## 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 .

```
References
```   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.

```
```  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
```