## AB01MD

### Controllable realization for single-input systems using orthogonal state and input transformations

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

Purpose

      To find a controllable realization for the linear time-invariant
      single-input system

              dX/dt = A * X + B * U,

      where A is an N-by-N matrix and B is an N element vector which
      are reduced by this routine to orthogonal canonical form using
      (and optionally accumulating) orthogonal similarity
      transformations.

Specification

          SUBROUTINE AB01MD( JOBZ, N, A, LDA, B, NCONT, Z, LDZ, TAU, TOL,
         $                   DWORK, LDWORK, INFO )
    C     .. Scalar Arguments ..
          CHARACTER         JOBZ
          INTEGER           INFO, LDA, LDZ, LDWORK, N, NCONT
          DOUBLE PRECISION  TOL
    C     .. Array Arguments ..
          DOUBLE PRECISION  A(LDA,*), B(*), DWORK(*), TAU(*), Z(LDZ,*)

Arguments

Mode Parameters

      JOBZ    CHARACTER*1
              Indicates whether the user wishes to accumulate in a
              matrix Z the orthogonal similarity transformations for
              reducing the system, as follows:
              = 'N':  Do not form Z and do not store the orthogonal
                      transformations;
              = 'F':  Do not form Z, but store the orthogonal
                      transformations in the factored form;
              = 'I':  Z is initialized to the unit matrix and the
                      orthogonal transformation matrix Z is returned.

Input/Output Parameters

      N       (input) INTEGER
              The order of the original state-space representation,
              i.e. the order of the matrix A.  N >= 0.

      A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
              On entry, the leading N-by-N part of this array must
              contain the original state dynamics matrix A.
              On exit, the leading NCONT-by-NCONT upper Hessenberg
              part of this array contains the canonical form of the
              state dynamics matrix, given by Z' * A * Z, of a
              controllable realization for the original system. The
              elements below the first subdiagonal are set to zero.

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

      B       (input/output) DOUBLE PRECISION array, dimension (N)
              On entry, the original input/state vector B.
              On exit, the leading NCONT elements of this array contain
              canonical form of the input/state vector, given by Z' * B,
              with all elements but B(1) set to zero.

      NCONT   (output) INTEGER
              The order of the controllable state-space representation.

      Z       (output) DOUBLE PRECISION array, dimension (LDZ,N)
              If JOBZ = 'I', then the leading N-by-N part of this array
              contains the matrix of accumulated orthogonal similarity
              transformations which reduces the given system to
              orthogonal canonical form.
              If JOBZ = 'F', the elements below the diagonal, with the
              array TAU, represent the orthogonal transformation matrix
              as a product of elementary reflectors. The transformation
              matrix can then be obtained by calling the LAPACK Library
              routine DORGQR.
              If JOBZ = 'N', the array Z is not referenced and can be
              supplied as a dummy array (i.e. set parameter LDZ = 1 and
              declare this array to be Z(1,1) in the calling program).

      LDZ     INTEGER
              The leading dimension of array Z. If JOBZ = 'I' or
              JOBZ = 'F', LDZ >= MAX(1,N); if JOBZ = 'N', LDZ >= 1.

      TAU     (output) DOUBLE PRECISION array, dimension (N)
              The elements of TAU contain the scalar factors of the
              elementary reflectors used in the reduction of B and A.

Tolerances

      TOL     DOUBLE PRECISION
              The tolerance to be used in determining the
              controllability of (A,B). If the user sets TOL > 0, then
              the given value of TOL is used as an absolute tolerance;
              elements with absolute value less than TOL are considered
              neglijible. If the user sets TOL <= 0, then an implicitly
              computed, default tolerance, defined by
              TOLDEF = N*EPS*MAX( NORM(A), NORM(B) ) is used instead,
              where EPS is the machine precision (see LAPACK Library
              routine DLAMCH).

Workspace

      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. LDWORK >= MAX(1,N).
              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

      The Householder matrix which reduces all but the first element
      of vector B to zero is found and this orthogonal similarity
      transformation is applied to the matrix A. The resulting A is then
      reduced to upper Hessenberg form by a sequence of Householder
      transformations. Finally, the order of the controllable state-
      space representation (NCONT) is determined by finding the position
      of the first sub-diagonal element of A which is below an
      appropriate zero threshold, either TOL or TOLDEF (see parameter
      TOL); if NORM(B) is smaller than this threshold, NCONT is set to
      zero, and no computations for reducing the system to orthogonal
      canonical form are performed.

References

      [1] Konstantinov, M.M., Petkov, P.Hr. and Christov, N.D.
          Orthogonal Invariants and Canonical Forms for Linear
          Controllable Systems.
          Proc. 8th IFAC World Congress, Kyoto, 1, pp. 49-54, 1981.

      [2] Hammarling, S.J.
          Notes on the use of orthogonal similarity transformations in
          control.
          NPL Report DITC 8/82, August 1982.

      [3] Paige, C.C
          Properties of numerical algorithms related to computing
          controllability.
          IEEE Trans. Auto. Contr., AC-26, pp. 130-138, 1981.

Numerical Aspects

                                3
      The algorithm requires 0(N ) operations and is backward stable.

Further Comments

      None

Example

Program Text

    *     AB01MD EXAMPLE PROGRAM TEXT
    *
    *     .. Parameters ..
          INTEGER          NIN, NOUT
          PARAMETER        ( NIN = 5, NOUT = 6 )
          INTEGER          NMAX
          PARAMETER        ( NMAX = 20 )
          INTEGER          LDA, LDZ
          PARAMETER        ( LDA = NMAX, LDZ = NMAX )
          INTEGER          LDWORK
          PARAMETER        ( LDWORK = NMAX )
    *     .. Local Scalars ..
          DOUBLE PRECISION TOL
          INTEGER          I, INFO, J, N, NCONT
          CHARACTER*1      JOBZ
    *     .. Local Arrays ..
          DOUBLE PRECISION A(LDA,NMAX), B(NMAX), DWORK(LDWORK), TAU(NMAX),
         $                 Z(LDZ,NMAX)
    *     .. External Functions ..
          LOGICAL          LSAME
          EXTERNAL         LSAME
    *     .. External Subroutines ..
          EXTERNAL         AB01MD, DORGQR
    *     .. Executable Statements ..
    *
          WRITE ( NOUT, FMT = 99999 )
    *     Skip the heading in the data file and read in the data.
          READ ( NIN, FMT = '()' )
          READ ( NIN, FMT = * ) N, TOL, JOBZ
          IF ( N.LE.0 .OR. N.GT.NMAX ) THEN
             WRITE ( NOUT, FMT = 99993 ) N
          ELSE
             READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
             READ ( NIN, FMT = * ) ( B(I), I = 1,N )
    *        Find a controllable realization for the given system.
             CALL AB01MD( JOBZ, N, A, LDA, B, NCONT, Z, LDZ, TAU, TOL,
         $                DWORK, LDWORK, INFO )
    *
             IF ( INFO.NE.0 ) THEN
                WRITE ( NOUT, FMT = 99998 ) INFO
             ELSE
                WRITE ( NOUT, FMT = 99997 ) NCONT
                DO 20 I = 1, NCONT
                   WRITE ( NOUT, FMT = 99994 ) ( A(I,J), J = 1,NCONT )
       20       CONTINUE
                WRITE ( NOUT, FMT = 99996 ) ( B(I), I = 1,NCONT )
                IF ( LSAME( JOBZ, 'F' ) )
         $         CALL DORGQR( N, N, N, Z, LDZ, TAU, DWORK, LDWORK, INFO )
                IF ( LSAME( JOBZ, 'F' ).OR.LSAME( JOBZ, 'I' ) ) THEN
                   WRITE ( NOUT, FMT = 99995 )
                   DO 40 I = 1, N
                      WRITE ( NOUT, FMT = 99994 ) ( Z(I,J), J = 1,N )
       40          CONTINUE
                END IF
             END IF
          END IF
          STOP
    *
    99999 FORMAT (' AB01MD EXAMPLE PROGRAM RESULTS',/1X)
    99998 FORMAT (' INFO on exit from AB01MD = ',I2)
    99997 FORMAT (' The order of the controllable state-space representati',
         $       'on = ',I2,//' The state dynamics matrix A of a controlla',
         $       'ble realization is ')
    99996 FORMAT (/' The input/state vector B of a controllable realizatio',
         $       'n is ',/(1X,F8.4))
    99995 FORMAT (/' The similarity transformation matrix Z is ')
    99994 FORMAT (20(1X,F8.4))
    99993 FORMAT (/' N is out of range.',/' N = ',I5)
          END

Program Data

     AB01MD EXAMPLE PROGRAM DATA
       3     0.0     I
       1.0   2.0   0.0
       4.0  -1.0   0.0
       0.0   0.0   1.0
       1.0   0.0   1.0

Program Results

     AB01MD EXAMPLE PROGRAM RESULTS

     The order of the controllable state-space representation =  3

     The state dynamics matrix A of a controllable realization is
       1.0000   1.4142   0.0000
       2.8284  -1.0000   2.8284
       0.0000   1.4142   1.0000

     The input/state vector B of a controllable realization is
      -1.4142
       0.0000
       0.0000

     The similarity transformation matrix Z is
      -0.7071   0.0000  -0.7071
       0.0000  -1.0000   0.0000
      -0.7071   0.0000   0.7071

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