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
*     Copyright (c) 2002-2017 NICONET e.V.
*
*     .. 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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