## TF01MX

### Output sequence of a linear time-invariant open-loop system given its system matrix

[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
      with an (N+P)-by-(N+M) general system matrix S,

             ( A  B )
         S = (      ) .
             ( C  D )

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

      The input and output trajectories are stored as in the SLICOT
      Library routine TF01MY.

Specification

          SUBROUTINE TF01MX( N, M, P, NY, S, LDS, U, LDU, X, Y, LDY,
         $                   DWORK, LDWORK, INFO )
    C     .. Scalar Arguments ..
          INTEGER           INFO, LDS, LDU, LDWORK, LDY, M, N, NY, P
    C     .. Array Arguments ..
          DOUBLE PRECISION  DWORK(*), S(LDS,*), 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.

      S       (input) DOUBLE PRECISION array, dimension (LDS,N+M)
              The leading (N+P)-by-(N+M) part of this array must contain
              the system matrix S.

      LDS     INTEGER
              The leading dimension of array S.  LDS >= MAX(1,N+P).

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

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

      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+1.

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

Workspace

      DWORK   DOUBLE PRECISION array, dimension (LDWORK)

      LDWORK  INTEGER
              The length of the array DWORK.
              LDWORK >= 0,        if MIN(N,P,NY) = 0;  otherwise,
              LDWORK >= N+P,      if M = 0;
              LDWORK >= 2*N+M+P,  if M > 0.
              For better performance, LDWORK should be larger.

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) )     ( x(k) )
         (        ) = S (      ) ,
         (  y(k)  )     ( 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

      The implementation exploits data locality as much as possible,
      given the workspace length.

Example

Program Text

      None

Program Data

      None

Program Results

      None

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