## TF01QD

### Markov parameters of a multivariable system from its transfer function matrix

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

Purpose

      To compute N Markov parameters M(1), M(2),..., M(N) from a
      multivariable system whose transfer function matrix G(z) is given.

Specification

          SUBROUTINE TF01QD( NC, NB, N, IORD, AR, MA, H, LDH, INFO )
    C     .. Scalar Arguments ..
          INTEGER           INFO, LDH, N, NB, NC
    C     .. Array Arguments ..
          INTEGER           IORD(*)
          DOUBLE PRECISION  AR(*), H(LDH,*), MA(*)

Arguments

Input/Output Parameters

      NC      (input) INTEGER
              The number of system outputs, i.e. the number of rows in
              the transfer function matrix G(z).  NC >= 0.

      NB      (input) INTEGER
              The number of system inputs, i.e. the number of columns in
              the transfer function matrix G(z).  NB >= 0.

      N       (input) INTEGER
              The number of Markov parameters M(k) to be computed.
              N >= 0.

      IORD    (input) INTEGER array, dimension (NC*NB)
              This array must contain the order r of the elements of the
              transfer function matrix G(z), stored row by row.
              For example, the order of the (i,j)-th element of G(z) is
              given by IORD((i-1)xNB+j).

      AR      (input) DOUBLE PRECISION array, dimension (NA), where
              NA = IORD(1) + IORD(2) + ... + IORD(NC*NB).
              The leading NA elements of this array must contain the
              denominator coefficients AR(1),...,AR(r) in equation (1)
              of the (i,j)-th element of the transfer function matrix
              G(z), stored row by row, i.e. in the order
              (1,1),(1,2),...,(1,NB), (2,1),(2,2),...,(2,NB), ...,
              (NC,1),(NC,2),...,(NC,NB). The coefficients must be given
              in decreasing order of powers of z; the coefficient of the
              highest order term is assumed to be equal to 1.

      MA      (input) DOUBLE PRECISION array, dimension (NA)
              The leading NA elements of this array must contain the
              numerator coefficients MA(1),...,MA(r) in equation (1)
              of the (i,j)-th element of the transfer function matrix
              G(z), stored row by row, i.e. in the order
              (1,1),(1,2),...,(1,NB), (2,1),(2,2),...,(2,NB), ...,
              (NC,1),(NC,2),...,(NC,NB). The coefficients must be given
              in decreasing order of powers of z.

      H       (output) DOUBLE PRECISION array, dimension (LDH,N*NB)
              The leading NC-by-N*NB part of this array contains the
              multivariable Markov parameter sequence M(k), where each
              parameter M(k) is an NC-by-NB matrix and k = 1,2,...,N.
              The Markov parameters are stored such that H(i,(k-1)xNB+j)
              contains the (i,j)-th element of M(k) for i = 1,2,...,NC
              and j = 1,2,...,NB.

      LDH     INTEGER
              The leading dimension of array H.  LDH >= MAX(1,NC).

Error Indicator

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

Method

      The (i,j)-th element of G(z), defining the particular I/O transfer
      between output i and input j, has the following form:

                           -1         -2               -r
                     MA(1)z   + MA(2)z   + ... + MA(r)z
          G  (z) = ----------------------------------------.         (1)
           ij                -1         -2               -r
                   1 + AR(1)z   + AR(2)z   + ... + AR(r)z

      The (i,j)-th element of G(z) is defined by its order r, its r
      moving average coefficients (= numerator) MA(1),...,MA(r) and its
      r autoregressive coefficients (= denominator) AR(1),...,AR(r). The
      coefficient of the constant term in the denominator is assumed to
      be equal to 1.

      The relationship between the (i,j)-th element of the Markov
      parameters M(1),M(2),...,M(N) and the corresponding element of the
      transfer function matrix G(z) is given by:

                                -1          -2                -k
       G  (z) = M  (0) + M  (1)z   + M  (2)z   + ... + M  (k)z  + ...(2)
        ij       ij       ij          ij                ij

      Equating (1) and (2), we find that the relationship between the
      (i,j)-th element of the Markov parameters M(k) and the ARMA
      parameters AR(1),...,AR(r) and MA(1),...,MA(r) of the (i,j)-th
      element of the transfer function matrix G(z) is as follows:

         M  (1)   = MA(1),
          ij
                            k-1
         M  (k)   = MA(k) - SUM AR(p) x M  (k-p) for 1 < k <= r and
          ij                p=1          ij
                       r
         M  (k+r) = - SUM AR(p) x M  (k+r-p) for k > 0.
          ij          p=1          ij

      From these expressions the Markov parameters M(k) are computed
      element by element.

References

      [1] Luenberger, D.G.
          Introduction to Dynamic Systems: Theory, Models and
          Applications.
          John Wiley & Sons, New York, 1979.

Numerical Aspects

      The computation of the (i,j)-th element of M(k) requires:
         (k-1) multiplications and k additions if k <= r;
           r   multiplications and r additions if k > r.

Further Comments

      None

Example

Program Text

    *     TF01QD EXAMPLE PROGRAM TEXT
    *
    *     .. Parameters ..
          INTEGER          NIN, NOUT
          PARAMETER        ( NIN = 5, NOUT = 6 )
          INTEGER          NMAX, NAMAX, NBMAX, NCMAX
          PARAMETER        ( NMAX = 20, NAMAX = 20, NBMAX = 20, NCMAX = 20 )
          INTEGER          LDH
          PARAMETER        ( LDH = NCMAX )
    *     .. Local Scalars ..
          INTEGER          I, INFO, J, K, L, N, NA, NASUM, NB, NC, NL, NORD
          LOGICAL          ERROR
    *     .. Local Arrays ..
          DOUBLE PRECISION AR(NAMAX), H(LDH,NMAX*NBMAX), MA(NAMAX)
          INTEGER          IORD(NCMAX*NBMAX)
    *     .. External Subroutines ..
          EXTERNAL         TF01QD
    *     .. Executable Statements ..
    *
          WRITE ( NOUT, FMT = 99999 )
    *     Skip the heading in the data file and read the data.
          READ ( NIN, FMT = '()' )
          READ ( NIN, FMT = * ) N, NA, NB, NC
          IF ( N.LE.0 .OR. N.GT.NMAX ) THEN
             WRITE ( NOUT, FMT = 99994 ) N
          ELSE IF ( NA.LE.0 .OR. NA.GT.NAMAX ) THEN
             WRITE ( NOUT, FMT = 99993 ) NA
          ELSE IF ( NB.LE.0 .OR. NB.GT.NBMAX ) THEN
             WRITE ( NOUT, FMT = 99992 ) NB
          ELSE IF ( NC.LE.0 .OR. NC.GT.NCMAX ) THEN
             WRITE ( NOUT, FMT = 99991 ) NC
          ELSE
             ERROR = .FALSE.
             NL = 0
             K = 1
             NASUM = 0
             DO 40 I = 1, NC
                DO 20 J = 1, NB
                   READ ( NIN, FMT = * ) NORD
                   NASUM = NASUM + NORD
                   IF ( NA.GE.NASUM ) THEN
                      READ ( NIN, FMT = * ) ( MA(NL+L), L = 1,NORD )
                      READ ( NIN, FMT = * ) ( AR(NL+L), L = 1,NORD )
                      IORD(K) = NORD
                      K = K + 1
                      NL = NL + NORD
                   ELSE
                      WRITE ( NOUT, FMT = 99993 ) NA
                      ERROR = .TRUE.
                   END IF
       20       CONTINUE
       40    CONTINUE
             IF ( .NOT. ERROR ) THEN
    *           Compute M(1),...,M(N) from the given transfer function
    *           matrix G(z).
                CALL TF01QD( NC, NB, N, IORD, AR, MA, H, LDH, INFO )
    *
                IF ( INFO.NE.0 ) THEN
                   WRITE ( NOUT, FMT = 99998 ) INFO
                ELSE
                   WRITE ( NOUT, FMT = 99997 ) N
                   DO 80 K = 1, N
                      WRITE ( NOUT, FMT = 99996 ) K,
         $                  ( H(1,(K-1)*NB+J), J = 1,NB )
                      DO 60 I = 2, NC
                         WRITE ( NOUT, FMT = 99995 )
         $                     ( H(I,(K-1)*NB+J), J = 1,NB )
       60             CONTINUE
       80          CONTINUE
                END IF
             END IF
          END IF
          STOP
    *
    99999 FORMAT (' TF01QD EXAMPLE PROGRAM RESULTS',/1X)
    99998 FORMAT (' INFO on exit from TF01QD = ',I2)
    99997 FORMAT (' The Markov Parameters M(1),...,M(',I1,') are ')
    99996 FORMAT (/' M(',I1,') : ',20(1X,F8.4))
    99995 FORMAT (8X,20(1X,F8.4))
    99994 FORMAT (/' N is out of range.',/' N = ',I5)
    99993 FORMAT (/' NA is out of range.',/' NA = ',I5)
    99992 FORMAT (/' NB is out of range.',/' NB = ',I5)
    99991 FORMAT (/' NC is out of range.',/' NC = ',I5)
          END

Program Data

     TF01QD EXAMPLE PROGRAM DATA
       8    10     2     2
       2
       1.0  -0.5
       0.6  -0.2
       1
       1.0
      -0.8
       3
       0.5  -0.4   0.3
       0.8   0.4   0.1
       4
       1.0   0.5  -0.5   0.0
      -0.8   0.6   0.0  -0.2

Program Results

     TF01QD EXAMPLE PROGRAM RESULTS

     The Markov Parameters M(1),...,M(8) are

     M(1) :    1.0000   1.0000
               0.5000   1.0000

     M(2) :   -1.1000   0.8000
              -0.8000   1.3000

     M(3) :    0.8600   0.6400
               0.7400  -0.0600

     M(4) :   -0.7360   0.5120
              -0.3220  -0.8280

     M(5) :    0.6136   0.4096
               0.0416  -0.4264

     M(6) :   -0.5154   0.3277
               0.0215   0.4157

     M(7) :    0.4319   0.2621
              -0.0017   0.5764

     M(8) :   -0.3622   0.2097
              -0.0114   0.0461

Return to index