## AB13BD

### H2 or L2 norm of a system

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

Purpose

      To compute the H2 or L2 norm of the transfer-function matrix G
      of the system (A,B,C,D). G must not have poles on the imaginary
      axis, for a continuous-time system, or on the unit circle, for
      a discrete-time system. If the H2-norm is computed, the system
      must be stable.

Specification

          DOUBLE PRECISION FUNCTION AB13BD( DICO, JOBN, N, M, P, A, LDA,
         $                                  B, LDB, C, LDC, D, LDD, NQ, TOL,
         $                                  DWORK, LDWORK, IWARN, INFO)
    C     .. Scalar Arguments ..
          CHARACTER         DICO, JOBN
          INTEGER           INFO, IWARN, LDA, LDB, LDC, LDD, LDWORK, M,
         $                  N, NQ, P
          DOUBLE PRECISION  TOL
    C     .. Array Arguments ..
          DOUBLE PRECISION  A(LDA,*), B(LDB,*), C(LDC,*), D(LDD,*), DWORK(*)

Function Value

      AB13BD   DOUBLE PRECISION
               The H2-norm of G, if JOBN = 'H', or the L2-norm of G,
               if JOBN = 'L' (if INFO = 0).

Arguments

Mode Parameters

      DICO    CHARACTER*1
              Specifies the type of the system as follows:
              = 'C':  continuous-time system;
              = 'D':  discrete-time system.

      JOBN    CHARACTER*1
              Specifies the norm to be computed as follows:
              = 'H':  the H2-norm;
              = 'L':  the L2-norm.

Input/Output Parameters

      N       (input) INTEGER
              The order of the matrix A, the number of rows of the
              matrix B, and the number of columns of the matrix C.
              N represents the dimension of the state vector.  N >= 0.

      M       (input) INTEGER
              The number of columns of the matrices B and D.
              M represents the dimension of input vector.  M >= 0.

      P       (input) INTEGER
              The number of rows of the matrices C and D.
              P represents the dimension of output vector.  P >= 0.

      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 of the system.
              On exit, the leading NQ-by-NQ part of this array contains
              the state dynamics matrix (in a real Schur form) of the
              numerator factor Q of the right coprime factorization with
              inner denominator of G (see METHOD).

      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 input/state matrix of the system.
              On exit, the leading NQ-by-M part of this array contains
              the input/state matrix of the numerator factor Q of the
              right coprime factorization with inner denominator of G
              (see METHOD).

      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 state/output matrix of the system.
              On exit, the leading P-by-NQ part of this array contains
              the state/output matrix of the numerator factor Q of the
              right coprime factorization with inner denominator of G
              (see METHOD).

      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 input/output matrix of the system.
              If DICO = 'C', D must be a null matrix.
              On exit, the leading P-by-M part of this array contains
              the input/output matrix of the numerator factor Q of
              the right coprime factorization with inner denominator
              of G (see METHOD).

      LDD     INTEGER
              The leading dimension of array D.  LDD >= MAX(1,P).

      NQ      (output) INTEGER
              The order of the resulting numerator Q of the right
              coprime factorization with inner denominator of G (see
              METHOD).
              Generally, NQ = N - NS, where NS is the number of
              uncontrollable unstable eigenvalues.

Tolerances

      TOL     DOUBLE PRECISION
              The absolute tolerance level below which the elements of
              B are considered zero (used for controllability tests).
              If the user sets TOL <= 0, then an implicitly computed,
              default tolerance, defined by  TOLDEF = N*EPS*NORM(B),
              is used instead, where EPS is the machine precision
              (see LAPACK Library routine DLAMCH) and NORM(B) denotes
              the 1-norm of B.

Workspace

      DWORK   DOUBLE PRECISION array, dimension (LDWORK)
              On exit, if INFO = 0, DWORK(1) returns the optimal value
              of LDWORK.

      LDWORK  INTEGER
              The dimension of working array DWORK.
              LDWORK >= MAX( 1, M*(N+M) + MAX( N*(N+5), M*(M+2), 4*P ),
                                N*( MAX( N, P ) + 4 ) + MIN( N, P ) ).
              For optimum performance LDWORK should be larger.

Warning Indicator

      IWARN   INTEGER
              = 0:  no warning;
              = K:  K violations of the numerical stability condition
                    occured during the assignment of eigenvalues in
                    computing the right coprime factorization with inner
                    denominator of G (see the SLICOT subroutine SB08DD).

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 reordering of the
                    real Schur form of A, or in the iterative process
                    for reordering the eigenvalues of Z'*(A + B*F)*Z
                    along the diagonal (see SLICOT routine SB08DD);
              = 3:  if DICO = 'C' and the matrix A has a controllable
                    eigenvalue on the imaginary axis, or DICO = 'D'
                    and A has a controllable eigenvalue on the unit
                    circle;
              = 4:  the solution of Lyapunov equation failed because
                    the equation is singular;
              = 5:  if DICO = 'C' and D is a nonzero matrix;
              = 6:  if JOBN = 'H' and the system is unstable.

Method

      The subroutine is based on the algorithms proposed in [1] and [2].

      If the given transfer-function matrix G is unstable, then a right
      coprime factorization with inner denominator of G is first
      computed
                -1
         G = Q*R  ,

      where Q and R are stable transfer-function matrices and R is
      inner. If G is stable, then Q = G and R = I.
      Let (AQ,BQ,CQ,DQ) be the state-space representation of Q.

      If DICO = 'C', then the L2-norm of G is computed as

         NORM2(G) = NORM2(Q) = SQRT(TRACE(BQ'*X*BQ)),

      where X satisfies the continuous-time Lyapunov equation

         AQ'*X + X*AQ + CQ'*CQ = 0.

      If DICO = 'D', then the l2-norm of G is computed as

         NORM2(G) = NORM2(Q) = SQRT(TRACE(BQ'*X*BQ+DQ'*DQ)),

      where X satisfies the discrete-time Lyapunov equation

         AQ'*X*AQ - X + CQ'*CQ = 0.

References

      [1] Varga A.
          On computing 2-norms of transfer-function matrices.
          Proc. 1992 ACC, Chicago, June 1992.

      [2] 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

                                             3
      The algorithm requires no more than 14N  floating point
      operations.

Further Comments

      None

Example

Program Text

    *     AB13BD 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          LDWORK
          PARAMETER        ( LDWORK = MAX( MMAX*( NMAX + MMAX ) +
         $                                 MAX( NMAX*( NMAX + 5 ),
         $                                      MMAX*( MMAX + 2 ), 4*PMAX ),
         $                                 NMAX*( MAX( NMAX, PMAX ) + 4 ) +
         $                                 MIN( NMAX, PMAX ) ) )
    *     .. Local Scalars ..
          DOUBLE PRECISION S2NORM, TOL
          INTEGER          I, INFO, IWARN, J, M, N, NQ, P
          CHARACTER*1      DICO, JOBN
    *     .. Local Arrays ..
          DOUBLE PRECISION A(LDA,NMAX), B(LDB,MMAX), C(LDC,NMAX),
         $                 D(LDD,MMAX), DWORK(LDWORK)
    *     .. External Functions ..
          LOGICAL          LSAME
          DOUBLE PRECISION AB13BD
          EXTERNAL         AB13BD, LSAME
    *     .. 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, TOL, DICO, JOBN
          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 )
    *              Compute the H2 or L2 norm of (A,B,C,D).
                   S2NORM = AB13BD( DICO, JOBN, N, M, P, A, LDA, B, LDB,
         *                          C, LDC, D, LDD, NQ, TOL, DWORK, LDWORK,
         *                          IWARN, INFO)
    *
                   IF ( INFO.NE.0 ) THEN
                      WRITE ( NOUT, FMT = 99998 ) INFO
                   ELSE
                      IF( LSAME( JOBN, 'H' ) ) THEN
                         WRITE ( NOUT, FMT = 99997 ) S2NORM
                      ELSE
                         WRITE ( NOUT, FMT = 99996 ) S2NORM
                      END IF
                   END IF
                END IF
             END IF
          END IF
          STOP
    *
    99999 FORMAT (' AB13BD EXAMPLE PROGRAM RESULTS',/1X)
    99998 FORMAT (' INFO on exit from AB13BD = ',I2)
    99997 FORMAT (' The H2-norm of the system = ',1PD14.5)
    99996 FORMAT (' The L2-norm of the system = ',1PD14.5)
    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)
          END

Program Data

     AB13BD EXAMPLE PROGRAM DATA (Continuous system)
      7  2  3   1.E-10 C L
     -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
      0.0000   0.0000
      0.0000   0.0000
      0.0000   0.0000

Program Results

     AB13BD EXAMPLE PROGRAM RESULTS

     The L2-norm of the system =    7.93948D+00

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