AB09AX

Balance & Truncate model reduction for stable systems with state matrix in real Schur canonical form

Purpose

  To compute a reduced order model (Ar,Br,Cr) for a stable original
  state-space representation (A,B,C) by using either the square-root
  or the balancing-free square-root Balance & Truncate model
  reduction method. The state dynamics matrix A of the original
  system is an upper quasi-triangular matrix in real Schur canonical
  form. The matrices of the reduced order system are computed using
  the truncation formulas:

       Ar = TI * A * T ,  Br = TI * B ,  Cr = C * T .

      SUBROUTINE AB09AX( DICO, JOB, ORDSEL, N, M, P, NR, A, LDA, B, LDB,
     $                   C, LDC, HSV, T, LDT, TI, LDTI, TOL, IWORK,
     $                   DWORK, LDWORK, IWARN, INFO )
C     .. Scalar Arguments ..
      CHARACTER         DICO, JOB, ORDSEL
      INTEGER           INFO, IWARN, LDA, LDB, LDC, LDT, LDTI, LDWORK,
     $                  M, N, NR, P
      DOUBLE PRECISION  TOL
C     .. Array Arguments ..
      INTEGER           IWORK(*)
      DOUBLE PRECISION  A(LDA,*), B(LDB,*), C(LDC,*), DWORK(*), HSV(*),
     $                  T(LDT,*), TI(LDTI,*)

Mode Parameters

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

  JOB     CHARACTER*1
          Specifies the model reduction approach to be used
          as follows:
          = 'B':  use the square-root Balance & Truncate method;
          = 'N':  use the balancing-free square-root
                  Balance & Truncate method.

  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 TOL.

  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(NR,NMIN), where 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(TOL,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.

  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).

  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.

  T       (output) DOUBLE PRECISION array, dimension (LDT,N)
          If INFO = 0 and NR > 0, the leading N-by-NR part of this
          array contains the right truncation matrix T.

  LDT     INTEGER
          The leading dimension of array T.  LDT >= MAX(1,N).

  TI      (output) DOUBLE PRECISION array, dimension (LDTI,N)
          If INFO = 0 and NR > 0, the leading NR-by-N part of this
          array contains the left truncation matrix TI.

  LDTI    INTEGER
          The leading dimension of array TI.  LDTI >= MAX(1,N).

  TOL     DOUBLE PRECISION
          If ORDSEL = 'A', TOL contains the tolerance for
          determining the order of reduced system.
          For model reduction, the recommended value is
          TOL = 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 TOL = N*EPS*HNORM(A,B,C), where EPS is the
          machine precision (see LAPACK Library Routine DLAMCH).
          This value is used by default if TOL <= 0 on entry.
          If ORDSEL = 'F', the value of TOL is ignored.

  IWORK   INTEGER array, dimension (LIWORK)
          LIWORK = 0, if JOB = 'B', or
          LIWORK = N, if JOB = '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*(MAX(N,M,P)+5) + N*(N+1)/2).
          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.

  Let be the stable linear system

       d[x(t)] = Ax(t) + Bu(t)
       y(t)    = Cx(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 AB09AX determines for
  the given system (1), the matrices of a reduced NR order system

       d[z(t)] = Ar*z(t) + Br*u(t)
       yr(t)   = Cr*z(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) and (Ar,Br,Cr), respectively, and INFNORM(G) is the
  infinity-norm of G.

  If JOB = 'B', the square-root Balance & Truncate method of [1]
  is used and, for DICO = 'C', the resulting model is balanced.
  By setting TOL <= 0, the routine can be used to compute balanced
  minimal state-space realizations of stable systems.

  If JOB = 'N', the balancing-free square-root version of the
  Balance & Truncate method [2] is used.
  By setting TOL <= 0, the routine can be used to compute minimal
  state-space realizations of stable systems.

  [1] 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.

  [2] Varga A.
      Efficient minimal realization procedure based on balancing.
      Proc. of IMACS/IFAC Symp. MCTS, Lille, France, May 1991,
      A. El Moudui, P. Borne, S. G. Tzafestas (Eds.),
      Vol. 2, pp. 42-46.

  The implemented methods rely on accuracy enhancing square-root or
  balancing-free square-root techniques.
                                      3
  The algorithms require less than 30N  floating point operations.

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Program Text

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