## TB04BD

### Transfer matrix of a given state-space representation (A,B,C,D), using the pole-zeros method

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

Purpose

      To compute the transfer function matrix G of a state-space
      representation (A,B,C,D) of a linear time-invariant multivariable
      system, using the pole-zeros method. Each element of the transfer
      function matrix is returned in a cancelled, minimal form, with
      numerator and denominator polynomials stored either in increasing
      or decreasing order of the powers of the indeterminate.

Specification

          SUBROUTINE TB04BD( JOBD, ORDER, EQUIL, N, M, P, MD, A, LDA, B,
         $                   LDB, C, LDC, D, LDD, IGN, LDIGN, IGD, LDIGD,
         $                   GN, GD, TOL, IWORK, DWORK, LDWORK, INFO )
    C     .. Scalar Arguments ..
          CHARACTER          EQUIL, JOBD, ORDER
          DOUBLE PRECISION   TOL
          INTEGER            INFO, LDA, LDB, LDC, LDD, LDIGD, LDIGN, LDWORK,
         $                   M, MD, N, P
    C     .. Array Arguments ..
          DOUBLE PRECISION   A(LDA,*), B(LDB,*), C(LDC,*), D(LDD,*),
         $                   DWORK(*), GD(*), GN(*)
          INTEGER            IGD(LDIGD,*), IGN(LDIGN,*), IWORK(*)

Arguments

Mode Parameters

      JOBD    CHARACTER*1
              Specifies whether or not a non-zero matrix D appears in
              the given state-space model:
              = 'D':  D is present;
              = 'Z':  D is assumed to be a zero matrix.

      ORDER   CHARACTER*1
              Specifies the order in which the polynomial coefficients
              are stored, as follows:
              = 'I':  Increasing order of powers of the indeterminate;
              = 'D':  Decreasing order of powers of the indeterminate.

      EQUIL   CHARACTER*1
              Specifies whether the user wishes to preliminarily
              equilibrate the triplet (A,B,C) as follows:
              = 'S':  perform equilibration (scaling);
              = 'N':  do not perform equilibration.

Input/Output Parameters

      N       (input) INTEGER
              The order of the system (A,B,C,D).  N >= 0.

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

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

      MD      (input) INTEGER
              The maximum degree of the polynomials in G, plus 1. An
              upper bound for MD is N+1.  MD >= 1.

      A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
              On entry, the leading N-by-N part of this array must
              contain the original state dynamics matrix A.
              On exit, if EQUIL = 'S', the leading N-by-N part of this
              array contains the balanced matrix inv(S)*A*S, as returned
              by SLICOT Library routine TB01ID.
              If EQUIL = 'N', this array is unchanged on exit.

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

      B       (input/output) DOUBLE PRECISION array, dimension (LDB,M)
              On entry, the leading N-by-M part of this array must
              contain the input matrix B.
              On exit, the contents of B are destroyed: all elements but
              those in the first row are set to zero.

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

      C       (input/output) DOUBLE PRECISION array, dimension (LDC,N)
              On entry, the leading P-by-N part of this array must
              contain the output matrix C.
              On exit, if EQUIL = 'S', the leading P-by-N part of this
              array contains the balanced matrix C*S, as returned by
              SLICOT Library routine TB01ID.
              If EQUIL = 'N', this array is unchanged on exit.

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

      D       (input) DOUBLE PRECISION array, dimension (LDD,M)
              If JOBD = 'D', the leading P-by-M part of this array must
              contain the matrix D.
              If JOBD = 'Z', the array D is not referenced.

      LDD     INTEGER
              The leading dimension of array D.
              LDD >= MAX(1,P), if JOBD = 'D';
              LDD >= 1,        if JOBD = 'Z'.

      IGN     (output) INTEGER array, dimension (LDIGN,M)
              The leading P-by-M part of this array contains the degrees
              of the numerator polynomials in the transfer function
              matrix G. Specifically, the (i,j) element of IGN contains
              the degree of the numerator polynomial of the transfer
              function G(i,j) from the j-th input to the i-th output.

      LDIGN   INTEGER
              The leading dimension of array IGN.  LDIGN >= max(1,P).

      IGD     (output) INTEGER array, dimension (LDIGD,M)
              The leading P-by-M part of this array contains the degrees
              of the denominator polynomials in the transfer function
              matrix G. Specifically, the (i,j) element of IGD contains
              the degree of the denominator polynomial of the transfer
              function G(i,j).

      LDIGD   INTEGER
              The leading dimension of array IGD.  LDIGD >= max(1,P).

      GN      (output) DOUBLE PRECISION array, dimension (P*M*MD)
              This array contains the coefficients of the numerator
              polynomials, Num(i,j), of the transfer function matrix G.
              The polynomials are stored in a column-wise order, i.e.,
              Num(1,1), Num(2,1), ..., Num(P,1), Num(1,2), Num(2,2),
              ..., Num(P,2), ..., Num(1,M), Num(2,M), ..., Num(P,M);
              MD memory locations are reserved for each polynomial,
              hence, the (i,j) polynomial is stored starting from the
              location ((j-1)*P+i-1)*MD+1. The coefficients appear in
              increasing or decreasing order of the powers of the
              indeterminate, according to ORDER.

      GD      (output) DOUBLE PRECISION array, dimension (P*M*MD)
              This array contains the coefficients of the denominator
              polynomials, Den(i,j), of the transfer function matrix G.
              The polynomials are stored in the same way as the
              numerator polynomials.

Tolerances

      TOL     DOUBLE PRECISION
              The tolerance to be used in determining the
              controllability of a single-input system (A,b) or (A',c'),
              where b and c' are columns in B and C' (C transposed). If
              the user sets TOL > 0, then the given value of TOL is used
              as an absolute tolerance; elements with absolute value
              less than TOL are considered neglijible. If the user sets
              TOL <= 0, then an implicitly computed, default tolerance,
              defined by TOLDEF = N*EPS*MAX( NORM(A), NORM(bc) ) is used
              instead, where EPS is the machine precision (see LAPACK
              Library routine DLAMCH), and bc denotes the currently used
              column in B or C' (see METHOD).

Workspace

      IWORK   INTEGER array, dimension (N)

      DWORK   DOUBLE PRECISION array, dimension (LDWORK)
              On exit, if INFO = 0, DWORK(1) returns the optimal value
              of LDWORK.

      LDWORK  INTEGER
              The length of the array DWORK.
              LDWORK >= MAX(1, N*(N+P) +
                               MAX( N + MAX( N,P ), N*(2*N+5)))
              If N >= P, N >= 1, the formula above can be written as
              LDWORK >= N*(3*N + P + 5).
              For optimum performance LDWORK should be larger.

Error Indicator

      INFO    INTEGER
              = 0:  successful exit;
              < 0:  if INFO = -i, the i-th argument had an illegal
                    value;
              = 1:  the QR algorithm failed to converge when trying to
                    compute the zeros of a transfer function;
              = 2:  the QR algorithm failed to converge when trying to
                    compute the poles of a transfer function.
                    The errors INFO = 1 or 2 are unlikely to appear.

Method

      The routine implements the pole-zero method proposed in [1].
      This method is based on an algorithm for computing the transfer
      function of a single-input single-output (SISO) system.
      Let (A,b,c,d) be a SISO system. Its transfer function is computed
      as follows:

      1) Find a controllable realization (Ac,bc,cc) of (A,b,c).
      2) Find an observable realization (Ao,bo,co) of (Ac,bc,cc).
      3) Compute the r eigenvalues of Ao (the poles of (Ao,bo,co)).
      4) Compute the zeros of (Ao,bo,co,d).
      5) Compute the gain of (Ao,bo,co,d).

      This algorithm can be implemented using only orthogonal
      transformations [1]. However, for better efficiency, the
      implementation in TB04BD uses one elementary transformation
      in Step 4 and r elementary transformations in Step 5 (to reduce
      an upper Hessenberg matrix to upper triangular form). These
      special elementary transformations are numerically stable
      in practice.

      In the multi-input multi-output (MIMO) case, the algorithm
      computes each element (i,j) of the transfer function matrix G,
      for i = 1 : P, and for j = 1 : M. For efficiency reasons, Step 1
      is performed once for each value of j (each column of B). The
      matrices Ac and Ao result in Hessenberg form.

References

      [1] Varga, A. and Sima, V.
          Numerically Stable Algorithm for Transfer Function Matrix
          Evaluation.
          Int. J. Control, vol. 33, nr. 6, pp. 1123-1133, 1981.

Numerical Aspects

      The algorithm is numerically stable in practice and requires about
      20*N**3 floating point operations at most, but usually much less.

Further Comments

      For maximum efficiency of index calculations, GN and GD are
      implemented as one-dimensional arrays.

Example

Program Text

    *     TB04BD EXAMPLE PROGRAM TEXT
    *
    *     .. Parameters ..
          INTEGER          NIN, NOUT
          PARAMETER        ( NIN = 5, NOUT = 6 )
          INTEGER          NMAX, MMAX, PMAX, MDMAX
          PARAMETER        ( NMAX = 20, MMAX = 20, PMAX = 20,
         $                   MDMAX = NMAX + 1 )
          INTEGER          PMNMAX
          PARAMETER        ( PMNMAX = PMAX*MMAX*MDMAX )
          INTEGER          LDA, LDB, LDC, LDD, LDIGD, LDIGN
          PARAMETER        ( LDA = NMAX, LDB = NMAX, LDC = PMAX,
         $                   LDD = PMAX, LDIGD = PMAX, LDIGN = PMAX )
          INTEGER          LIWORK
          PARAMETER        ( LIWORK = NMAX )
          INTEGER          LDWORK
          PARAMETER        ( LDWORK = NMAX*( NMAX + PMAX ) +
         $                            MAX( NMAX + MAX( NMAX, PMAX ),
         $                                 NMAX*( 2*NMAX + 5 ) ) )
    *     .. Local Scalars ..
          DOUBLE PRECISION TOL
          INTEGER          I, IJ, INFO, J, K, M, MD, N, P
          CHARACTER*1      JOBD, ORDER, EQUIL
          CHARACTER*132    ULINE
    *     .. Local Arrays ..
          DOUBLE PRECISION A(LDA,NMAX), B(LDB,MMAX), C(LDC,NMAX),
         $                 D(LDD,MMAX), DWORK(LDWORK), GD(PMNMAX),
         $                 GN(PMNMAX)
          INTEGER          IGD(LDIGD,MMAX), IGN(LDIGN,MMAX), IWORK(LIWORK)
    *     .. External Functions ..
          LOGICAL          LSAME
          EXTERNAL         LSAME
    *     .. External Subroutines ..
          EXTERNAL         TB04BD
    *     .. Intrinsic Functions ..
          INTRINSIC        MAX
    *     .. 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, TOL, JOBD, ORDER, EQUIL
          MD = N + 1
          ULINE(1:20) = ' '
          DO 20 I = 21, 132
             ULINE(I:I) = '-'
       20 CONTINUE
          IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
             WRITE ( NOUT, FMT = 99991 ) N
          ELSE
             READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
             IF ( M.LT.0 .OR. M.GT.MMAX ) THEN
                WRITE ( NOUT, FMT = 99990 ) M
             ELSE
                READ ( NIN, FMT = * ) ( ( B(I,J), I = 1,N ), J = 1,M )
                IF ( P.LT.0 .OR. P.GT.PMAX ) THEN
                   WRITE ( NOUT, FMT = 99989 ) P
                ELSE
                   READ ( NIN, FMT = * ) ( ( C(I,J), J = 1,N ), I = 1,P )
                   READ ( NIN, FMT = * ) ( ( D(I,J), J = 1,M ), I = 1,P )
    *              Find the transfer matrix T(s) of (A,B,C,D).
                   CALL TB04BD( JOBD, ORDER, EQUIL, N, M, P, MD, A, LDA, B,
         $                      LDB, C, LDC, D, LDD, IGN, LDIGN, IGD, LDIGD,
         $                      GN, GD, TOL, IWORK, DWORK, LDWORK, INFO )
    *
                   IF ( INFO.NE.0 ) THEN
                      WRITE ( NOUT, FMT = 99998 ) INFO
                   ELSE
                      IF ( LSAME( ORDER, 'I' ) ) THEN
                         WRITE ( NOUT, FMT = 99997 )
                      ELSE
                         WRITE ( NOUT, FMT = 99996 )
                      END IF
                      WRITE ( NOUT, FMT = 99995 )
                      DO 60 J = 1, M
                         DO 40 I = 1, P
                            IJ = ( (J-1)*P + I-1 )*MD + 1
                            WRITE ( NOUT, FMT = 99994 ) I, J,
         $                    ( GN(K), K = IJ,IJ+IGN(I,J) )
                            WRITE ( NOUT, FMT = 99993 )
         $                          ULINE(1:7*(IGD(I,J)+1)+21)
                            WRITE ( NOUT, FMT = 99992 )
         $                        ( GD(K), K = IJ,IJ+IGD(I,J) )
       40                CONTINUE
       60             CONTINUE
                   END IF
                END IF
             END IF
          END IF
          STOP
    *
    99999 FORMAT (' TB04BD EXAMPLE PROGRAM RESULTS',/1X)
    99998 FORMAT (' INFO on exit from TB04BD = ',I2)
    99997 FORMAT (/' The polynomial coefficients appear in increasing',
         $         ' order'/' of the powers of the indeterminate')
    99996 FORMAT (/' The polynomial coefficients appear in decreasing',
         $         ' order'/' of the powers of the indeterminate')
    99995 FORMAT (/' The coefficients of polynomials in the transfer matri',
         $       'x T(s) are ')
    99994 FORMAT (/' element (',I2,',',I2,') is ',20(1X,F6.2))
    99993 FORMAT (1X,A)
    99992 FORMAT (20X,20(1X,F6.2))
    99991 FORMAT (/' N is out of range.',/' N = ',I5)
    99990 FORMAT (/' M is out of range.',/' M = ',I5)
    99989 FORMAT (/' P is out of range.',/' P = ',I5)
          END

Program Data

     TB04BD EXAMPLE PROGRAM DATA
       3     2     2  0.0         D     I     N
      -1.0   0.0   0.0
       0.0  -2.0   0.0
       0.0   0.0  -3.0
       0.0   1.0  -1.0
       1.0   1.0   0.0
       0.0   1.0   1.0
       1.0   1.0   1.0
       1.0   0.0
       0.0   1.0

Program Results

     TB04BD EXAMPLE PROGRAM RESULTS

     The polynomial coefficients appear in increasing order
     of the powers of the indeterminate

     The coefficients of polynomials in the transfer matrix T(s) are

     element ( 1, 1) is    7.00   5.00   1.00
                         ----------------------
                           6.00   5.00   1.00

     element ( 2, 1) is    1.00
                         ----------------------
                           6.00   5.00   1.00

     element ( 1, 2) is    1.00
                         ---------------
                           2.00   1.00

     element ( 2, 2) is    5.00   5.00   1.00
                         ----------------------
                           2.00   3.00   1.00

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