## TF01MD

### Output response sequence of a linear time-invariant discrete-time system

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

Purpose

      To compute the output sequence of a linear time-invariant
      open-loop system given by its discrete-time state-space model
      (A,B,C,D), where A is an N-by-N general matrix.

      The initial state vector x(1) must be supplied by the user.

Specification

          SUBROUTINE TF01MD( N, M, P, NY, A, LDA, B, LDB, C, LDC, D, LDD,
         $                   U, LDU, X, Y, LDY, DWORK, INFO )
    C     .. Scalar Arguments ..
          INTEGER           INFO, LDA, LDB, LDC, LDD, LDU, LDY, M, N, NY, P
    C     .. Array Arguments ..
          DOUBLE PRECISION  A(LDA,*), B(LDB,*), C(LDC,*), D(LDD,*),
         $                  DWORK(*), U(LDU,*), X(*), Y(LDY,*)

Arguments

Input/Output Parameters

      N       (input) INTEGER
              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.

      NY      (input) INTEGER
              The number of output vectors y(k) to be computed.
              NY >= 0.

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

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

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

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

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

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

      D       (input) DOUBLE PRECISION array, dimension (LDD,M)
              The leading P-by-M part of this array must contain the
              direct link matrix D of the system.

      LDD     INTEGER
              The leading dimension of array D.  LDD >= MAX(1,P).

      U       (input) DOUBLE PRECISION array, dimension (LDU,NY)
              The leading M-by-NY part of this array must contain the
              input vector sequence u(k), for k = 1,2,...,NY.
              Specifically, the k-th column of U must contain u(k).

      LDU     INTEGER
              The leading dimension of array U.  LDU >= MAX(1,M).

      X       (input/output) DOUBLE PRECISION array, dimension (N)
              On entry, this array must contain the initial state vector
              x(1) which consists of the N initial states of the system.
              On exit, this array contains the final state vector
              x(NY+1) of the N states of the system at instant NY.

      Y       (output) DOUBLE PRECISION array, dimension (LDY,NY)
              The leading P-by-NY part of this array contains the output
              vector sequence y(1),y(2),...,y(NY) such that the k-th
              column of Y contains y(k) (the outputs at instant k),
              for k = 1,2,...,NY.

      LDY     INTEGER
              The leading dimension of array Y.  LDY >= MAX(1,P).

Workspace

      DWORK   DOUBLE PRECISION array, dimension (N)

Error Indicator

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

Method

      Given an initial state vector x(1), the output vector sequence
      y(1), y(2),..., y(NY) is obtained via the formulae

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

      where each element y(k) is a vector of length P containing the
      outputs at instant k and k = 1,2,...,NY.

References

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

Numerical Aspects

      The algorithm requires approximately (N + M) x (N + P) x NY
      multiplications and additions.

Further Comments

      None

Example

Program Text

    *     TF01MD EXAMPLE PROGRAM TEXT
    *
    *     .. Parameters ..
          INTEGER          NIN, NOUT
          PARAMETER        ( NIN = 5, NOUT = 6 )
          INTEGER          NMAX, MMAX, PMAX, NYMAX
          PARAMETER        ( NMAX = 20, MMAX = 20, PMAX = 20, NYMAX = 20 )
          INTEGER          LDA, LDB, LDC, LDD, LDU, LDY
          PARAMETER        ( LDA = NMAX, LDB = NMAX, LDC = PMAX, LDD = PMAX,
         $                   LDU = MMAX, LDY = PMAX )
          INTEGER          LDWORK
          PARAMETER        ( LDWORK = NMAX )
    *     .. Local Scalars ..
          INTEGER          I, INFO, J, K, M, N, NY, P
    *     .. Local Arrays ..
          DOUBLE PRECISION A(LDA,NMAX), B(LDB,MMAX), C(LDC,NMAX),
         $                 D(LDD,MMAX), DWORK(LDWORK), U(LDU,NYMAX),
         $                 X(NMAX), Y(LDY,NYMAX)
    *     .. External Subroutines ..
          EXTERNAL         TF01MD
    *     .. 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, NY
          IF ( N.LE.0 .OR. N.GT.NMAX ) THEN
             WRITE ( NOUT, FMT = 99994 ) N
          ELSE
             READ ( NIN, FMT = * ) ( ( A(I,J), I = 1,N ), J = 1,N )
             IF ( M.LE.0 .OR. M.GT.MMAX ) THEN
                WRITE ( NOUT, FMT = 99993 ) M
             ELSE
                READ ( NIN, FMT = * ) ( ( B(I,J), I = 1,N ), J = 1,M )
                IF ( P.LE.0 .OR. P.GT.PMAX ) THEN
                   WRITE ( NOUT, FMT = 99992 ) P
                ELSE
                   READ ( NIN, FMT = * ) ( ( C(I,J), I = 1,P ), J = 1,N )
                   READ ( NIN, FMT = * ) ( ( D(I,J), I = 1,P ), J = 1,M )
                   READ ( NIN, FMT = * ) ( X(I), I = 1,N )
                   IF ( NY.LE.0 .OR. NY.GT.NYMAX ) THEN
                      WRITE ( NOUT, FMT = 99991 ) NY
                   ELSE
                      READ ( NIN, FMT = * )
         $                 ( ( U(I,J), I = 1,M ), J = 1,NY )
    *                 Compute y(1),...,y(NY) of the given system.
                      CALL TF01MD( N, M, P, NY, A, LDA, B, LDB, C, LDC, D,
         $                         LDD, U, LDU, X, Y, LDY, DWORK, INFO )
    *
                      IF ( INFO.NE.0 ) THEN
                         WRITE ( NOUT, FMT = 99998 ) INFO
                      ELSE
                         WRITE ( NOUT, FMT = 99997 ) NY
                         DO 20 K = 1, NY
                            WRITE ( NOUT, FMT = 99996 ) K, Y(1,K)
                            WRITE ( NOUT, FMT = 99995 ) ( Y(J,K), J = 2,P )
       20                CONTINUE
                      END IF
                   END IF
                END IF
             END IF
          END IF
          STOP
    *
    99999 FORMAT (' TF01MD EXAMPLE PROGRAM RESULTS',/1X)
    99998 FORMAT (' INFO on exit from TF01MD = ',I2)
    99997 FORMAT (' The output sequence Y(1),...,Y(',I2,') is',/)
    99996 FORMAT (' Y(',I2,') : ',F8.4)
    99995 FORMAT (9X,F8.4,/)
    99994 FORMAT (/' N is out of range.',/' N = ',I5)
    99993 FORMAT (/' M is out of range.',/' M = ',I5)
    99992 FORMAT (/' P is out of range.',/' P = ',I5)
    99991 FORMAT (/' NY is out of range.',/' NY = ',I5)
          END

Program Data

     TF01MD EXAMPLE PROGRAM DATA
       3     2     2     10
       0.0000 -0.0700  0.0150
       1.0000  0.8000 -0.1500
       0.0000  0.0000  0.5000
       0.0000  2.0000  1.0000
      -1.0000 -0.1000  1.0000
       0.0000  1.0000
       0.0000  0.0000
       1.0000  0.0000
       1.0000  0.5000
       0.0000  0.5000
       1.0000  1.0000  1.0000
      -0.6922 -1.4934  0.3081 -2.7726  2.0039
       0.2614 -0.9160 -0.6030  1.2556  0.2951
      -1.5734  1.5639 -0.9942  1.8957  0.8988
       0.4118 -1.4893 -0.9344  1.2506 -0.0701

Program Results

     TF01MD EXAMPLE PROGRAM RESULTS

     The output sequence Y(1),...,Y(10) is

     Y( 1) :   0.3078
              -0.0928

     Y( 2) :  -1.5125
               1.2611

     Y( 3) :  -1.2577
               3.4002

     Y( 4) :  -0.2947
              -0.7060

     Y( 5) :  -0.5632
               5.4532

     Y( 6) :  -1.0846
               1.1846

     Y( 7) :  -1.2427
               2.2286

     Y( 8) :   1.8097
              -1.9534

     Y( 9) :   0.6685
              -4.4965

     Y(10) :  -0.0896
               1.1654

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