## AB09AX

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

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

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 .

Specification

          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,*)

Arguments

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.

Input/Output Parameters

      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).

Tolerances

      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.

Workspace

      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.

Warning Indicator

      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.

Error Indicator

      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.

Method

      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.

References

      [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.

Numerical Aspects

      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.

Further Comments

      None

Example

Program Text

      None

Program Data

      None

Program Results

      None

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