## AB09MD

### Balance & Truncate model reduction for the stable part of a system

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

Purpose

      To compute a reduced order model (Ar,Br,Cr) for an original
      state-space representation (A,B,C) by using either the square-root
      or the balancing-free square-root Balance & Truncate (B & T)
      model reduction method for the ALPHA-stable part of the system.

Specification

          SUBROUTINE AB09MD( DICO, JOB, EQUIL, ORDSEL, N, M, P, NR, ALPHA,
         $                   A, LDA, B, LDB, C, LDC, NS, HSV, TOL, IWORK,
         $                   DWORK, LDWORK, IWARN, INFO )
    C     .. Scalar Arguments ..
          CHARACTER         DICO, EQUIL, JOB, ORDSEL
          INTEGER           INFO, IWARN, LDA, LDB, LDC, LDWORK, M, N, NR,
         $                  NS, P
          DOUBLE PRECISION  ALPHA, TOL
    C     .. Array Arguments ..
          INTEGER           IWORK(*)
          DOUBLE PRECISION  A(LDA,*), B(LDB,*), C(LDC,*), 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.

      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.

      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 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. 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),NMIN), where 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 given system (computed in HSV(1));
              if ORDSEL = 'A', NR is the sum of NU and the number of
              Hankel singular values greater than
              MAX(TOL,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.
              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 in an
              upper real Schur form.
              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).

      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 HSV contain the
              Hankel singular values of the ALPHA-stable part of the
              original system ordered decreasingly.
              HSV(1) is the Hankel norm of the ALPHA-stable subsystem.

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(As,Bs,Cs), where c is a constant in the
              interval [0.00001,0.001], and HNORM(As,Bs,Cs) is the
              Hankel-norm of the ALPHA-stable part of the given system
              (computed in HSV(1)).
              If TOL <= 0 on entry, the used default value is
              TOL = NS*EPS*HNORM(As,Bs,Cs), where NS is the number of
              ALPHA-stable eigenvalues of A and EPS is the machine
              precision (see LAPACK Library Routine DLAMCH).
              This value is appropriate to compute a minimal realization
              of the ALPHA-stable part.
              If ORDSEL = 'F', the value of TOL is ignored.

Workspace

      IWORK   INTEGER array, dimension (LIWORK)
              LIWORK = 0, if JOB = 'B';
              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*(2*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 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 computation of Hankel singular values failed.

Method

      Let be the following 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 AB09MD 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)                             (2)

      such that

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

      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.

      The following procedure is used to reduce a given G:

      1) Decompose additively G as

           G = G1 + G2

         such that G1 = (As,Bs,Cs) has only ALPHA-stable poles and
         G2 = (Au,Bu,Cu) has only ALPHA-unstable poles.

      2) Determine G1r, a reduced order approximation of the
         ALPHA-stable part G1.

      3) Assemble the reduced model Gr as

            Gr = G1r + G2.

      To reduce the ALPHA-stable part G1, if JOB = 'B', the square-root
      Balance & Truncate method of [1] is used, and for an ALPHA-stable
      continuous-time system (DICO = 'C'), the resulting reduced model
      is balanced. For ALPHA-stable systems, setting TOL < 0, the
      routine can be used to compute balanced minimal state-space
      realizations.

      If JOB = 'N', the balancing-free square-root version of the
      Balance & Truncate method [2] is used to reduce the ALPHA-stable
      part G1.

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

    *     AB09MD EXAMPLE PROGRAM TEXT
    *
    *     .. Parameters ..
          INTEGER          NIN, NOUT
          PARAMETER        ( NIN = 5, NOUT = 6 )
          INTEGER          NMAX, MMAX, PMAX
          PARAMETER        ( NMAX = 20, MMAX = 20, PMAX = 20 )
          INTEGER          LDA, LDB, LDC
          PARAMETER        ( LDA = NMAX, LDB = NMAX, LDC = PMAX )
          INTEGER          LIWORK
          PARAMETER        ( LIWORK = NMAX )
          INTEGER          LDWORK
          PARAMETER        ( LDWORK = NMAX*( 2*NMAX +
         $                            MAX( NMAX, MMAX, PMAX ) + 5 ) +
         $                            ( NMAX*( NMAX + 1 ) )/2 )
    *     .. Local Scalars ..
          DOUBLE PRECISION ALPHA, TOL
          INTEGER          I, INFO, IWARN, J, M, N, NR, NS, P
          CHARACTER*1      DICO, EQUIL, JOB, ORDSEL
    *     .. Local Arrays ..
          DOUBLE PRECISION A(LDA,NMAX), B(LDB,MMAX), C(LDC,NMAX),
         $                 DWORK(LDWORK), HSV(NMAX)
          INTEGER          IWORK(LIWORK)
    *     .. External Subroutines ..
          EXTERNAL         AB09MD
    *     .. Intrinsic Functions ..
          INTRINSIC        MAX
    *     .. 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, NR, ALPHA, TOL, DICO, JOB, EQUIL,
         $                      ORDSEL
          IF ( N.LT.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.LT.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.LT.0 .OR. P.GT.PMAX ) THEN
                   WRITE ( NOUT, FMT = 99988 ) P
                ELSE
                   READ ( NIN, FMT = * ) ( ( C(I,J), J = 1,N ), I = 1,P )
    *              Find a reduced ssr for (A,B,C).
                   CALL AB09MD( DICO, JOB, EQUIL, ORDSEL, N, M, P, NR,
         $                      ALPHA, A, LDA, B, LDB, C, LDC, NS, HSV,
         $                      TOL, IWORK, DWORK, LDWORK, IWARN, INFO)
    *
                   IF ( INFO.NE.0 ) THEN
                      WRITE ( NOUT, FMT = 99998 ) INFO
                   ELSE
                      WRITE ( NOUT, FMT = 99997 ) NR
                      WRITE ( NOUT, FMT = 99987 )
                      WRITE ( NOUT, FMT = 99995 ) ( HSV(J), J = 1,NS )
                      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
                END IF
             END IF
          END IF
          STOP
    *
    99999 FORMAT (' AB09MD EXAMPLE PROGRAM RESULTS',/1X)
    99998 FORMAT (' INFO on exit from AB09MD = ',I2)
    99997 FORMAT (' The order of reduced model = ',I2)
    99996 FORMAT (/' The reduced state dynamics matrix Ar is ')
    99995 FORMAT (20(1X,F8.4))
    99993 FORMAT (/' The reduced input/state matrix Br is ')
    99992 FORMAT (/' The reduced state/output matrix Cr 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 ALPHA-stable part are')
          END

Program Data

     AB09MD EXAMPLE PROGRAM DATA (Continuous system)
      7  2   3   0   -.6D0 1.D-1  C  N  N  A
     -0.04165  0.0000  4.9200  -4.9200  0.0000  0.0000  0.0000
     -5.2100  -12.500  0.0000   0.0000  0.0000  0.0000  0.0000
      0.0000   3.3300 -3.3300   0.0000  0.0000  0.0000  0.0000
      0.5450   0.0000  0.0000   0.0000 -0.5450  0.0000  0.0000
      0.0000   0.0000  0.0000   4.9200 -0.04165 0.0000  4.9200
      0.0000   0.0000  0.0000   0.0000 -5.2100 -12.500  0.0000
      0.0000   0.0000  0.0000   0.0000  0.0000  3.3300 -3.3300
      0.0000   0.0000
      12.500   0.0000
      0.0000   0.0000
      0.0000   0.0000
      0.0000   0.0000
      0.0000   12.500
      0.0000   0.0000
      1.0000   0.0000  0.0000   0.0000  0.0000  0.0000  0.0000
      0.0000   0.0000  0.0000   1.0000  0.0000  0.0000  0.0000
      0.0000   0.0000  0.0000   0.0000  1.0000  0.0000  0.0000

Program Results

     AB09MD EXAMPLE PROGRAM RESULTS

     The order of reduced model =  5

     The Hankel singular values of ALPHA-stable part are
       1.9178   0.8621   0.7666   0.0336   0.0246

     The reduced state dynamics matrix Ar is
      -0.5181  -1.1084   0.0000   0.0000   0.0000
       8.8157  -0.5181   0.0000   0.0000   0.0000
       0.0000   0.0000   0.5124   0.0000   1.7910
       0.0000   0.0000   0.0000  -1.4460   0.0000
       0.0000   0.0000  -4.2167   0.0000  -2.9900

     The reduced input/state matrix Br is
      -1.2837   1.2837
      -0.7522   0.7522
      -0.7447  -0.7447
       1.9275  -1.9275
      -3.6872  -3.6872

     The reduced state/output matrix Cr is
      -0.1380  -0.6445  -0.6582  -0.5771   0.2222
       0.6246   0.0196   0.0000   0.4131   0.0000
       0.1380   0.6445  -0.6582   0.5771   0.2222

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