## AB09HD

### Stochastic balancing based model reduction

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

Purpose

      To compute a reduced order model (Ar,Br,Cr,Dr) for an original
      state-space representation (A,B,C,D) by using the stochastic
      balancing approach in conjunction with the square-root or
      the balancing-free square-root Balance & Truncate (B&T)
      or Singular Perturbation Approximation (SPA) model reduction
      methods for the ALPHA-stable part of the system.

Specification

          SUBROUTINE AB09HD( DICO, JOB, EQUIL, ORDSEL, N, M, P, NR, ALPHA,
         $                   BETA, A, LDA, B, LDB, C, LDC, D, LDD, NS, HSV,
         $                   TOL1, TOL2, IWORK, DWORK, LDWORK, BWORK, IWARN,
         $                   INFO )
    C     .. Scalar Arguments ..
          CHARACTER         DICO, EQUIL, JOB, ORDSEL
          INTEGER           INFO, IWARN, LDA, LDB, LDC, LDD, LDWORK,
         $                  M, N, NR, NS, P
          DOUBLE PRECISION  ALPHA, BETA, TOL1, TOL2
    C     .. Array Arguments ..
          INTEGER           IWORK(*)
          DOUBLE PRECISION  A(LDA,*), B(LDB,*), C(LDC,*), D(LDD,*),
         $                  DWORK(*), HSV(*)
          LOGICAL           BWORK(*)

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;
              = 'F':  use the balancing-free square-root
                      Balance & Truncate method;
              = 'S':  use the square-root Singular Perturbation
                      Approximation method;
              = 'P':  use the balancing-free square-root
                      Singular Perturbation Approximation 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.
              P <= M if BETA = 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, where EPS is the machine precision
              (see LAPACK Library Routine DLAMCH);
              if ORDSEL = 'A', NR is the sum of NU and the number of
              Hankel singular values greater than MAX(TOL1,NS*EPS);
              NR can be further reduced to ensure that
              HSV(NR-NU) > HSV(NR+1-NU).

      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.

      BETA    (input) DOUBLE PRECISION
              BETA > 0 specifies the absolute/relative error weighting
              parameter. A large positive value of BETA favours the
              minimization of the absolute approximation error, while a
              small value of BETA is appropriate for the minimization
              of the relative error.
              BETA = 0 means a pure relative error method and can be
              used only if rank(D) = P.

      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).

      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).

      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 phase system corresponding
              to the ALPHA-stable part of the original system.
              The Hankel singular values are ordered decreasingly.

Tolerances

      TOL1    DOUBLE PRECISION
              If ORDSEL = 'A', TOL1 contains the tolerance for
              determining the order of reduced system.
              For model reduction, the recommended value of TOL1 lies
              in the interval [0.00001,0.001].
              If TOL1 <= 0 on entry, the used default value is
              TOL1 = NS*EPS, where NS is the number of
              ALPHA-stable eigenvalues of A and EPS is the machine
              precision (see LAPACK Library Routine DLAMCH).
              If ORDSEL = 'F', the value of TOL1 is ignored.
              TOL1 < 1.

      TOL2    DOUBLE PRECISION
              The tolerance for determining the order of a minimal
              realization of the phase system (see METHOD) corresponding
              to the ALPHA-stable part of the given system.
              The recommended value is TOL2 = NS*EPS.
              This value is used by default if TOL2 <= 0 on entry.
              If TOL2 > 0 and ORDSEL = 'A', then TOL2 <= TOL1.
              TOL2 < 1.

Workspace

      IWORK   INTEGER array, dimension (MAX(1,2*N))
              On exit with INFO = 0, IWORK(1) contains the order of the
              minimal realization of the system.

      DWORK   DOUBLE PRECISION array, dimension (LDWORK)
              On exit, if INFO = 0, DWORK(1) returns the optimal value
              of LDWORK and DWORK(2) contains RCOND, the reciprocal
              condition number of the U11 matrix from the expression
              used to compute the solution X = U21*inv(U11) of the
              Riccati equation for spectral factorization.
              A small value RCOND indicates possible ill-conditioning
              of the respective Riccati equation.

      LDWORK  INTEGER
              The length of the array DWORK.
              LDWORK >= 2*N*N + MB*(N+P) + MAX( 2, N*(MAX(N,MB,P)+5),
                                     2*N*P+MAX(P*(MB+2),10*N*(N+1) ) ),
              where MB = M if BETA = 0 and MB = M+P if BETA > 0.
              For optimum performance LDWORK should be larger.

      BWORK   LOGICAL array, dimension 2*N

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 corresponds
                    to repeated singular values for the ALPHA-stable
                    part, which are neither all included nor all
                    excluded from the reduced model; in this case, the
                    resulting NR is automatically decreased to exclude
                    all repeated singular values;
              = 3:  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 reduction of the Hamiltonian matrix to real
                    Schur form failed;
              = 3:  the reordering of the real Schur form of the
                    Hamiltonian matrix failed;
              = 4:  the Hamiltonian matrix has less than N stable
                    eigenvalues;
              = 5:  the coefficient matrix U11 in the linear system
                    X*U11 = U21 to determine X is singular to working
                    precision;
              = 6:  BETA = 0 and D has not a maximal row rank;
              = 7:  the computation of Hankel singular values failed;
              = 8:  the separation of the ALPHA-stable/unstable diagonal
                    blocks failed because of very close eigenvalues;
              = 9:  the resulting order of reduced stable part is less
                    than the number of unstable zeros of the stable
                    part.

Method

      Let be the following 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 AB09HD 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

           INFNORM[inv(conj(W))*(G-Gr)] <=
                        (1+HSV(NR+NS-N+1)) / (1-HSV(NR+NS-N+1)) + ...
                        + (1+HSV(NS)) / (1-HSV(NS)) - 1,

      where G and Gr are transfer-function matrices of the systems
      (A,B,C,D) and (Ar,Br,Cr,Dr), respectively, W is the right, minimum
      phase spectral factor satisfying

          G1*conj(G1) = conj(W)* W,                      (3)

      G1 is the NS-order ALPHA-stable part of G, and INFNORM(G) is the
      infinity-norm of G. HSV(1), ... , HSV(NS) are the Hankel-singular
      values of the stable part of the phase system (Ap,Bp,Cp)
      with the transfer-function matrix

           P = inv(conj(W))*G1.

      If BETA > 0, then the model reduction is performed on [G BETA*I]
      instead of G. This is the recommended approach to be used when D
      has not a maximal row rank or when a certain balance between
      relative and absolute approximation errors is desired. For
      increasingly large values of BETA, the obtained reduced system
      assymptotically approaches that computed by using the
      Balance & Truncate or Singular Perturbation Approximation methods.

      Note: conj(G)  denotes either G'(-s) for a continuous-time system
            or G'(1/z) for a discrete-time system.
            inv(G) is the inverse 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,D) 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 using the balancing stochastic method
         in conjunction with either the B&T [1,2] or SPA methods [3].

      3) Assemble the reduced model Gr as

            Gr = G1r + G2.

      Note: The employed stochastic truncation algorithm [2,3] has the
      property that right half plane zeros of G1 remain as right half
      plane zeros of G1r. Thus, the order can not be chosen smaller than
      the sum of the number of unstable poles of G and the number of
      unstable zeros of G1.

      The reduction of the ALPHA-stable part G1 is done as follows.

      If JOB = 'B', the square-root stochastic Balance & Truncate
      method of [1] is used.
      For an ALPHA-stable continuous-time system (DICO = 'C'),
      the resulting reduced model is stochastically balanced.

      If JOB = 'F', the balancing-free square-root version of the
      stochastic Balance & Truncate method [1] is used to reduce
      the ALPHA-stable part G1.

      If JOB = 'S', the stochastic balancing method is used to reduce
      the ALPHA-stable part G1, in conjunction with the square-root
      version of the Singular Perturbation Approximation method [3,4].

      If JOB = 'P', the stochastic balancing method is used to reduce
      the ALPHA-stable part G1, in conjunction with the balancing-free
      square-root version of the Singular Perturbation Approximation
      method [3,4].

References

      [1] Varga A. and Fasol K.H.
          A new square-root balancing-free stochastic truncation model
          reduction algorithm.
          Proc. 12th IFAC World Congress, Sydney, 1993.

      [2] Safonov M. G. and Chiang R. Y.
          Model reduction for robust control: a Schur relative error
          method.
          Int. J. Adapt. Contr. Sign. Proc., vol. 2, pp. 259-272, 1988.

      [3] Green M. and Anderson B. D. O.
          Generalized balanced stochastic truncation.
          Proc. 29-th CDC, Honolulu, Hawaii, pp. 476-481, 1990.

      [4] Varga A.
          Balancing-free square-root algorithm for computing
          singular perturbation approximations.
          Proc. 30-th IEEE CDC,  Brighton, Dec. 11-13, 1991,
          Vol. 2, pp. 1062-1065.

Numerical Aspects

      The implemented methods rely on accuracy enhancing square-root or
      balancing-free square-root techniques. The effectiveness of the
      accuracy enhancing technique depends on the accuracy of the
      solution of a Riccati equation. An ill-conditioned Riccati
      solution typically results when [D BETA*I] is nearly
      rank deficient.
                                       3
      The algorithm requires about 100N  floating point operations.

Further Comments

      None

Example

Program Text

    *     AB09HD 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, LDD
          PARAMETER        ( LDA = NMAX, LDB = NMAX, LDC = PMAX,
         $                 LDD = PMAX )
          INTEGER          LBWORK, LIWORK
          PARAMETER        ( LBWORK = 2*NMAX, LIWORK = 2*NMAX )
          INTEGER          LDWORK, MBMAX
          PARAMETER        ( MBMAX = MMAX + PMAX )
          PARAMETER        ( LDWORK = 2*NMAX*NMAX + MBMAX*(NMAX+PMAX) +
         $                      MAX( NMAX*(MAX( NMAX, MMAX, PMAX) + 5),
         $                     2*NMAX*PMAX + MAX( PMAX*(MBMAX+2),
         $                                        10*NMAX*(NMAX+1) ) ) )
    *     .. Local Scalars ..
          DOUBLE PRECISION ALPHA, BETA, TOL1, TOL2
          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),
         $                 D(LDD,MMAX), DWORK(LDWORK), HSV(NMAX)
          LOGICAL          BWORK(LBWORK)
          INTEGER          IWORK(LIWORK)
    *     .. External Subroutines ..
          EXTERNAL         AB09HD
    *     .. 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, BETA, TOL1, TOL2,
         $                      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 )
                   READ ( NIN, FMT = * ) ( ( D(I,J), J = 1,M ), I = 1,P )
    *              Find a reduced ssr for (A,B,C,D).
                   CALL AB09HD( DICO, JOB, EQUIL, ORDSEL, N, M, P, NR,
         $                      ALPHA, BETA, A, LDA, B, LDB, C, LDC, D, LDD,
         $                      NS, HSV, TOL1, TOL2, IWORK, DWORK, LDWORK,
         $                      BWORK, 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 )
                      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
                      WRITE ( NOUT, FMT = 99991 )
                      DO 70 I = 1, P
                         WRITE ( NOUT, FMT = 99995 ) ( D(I,J), J = 1,M )
       70             CONTINUE
                   END IF
                END IF
             END IF
          END IF
          STOP
    *
    99999 FORMAT (' AB09HD EXAMPLE PROGRAM RESULTS',/1X)
    99998 FORMAT (' INFO on exit from AB09HD = ',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 ')
    99991 FORMAT (/' The reduced input/output matrix Dr 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 stochastic Hankel singular values of ALPHA-stable'
         $        ,' part are')
          END

Program Data

     AB09HD EXAMPLE PROGRAM DATA (Continuous system)
      7  2   3   0   0.0   1.0  0.1E0  0.0    C     F     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
      0.0000   0.0000
      0.0000   0.0000
      0.0000   0.0000

Program Results

     AB09HD EXAMPLE PROGRAM RESULTS

     The order of reduced model =  5

     The stochastic Hankel singular values of ALPHA-stable part are
       0.8803   0.8506   0.8038   0.4494   0.3973   0.0214   0.0209

     The reduced state dynamics matrix Ar is
       1.2729   0.0000   6.5947   0.0000  -3.4229
       0.0000   0.8169   0.0000   2.4821   0.0000
      -2.9889   0.0000  -2.9028   0.0000  -0.3692
       0.0000  -3.3921   0.0000  -3.1126   0.0000
      -1.4767   0.0000  -2.0339   0.0000  -0.6107

     The reduced input/state matrix Br is
       0.1331  -0.1331
      -0.0862  -0.0862
      -2.6777   2.6777
      -3.5767  -3.5767
      -2.3033   2.3033

     The reduced state/output matrix Cr is
      -0.6907  -0.6882   0.0779   0.0958  -0.0038
       0.0676   0.0000   0.6532   0.0000  -0.7522
       0.6907  -0.6882  -0.0779   0.0958   0.0038

     The reduced input/output matrix Dr is
       0.0000   0.0000
       0.0000   0.0000
       0.0000   0.0000

Return to index