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
NoneExample
Program Text
*     TF01RD EXAMPLE PROGRAM TEXT
*     Copyright (c) 2002-2010 NICONET e.V.
*
*     .. 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 0.000Program 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
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