## AB09JD

### Frequency-weighted Hankel norm approximation with invertible weights

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

Purpose

      To compute a reduced order model (Ar,Br,Cr,Dr) for an original
      state-space representation (A,B,C,D) by using the frequency
      weighted optimal Hankel-norm approximation method.
      The Hankel norm of the weighted error

            op(V)*(G-Gr)*op(W)

      is minimized, where G and Gr are the transfer-function matrices
      of the original and reduced systems, respectively, V and W are
      invertible transfer-function matrices representing the left and
      right frequency weights, and op(X) denotes X, inv(X), conj(X) or
      conj(inv(X)). V and W are specified by their state space
      realizations (AV,BV,CV,DV) and (AW,BW,CW,DW), respectively.
      When minimizing ||V*(G-Gr)*W||, V and W must be antistable.
      When minimizing inv(V)*(G-Gr)*inv(W), V and W must have only
      antistable zeros.
      When minimizing conj(V)*(G-Gr)*conj(W), V and W must be stable.
      When minimizing conj(inv(V))*(G-Gr)*conj(inv(W)), V and W must
      be minimum-phase.
      If the original system is unstable, then the frequency weighted
      Hankel-norm approximation is computed only for the
      ALPHA-stable part of the system.

      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 AB09JD( JOBV, JOBW, JOBINV, DICO, EQUIL, ORDSEL,
         $                   N, NV, NW, M, P, NR, ALPHA, A, LDA, B, LDB,
         $                   C, LDC, D, LDD, AV, LDAV, BV, LDBV,
         $                   CV, LDCV, DV, LDDV, AW, LDAW, BW, LDBW,
         $                   CW, LDCW, DW, LDDW, NS, HSV, TOL1, TOL2,
         $                   IWORK, DWORK, LDWORK, IWARN, INFO )
    C     .. Scalar Arguments ..
          CHARACTER         DICO, EQUIL, JOBINV, JOBV, JOBW, ORDSEL
          INTEGER           INFO, IWARN, LDA, LDAV, LDAW, LDB, LDBV, LDBW,
         $                  LDC, LDCV, LDCW, LDD, LDDV, LDDW, LDWORK, M, N,
         $                  NR, NS, NV, NW, P
          DOUBLE PRECISION  ALPHA, TOL1, TOL2
    C     .. Array Arguments ..
          INTEGER           IWORK(*)
          DOUBLE PRECISION  A(LDA,*), AV(LDAV,*), AW(LDAW,*),
         $                  B(LDB,*), BV(LDBV,*), BW(LDBW,*),
         $                  C(LDC,*), CV(LDCV,*), CW(LDCW,*),
         $                  D(LDD,*), DV(LDDV,*), DW(LDDW,*), DWORK(*),
         $                  HSV(*)

Arguments

Mode Parameters

      JOBV    CHARACTER*1
              Specifies the left frequency-weighting as follows:
              = 'N':  V = I;
              = 'V':  op(V) = V;
              = 'I':  op(V) = inv(V);
              = 'C':  op(V) = conj(V);
              = 'R':  op(V) = conj(inv(V)).

      JOBW    CHARACTER*1
              Specifies the right frequency-weighting as follows:
              = 'N':  W = I;
              = 'W':  op(W) = W;
              = 'I':  op(W) = inv(W);
              = 'C':  op(W) = conj(W);
              = 'R':  op(W) = conj(inv(W)).

      JOBINV  CHARACTER*1
              Specifies the computational approach to be used as
              follows:
              = 'N':  use the inverse free descriptor system approach;
              = 'I':  use the inversion based standard approach;
              = 'A':  switch automatically to the inverse free
                      descriptor approach in case of badly conditioned
                      feedthrough matrices in V or W (see METHOD).

      DICO    CHARACTER*1
              Specifies the type of the original system as follows:
              = 'C':  continuous-time system;
              = 'D':  discrete-time system.

      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.

      ORDSEL  CHARACTER*1
              Specifies the order selection method as follows:
              = 'F':  the resulting order NR is fixed;
              = 'A':  the resulting order NR is automatically determined
                      on basis of the given tolerance TOL1.

Input/Output Parameters

      N       (input) INTEGER
              The order of the original state-space representation,
              i.e., the order of the matrix A.  N >= 0.

      NV      (input) INTEGER
              The order of the realization of the left frequency
              weighting V, i.e., the order of the matrix AV.  NV >= 0.

      NW      (input) INTEGER
              The order of the realization of the right frequency
              weighting W, i.e., the order of the matrix AW.  NW >= 0.

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

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

      NR      (input/output) INTEGER
              On entry with ORDSEL = 'F', NR is the desired order of
              the resulting reduced order system.  0 <= NR <= N.
              On exit, if INFO = 0, NR is the order of the resulting
              reduced order model. For a system with NU ALPHA-unstable
              eigenvalues and NS ALPHA-stable eigenvalues (NU+NS = N),
              NR is set as follows: if ORDSEL = 'F', NR is equal to
              NU+MIN(MAX(0,NR-NU-KR+1),NMIN), where KR is the
              multiplicity of the Hankel singular value HSV(NR-NU+1),
              NR is the desired order on entry, and NMIN is the order
              of a minimal realization of the ALPHA-stable part of the
              given system; NMIN is determined as the number of Hankel
              singular values greater than NS*EPS*HNORM(As,Bs,Cs), where
              EPS is the machine precision (see LAPACK Library Routine
              DLAMCH) and HNORM(As,Bs,Cs) is the Hankel norm of the
              ALPHA-stable part of the weighted system (computed in
              HSV(1));
              if ORDSEL = 'A', NR is the sum of NU and the number of
              Hankel singular values greater than
              MAX(TOL1,NS*EPS*HNORM(As,Bs,Cs)).

      ALPHA   (input) DOUBLE PRECISION
              Specifies the ALPHA-stability boundary for the eigenvalues
              of the state dynamics matrix A. For a continuous-time
              system (DICO = 'C'), ALPHA <= 0 is the boundary value for
              the real parts of eigenvalues, while for a discrete-time
              system (DICO = 'D'), 0 <= ALPHA <= 1 represents the
              boundary value for the moduli of eigenvalues.
              The ALPHA-stability domain does not include the boundary.

      A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
              On entry, the leading N-by-N part of this array must
              contain the state dynamics matrix A.
              On exit, if INFO = 0, the leading NR-by-NR part of this
              array contains the state dynamics matrix Ar of the
              reduced order system in a real Schur form.
              The resulting A has a block-diagonal form with two blocks.
              For a system with NU ALPHA-unstable eigenvalues and
              NS ALPHA-stable eigenvalues (NU+NS = N), the leading
              NU-by-NU block contains the unreduced part of A
              corresponding to ALPHA-unstable eigenvalues.
              The trailing (NR+NS-N)-by-(NR+NS-N) block contains
              the reduced part of A corresponding to ALPHA-stable
              eigenvalues.

      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 original input/state matrix B.
              On exit, if INFO = 0, the leading NR-by-M part of this
              array contains the input/state matrix Br of the reduced
              order system.

      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 original state/output matrix C.
              On exit, if INFO = 0, the leading P-by-NR part of this
              array contains the state/output matrix Cr of the reduced
              order system.

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

      D       (input/output) DOUBLE PRECISION array, dimension (LDD,M)
              On entry, the leading P-by-M part of this array must
              contain the original input/output matrix D.
              On exit, if INFO = 0, the leading P-by-M part of this
              array contains the input/output matrix Dr of the reduced
              order system.

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

      AV      (input/output) DOUBLE PRECISION array, dimension (LDAV,NV)
              On entry, if JOBV <> 'N', the leading NV-by-NV part of
              this array must contain the state matrix AV of a state
              space realization of the left frequency weighting V.
              On exit, if JOBV <> 'N', and INFO = 0, the leading
              NV-by-NV part of this array contains the real Schur form
              of AV.
              AV is not referenced if JOBV = 'N'.

      LDAV    INTEGER
              The leading dimension of the array AV.
              LDAV >= MAX(1,NV), if JOBV <> 'N';
              LDAV >= 1,         if JOBV =  'N'.

      BV      (input/output) DOUBLE PRECISION array, dimension (LDBV,P)
              On entry, if JOBV <> 'N', the leading NV-by-P part of
              this array must contain the input matrix BV of a state
              space realization of the left frequency weighting V.
              On exit, if JOBV <> 'N', and INFO = 0, the leading
              NV-by-P part of this array contains the transformed
              input matrix BV corresponding to the transformed AV.
              BV is not referenced if JOBV = 'N'.

      LDBV    INTEGER
              The leading dimension of the array BV.
              LDBV >= MAX(1,NV), if JOBV <> 'N';
              LDBV >= 1,         if JOBV =  'N'.

      CV      (input/output) DOUBLE PRECISION array, dimension (LDCV,NV)
              On entry, if JOBV <> 'N', the leading P-by-NV part of
              this array must contain the output matrix CV of a state
              space realization of the left frequency weighting V.
              On exit, if JOBV <> 'N', and INFO = 0, the leading
              P-by-NV part of this array contains the transformed output
              matrix CV corresponding to the transformed AV.
              CV is not referenced if JOBV = 'N'.

      LDCV    INTEGER
              The leading dimension of the array CV.
              LDCV >= MAX(1,P), if JOBV <> 'N';
              LDCV >= 1,        if JOBV =  'N'.

      DV      (input) DOUBLE PRECISION array, dimension (LDDV,P)
              If JOBV <> 'N', the leading P-by-P part of this array
              must contain the feedthrough matrix DV of a state space
              realization of the left frequency weighting V.
              DV is not referenced if JOBV = 'N'.

      LDDV    INTEGER
              The leading dimension of the array DV.
              LDDV >= MAX(1,P), if JOBV <> 'N';
              LDDV >= 1,        if JOBV =  'N'.

      AW      (input/output) DOUBLE PRECISION array, dimension (LDAW,NW)
              On entry, if JOBW <> 'N', the leading NW-by-NW part of
              this array must contain the state matrix AW of a state
              space realization of the right frequency weighting W.
              On exit, if JOBW <> 'N', and INFO = 0, the leading
              NW-by-NW part of this array contains the real Schur form
              of AW.
              AW is not referenced if JOBW = 'N'.

      LDAW    INTEGER
              The leading dimension of the array AW.
              LDAW >= MAX(1,NW), if JOBW <> 'N';
              LDAW >= 1,         if JOBW =  'N'.

      BW      (input/output) DOUBLE PRECISION array, dimension (LDBW,M)
              On entry, if JOBW <> 'N', the leading NW-by-M part of
              this array must contain the input matrix BW of a state
              space realization of the right frequency weighting W.
              On exit, if JOBW <> 'N', and INFO = 0, the leading
              NW-by-M part of this array contains the transformed
              input matrix BW corresponding to the transformed AW.
              BW is not referenced if JOBW = 'N'.

      LDBW    INTEGER
              The leading dimension of the array BW.
              LDBW >= MAX(1,NW), if JOBW <> 'N';
              LDBW >= 1,         if JOBW =  'N'.

      CW      (input/output) DOUBLE PRECISION array, dimension (LDCW,NW)
              On entry, if JOBW <> 'N', the leading M-by-NW part of
              this array must contain the output matrix CW of a state
              space realization of the right frequency weighting W.
              On exit, if JOBW <> 'N', and INFO = 0, the leading
              M-by-NW part of this array contains the transformed output
              matrix CW corresponding to the transformed AW.
              CW is not referenced if JOBW = 'N'.

      LDCW    INTEGER
              The leading dimension of the array CW.
              LDCW >= MAX(1,M), if JOBW <> 'N';
              LDCW >= 1,        if JOBW =  'N'.

      DW      (input) DOUBLE PRECISION array, dimension (LDDW,M)
              If JOBW <> 'N', the leading M-by-M part of this array
              must contain the feedthrough matrix DW of a state space
              realization of the right frequency weighting W.
              DW is not referenced if JOBW = 'N'.

      LDDW    INTEGER
              The leading dimension of the array DW.
              LDDW >= MAX(1,M), if JOBW <> 'N';
              LDDW >= 1,        if JOBW =  'N'.

      NS      (output) INTEGER
              The dimension of the ALPHA-stable subsystem.

      HSV     (output) DOUBLE PRECISION array, dimension (N)
              If INFO = 0, the leading NS elements of this array contain
              the Hankel singular values, ordered decreasingly, of the
              projection G1s of op(V)*G1*op(W) (see METHOD), where G1
              is the ALPHA-stable part of the original system.

Tolerances

      TOL1    DOUBLE PRECISION
              If ORDSEL = 'A', TOL1 contains the tolerance for
              determining the order of reduced system.
              For model reduction, the recommended value is
              TOL1 = c*HNORM(G1s), where c is a constant in the
              interval [0.00001,0.001], and HNORM(G1s) is the
              Hankel-norm of the projection G1s of op(V)*G1*op(W)
              (see METHOD), computed in HSV(1).
              If TOL1 <= 0 on entry, the used default value is
              TOL1 = NS*EPS*HNORM(G1s), where NS is the number of
              ALPHA-stable eigenvalues of A and EPS is the machine
              precision (see LAPACK Library Routine DLAMCH).
              If ORDSEL = 'F', the value of TOL1 is ignored.
              TOL1 < 1.

      TOL2    DOUBLE PRECISION
              The tolerance for determining the order of a minimal
              realization of the ALPHA-stable part of the given system.
              The recommended value is TOL2 = NS*EPS*HNORM(G1s).
              This value is used by default if TOL2 <= 0 on entry.
              If TOL2 > 0 and ORDSEL = 'A', then TOL2 <= TOL1.
              TOL2 < 1.

Workspace

      IWORK   INTEGER array, dimension (LIWORK)
              LIWORK = MAX(1,M,c,d),    if DICO = 'C',
              LIWORK = MAX(1,N,M,c,d),  if DICO = 'D', where
                 c = 0,                          if JOBV =  'N',
                 c = MAX(2*P,NV+P+N+6,2*NV+P+2), if JOBV <> 'N',
                 d = 0,                          if JOBW =  'N',
                 d = MAX(2*M,NW+M+N+6,2*NW+M+2), if JOBW <> '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( LDW1, LDW2, LDW3, LDW4 ), where
              for NVP = NV+P and NWM = NW+M we have
              LDW1 = 0 if JOBV =  'N' and
              LDW1 = 2*NVP*(NVP+P) + P*P +
                     MAX( 2*NVP*NVP + MAX( 11*NVP+16, P*NVP ),
                           NVP*N + MAX( NVP*N+N*N, P*N, P*M ) )
                       if JOBV <> 'N',
              LDW2 = 0 if JOBW =  'N' and
              LDW2 = 2*NWM*(NWM+M) + M*M +
                     MAX( 2*NWM*NWM + MAX( 11*NWM+16, M*NWM ),
                           NWM*N + MAX( NWM*N+N*N, M*N, P*M ) )
                       if JOBW <> 'N',
              LDW3 = N*(2*N + MAX(N,M,P) + 5) + N*(N+1)/2,
              LDW4 = N*(M+P+2) + 2*M*P + MIN(N,M) +
                     MAX( 3*M+1, MIN(N,M)+P ).
              For optimum performance LDWORK should be larger.

Warning Indicator

      IWARN   INTEGER
              = 0:  no warning;
              = 1:  with ORDSEL = 'F', the selected order NR is greater
                    than NSMIN, the sum of the order of the
                    ALPHA-unstable part and the order of a minimal
                    realization of the ALPHA-stable part of the given
                    system. In this case, the resulting NR is set equal
                    to NSMIN.
              = 2:  with ORDSEL = 'F', the selected order NR is less
                    than the order of the ALPHA-unstable part of the
                    given system. In this case NR is set equal to the
                    order of the ALPHA-unstable part.

Error Indicator

      INFO    INTEGER
              =  0:  successful exit;
              <  0:  if INFO = -i, the i-th argument had an illegal
                     value;
              =  1:  the computation of the ordered real Schur form of A
                     failed;
              =  2:  the separation of the ALPHA-stable/unstable
                     diagonal blocks failed because of very close
                     eigenvalues;
              =  3:  the reduction of AV to a real Schur form failed;
              =  4:  the reduction of AW to a real Schur form failed;
              =  5:  the reduction to generalized Schur form of the
                     descriptor pair corresponding to the inverse of V
                     failed;
              =  6:  the reduction to generalized Schur form of the
                     descriptor pair corresponding to the inverse of W
                     failed;
              =  7:  the computation of Hankel singular values failed;
              =  8:  the computation of stable projection in the
                     Hankel-norm approximation algorithm failed;
              =  9:  the order of computed stable projection in the
                     Hankel-norm approximation algorithm differs
                     from the order of Hankel-norm approximation;
              = 10:  the reduction of AV-BV*inv(DV)*CV to a
                     real Schur form failed;
              = 11:  the reduction of AW-BW*inv(DW)*CW to a
                     real Schur form failed;
              = 12:  the solution of the Sylvester equation failed
                     because the poles of V (if JOBV = 'V') or of
                     conj(V) (if JOBV = 'C') are not distinct from
                     the poles of G1 (see METHOD);
              = 13:  the solution of the Sylvester equation failed
                     because the poles of W (if JOBW = 'W') or of
                     conj(W) (if JOBW = 'C') are not distinct from
                     the poles of G1 (see METHOD);
              = 14:  the solution of the Sylvester equation failed
                     because the zeros of V (if JOBV = 'I') or of
                     conj(V) (if JOBV = 'R') are not distinct from
                     the poles of G1sr (see METHOD);
              = 15:  the solution of the Sylvester equation failed
                     because the zeros of W (if JOBW = 'I') or of
                     conj(W) (if JOBW = 'R') are not distinct from
                     the poles of G1sr (see METHOD);
              = 16:  the solution of the generalized Sylvester system
                     failed because the zeros of V (if JOBV = 'I') or
                     of conj(V) (if JOBV = 'R') are not distinct from
                     the poles of G1sr (see METHOD);
              = 17:  the solution of the generalized Sylvester system
                     failed because the zeros of W (if JOBW = 'I') or
                     of conj(W) (if JOBW = 'R') are not distinct from
                     the poles of G1sr (see METHOD);
              = 18:  op(V) is not antistable;
              = 19:  op(W) is not antistable;
              = 20:  V is not invertible;
              = 21:  W is not invertible.

Method

      Let G be the transfer-function matrix of the original
      linear system

           d[x(t)] = Ax(t) + Bu(t)
           y(t)    = Cx(t) + Du(t),                          (1)

      where d[x(t)] is dx(t)/dt for a continuous-time system and x(t+1)
      for a discrete-time system. The subroutine AB09JD determines
      the matrices of a reduced order system

           d[z(t)] = Ar*z(t) + Br*u(t)
           yr(t)   = Cr*z(t) + Dr*u(t),                      (2)

      such that the corresponding transfer-function matrix Gr minimizes
      the Hankel-norm of the frequency-weighted error

              op(V)*(G-Gr)*op(W).                            (3)

      For minimizing (3) with op(V) = V and op(W) = W, V and W are
      assumed to have poles distinct from those of G, while with
      op(V) = conj(V) and op(W) = conj(W), conj(V) and conj(W) are
      assumed to have poles distinct from those of G. For minimizing (3)
      with op(V) = inv(V) and op(W) = inv(W), V and W are assumed to
      have zeros distinct from the poles of G, while with
      op(V) = conj(inv(V)) and op(W) = conj(inv(W)), conj(V) and conj(W)
      are assumed to have zeros distinct from the poles of G.

      Note: conj(G) = G'(-s) for a continuous-time system and
            conj(G) = G'(1/z) for a discrete-time system.

      The following procedure is used to reduce G (see [1]):

      1) Decompose additively G as

           G = G1 + G2,

         such that G1 = (A1,B1,C1,D) has only ALPHA-stable poles and
         G2 = (A2,B2,C2,0) has only ALPHA-unstable poles.

      2) Compute G1s, the projection of op(V)*G1*op(W) containing the
         poles of G1, using explicit formulas [4] or the inverse-free
         descriptor system formulas of [5].

      3) Determine G1sr, the optimal Hankel-norm approximation of G1s,
         of order r.

      4) Compute G1r, the projection of inv(op(V))*G1sr*inv(op(W))
         containing the poles of G1sr, using explicit formulas [4]
         or the inverse-free descriptor system formulas of [5].

      5) Assemble the reduced model Gr as

            Gr = G1r + G2.

      To reduce the weighted ALPHA-stable part G1s at step 3, the
      optimal Hankel-norm approximation method of [2], based on the
      square-root balancing projection formulas of [3], is employed.

      The optimal weighted approximation error satisfies

           HNORM[op(V)*(G-Gr)*op(W)] >= S(r+1),

      where S(r+1) is the (r+1)-th Hankel singular value of G1s, the
      transfer-function matrix computed at step 2 of the above
      procedure, and HNORM(.) denotes the Hankel-norm.

References

      [1] Latham, G.A. and Anderson, B.D.O.
          Frequency-weighted optimal Hankel-norm approximation of stable
          transfer functions.
          Systems & Control Letters, Vol. 5, pp. 229-236, 1985.

      [2] Glover, K.
          All optimal Hankel norm approximation of linear
          multivariable systems and their L-infinity error bounds.
          Int. J. Control, Vol. 36, pp. 1145-1193, 1984.

      [3] Tombs, M.S. and Postlethwaite, I.
          Truncated balanced realization of stable, non-minimal
          state-space systems.
          Int. J. Control, Vol. 46, pp. 1319-1330, 1987.

      [4] Varga, A.
          Explicit formulas for an efficient implementation
          of the frequency-weighting model reduction approach.
          Proc. 1993 European Control Conference, Groningen, NL,
          pp. 693-696, 1993.

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

Numerical Aspects

      The implemented methods rely on an accuracy enhancing square-root
      technique.

Further Comments

      None

Example

Program Text

    *     AB09JD EXAMPLE PROGRAM TEXT
    *
    *     .. Parameters ..
          INTEGER          NIN, NOUT
          PARAMETER        ( NIN = 5, NOUT = 6 )
          INTEGER          MMAX, NMAX, NVMAX, NVPMAX, NWMAX, NWMMAX, PMAX
          PARAMETER        ( MMAX = 20, NMAX = 20, NVMAX = 10, NWMAX = 10,
         $                   PMAX = 20, NVPMAX = NVMAX + PMAX,
         $                   NWMMAX = NWMAX + MMAX )
          INTEGER          LDA, LDAV, LDAW, LDB, LDBV, LDBW,
         $                 LDC, LDCV, LDCW, LDD, LDDV, LDDW
          PARAMETER        ( LDA = NMAX, LDAV = NVMAX, LDAW = NWMAX,
         $                   LDB = NMAX, LDBV = NVMAX, LDBW = NWMAX,
         $                   LDC = PMAX, LDCV = PMAX,  LDCW = MMAX,
         $                   LDD = PMAX, LDDV = PMAX,  LDDW = MMAX )
          INTEGER          LIW1, LIW2, LIW3, LIWORK
          PARAMETER        ( LIW1 = 2*MAX( MMAX, PMAX ),
         $                   LIW2 = MAX( NVPMAX, NWMMAX ) + NMAX + 6,
         $                   LIW3 = MAX( 2*NVMAX + PMAX + 2,
         $                               2*NWMAX + MMAX + 2 ) )
          PARAMETER        ( LIWORK = MAX( LIW1, LIW2, LIW3 ) )
          INTEGER          LDW1, LDW2, LDW3, LDW4, LDWORK
          PARAMETER        ( LDW1 = 2*NVPMAX*( NVPMAX + PMAX ) + PMAX*PMAX +
         $                          MAX( 2*NVPMAX*NVPMAX +
         $                               MAX( 11*NVPMAX + 16, PMAX*NVPMAX ),
         $                               NVPMAX*NMAX +
         $                               MAX( NVPMAX*NMAX + NMAX*NMAX,
         $                                    PMAX*NMAX, PMAX*MMAX ) ) )
          PARAMETER        ( LDW2 = 2*NWMMAX*( NWMMAX + MMAX ) + MMAX*MMAX +
         $                          MAX( 2*NWMMAX*NWMMAX +
         $                               MAX( 11*NWMMAX + 16, MMAX*NWMMAX ),
         $                               NWMMAX*NMAX +
         $                               MAX( NWMMAX*NMAX + NMAX*NMAX,
         $                                    MMAX*NMAX, PMAX*MMAX ) ) )
          PARAMETER        ( LDW3 = NMAX*( 2*NMAX + MAX( NMAX, MMAX, PMAX )
         $                                 + 5 ) + ( NMAX*( NMAX + 1 ) )/2 )
          PARAMETER        ( LDW4 = NMAX*( MMAX + PMAX + 2 ) + 2*MMAX*PMAX +
         $                          MIN( NMAX, MMAX ) +
         $                          MAX( 3*MMAX + 1,
         $                               MIN( NMAX, MMAX ) + PMAX ) )
          PARAMETER        ( LDWORK = MAX( LDW1, LDW2, LDW3, LDW4 ) )
    *     .. Local Scalars ..
          LOGICAL          LEFTW, RIGHTW
          DOUBLE PRECISION ALPHA, TOL1, TOL2
          INTEGER          I, INFO, IWARN, J, M, N, NR, NS, NV, NW, P
          CHARACTER*1      DICO, EQUIL, JOBINV, JOBV, JOBW, ORDSEL
    *     .. Local Arrays ..
          DOUBLE PRECISION A(LDA,NMAX), AV(LDAV,NVMAX), AW(LDAW,NWMAX),
         $                 B(LDB,MMAX), BV(LDBV,PMAX),  BW(LDBW,MMAX),
         $                 C(LDC,NMAX), CV(LDCV,NVMAX), CW(LDCW,NWMAX),
         $                 D(LDD,MMAX), DV(LDDV,PMAX),  DW(LDDW,MMAX),
         $                 DWORK(LDWORK), HSV(NMAX)
          INTEGER          IWORK(LIWORK)
    *     .. External Functions ..
          LOGICAL          LSAME
          EXTERNAL         LSAME
    *     .. External Subroutines ..
          EXTERNAL         AB09JD
    *     .. Intrinsic Functions ..
          INTRINSIC        MAX, MIN
    *     .. 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, NV, NW, NR, ALPHA, TOL1, TOL2,
         $                      JOBV, JOBW, JOBINV, DICO, EQUIL, ORDSEL
          LEFTW  = .NOT.LSAME( JOBV, 'N' )
          RIGHTW = .NOT.LSAME( JOBW, 'N' )
          IF( N.LE.0 .OR. N.GT.NMAX ) THEN
             WRITE ( NOUT, FMT = 99990 ) N
          ELSE
             READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
             IF( M.LE.0 .OR. M.GT.MMAX ) THEN
                WRITE ( NOUT, FMT = 99989 ) M
             ELSE
                READ ( NIN, FMT = * ) ( ( B(I,J), J = 1,M ), I = 1,N )
                IF( P.LE.0 .OR. P.GT.PMAX ) THEN
                   WRITE ( NOUT, FMT = 99988 ) 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 )
                   IF( LEFTW ) THEN
                      IF( NV.LT.0 .OR. NV.GT.NVMAX ) THEN
                         WRITE ( NOUT, FMT = 99986 ) NV
                      ELSE
                         IF( NV.GT.0 ) THEN
                            READ ( NIN, FMT = * )
         $                     ( ( AV(I,J), J = 1,NV ), I = 1,NV )
                            READ ( NIN, FMT = * )
         $                     ( ( BV(I,J), J = 1,P  ), I = 1,NV )
                            READ ( NIN, FMT = * )
         $                     ( ( CV(I,J), J = 1,NV ), I = 1,P )
                         END IF
                         IF( LEFTW )
         $                  READ ( NIN, FMT = * )
         $                     ( ( DV(I,J), J = 1,P ), I = 1,P )
                      END IF
                   END IF
                   IF( RIGHTW ) THEN
                      IF( NW.LT.0 .OR. NW.GT.NWMAX ) THEN
                         WRITE ( NOUT, FMT = 99985 ) NW
                      ELSE
                         IF( NW.GT.0 ) THEN
                            READ ( NIN, FMT = * )
         $                     ( ( AW(I,J), J = 1,NW ), I = 1,NW )
                            READ ( NIN, FMT = * )
         $                     ( ( BW(I,J), J = 1,M  ), I = 1,NW )
                            READ ( NIN, FMT = * )
         $                     ( ( CW(I,J), J = 1,NW ), I = 1,M )
                         END IF
                         READ ( NIN, FMT = * )
         $                     ( ( DW(I,J), J = 1,M  ), I = 1,M )
                      END IF
                   END IF
    *              Find a reduced ssr for (A,B,C,D).
                   CALL AB09JD( JOBV, JOBW, JOBINV, DICO, EQUIL, ORDSEL, N,
         $                      NV, NW, M, P, NR, ALPHA,  A, LDA, B, LDB,
         $                      C, LDC, D, LDD, AV, LDAV, BV, LDBV,
         $                      CV, LDCV, DV, LDDV, AW, LDAW, BW, LDBW,
         $                      CW, LDCW, DW, LDDW, NS, HSV, TOL1, TOL2,
         $                      IWORK, DWORK, LDWORK, IWARN, INFO )
    *
                   IF ( INFO.NE.0 ) THEN
                      WRITE ( NOUT, FMT = 99998 ) INFO
                   ELSE
                      IF( IWARN.NE.0 ) WRITE ( NOUT, FMT = 99994 ) IWARN
                      WRITE ( NOUT, FMT = 99997 ) NR
                      WRITE ( NOUT, FMT = 99987 )
                      WRITE ( NOUT, FMT = 99995 ) ( HSV(J), J = 1, NS )
                      IF( NR.GT.0 ) THEN
                         WRITE ( NOUT, FMT = 99996 )
                         DO 20 I = 1, NR
                            WRITE ( NOUT, FMT = 99995 ) ( A(I,J), J = 1,NR )
       20                CONTINUE
                         WRITE ( NOUT, FMT = 99993 )
                         DO 40 I = 1, NR
                            WRITE ( NOUT, FMT = 99995 ) ( B(I,J), J = 1,M )
       40                CONTINUE
                         WRITE ( NOUT, FMT = 99992 )
                         DO 60 I = 1, P
                            WRITE ( NOUT, FMT = 99995 ) ( C(I,J), J = 1,NR )
       60                CONTINUE
                      END IF
                      WRITE ( NOUT, FMT = 99991 )
                      DO 70 I = 1, P
                         WRITE ( NOUT, FMT = 99995 ) ( D(I,J), J = 1,M )
       70             CONTINUE
                   END IF
                END IF
             END IF
          END IF
          STOP
    *
    99999 FORMAT (' AB09JD EXAMPLE PROGRAM RESULTS',/1X)
    99998 FORMAT (' INFO on exit from AB09JD = ',I2)
    99997 FORMAT (/' The order of reduced model = ',I2)
    99996 FORMAT (/' The reduced state dynamics matrix Ar is ')
    99995 FORMAT (20(1X,F8.4))
    99994 FORMAT (' IWARN on exit from AB09JD = ',I2)
    99993 FORMAT (/' The reduced input/state matrix Br is ')
    99992 FORMAT (/' The reduced state/output matrix Cr is ')
    99991 FORMAT (/' The reduced input/output matrix Dr is ')
    99990 FORMAT (/' N is out of range.',/' N = ',I5)
    99989 FORMAT (/' M is out of range.',/' M = ',I5)
    99988 FORMAT (/' P is out of range.',/' P = ',I5)
    99987 FORMAT (/' The Hankel singular values of weighted ALPHA-stable',
         $         ' part are')
    99986 FORMAT (/' NV is out of range.',/' NV = ',I5)
    99985 FORMAT (/' NW is out of range.',/' NW = ',I5)
          END

Program Data

     AB09JD EXAMPLE PROGRAM DATA (Continuous system)
      6     1     1     2   0     0   0.0  1.E-1  1.E-14    V   N   I   C    S   A
       -3.8637   -7.4641   -9.1416   -7.4641   -3.8637   -1.0000
        1.0000         0         0         0         0         0
             0    1.0000         0         0         0         0
             0         0    1.0000         0         0         0
             0         0         0    1.0000         0         0
             0         0         0         0    1.0000         0
             1
             0
             0
             0
             0
             0
             0         0         0         0         0         1
             0
        0.2000   -1.0000
        1.0000         0
         1
         0
       -1.8000         0
         1

Program Results

     AB09JD EXAMPLE PROGRAM RESULTS

     The order of reduced model =  4

     The Hankel singular values of weighted ALPHA-stable part are
       2.6790   2.1589   0.8424   0.1929   0.0219   0.0011

     The reduced state dynamics matrix Ar is
      -0.2391   0.3072   1.1630   1.1967
      -2.9709  -0.2391   2.6270   3.1027
       0.0000   0.0000  -0.5137  -1.2842
       0.0000   0.0000   0.1519  -0.5137

     The reduced input/state matrix Br is
      -1.0497
      -3.7052
       0.8223
       0.7435

     The reduced state/output matrix Cr is
      -0.4466   0.0143  -0.4780  -0.2013

     The reduced input/output matrix Dr is
       0.0219

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