## 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
*     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, 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
```