## AB09FD

### Balance & Truncate model reduction for unstable systems in conjunction with coprime factorization

[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 in conjunction with stable coprime
      factorization techniques.

Specification

          SUBROUTINE AB09FD( DICO, JOBCF, FACT, JOBMR, EQUIL, ORDSEL, N, M,
         $                   P, NR, ALPHA, A, LDA, B, LDB, C, LDC, NQ, HSV,
         $                   TOL1, TOL2, IWORK, DWORK, LDWORK, IWARN, INFO )
    C     .. Scalar Arguments ..
          CHARACTER         DICO, EQUIL, FACT, JOBCF, JOBMR, ORDSEL
          INTEGER           INFO, IWARN, LDA, LDB, LDC, LDWORK, M, N, NQ,
         $                  NR, P
          DOUBLE PRECISION  ALPHA, TOL1, TOL2
    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.

      JOBCF   CHARACTER*1
              Specifies whether left or right coprime factorization is
              to be used as follows:
              = 'L':  use left coprime factorization;
              = 'R':  use right coprime factorization.

      FACT    CHARACTER*1
              Specifies the type of coprime factorization to be computed
              as follows:
              = 'S':  compute a coprime factorization with prescribed
                      stability degree ALPHA;
              = 'I':  compute a coprime factorization with inner
                      denominator.

      JOBMR   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 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(NR,NQ,NMIN), where NR
              is the desired order on entry, NQ is the order of the
              computed coprime factorization of the given system, and
              NMIN is the order of a minimal realization of the extended
              system (see METHOD); NMIN is determined as the number of
              Hankel singular values greater than NQ*EPS*HNORM(Ge),
              where EPS is the machine precision (see LAPACK Library
              Routine DLAMCH) and HNORM(Ge) is the Hankel norm of the
              extended system (computed in HSV(1));
              if ORDSEL = 'A', NR is equal to the number of Hankel
              singular values greater than MAX(TOL1,NQ*EPS*HNORM(Ge)).

      ALPHA   (input) DOUBLE PRECISION
              If FACT = 'S', the desired stability degree for the
              factors of the coprime factorization (see SLICOT Library
              routines SB08ED/SB08FD).
              ALPHA < 0 for a continuous-time system (DICO = 'C'), and
              0 <= ALPHA < 1 for a discrete-time system (DICO = 'D').
              If FACT = 'I', ALPHA is not used.

      A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
              On entry, the leading N-by-N part of this array must
              contain the original 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.

      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).

      NQ      (output) INTEGER
              The order of the computed extended system Ge (see METHOD).

      HSV     (output) DOUBLE PRECISION array, dimension (N)
              If INFO = 0, it contains the NQ Hankel singular values of
              the extended system Ge ordered decreasingly (see METHOD).

Tolerances

      TOL1    DOUBLE PRECISION
              If ORDSEL = 'A', TOL1 contains the tolerance for
              determining the order of reduced extended system.
              For model reduction, the recommended value is
              TOL1 = c*HNORM(Ge), where c is a constant in the
              interval [0.00001,0.001], and HNORM(Ge) is the
              Hankel-norm of the extended system (computed in HSV(1)).
              The value TOL1 = NQ*EPS*HNORM(Ge) is used by default if
              TOL1 <= 0 on entry, where EPS is the machine precision
              (see LAPACK Library Routine DLAMCH).
              If ORDSEL = 'F', the value of TOL1 is ignored.

      TOL2    DOUBLE PRECISION
              The absolute tolerance level below which the elements of
              B or C are considered zero (used for controllability or
              observability tests).
              If the user sets TOL2 <= 0, then an implicitly computed,
              default tolerance TOLDEF is used:
              TOLDEF = N*EPS*NORM(C'), if JOBCF = 'L', or
              TOLDEF = N*EPS*NORM(B),  if JOBCF = 'R',
              where EPS is the machine precision, and NORM(.) denotes
              the 1-norm of a matrix.

Workspace

      IWORK   INTEGER array, dimension (LIWORK)
              LIWORK = PM,        if JOBMR = 'B',
              LIWORK = MAX(N,PM), if JOBMR = 'N', where
              PM = P, if JOBCF = 'L',
              PM = M, if JOBCF = 'R'.

      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,LW1) if JOBCF = 'L' and FACT = 'S',
              LDWORK >= MAX(1,LW2) if JOBCF = 'L' and FACT = 'I',
              LDWORK >= MAX(1,LW3) if JOBCF = 'R' and FACT = 'S',
              LDWORK >= MAX(1,LW4) if JOBCF = 'R' and FACT = 'I', where
              LW1 = N*(2*MAX(M,P) + P) + MAX(M,P)*(MAX(M,P) + P) +
                    MAX( N*P+MAX(N*(N+5), 5*P, 4*M), LWR ),
              LW2 = N*(2*MAX(M,P) + P) + MAX(M,P)*(MAX(M,P) + P) +
                    MAX( N*P+MAX(N*(N+5), P*(P+2), 4*P, 4*M), LWR ),
              LW3 = (N+M)*(M+P) + MAX( 5*M, 4*P, LWR ),
              LW4 = (N+M)*(M+P) + MAX( M*(M+2), 4*M, 4*P, LWR ), and
              LWR = 2*N*N + N*(MAX(N,M+P)+5) + N*(N+1)/2.
              For optimum performance LDWORK should be larger.

Warning Indicator

      IWARN   INTEGER
              = 0:  no warning;
              = 10*K+I:
                I = 1:  with ORDSEL = 'F', the selected order NR is
                        greater than the order of the computed coprime
                        factorization 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;
                K > 0:  K violations of the numerical stability
                        condition occured when computing the coprime
                        factorization using pole assignment (see SLICOT
                        Library routines SB08CD/SB08ED, SB08DD/SB08FD).

Error Indicator

      INFO    INTEGER
              = 0:  successful exit;
              < 0:  if INFO = -i, the i-th argument had an illegal
                    value;
              = 1:  the reduction of A to a real Schur form failed;
              = 2:  a failure was detected during the ordering of the
                    real Schur form of A, or in the iterative process
                    for reordering the eigenvalues of Z'*(A + H*C)*Z
                    (or Z'*(A + B*F)*Z) along the diagonal; see SLICOT
                    Library routines SB08CD/SB08ED (or SB08DD/SB08FD);
              = 3:  the matrix A has an observable or controllable
                    eigenvalue on the imaginary axis if DICO = 'C' or
                    on the unit circle if DICO = 'D';
              = 4:  the computation of Hankel singular values failed.

Method

      Let be the 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, and let G be the corresponding
      transfer-function matrix. The subroutine AB09FD determines
      the matrices of a reduced order system

           d[z(t)] = Ar*z(t) + Br*u(t)
           yr(t)   = Cr*z(t)                             (2)

      with the transfer-function matrix Gr, by using the
      balanced-truncation model reduction method in conjunction with
      a left coprime factorization (LCF) or a right coprime
      factorization (RCF) technique:

      1. Compute the appropriate stable coprime factorization of G:
                      -1                   -1
                 G = R  *Q (LCF) or G = Q*R   (RCF).

      2. Perform the model reduction algorithm on the extended system
                                            ( Q )
                 Ge = ( Q R ) (LCF) or Ge = ( R )  (RCF)

         to obtain a reduced extended system with reduced factors
                                                ( Qr )
                 Ger = ( Qr Rr ) (LCF) or Ger = ( Rr )  (RCF).

      3. Recover the reduced system from the reduced factors as
                        -1                       -1
                 Gr = Rr  *Qr (LCF) or Gr = Qr*Rr   (RCF).

      The approximation error for the extended system satisfies

         HSV(NR) <= INFNORM(Ge-Ger) <= 2*[HSV(NR+1) + ... + HSV(NQ)],

      where INFNORM(G) is the infinity-norm of G.

      If JOBMR = 'B', the square-root Balance & Truncate method of [1]
      is used for model reduction.
      If JOBMR = 'N', the balancing-free square-root version of the
      Balance & Truncate method [2] is used for model reduction.

      If FACT = 'S', the stable coprime factorization with prescribed
      stability degree ALPHA is computed by using the algorithm of [3].
      If FACT = 'I', the stable coprime factorization with inner
      denominator is computed by using the algorithm of [4].

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, 1991.

      [3] Varga A.
          Coprime factors model reduction method based on square-root
          balancing-free techniques.
          System Analysis, Modelling and Simulation, Vol. 11,
          pp. 303-311, 1993.

      [4] Varga A.
          A Schur method for computing coprime factorizations with
          inner denominators and applications in model reduction.
          Proc. ACC'93, San Francisco, CA, pp. 2130-2131, 1993.

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

    *     AB09FD 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 = MAX( NMAX, MMAX, PMAX ) )
    *     The formula below uses that NMAX = MMAX = PMAX.
          INTEGER          LDWORK
          PARAMETER        ( LDWORK = 10*NMAX*NMAX + 5*NMAX )
    *     .. Local Scalars ..
          DOUBLE PRECISION ALPHA, TOL1, TOL2
          INTEGER          I, INFO, IWARN, J, M, N, NQ, NR, P
          CHARACTER*1      DICO, EQUIL, FACT, JOBCF, JOBMR, ORDSEL
    *     .. Local Arrays ..
          DOUBLE PRECISION A(LDA,NMAX), B(LDB,MMAX), C(LDC,NMAX),
         $                 DWORK(LDWORK), HSV(NMAX)
          INTEGER          IWORK(LIWORK)
    *     .. External Subroutines ..
          EXTERNAL         AB09FD
    *     .. 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, TOL1, TOL2,
         $                      DICO, JOBCF, FACT, JOBMR, 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 AB09FD( DICO, JOBCF, FACT, JOBMR, EQUIL, ORDSEL,
         $                      N, M, P, NR, ALPHA, A, LDA, B, LDB, C, LDC,
         $                      NQ, HSV, TOL1, TOL2, 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, NQ )
                      IF( NR.GT.0 ) WRITE ( NOUT, FMT = 99996 )
                      DO 20 I = 1, NR
                         WRITE ( NOUT, FMT = 99995 ) ( A(I,J), J = 1, NR )
       20             CONTINUE
                      IF( NR.GT.0 ) WRITE ( NOUT, FMT = 99993 )
                      DO 40 I = 1, NR
                         WRITE ( NOUT, FMT = 99995 ) ( B(I,J), J = 1, M )
       40             CONTINUE
                      IF( NR.GT.0 ) 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 (' AB09FD EXAMPLE PROGRAM RESULTS',/1X)
    99998 FORMAT (' INFO on exit from AB09FD = ',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 coprime factors are')
          END

Program Data

     AB08FD EXAMPLE PROGRAM DATA (Continuous system)
      7  2  3  0 -1.e-1  .1 1.E-10 C L I B S A
     -0.04165  0.0000  4.9200   0.4920  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  0.0545   0.0000  0.0000
      0.0000   0.0000  0.0000  -0.49200 0.004165 0.0000  4.9200
      0.0000   0.0000  0.0000   0.0000  0.5210  -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

     AB09FD EXAMPLE PROGRAM RESULTS

     The order of reduced model =  5

     The Hankel singular values of coprime factors are
      13.6047   9.4106   1.7684   0.7456   0.6891   0.0241   0.0230

     The reduced state dynamics matrix Ar is
       0.0520  -0.1491   0.0037  -0.0232   0.0168
       0.2340   0.2618   0.0010  -0.0153  -0.0318
       0.1197   0.0075  -0.5752   2.0119  -0.7779
       0.1571  -0.2019  -2.1282  -2.1192  -0.3618
       0.0368  -0.4810   0.8395  -0.2790  -2.8796

     The reduced input/state matrix Br is
       1.0454   0.5860
      -0.0489  -1.9194
      -1.4282   0.0541
      -1.6144  -0.7533
       0.5916  -1.9242

     The reduced state/output matrix Cr is
       0.4368   0.1122  -1.2917   1.5888  -0.6354
       1.1170   0.3963   0.6115   0.1249  -0.0859
       0.0756  -1.8904   0.0144   0.7964   1.9085

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