AB09CX

Optimal Hankel-norm approximation based model reduction for stable systems with state matrix in real Schur canonical form

Purpose

  To compute a reduced order model (Ar,Br,Cr,Dr) for a stable
  original state-space representation (A,B,C,D) by using the optimal
  Hankel-norm approximation method in conjunction with square-root
  balancing. The state dynamics matrix A of the original system is
  an upper quasi-triangular matrix in real Schur canonical form.

      SUBROUTINE AB09CX( DICO, ORDSEL, N, M, P, NR, A, LDA, B, LDB,
     $                   C, LDC, D, LDD, HSV, TOL1, TOL2, IWORK,
     $                   DWORK, LDWORK, IWARN, INFO )
C     .. Scalar Arguments ..
      CHARACTER         DICO, ORDSEL
      INTEGER           INFO, IWARN, LDA, LDB, LDC, LDD, LDWORK,
     $                  M, N, NR, P
      DOUBLE PRECISION  TOL1, TOL2
C     .. Array Arguments ..
      INTEGER           IWORK(*)
      DOUBLE PRECISION  A(LDA,*), B(LDB,*), C(LDC,*), D(LDD,*),
     $                  DWORK(*), HSV(*)

Mode Parameters

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

  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.

  N       (input) INTEGER
          The order of the original state-space representation, i.e.
          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.

  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. NR is set as follows:
          if ORDSEL = 'F', NR is equal to MIN(MAX(0,NR-KR+1),NMIN),
          where KR is the multiplicity of the Hankel singular value
          HSV(NR+1), NR is the desired order on entry, and NMIN is
          the order of a minimal realization of the given system;
          NMIN is determined as the number of Hankel singular values
          greater than N*EPS*HNORM(A,B,C), where EPS is the machine
          precision (see LAPACK Library Routine DLAMCH) and
          HNORM(A,B,C) is the Hankel norm of the system (computed
          in HSV(1));
          if ORDSEL = 'A', NR is equal to the number of Hankel
          singular values greater than MAX(TOL1,N*EPS*HNORM(A,B,C)).

  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 in a real Schur
          canonical form.
          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.

  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).

  HSV     (output) DOUBLE PRECISION array, dimension (N)
          If INFO = 0, it contains the Hankel singular values of
          the original system ordered decreasingly. HSV(1) is the
          Hankel norm of the system.

  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(A,B,C), where c is a constant in the
          interval [0.00001,0.001], and HNORM(A,B,C) is the
          Hankel-norm of the given system (computed in HSV(1)).
          For computing a minimal realization, the recommended
          value is TOL1 = N*EPS*HNORM(A,B,C), where EPS is the
          machine precision (see LAPACK Library Routine DLAMCH).
          This value is used by default if TOL1 <= 0 on entry.
          If ORDSEL = 'F', the value of TOL1 is ignored.

  TOL2    DOUBLE PRECISION
          The tolerance for determining the order of a minimal
          realization of the given system. The recommended value is
          TOL2 = N*EPS*HNORM(A,B,C). This value is used by default
          if TOL2 <= 0 on entry.
          If TOL2 > 0, then TOL2 <= TOL1.

  IWORK   INTEGER array, dimension (LIWORK)
          LIWORK = MAX(1,M),   if DICO = 'C';
          LIWORK = MAX(1,N,M), if DICO = 'D'.
          On exit, if INFO = 0, IWORK(1) contains NMIN, the order of
          the computed minimal realization.

  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 ), where
          LDW1 = N*(2*N+MAX(N,M,P)+5) + N*(N+1)/2,
          LDW2 = 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.

  IWARN   INTEGER
          = 0:  no warning;
          = 1:  with ORDSEL = 'F', the selected order NR is greater
                than the order of a minimal realization of the
                given system. In this case, the resulting NR is set
                automatically to a value corresponding to the order
                of a minimal realization of the system.

  INFO    INTEGER
          = 0:  successful exit;
          < 0:  if INFO = -i, the i-th argument had an illegal
                value;
          = 1:  the state matrix A is not stable (if DICO = 'C')
                or not convergent (if DICO = 'D');
          = 2:  the computation of Hankel singular values failed;
          = 3:  the computation of stable projection failed;
          = 4:  the order of computed stable projection differs
                from the order of Hankel-norm approximation.

  Let be the stable 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 AB09CX determines for
  the given system (1), 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

        HSV(NR) <= INFNORM(G-Gr) <= 2*[HSV(NR+1) + ... + HSV(N)],

  where G and Gr are transfer-function matrices of the systems
  (A,B,C,D) and (Ar,Br,Cr,Dr), respectively, and INFNORM(G) is the
  infinity-norm of G.

  The optimal Hankel-norm approximation method of [1], based on the
  square-root balancing projection formulas of [2], is employed.

  [1] 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.

  [2] 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.

  The implemented methods rely on an accuracy enhancing square-root
  technique.
                                      3
  The algorithms require less than 30N  floating point operations.

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