## AB09JW

### State-space representation of a projection of a right weighted transfer-function matrix

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

Purpose

      To construct a state-space representation (A,BS,CS,DS) of the
      projection of G*W or G*conj(W) containing the poles of G, from the
      state-space representations (A,B,C,D) and (AW-lambda*EW,BW,CW,DW),
      of the transfer-function matrices G and W, respectively.
      G is assumed to be a stable transfer-function matrix and
      the state matrix A must be in a real Schur form.
      When computing the stable projection of G*W, it is assumed
      that G and W have completely distinct poles.
      When computing the stable projection of G*conj(W), it is assumed
      that G and conj(W) have completely distinct poles.

      Note: For a transfer-function matrix G, conj(G) denotes the
      conjugate of G given by G'(-s) for a continuous-time system or
      G'(1/z) for a discrete-time system.

Specification

          SUBROUTINE AB09JW( JOB, DICO, JOBEW, STBCHK, N, M, P, NW, MW,
         $                   A, LDA, B, LDB, C, LDC, D, LDD, AW, LDAW,
         $                   EW, LDEW, BW, LDBW, CW, LDCW, DW, LDDW, IWORK,
         $                   DWORK, LDWORK, INFO )
    C     .. Scalar Arguments ..
          CHARACTER         DICO, JOB, JOBEW, STBCHK
          INTEGER           INFO, LDA, LDAW, LDB, LDBW, LDC, LDCW,
         $                  LDD, LDDW, LDEW, LDWORK, M, MW, N, NW, P
    C     .. Array Arguments ..
          INTEGER           IWORK(*)
          DOUBLE PRECISION  A(LDA,*), AW(LDAW,*), B(LDB,*), BW(LDBW,*),
         $                  C(LDC,*), CW(LDCW,*), D(LDD,*), DW(LDDW,*),
         $                  DWORK(*), EW(LDEW,*)

Arguments

Mode Parameters

      JOB     CHARACTER*1
              Specifies the projection to be computed as follows:
              = 'W':  compute the projection of G*W containing
                      the poles of G;
              = 'C':  compute the projection of G*conj(W) containing
                      the poles of G.

      DICO    CHARACTER*1
              Specifies the type of the systems as follows:
              = 'C':  G and W are continuous-time systems;
              = 'D':  G and W are discrete-time systems.

      JOBEW   CHARACTER*1
              Specifies whether EW is a general square or an identity
              matrix as follows:
              = 'G':  EW is a general square matrix;
              = 'I':  EW is the identity matrix.

      STBCHK  CHARACTER*1
              Specifies whether stability/antistability of W is to be
              checked as follows:
              = 'C':  check stability if JOB = 'C' or antistability if
                      JOB = 'W';
              = 'N':  do not check stability or antistability.

Input/Output Parameters

      N       (input) INTEGER
              The dimension of the state vector of the system with
              the transfer-function matrix G.  N >= 0.

      M       (input) INTEGER
              The dimension of the input vector of the system with
              the transfer-function matrix G, and also the dimension
              of the output vector if JOB = 'W', or of the input vector
              if JOB = 'C', of the system with the transfer-function
              matrix W.  M >= 0.

      P       (input) INTEGER
              The dimension of the output vector of the system with the
              transfer-function matrix G.  P >= 0.

      NW      (input) INTEGER
              The dimension of the state vector of the system with the
              transfer-function matrix W.  NW >= 0.

      MW      (input) INTEGER
              The dimension of the input vector, if JOB = 'W', or of
              the output vector, if JOB = 'C', of the system with the
              transfer-function matrix W.  MW >= 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 with the transfer-function
              matrix G in a real Schur form.

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

      B       (input/output) DOUBLE PRECISION array,
              dimension (LDB,MAX(M,MW))
              On entry, the leading N-by-M part of this array must
              contain the input matrix B of the system with the
              transfer-function matrix G.
              On exit, if INFO = 0, the leading N-by-MW part of this
              array contains the input matrix BS of the projection of
              G*W, if JOB = 'W', or of G*conj(W), if JOB = 'C'.

      LDB     INTEGER
              The leading dimension of the 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/state matrix C of the system with the
              transfer-function matrix G. The matrix CS is equal to C.

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

      D       (input/output) DOUBLE PRECISION array,
              dimension (LDB,MAX(M,MW))
              On entry, the leading P-by-M part of this array must
              contain the feedthrough matrix D of the system with
              the transfer-function matrix G.
              On exit, if INFO = 0, the leading P-by-MW part of
              this array contains the feedthrough matrix DS of the
              projection of G*W, if JOB = 'W', or of G*conj(W),
              if JOB = 'C'.

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

      AW      (input/output) DOUBLE PRECISION array, dimension (LDAW,NW)
              On entry, the leading NW-by-NW part of this array must
              contain the state matrix AW of the system with the
              transfer-function matrix W.
              On exit, if INFO = 0, the leading NW-by-NW part of this
              array contains a condensed matrix as follows:
              if JOBEW = 'I', it contains the real Schur form of AW;
              if JOBEW = 'G' and JOB = 'W', it contains a quasi-upper
              triangular matrix representing the real Schur matrix
              in the real generalized Schur form of the pair (AW,EW);
              if JOBEW = 'G', JOB = 'C' and DICO = 'C', it contains a
              quasi-upper triangular matrix corresponding to the
              generalized real Schur form of the pair (AW',EW');
              if JOBEW = 'G', JOB = 'C' and DICO = 'D', it contains an
              upper triangular matrix corresponding to the generalized
              real Schur form of the pair (EW',AW').

      LDAW    INTEGER
              The leading dimension of the array AW.  LDAW >= MAX(1,NW).

      EW      (input/output) DOUBLE PRECISION array, dimension (LDEW,NW)
              On entry, if JOBEW = 'G', the leading NW-by-NW part of
              this array must contain the descriptor matrix EW of the
              system with the transfer-function matrix W.
              If JOBEW = 'I', EW is assumed to be an identity matrix
              and is not referenced.
              On exit, if INFO = 0 and JOBEW = 'G', the leading NW-by-NW
              part of this array contains a condensed matrix as follows:
              if JOB = 'W', it contains an upper triangular matrix
              corresponding to the real generalized Schur form of the
              pair (AW,EW);
              if JOB = 'C' and DICO = 'C', it contains an upper
              triangular matrix corresponding to the generalized real
              Schur form of the pair (AW',EW');
              if JOB = 'C' and DICO = 'D', it contains a quasi-upper
              triangular matrix corresponding to the generalized
              real Schur form of the pair (EW',AW').

      LDEW    INTEGER
              The leading dimension of the array EW.
              LDEW >= MAX(1,NW), if JOBEW = 'G';
              LDEW >= 1,         if JOBEW = 'I'.

      BW      (input/output) DOUBLE PRECISION array,
              dimension (LDBW,MBW), where MBW = MW, if JOB = 'W', and
              MBW = M, if JOB = 'C'.
              On entry, the leading NW-by-MBW part of this array must
              contain the input matrix BW of the system with the
              transfer-function matrix W.
              On exit, if INFO = 0, the leading NW-by-MBW part of this
              array contains Q'*BW, where Q is the orthogonal matrix
              that reduces AW to the real Schur form or the left
              orthogonal matrix used to reduce the pair (AW,EW),
              (AW',EW') or (EW',AW') to the generalized real Schur form.

      LDBW    INTEGER
              The leading dimension of the array BW.  LDBW >= MAX(1,NW).

      CW      (input/output) DOUBLE PRECISION array, dimension (LDCW,NW)
              On entry, the leading PCW-by-NW part of this array must
              contain the output matrix CW of the system with the
              transfer-function matrix W, where PCW = M if JOB = 'W' or
              PCW = MW if JOB = 'C'.
              On exit, if INFO = 0, the leading PCW-by-NW part of this
              array contains CW*Q, where Q is the orthogonal matrix that
              reduces AW to the real Schur form, or CW*Z, where Z is the
              right orthogonal matrix used to reduce the pair (AW,EW),
              (AW',EW') or (EW',AW') to the generalized real Schur form.

      LDCW    INTEGER
              The leading dimension of the array CW.
              LDCW >= MAX(1,PCW), where PCW = M if JOB = 'W', or
              PCW = MW if JOB = 'C'.

      DW      (input) DOUBLE PRECISION array,
              dimension (LDDW,MBW), where MBW = MW if JOB = 'W', and
              MBW = M if JOB = 'C'.
              The leading PCW-by-MBW part of this array must contain
              the feedthrough matrix DW of the system with the
              transfer-function matrix W, where PCW = M if JOB = 'W',
              or PCW = MW if JOB = 'C'.

      LDDW    INTEGER
              LDDW >= MAX(1,PCW), where PCW = M if JOB = 'W', or
              PCW = MW if JOB = 'C'.

Workspace

      IWORK   INTEGER array, dimension (LIWORK)
              LIWORK =   0,    if JOBEW = 'I';
              LIWORK = NW+N+6, if JOBEW = 'G'.

      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 >= LW1, if JOBEW = 'I',
              LDWORK >= LW2, if JOBEW = 'G', where
                LW1 = MAX( 1, NW*(NW+5), NW*N + MAX( a, N*MW, P*MW ) )
                      a = 0,    if DICO = 'C' or  JOB = 'W',
                      a = 2*NW, if DICO = 'D' and JOB = 'C';
                LW2 = MAX( 2*NW*NW + MAX( 11*NW+16, NW*M, MW*NW ),
                           NW*N + MAX( NW*N+N*N, MW*N, P*MW ) ).
              For good 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 reduction of the pair (AW,EW) to the real
                     generalized Schur form failed (JOBEW = 'G'),
                     or the reduction of the matrix AW to the real
                     Schur form failed (JOBEW = 'I);
              =  2:  the solution of the Sylvester equation failed
                     because the matrix A and the pencil AW-lambda*EW
                     have common eigenvalues (if JOB = 'W'), or the
                     pencil -AW-lambda*EW and A have common eigenvalues
                     (if JOB = 'C' and DICO = 'C'), or the pencil
                     AW-lambda*EW has an eigenvalue which is the
                     reciprocal of one of eigenvalues of A
                     (if JOB = 'C' and DICO = 'D');
              =  3:  the solution of the Sylvester equation failed
                     because the matrices A and AW have common
                     eigenvalues (if JOB = 'W'), or the matrices A
                     and -AW have common eigenvalues (if JOB = 'C' and
                     DICO = 'C'), or the matrix A has an eigenvalue
                     which is the reciprocal of one of eigenvalues of AW
                     (if JOB = 'C' and DICO = 'D');
              =  4:  JOB = 'W' and the pair (AW,EW) has not completely
                     unstable generalized eigenvalues, or JOB = 'C' and
                     the pair (AW,EW) has not completely stable
                     generalized eigenvalues.

Method

      If JOB = 'W', the matrices of the stable projection of G*W are
      computed as

        BS = B*DW + Y*BW,  CS = C,  DS = D*DW,

      where Y satisfies the generalized Sylvester equation

        -A*Y*EW + Y*AW + B*CW = 0.

      If JOB = 'C', the matrices of the stable projection of G*conj(W)
      are computed using the following formulas:

      - for a continuous-time system, the matrices BS, CS and DS of
        the stable projection are computed as

          BS = B*DW' + Y*CW',  CS = C,  DS = D*DW',

        where Y satisfies the generalized Sylvester equation

          A*Y*EW' + Y*AW' + B*BW' = 0.

      - for a discrete-time system, the matrices BS, CS and DS of
        the stable projection are computed as

          BS = B*DW' + A*Y*CW',  CS = C,  DS = D*DW' + C*Y*CW',

        where Y satisfies the generalized Sylvester equation

          Y*EW' - A*Y*AW' = B*BW'.

References

      [1] Varga, A.
          Efficient and numerically reliable implementation of the
          frequency-weighted Hankel-norm approximation model reduction
          approach.
          Proc. 2001 ECC, Porto, Portugal, 2001.

      [2] Zhou, K.
          Frequency-weighted H-infinity norm and optimal Hankel norm
          model reduction.
          IEEE Trans. Autom. Control, vol. 40, pp. 1687-1699, 1995.

Numerical Aspects

      The implemented methods rely on numerically stable algorithms.

Further Comments

      None

Example

Program Text

      None

Program Data

      None

Program Results

      None

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