## AB09ED

### Optimal Hankel-norm approximation based model reduction for unstable systems

[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 optimal
      Hankel-norm approximation method in conjunction with square-root
      balancing for the ALPHA-stable part of the system.

Specification

          SUBROUTINE AB09ED( DICO, EQUIL, ORDSEL, N, M, P, NR, ALPHA,
         $                   A, LDA, B, LDB, C, LDC, D, LDD, NS, HSV, TOL1,
         $                   TOL2, IWORK, DWORK, LDWORK, IWARN, INFO )
    C     .. Scalar Arguments ..
          CHARACTER         DICO, EQUIL, ORDSEL
          INTEGER           INFO, IWARN, LDA, LDB, LDC, LDD, LDWORK,
         $                  M, N, NR, NS, P
          DOUBLE PRECISION  ALPHA, 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.

      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. 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-KR+1),NMIN), where KR is the
              multiplicity of the Hankel singular value HSV(NR-NU+1),
              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(TOL1,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 in a real Schur form.
              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.
              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 ALPHA-stable part of the
              original system ordered decreasingly.
              HSV(1) is the Hankel norm of the ALPHA-stable subsystem.

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(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 TOL1 <= 0 on entry, the used default value is
              TOL1 = 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 TOL1 is ignored.

      TOL2    DOUBLE PRECISION
              The tolerance for determining the order of a minimal
              realization of the ALPHA-stable part of the given system.
              The recommended value is TOL2 = NS*EPS*HNORM(As,Bs,Cs).
              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 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 computed ALPHA-stable part is just stable,
                    having stable eigenvalues very near to the imaginary
                    axis (if DICO = 'C') or to the unit circle
                    (if DICO = 'D');
              = 4:  the computation of Hankel singular values failed;
              = 5:  the computation of stable projection in the
                    Hankel-norm approximation algorithm failed;
              = 6:  the order of computed stable projection in the
                    Hankel-norm approximation algorithm differs
                    from the order of Hankel-norm approximation.

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 AB09ED 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+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,D) and (Ar,Br,Cr,Dr), 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,D) has only ALPHA-stable poles and
         G2 = (Au,Bu,Cu,0) 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, 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

    *     AB09ED 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          LIWORK
          PARAMETER        ( LIWORK = MAX( NMAX, MMAX ) )
          INTEGER          LDWORK
          PARAMETER        ( LDWORK = MAX( NMAX*( 2*NMAX +
         $                                        MAX( NMAX, MMAX, PMAX ) +
         $                                 5 ) + ( NMAX*( NMAX + 1 ) )/2,
         $                                 NMAX*( MMAX + PMAX + 2 ) +
         $                                 2*MMAX*PMAX + MIN( NMAX, MMAX ) +
         $                                 MAX( 3*MMAX + 1,
         $                                      MIN( NMAX, MMAX ) +
         $                                      PMAX ) ) )
    *     .. Local Scalars ..
          DOUBLE PRECISION ALPHA, TOL1, TOL2
          INTEGER          I, INFO, IWARN, J, M, N, NR, NS, P
          CHARACTER*1      DICO, EQUIL, ORDSEL
    *     .. Local Arrays ..
          DOUBLE PRECISION A(LDA,NMAX), B(LDB,MMAX), C(LDC,NMAX),
         $                 D(LDD,MMAX), DWORK(LDWORK), HSV(NMAX)
          INTEGER          IWORK(LIWORK)
    *     .. External Subroutines ..
          EXTERNAL         AB09ED
    *     .. Intrinsic Functions ..
          INTRINSIC        MAX, MIN
    *     .. 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, 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 AB09ED( DICO, EQUIL, ORDSEL, N, M, P, NR, ALPHA,
         $                      A, LDA, B, LDB, C, LDC, D, LDD, NS, 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, 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 (' AB09ED EXAMPLE PROGRAM RESULTS',/1X)
    99998 FORMAT (' INFO on exit from AB09ED = ',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 Hankel singular values of ALPHA-stable part are')
          END

Program Data

     AB09ED EXAMPLE PROGRAM DATA (Continuous system)
      7  2   3   0   -0.6D0 1.E-1  1.E-14 C  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

     AB09ED 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  -1.2769   7.3264   0.0000
       0.0000   0.0000  -0.6203  -1.2769   0.0000
       0.0000   0.0000   0.0000   0.0000  -1.5496

     The reduced input/state matrix Br is
      -1.2837   1.2837
      -0.7522   0.7522
       3.2016   3.2016
      -0.7640  -0.7640
       1.3415  -1.3415

     The reduced state/output matrix Cr is
      -0.1380  -0.6445  -0.6247  -2.0857  -0.8964
       0.6246   0.0196   0.0000   0.0000   0.6131
       0.1380   0.6445  -0.6247  -2.0857   0.8964

     The reduced input/output matrix Dr is
       0.0168  -0.0168
       0.0008  -0.0008
      -0.0168   0.0168

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