## TF01RD

### Markov parameters of a multivariable system from the state-space representation

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

Purpose

      To compute N Markov parameters M(1), M(2),..., M(N) from the
      parameters (A,B,C) of a linear time-invariant system, where each
      M(k) is an NC-by-NB matrix and k = 1,2,...,N.

      All matrices are treated as dense, and hence TF01RD is not
      intended for large sparse problems.

Specification

          SUBROUTINE TF01RD( NA, NB, NC, N, A, LDA, B, LDB, C, LDC, H, LDH,
         $                   DWORK, LDWORK, INFO )
    C     .. Scalar Arguments ..
          INTEGER           INFO, LDA, LDB, LDC, LDH, LDWORK, N, NA, NB, NC
    C     .. Array Arguments ..
          DOUBLE PRECISION  A(LDA,*), B(LDB,*), C(LDC,*), DWORK(*), H(LDH,*)

Arguments

Input/Output Parameters

      NA      (input) INTEGER
              The order of the matrix A.  NA >= 0.

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

      NC      (input) INTEGER
              The number of system outputs.  NC >= 0.

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

      A       (input) DOUBLE PRECISION array, dimension (LDA,NA)
              The leading NA-by-NA part of this array must contain the
              state matrix A of the system.

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

      B       (input) DOUBLE PRECISION array, dimension (LDB,NB)
              The leading NA-by-NB part of this array must contain the
              input matrix B of the system.

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

      C       (input) DOUBLE PRECISION array, dimension (LDC,NA)
              The leading NC-by-NA part of this array must contain the
              output matrix C of the system.

      LDC     INTEGER
              The leading dimension of array C.  LDC >= MAX(1,NC).

      H       (output) DOUBLE PRECISION array, dimension (LDH,N*NB)
              The leading NC-by-N*NB part of this array contains the
              multivariable parameters 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).

Workspace

      DWORK   DOUBLE PRECISION array, dimension (LDWORK)

      LDWORK  INTEGER
              The length of the array DWORK.
              LDWORK >= MAX(1, 2*NA*NC).

Error Indicator

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

Method

      For the linear time-invariant discrete-time system

             x(k+1) = A x(k) + B u(k)
              y(k)  = C x(k) + D u(k),

      the transfer function matrix G(z) is given by
                             -1
               G(z) = C(zI-A)  B + D
                              -1        -2     2   -3
                    = D + CB z   + CAB z   + CA B z   + ...          (1)

      Using Markov parameters, G(z) can also be written as
                                  -1        -2        -3
               G(z) = M(0) + M(1)z   + M(2)z   + M(3)z   + ...       (2)

                                                                k-1
      Equating (1) and (2), we find that M(0) = D and M(k) = C A    B
      for k > 0, from which the Markov parameters M(1),M(2)...,M(N) are
      computed.

References

      [1] Chen, C.T.
          Introduction to Linear System Theory.
          H.R.W. Series in Electrical Engineering, Electronics and
          Systems, Holt, Rinehart and Winston Inc., London, 1970.

Numerical Aspects

      The algorithm requires approximately (NA + NB) x NA x NC x N
      multiplications and additions.

Further Comments

      None

Example

Program Text

    *     TF01RD 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          LDA, LDB, LDC, LDH
          PARAMETER        ( LDA = NAMAX, LDB = NAMAX, LDC = NCMAX,
         $                   LDH = NCMAX )
          INTEGER          LDWORK
          PARAMETER        ( LDWORK = 2*NAMAX*NCMAX )
    *     .. Local Scalars ..
          INTEGER          I, INFO, J, K, N, NA, NB, NC
    *     .. Local Arrays ..
          DOUBLE PRECISION A(LDA,NAMAX), B(LDB,NBMAX), C(LDC,NAMAX),
         $                 H(LDH,NMAX*NBMAX), DWORK(LDWORK)
    *     .. External Subroutines ..
          EXTERNAL         TF01RD
    *     .. 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
             READ ( NIN, FMT = * ) ( ( A(I,J), I = 1,NA ), J = 1,NA )
             IF ( NB.LE.0 .OR. NB.GT.NBMAX ) THEN
                WRITE ( NOUT, FMT = 99992 ) NB
             ELSE
                READ ( NIN, FMT = * ) ( ( B(I,J), I = 1,NA ), J = 1,NB )
                IF ( NC.LE.0 .OR. NC.GT.NCMAX ) THEN
                   WRITE ( NOUT, FMT = 99991 ) NC
                ELSE
                   READ ( NIN, FMT = * ) ( ( C(I,J), I = 1,NC ), J = 1,NA )
    *              Compute M(1),...,M(N) from the system (A,B,C).
                   CALL TF01RD( NA, NB, NC, N, A, LDA, B, LDB, C, LDC, H,
         $                      LDH, DWORK, LDWORK, INFO )
    *
                   IF ( INFO.NE.0 ) THEN
                      WRITE ( NOUT, FMT = 99998 ) INFO
                   ELSE
                      WRITE ( NOUT, FMT = 99997 ) N
                      DO 40 K = 1, N
                         WRITE ( NOUT, FMT = 99996 ) K,
         $                     ( H(1,(K-1)*NB+J), J = 1,NB )
                         DO 20 I = 2, NC
                            WRITE ( NOUT, FMT = 99995 )
         $                        ( H(I,(K-1)*NB+J), J = 1,NB )
       20                CONTINUE
       40             CONTINUE
                   END IF
                END IF
             END IF
          END IF
          STOP
    *
    99999 FORMAT (' TF01RD EXAMPLE PROGRAM RESULTS',/1X)
    99998 FORMAT (' INFO on exit from TF01RD = ',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

     TF01RD EXAMPLE PROGRAM DATA
       5     3     2     2
       0.000 -0.070  0.015
       1.000  0.800 -0.150
       0.000  0.000  0.500
       0.000  2.000  1.000
      -1.000 -0.100  1.000
       0.000  1.000  0.000
       0.000  1.000  0.000

Program Results

     TF01RD EXAMPLE PROGRAM RESULTS

     The Markov Parameters M(1),...,M(5) are

     M(1) :    1.0000   1.0000
               0.0000  -1.0000

     M(2) :    0.2000   0.5000
               2.0000  -0.1000

     M(3) :   -0.1100   0.2500
               1.6000  -0.0100

     M(4) :   -0.2020   0.1250
               1.1400  -0.0010

     M(5) :   -0.2039   0.0625
               0.8000  -0.0001

Return to index