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
*     Copyright (c) 2002-2017 NICONET e.V.
*
*     .. 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

Return to index