBU, JOBV
          IF ( N.LE.0 .OR. N.GT.NMAX ) THEN
             WRITE ( NOUT, FMT = 99992 ) N
          ELSE
             READ ( NIN, FMT = * ) ( ( A(I,J), I = 1,N ), J = 1,N )
             IF ( M.LE.0 .OR. M.GT.MMAX ) THEN
                WRITE ( NOUT, FMT = 99991 ) M
             ELSE
                READ ( NIN, FMT = * ) ( ( B(I,J), J = 1,M ), I = 1,N )
    *           Reduce the matrices A and B to upper "staircase" form.
                CALL AB01OD( STAGES, JOBU, JOBV, N, M, A, LDA, B, LDB, U,
         $                   LDU, V, LDV, NCONT, INDCON, KSTAIR, TOL, IWORK,
         $                   DWORK, LDWORK, INFO )
    *
                IF ( INFO.NE.0 ) THEN
                   WRITE ( NOUT, FMT = 99998 ) INFO
                ELSE
                   WRITE ( NOUT, FMT = 99997 )
                   DO 20 I = 1, N
                      WRITE ( NOUT, FMT = 99995 ) ( A(I,J), J = 1,N )
       20          CONTINUE
                   WRITE ( NOUT, FMT = 99996 )
                   DO 40 I = 1, N
                      WRITE ( NOUT, FMT = 99995 ) ( B(I,J), J = 1,M )
       40          CONTINUE
                   WRITE ( NOUT, FMT = 99994 ) INDCON
                   WRITE ( NOUT, FMT = 99993 ) ( KSTAIR(I), I = 1,INDCON )
                END IF
             END IF
          END IF
          STOP
    *
    99999 FORMAT (' AB01OD EXAMPLE PROGRAM RESULTS',/1X)
    99998 FORMAT (' INFO on exit from AB01OD = ',I2)
    99997 FORMAT (' The transformed state transition matrix is ')
    99996 FORMAT (/' The transformed input matrix is ')
    99995 FORMAT (20(1X,F8.4))
    99994 FORMAT (/' The number of stairs in the staircase form = ',I3,/)
    99993 FORMAT (' The dimensions of the stairs are ',/(20(I3,2X)))
    99992 FORMAT (/' N is out of range.',/' N = ',I5)
    99991 FORMAT (/' M is out of range.',/' M = ',I5)
          END

Program Data

     AB01OD EXAMPLE PROGRAM DATA
       5     2     0.0     F     N     N
       17.0   24.0    1.0    8.0   15.0
       23.0    5.0    7.0   14.0   16.0
        4.0    6.0   13.0   20.0   22.0
       10.0   12.0   19.0   21.0    3.0
       11.0   18.0   25.0    2.0    9.0
       -1.0   -4.0
        4.0    9.0
       -9.0  -16.0
       16.0   25.0
      -25.0  -36.0

Program Results

     AB01OD EXAMPLE PROGRAM RESULTS

     The transformed state transition matrix is
      12.8848   3.2345  11.8211   3.3758  -0.8982
       4.4741 -12.5544   5.3509   5.9403   1.4360
      14.4576   7.6855  23.1452  26.3872 -29.9557
       0.0000   1.4805  27.4668  22.6564  -0.0072
       0.0000   0.0000 -30.4822   0.6745  18.8680

     The transformed input matrix is
      31.1199  47.6865
       3.2480   0.0000
       0.0000   0.0000
       0.0000   0.0000
       0.0000   0.0000

     The number of stairs in the staircase form =   3

     The dimensions of the stairs are
      2    2    1

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