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
NoneExample
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 0.000000001Program Results
AB13CD EXAMPLE PROGRAM RESULTS The H_infty norm of the system is 0.5000000006D+06 The peak frequency is 0.1414213562D+01