## AB13DD

### L-infinity norm of a state space system

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

Purpose

      To compute the L-infinity norm of a continuous-time or
      discrete-time system, either standard or in the descriptor form,

                                      -1
         G(lambda) = C*( lambda*E - A ) *B + D .

      The norm is finite if and only if the matrix pair (A,E) has no
      eigenvalue on the boundary of the stability domain, i.e., the
      imaginary axis, or the unit circle, respectively. It is assumed
      that the matrix E is nonsingular.

Specification

          SUBROUTINE AB13DD( DICO, JOBE, EQUIL, JOBD, N, M, P, FPEAK,
         $                   A, LDA, E, LDE, B, LDB, C, LDC, D, LDD, GPEAK,
         $                   TOL, IWORK, DWORK, LDWORK, CWORK, LCWORK,
         $                   INFO )
    C     .. Scalar Arguments ..
          CHARACTER          DICO, EQUIL, JOBD, JOBE
          INTEGER            INFO, LCWORK, LDA, LDB, LDC, LDD, LDE, LDWORK,
         $                   M, N, P
          DOUBLE PRECISION   TOL
    C     .. Array Arguments ..
          COMPLEX*16         CWORK(  * )
          DOUBLE PRECISION   A( LDA, * ), B( LDB, * ), C( LDC, * ),
         $                   D( LDD, * ), DWORK(  * ), E( LDE, * ),
         $                   FPEAK(  2 ), GPEAK(  2 )
          INTEGER            IWORK(  * )

Arguments

Mode Parameters

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

      JOBE    CHARACTER*1
              Specifies whether E is a general square or an identity
              matrix, as follows:
              = 'G':  E is a general square matrix;
              = 'I':  E is the identity matrix.

      EQUIL   CHARACTER*1
              Specifies whether the user wishes to preliminarily
              equilibrate the system (A,E,B,C) or (A,B,C), as follows:
              = 'S':  perform equilibration (scaling);
              = 'N':  do not perform equilibration.

      JOBD    CHARACTER*1
              Specifies whether or not a non-zero matrix D appears in
              the given state space model:
              = 'D':  D is present;
              = 'Z':  D is assumed a zero matrix.

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.

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

      FPEAK   (input/output) DOUBLE PRECISION array, dimension (2)
              On entry, this parameter must contain an estimate of the
              frequency where the gain of the frequency response would
              achieve its peak value. Setting FPEAK(2) = 0 indicates an
              infinite frequency. An accurate estimate could reduce the
              number of iterations of the iterative algorithm. If no
              estimate is available, set FPEAK(1) = 0, and FPEAK(2) = 1.
              FPEAK(1) >= 0, FPEAK(2) >= 0.
              On exit, if INFO = 0, this array contains the frequency
              OMEGA, where the gain of the frequency response achieves
              its peak value GPEAK, i.e.,

                  || G ( j*OMEGA ) || = GPEAK ,  if DICO = 'C', or

                          j*OMEGA
                  || G ( e       ) || = GPEAK ,  if DICO = 'D',

              where OMEGA = FPEAK(1), if FPEAK(2) > 0, and OMEGA is
              infinite, if FPEAK(2) = 0.

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

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

      E       (input) DOUBLE PRECISION array, dimension (LDE,N)
              If JOBE = 'G', the leading N-by-N part of this array must
              contain the descriptor matrix E of the system.
              If JOBE = 'I', then E is assumed to be the identity
              matrix and is not referenced.

      LDE     INTEGER
              The leading dimension of the array E.
              LDE >= MAX(1,N), if JOBE = 'G';
              LDE >= 1,        if JOBE = 'I'.

      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 P-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,P).

      D       (input) DOUBLE PRECISION array, dimension (LDD,M)
              If JOBD = 'D', the leading P-by-M part of this array must
              contain the direct transmission matrix D.
              The array D is not referenced if JOBD = 'Z'.

      LDD     INTEGER
              The leading dimension of array D.
              LDD >= MAX(1,P), if JOBD = 'D';
              LDD >= 1,        if JOBD = 'Z'.

      GPEAK   (output) DOUBLE PRECISION array, dimension (2)
              The L-infinity norm of the system, i.e., the peak gain
              of the frequency response (as measured by the largest
              singular value in the MIMO case), coded in the same way
              as FPEAK.

Tolerances

      TOL     DOUBLE PRECISION
              Tolerance used to set the accuracy in determining the
              norm.  0 <= TOL < 1.

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.

      LDWORK  INTEGER
              The dimension of the array DWORK.
              LDWORK >= K, where K can be computed using the following
              pseudo-code (or the Fortran code included in the routine)

                 d = 6*MIN(P,M);
                 c = MAX( 4*MIN(P,M) + MAX(P,M), d );
                 if ( MIN(P,M) = 0 ) then
                    K = 1;
                 else if( N = 0 or B = 0 or C = 0 ) then
                    if( JOBD = 'D' ) then
                       K = P*M + c;
                    else
                       K = 1;
                    end
                 else
                    if ( DICO = 'D' ) then
                       b = 0;  e = d;
                    else
                       b = N*(N+M);  e = c;
                       if ( JOBD = Z' ) then  b = b + P*M;  end
                    end
                    if ( JOBD = 'D' ) then
                       r = P*M;
                       if ( JOBE = 'I', DICO = 'C',
                            N > 0, B <> 0, C <> 0 ) then
                          K = P*P + M*M;
                          r = r + N*(P+M);
                       else
                          K = 0;
                       end
                       K = K + r + c;  r = r + MIN(P,M);
                    else
                       r = 0;  K = 0;
                    end
                    r = r + N*(N+P+M);
                    if ( JOBE = 'G' ) then
                       r = r + N*N;
                       if ( EQUIL = 'S' ) then
                          K = MAX( K, r + 9*N );
                       end
                       K = MAX( K, r + 4*N + MAX( M, 2*N*N, N+b+e ) );
                    else
                       K = MAX( K, r + N +
                                   MAX( M, P, N*N+2*N, 3*N+b+e ) );
                    end
                    w = 0;
                    if ( JOBE = 'I', DICO = 'C' ) then
                       w = r + 4*N*N + 11*N;
                       if ( JOBD = 'D' ) then
                          w = w + MAX(M,P) + N*(P+M);
                       end
                    end
                    if ( JOBE = 'E' or DICO = 'D' or JOBD = 'D' ) then
                       w = MAX( w, r + 6*N + (2*N+P+M)*(2*N+P+M) +
                                MAX( 2*(N+P+M), 8*N*N + 16*N ) );
                    end
                    K = MAX( 1, K, w, r + 2*N + e );
                 end

              For good performance, LDWORK must generally be larger.

              An easily computable upper bound is

              K = MAX( 1, 15*N*N + P*P + M*M + (6*N+3)*(P+M) + 4*P*M +
                          N*M + 22*N + 7*MIN(P,M) ).

              The smallest workspace is obtained for DICO = 'C',
              JOBE = 'I', and JOBD = 'Z', namely

              K = MAX( 1, N*N + N*P + N*M + N +
                          MAX( N*N + N*M + P*M + 3*N + c,
                               4*N*N + 10*N ) ).

              for which an upper bound is

              K = MAX( 1, 6*N*N + N*P + 2*N*M + P*M + 11*N + MAX(P,M) +
                          6*MIN(P,M) ).

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

      LCWORK  INTEGER
              The dimension of the array CWORK.
              LCWORK >= 1,  if N = 0, or B = 0, or C = 0;
              LCWORK >= MAX(1, (N+M)*(N+P) + 2*MIN(P,M) + MAX(P,M)),
                            otherwise.
              For good performance, LCWORK must generally be larger.

Error Indicator

      INFO    INTEGER
              = 0:  successful exit;
              < 0:  if INFO = -i, the i-th argument had an illegal
                    value;
              = 1:  the matrix E is (numerically) singular;
              = 2:  the (periodic) QR (or QZ) algorithm for computing
                    eigenvalues did not converge;
              = 3:  the SVD algorithm for computing singular values did
                    not converge;
              = 4:  the tolerance is too small and the algorithm did
                    not converge.

Method

      The routine implements the method presented in [1], with
      extensions and refinements for improving numerical robustness and
      efficiency. Structure-exploiting eigenvalue computations for
      Hamiltonian matrices are used if JOBE = 'I', DICO = 'C', and the
      symmetric matrices to be implicitly inverted are not too ill-
      conditioned. Otherwise, generalized eigenvalue computations are
      used in the iterative algorithm of [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 in MAXIT = 30 iterations
      (INFO = 4), the tolerance must be increased.

Further Comments

      If the matrix E is singular, other SLICOT Library routines
      could be used before calling AB13DD, for removing the singular
      part of the system.

Example

Program Text

    *     AB13DD 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, LDE
          PARAMETER        ( LDA = NMAX, LDB = NMAX, LDC = PMAX,
         $                   LDD = PMAX, LDE = NMAX )
          INTEGER          LIWORK
          PARAMETER        ( LIWORK = NMAX )
          INTEGER          LCWORK
          PARAMETER        ( LCWORK = ( NMAX + MMAX )*( NMAX + PMAX ) +
         $                             2*MIN( PMAX, MMAX ) +
         $                             MAX( PMAX, MMAX ) )
          INTEGER          LDWORK
          PARAMETER        ( LDWORK = 15*NMAX*NMAX + PMAX*PMAX + MMAX*MMAX +
         $                            ( 6*NMAX + 3 )*( PMAX + MMAX ) +
         $                            4*PMAX*MMAX + NMAX*MMAX + 22*NMAX +
         $                            7*MIN( PMAX, MMAX ) )
          DOUBLE PRECISION   ZERO
          PARAMETER          ( ZERO = 0.0D+0 )

    *     .. Local Scalars ..
          DOUBLE PRECISION TOL
          INTEGER          I, INFO, J, M, N, P
          CHARACTER        DICO, EQUIL, JOBD, JOBE
    *     .. Local Arrays ..
          INTEGER          IWORK( LIWORK )
          DOUBLE PRECISION A( LDA, NMAX ), B( LDB, MMAX ),  C( LDC, NMAX ),
         $                 D( LDD, MMAX ), DWORK( LDWORK ), E( LDE, NMAX ),
         $                 FPEAK( 2 ), GPEAK( 2 )
          COMPLEX*16       CWORK( LCWORK )
    *     .. External Functions ..
          LOGICAL          LSAME
          EXTERNAL         LSAME
    *     .. External Subroutines ..
          EXTERNAL         AB13DD
    *     .. 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, FPEAK, TOL, DICO, JOBE, EQUIL, JOBD
          IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
             WRITE ( NOUT, FMT = 99994 ) N
          ELSE IF ( M.LT.0 .OR. M.GT.MMAX ) THEN
             WRITE ( NOUT, FMT = 99993 ) M
          ELSE IF ( P.LT.0 .OR. P.GT.PMAX ) THEN
             WRITE ( NOUT, FMT = 99992 ) P
          ELSE
             READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
             IF ( LSAME( JOBE, 'G' ) )
         $      READ ( NIN, FMT = * ) ( ( E(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,P )
             IF ( LSAME( JOBD, 'D' ) )
         $      READ ( NIN, FMT = * ) ( ( D(I,J), J = 1,M ), I = 1,P )
    *        Computing the Linf norm.
             CALL AB13DD( DICO, JOBE, EQUIL, JOBD, N, M, P, FPEAK, A, LDA,
         $                E, LDE, B, LDB, C, LDC, D, LDD, GPEAK, TOL, IWORK,
         $                DWORK, LDWORK, CWORK, LCWORK, INFO )
    *
             IF ( INFO.EQ.0 ) THEN
                IF ( GPEAK( 2 ).EQ.ZERO ) THEN
                   WRITE ( NOUT, FMT = 99991 )
                ELSE
                   WRITE ( NOUT, FMT = 99997 )
                   WRITE ( NOUT, FMT = 99995 ) GPEAK( 1 )
                END IF
                IF ( FPEAK( 2 ).EQ.ZERO ) THEN
                   WRITE ( NOUT, FMT = 99990 )
                ELSE
                   WRITE ( NOUT, FMT = 99996 )
                   WRITE ( NOUT, FMT = 99995 ) FPEAK( 1 )
                END IF
             ELSE
                WRITE( NOUT, FMT = 99998 ) INFO
             END IF
          END IF
          STOP
    *
    99999 FORMAT (' AB13DD EXAMPLE PROGRAM RESULTS',/1X)
    99998 FORMAT (/' INFO on exit from AB13DD =',I2)
    99997 FORMAT (/' The L_infty norm of the system is'/)
    99996 FORMAT (/' The peak frequency is'/)
    99995 FORMAT (D17.10)
    99994 FORMAT (/' N is out of range.',/' N = ',I5)
    99993 FORMAT (/' M is out of range.',/' M = ',I5)
    99992 FORMAT (/' P is out of range.',/' P = ',I5)
    99991 FORMAT (/' The L_infty norm of the system is infinite')
    99990 FORMAT (/' The peak frequency is infinite'/)
          END

Program Data

     AB13CD EXAMPLE PROGRAM DATA
       6     1     1     0.0     1.0   0.000000001     C     I     N     D
       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

Program Results

     AB13DD EXAMPLE PROGRAM RESULTS

     The L_infty norm of the system is

     0.5000000001D+06

     The peak frequency is

     0.1414213562D+01

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