## AB09CX

### Optimal Hankel-norm approximation based 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,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.

Specification

          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(*)

Arguments

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.

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(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.

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(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.

Workspace

      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.

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;
              = 3:  the computation of stable projection failed;
              = 4:  the order of computed stable projection differs
                    from the order of Hankel-norm approximation.

Method

      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.

References

      [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.

Numerical Aspects

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