## AB13CD

### H-infinity norm of a continuous-time stable system

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

Purpose

      To compute the H-infinity norm of the continuous-time stable
      system

                           | A | B |
                    G(s) = |---|---| .
                           | C | D |

Specification

          DOUBLE PRECISION FUNCTION AB13CD( N, M, NP, A, LDA, B, LDB, C,
         $                                  LDC, D, LDD, TOL, IWORK, DWORK,
         $                                  LDWORK, CWORK, LCWORK, BWORK,
         $                                  INFO )
    C     .. Scalar Arguments ..
          INTEGER            INFO, LDA, LDB, LDC, LCWORK, LDD, LDWORK, M, N,
         $                   NP
          DOUBLE PRECISION   TOL
    C     .. Array Arguments ..
          INTEGER            IWORK( * )
          COMPLEX*16         CWORK( * )
          DOUBLE PRECISION   A( LDA, * ), B( LDB, * ), C( LDC, * ),
         $                   D( LDD, * ), DWORK( * )
          LOGICAL            BWORK( * )

Function Value

      AB13CD  DOUBLE PRECISION
              If INFO = 0, the H-infinity norm of the system, HNORM,
              i.e., the peak gain of the frequency response (as measured
              by the largest singular value in the MIMO case).

Arguments

Input/Output Parameters

      N       (input) INTEGER
              The order of the system.  N >= 0.

      M       (input) INTEGER
              The column size of the matrix B.  M >= 0.

      NP      (input) INTEGER
              The row size of the matrix C.  NP >= 0.

      A       (input) DOUBLE PRECISION array, dimension (LDA,N)
              The leading N-by-N part of this array must contain the
              system state matrix A.

      LDA     INTEGER
              The leading dimension of the array A.  LDA >= max(1,N).

      B       (input) DOUBLE PRECISION array, dimension (LDB,M)
              The leading N-by-M part of this array must contain the
              system input matrix B.

      LDB     INTEGER
              The leading dimension of the array B.  LDB >= max(1,N).

      C       (input) DOUBLE PRECISION array, dimension (LDC,N)
              The leading NP-by-N part of this array must contain the
              system output matrix C.

      LDC     INTEGER
              The leading dimension of the array C.  LDC >= max(1,NP).

      D       (input) DOUBLE PRECISION array, dimension (LDD,M)
              The leading NP-by-M part of this array must contain the
              system input/output matrix D.

      LDD     INTEGER
              The leading dimension of the array D.  LDD >= max(1,NP).

Tolerances

      TOL     DOUBLE PRECISION
              Tolerance used to set the accuracy in determining the
              norm.

Workspace

      IWORK   INTEGER array, dimension (N)

      DWORK   DOUBLE PRECISION array, dimension (LDWORK)
              On exit, if INFO = 0, DWORK(1) contains the optimal value
              of LDWORK, and DWORK(2) contains the frequency where the
              gain of the frequency response achieves its peak value
              HNORM.

      LDWORK  INTEGER
              The dimension of the array DWORK.
              LDWORK >= max(2,4*N*N+2*M*M+3*M*N+M*NP+2*(N+NP)*NP+10*N+
                              6*max(M,NP)).
              For good performance, LDWORK must generally be larger.

      CWORK   COMPLEX*16 array, dimension (LCWORK)
              On exit, if INFO = 0, CWORK(1) contains the optimal value
              of LCWORK.

      LCWORK  INTEGER
              The dimension of the array CWORK.
              LCWORK >= max(1,(N+M)*(N+NP)+3*max(M,NP)).
              For good performance, LCWORK must generally be larger.

      BWORK   LOGICAL array, dimension (2*N)

Error Indicator

      INFO    INTEGER
              = 0:  successful exit;
              < 0:  if INFO = -i, the i-th argument had an illegal
                    value;
              = 1:  the system is unstable;
              = 2:  the tolerance is too small (the algorithm for
                    computing the H-infinity norm did not converge);
              = 3:  errors in computing the eigenvalues of A or of the
                    Hamiltonian matrix (the QR algorithm did not
                    converge);
              = 4:  errors in computing singular values.

Method

      The routine implements the method presented in [1].

References

      [1] Bruinsma, N.A. and Steinbuch, M.
          A fast algorithm to compute the Hinfinity-norm of a transfer
          function matrix.
          Systems & Control Letters, vol. 14, pp. 287-293, 1990.

Numerical Aspects

      If the algorithm does not converge (INFO = 2), the tolerance must
      be increased.

Further Comments

      None

Example

Program Text

    *     AB13CD EXAMPLE PROGRAM TEXT
    *
    *     .. Parameters ..
          INTEGER          NIN, NOUT
          PARAMETER        ( NIN = 5, NOUT = 6 )
          INTEGER          NMAX, MMAX, PMAX
          PARAMETER        ( NMAX = 10, MMAX = 10, PMAX = 10 )
          INTEGER          LDA, LDB, LDC, LDD
          PARAMETER        ( LDA = NMAX, LDB = NMAX, LDC = PMAX,
         $                   LDD = PMAX )
          INTEGER          LIWORK
          PARAMETER        ( LIWORK = NMAX )
          INTEGER          LCWORK
          PARAMETER        ( LCWORK = ( NMAX + MMAX )*( NMAX + PMAX ) +
         $                              3*MAX( MMAX, PMAX ) )
          INTEGER          LDWORK
          PARAMETER        ( LDWORK = 4*NMAX*NMAX + 2*MMAX*MMAX +
         $                            2*PMAX*PMAX + 3*NMAX*MMAX +
         $                            2*NMAX*PMAX + MMAX*PMAX + 10*NMAX +
         $                            6*MAX( MMAX, PMAX ) )
    *     .. Local Scalars ..
          DOUBLE PRECISION FPEAK, HNORM, TOL
          INTEGER          I, INFO, J, M, N, NP
    *     .. Local Arrays ..
          LOGICAL          BWORK(2*NMAX)
          INTEGER          IWORK(LIWORK)
          DOUBLE PRECISION A(LDA,NMAX), B(LDB,MMAX), C(LDC,NMAX),
         $                 D(LDD,MMAX), DWORK(LDWORK)
          COMPLEX*16       CWORK( LCWORK )
    *     .. External Functions ..
          DOUBLE PRECISION AB13CD
          EXTERNAL         AB13CD
    *     .. 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, NP
          IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
             WRITE ( NOUT, FMT = 99990 ) N
          ELSE IF ( M.LT.0 .OR. M.GT.MMAX ) THEN
             WRITE ( NOUT, FMT = 99989 ) M
          ELSE IF ( NP.LT.0 .OR. NP.GT.PMAX ) THEN
             WRITE ( NOUT, FMT = 99988 ) NP
          ELSE
             READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
             READ ( NIN, FMT = * ) ( ( B(I,J), J = 1,M ), I = 1,N )
             READ ( NIN, FMT = * ) ( ( C(I,J), J = 1,N ), I = 1,NP )
             READ ( NIN, FMT = * ) ( ( D(I,J), J = 1,M ), I = 1,NP )
             READ ( NIN, FMT = * ) TOL
    *        Computing the Hinf norm
             HNORM = AB13CD( N, M, NP, A, LDA, B, LDB, C, LDC, D, LDD, TOL,
         $                   IWORK, DWORK, LDWORK, CWORK, LCWORK, BWORK,
         $                   INFO )
    *
             IF ( INFO.EQ.0 ) THEN
                WRITE ( NOUT, FMT = 99997 )
                WRITE ( NOUT, FMT = 99991 ) HNORM
                FPEAK = DWORK(2)
                WRITE ( NOUT, FMT = 99996 )
                WRITE ( NOUT, FMT = 99991 ) FPEAK
             ELSE
                WRITE( NOUT, FMT = 99998 ) INFO
             END IF
          END IF
          STOP
    *
    99999 FORMAT (' AB13CD EXAMPLE PROGRAM RESULTS',/1X)
    99998 FORMAT (/' INFO on exit from AB13CD =',I2)
    99997 FORMAT (/' The H_infty norm of the system is'/)
    99996 FORMAT (/' The peak frequency is'/)
    99992 FORMAT (10(1X,F8.4))
    99991 FORMAT (D17.10)
    99990 FORMAT (/' N is out of range.',/' N = ',I5)
    99989 FORMAT (/' M is out of range.',/' M = ',I5)
    99988 FORMAT (/' NP is out of range.',/' NP = ',I5)
          END

Program Data

     AB13CD EXAMPLE PROGRAM DATA
       6     1     1
       0.0  1.0     0.0   0.0      0.0  0.0
      -0.5 -0.0002  0.0   0.0      0.0  0.0
       0.0  0.0     0.0   1.0      0.0  0.0
       0.0  0.0    -1.0  -0.00002  0.0  0.0
       0.0  0.0     0.0   0.0      0.0  1.0
       0.0  0.0     0.0   0.0     -2.0 -0.000002
       1.0
       0.0
       1.0
       0.0
       1.0
       0.0
       1.0  0.0  1.0  0.0  1.0  0.0
       0.0
     0.000000001

Program Results

     AB13CD EXAMPLE PROGRAM RESULTS

     The H_infty norm of the system is

     0.5000000006D+06

     The peak frequency is

     0.1414213562D+01

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