## TB01PD

### Minimal, controllable or observable block Hessenberg realization for a given state-space representation

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

Purpose

      To find a reduced (controllable, observable, or minimal) state-
      space representation (Ar,Br,Cr) for any original state-space
      representation (A,B,C). The matrix Ar is in upper block
      Hessenberg form.

Specification

          SUBROUTINE TB01PD( JOB, EQUIL, N, M, P, A, LDA, B, LDB, C, LDC,
         $                   NR, TOL, IWORK, DWORK, LDWORK, INFO )
    C     .. Scalar Arguments ..
          CHARACTER         EQUIL, JOB
          INTEGER           INFO, LDA, LDB, LDC, LDWORK, M, N, NR, P
          DOUBLE PRECISION  TOL
    C     .. Array Arguments ..
          INTEGER           IWORK(*)
          DOUBLE PRECISION  A(LDA,*), B(LDB,*), C(LDC,*), DWORK(*)

Arguments

Mode Parameters

      JOB     CHARACTER*1
              Indicates whether the user wishes to remove the
              uncontrollable and/or unobservable parts as follows:
              = 'M':  Remove both the uncontrollable and unobservable
                      parts to get a minimal state-space representation;
              = 'C':  Remove the uncontrollable part only to get a
                      controllable state-space representation;
              = 'O':  Remove the unobservable part only to get an
                      observable state-space representation.

      EQUIL   CHARACTER*1
              Specifies whether the user wishes to preliminarily balance
              the triplet (A,B,C) as follows:
              = 'S':  Perform balancing (scaling);
              = 'N':  Do not perform balancing.

Input/Output Parameters

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

      M       (input) INTEGER
              The number of system inputs.  M >= 0.

      P       (input) INTEGER
              The number of system outputs.   P >= 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 NR-by-NR part of this array contains
              the upper block Hessenberg state dynamics matrix Ar of a
              minimal, controllable, or observable realization for the
              original system, depending on the value of JOB, JOB = 'M',
              JOB = 'C', or JOB = 'O', respectively.

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

      B       (input/output) DOUBLE PRECISION array, dimension (LDB,M),
              if JOB = 'C', or (LDB,MAX(M,P)), otherwise.
              On entry, the leading N-by-M part of this array must
              contain the original input/state matrix B; if JOB = 'M',
              or JOB = 'O', the remainder of the leading N-by-MAX(M,P)
              part is used as internal workspace.
              On exit, the leading NR-by-M part of this array contains
              the transformed input/state matrix Br of a minimal,
              controllable, or observable realization for the original
              system, depending on the value of JOB, JOB = 'M',
              JOB = 'C', or JOB = 'O', respectively.
              If JOB = 'C', only the first IWORK(1) rows of B are
              nonzero.

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

      C       (input/output) DOUBLE PRECISION array, dimension (LDC,N)
              On entry, the leading P-by-N part of this array must
              contain the original state/output matrix C; if JOB = 'M',
              or JOB = 'O', the remainder of the leading MAX(M,P)-by-N
              part is used as internal workspace.
              On exit, the leading P-by-NR part of this array contains
              the transformed state/output matrix Cr of a minimal,
              controllable, or observable realization for the original
              system, depending on the value of JOB, JOB = 'M',
              JOB = 'C', or JOB = 'O', respectively.
              If JOB = 'M', or JOB = 'O', only the last IWORK(1) columns
              (in the first NR columns) of C are nonzero.

      LDC     INTEGER
              The leading dimension of array C.
              LDC >= MAX(1,M,P) if N > 0.
              LDC >= 1          if N = 0.

      NR      (output) INTEGER
              The order of the reduced state-space representation
              (Ar,Br,Cr) of a minimal, controllable, or observable
              realization for the original system, depending on
              JOB = 'M', JOB = 'C', or JOB = 'O'.

Tolerances

      TOL     DOUBLE PRECISION
              The tolerance to be used in rank determination when
              transforming (A, B, C). 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
              (determined by the SLICOT routine TB01UD) is used instead.

Workspace

      IWORK   INTEGER array, dimension (N+MAX(M,P))
              On exit, if INFO = 0, the first nonzero elements of
              IWORK(1:N) return the orders of the diagonal blocks of A.

      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 + MAX(N, 3*M, 3*P)).
              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

      If JOB = 'M', the matrices A and B are operated on by orthogonal
      similarity transformations (made up of products of Householder
      transformations) so as to produce an upper block Hessenberg matrix
      A1 and a matrix B1 with all but its first rank(B) rows zero; this
      separates out the controllable part of the original system.
      Applying the same algorithm to the dual of this subsystem,
      therefore separates out the controllable and observable (i.e.
      minimal) part of the original system representation, with the
      final Ar upper block Hessenberg (after using pertransposition).
      If JOB = 'C', or JOB = 'O', only the corresponding part of the
      above procedure is applied.

References

      [1] Van Dooren, P.
          The Generalized Eigenstructure Problem in Linear System
          Theory. (Algorithm 1)
          IEEE Trans. Auto. Contr., AC-26, pp. 111-129, 1981.

Numerical Aspects

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

Further Comments

      None

Example

Program Text

    *     TB01PD EXAMPLE PROGRAM TEXT
    *
    *     .. Parameters ..
          INTEGER          NIN, NOUT
          PARAMETER        ( NIN = 5, NOUT = 6 )
          INTEGER          NMAX, MMAX, PMAX
          PARAMETER        ( NMAX = 20, MMAX = 20, PMAX = 20 )
          INTEGER          MAXMP
          PARAMETER        ( MAXMP = MAX( MMAX, PMAX ) )
          INTEGER          LDA, LDB, LDC
          PARAMETER        ( LDA = NMAX, LDB = NMAX, LDC = MAXMP )
          INTEGER          LIWORK
          PARAMETER        ( LIWORK = NMAX+MAXMP )
          INTEGER          LDWORK
          PARAMETER        ( LDWORK = NMAX+MAX( NMAX, 3*MAXMP ) )
    *     .. Local Scalars ..
          DOUBLE PRECISION TOL
          INTEGER          I, INFO, J, M, N, NR, P
          CHARACTER        JOB, EQUIL
    *     .. Local Arrays ..
          DOUBLE PRECISION A(LDA,NMAX), B(LDB,MAXMP), C(LDC,NMAX),
         $                 DWORK(LDWORK)
          INTEGER          IWORK(LIWORK)
    *     .. External Subroutines ..
          EXTERNAL         TB01PD
    *     .. 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, P, TOL, JOB, EQUIL
          IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
             WRITE ( NOUT, FMT = 99990 ) N
          ELSE
             READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
             IF ( M.LT.0 .OR. M.GT.MMAX ) THEN
                WRITE ( NOUT, FMT = 99989 ) M
             ELSE
                READ ( NIN, FMT = * ) ( ( B(I,J), I = 1,N ), J = 1,M )
                IF ( P.LT.0 .OR. P.GT.PMAX ) THEN
                   WRITE ( NOUT, FMT = 99988 ) P
                ELSE
                   READ ( NIN, FMT = * ) ( ( C(I,J), J = 1,N ), I = 1,P )
    *              Find a minimal ssr for (A,B,C).
                   CALL TB01PD( JOB, EQUIL, N, M, P, A, LDA, B, LDB, C, LDC,
         $                      NR, TOL, IWORK, DWORK, LDWORK, INFO )
    *
                   IF ( INFO.NE.0 ) THEN
                      WRITE ( NOUT, FMT = 99998 ) INFO
                   ELSE
                      WRITE ( NOUT, FMT = 99997 ) NR
                      WRITE ( NOUT, FMT = 99996 )
                      DO 20 I = 1, NR
                         WRITE ( NOUT, FMT = 99995 ) ( A(I,J), J = 1,NR )
       20             CONTINUE
                      WRITE ( NOUT, FMT = 99993 )
                      DO 40 I = 1, NR
                         WRITE ( NOUT, FMT = 99995 ) ( B(I,J), J = 1,M )
       40             CONTINUE
                      WRITE ( NOUT, FMT = 99992 )
                      DO 60 I = 1, P
                         WRITE ( NOUT, FMT = 99995 ) ( C(I,J), J = 1,NR )
       60             CONTINUE
                   END IF
                END IF
             END IF
          END IF
          STOP
    *
    99999 FORMAT (' TB01PD EXAMPLE PROGRAM RESULTS',/1X)
    99998 FORMAT (' INFO on exit from TB01PD = ',I2)
    99997 FORMAT (' The order of the minimal realization = ',I2)
    99996 FORMAT (/' The transformed state dynamics matrix of a minimal re',
         $       'alization is ')
    99995 FORMAT (20(1X,F8.4))
    99993 FORMAT (/' The transformed input/state matrix of a minimal reali',
         $       'zation is ')
    99992 FORMAT (/' The transformed state/output matrix of a minimal real',
         $       'ization is ')
    99990 FORMAT (/' N is out of range.',/' N = ',I5)
    99989 FORMAT (/' M is out of range.',/' M = ',I5)
    99988 FORMAT (/' P is out of range.',/' P = ',I5)
          END

Program Data

     TB01PD EXAMPLE PROGRAM DATA
       3     1     2     0.0     M     N
       1.0   2.0   0.0
       4.0  -1.0   0.0
       0.0   0.0   1.0
       1.0   0.0   1.0
       0.0   1.0  -1.0
       0.0   0.0   1.0

Program Results

     TB01PD EXAMPLE PROGRAM RESULTS

     The order of the minimal realization =  3

     The transformed state dynamics matrix of a minimal realization is
       1.0000  -1.4142   1.4142
      -2.8284   0.0000   1.0000
       2.8284   1.0000   0.0000

     The transformed input/state matrix of a minimal realization is
      -1.0000
       0.7071
       0.7071

     The transformed state/output matrix of a minimal realization is
       0.0000   0.0000  -1.4142
       0.0000   0.7071   0.7071

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