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
*     Copyright (c) 2002-2017 NICONET e.V.
*
*     .. 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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