Purpose
To compute the H2 or L2 norm of the transfer-function matrix G of the system (A,B,C,D). G must not have poles on the imaginary axis, for a continuous-time system, or on the unit circle, for a discrete-time system. If the H2-norm is computed, the system must be stable.Specification
DOUBLE PRECISION FUNCTION AB13BD( DICO, JOBN, N, M, P, A, LDA,
$ B, LDB, C, LDC, D, LDD, NQ, TOL,
$ DWORK, LDWORK, IWARN, INFO)
C .. Scalar Arguments ..
CHARACTER DICO, JOBN
INTEGER INFO, IWARN, LDA, LDB, LDC, LDD, LDWORK, M,
$ N, NQ, P
DOUBLE PRECISION TOL
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), B(LDB,*), C(LDC,*), D(LDD,*), DWORK(*)
Function Value
AB13BD DOUBLE PRECISION
The H2-norm of G, if JOBN = 'H', or the L2-norm of G,
if JOBN = 'L' (if INFO = 0).
Arguments
Mode Parameters
DICO CHARACTER*1
Specifies the type of the system as follows:
= 'C': continuous-time system;
= 'D': discrete-time system.
JOBN CHARACTER*1
Specifies the norm to be computed as follows:
= 'H': the H2-norm;
= 'L': the L2-norm.
Input/Output Parameters
N (input) INTEGER
The order of the matrix A, the number of rows of the
matrix B, and the number of columns of the matrix C.
N represents the dimension of the state vector. N >= 0.
M (input) INTEGER
The number of columns of the matrices B and D.
M represents the dimension of input vector. M >= 0.
P (input) INTEGER
The number of rows of the matrices C and D.
P represents the dimension of output vector. P >= 0.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading N-by-N part of this array must
contain the state dynamics matrix of the system.
On exit, the leading NQ-by-NQ part of this array contains
the state dynamics matrix (in a real Schur form) of the
numerator factor Q of the right coprime factorization with
inner denominator of G (see METHOD).
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,N).
B (input/output) DOUBLE PRECISION array, dimension (LDB,M)
On entry, the leading N-by-M part of this array must
contain the input/state matrix of the system.
On exit, the leading NQ-by-M part of this array contains
the input/state matrix of the numerator factor Q of the
right coprime factorization with inner denominator of G
(see METHOD).
LDB INTEGER
The leading dimension of array B. LDB >= MAX(1,N).
C (input/output) DOUBLE PRECISION array, dimension (LDC,N)
On entry, the leading P-by-N part of this array must
contain the state/output matrix of the system.
On exit, the leading P-by-NQ part of this array contains
the state/output matrix of the numerator factor Q of the
right coprime factorization with inner denominator of G
(see METHOD).
LDC INTEGER
The leading dimension of array C. LDC >= MAX(1,P).
D (input/output) DOUBLE PRECISION array, dimension (LDD,M)
On entry, the leading P-by-M part of this array must
contain the input/output matrix of the system.
If DICO = 'C', D must be a null matrix.
On exit, the leading P-by-M part of this array contains
the input/output matrix of the numerator factor Q of
the right coprime factorization with inner denominator
of G (see METHOD).
LDD INTEGER
The leading dimension of array D. LDD >= MAX(1,P).
NQ (output) INTEGER
The order of the resulting numerator Q of the right
coprime factorization with inner denominator of G (see
METHOD).
Generally, NQ = N - NS, where NS is the number of
uncontrollable unstable eigenvalues.
Tolerances
TOL DOUBLE PRECISION
The absolute tolerance level below which the elements of
B are considered zero (used for controllability tests).
If the user sets TOL <= 0, then an implicitly computed,
default tolerance, defined by TOLDEF = N*EPS*NORM(B),
is used instead, where EPS is the machine precision
(see LAPACK Library routine DLAMCH) and NORM(B) denotes
the 1-norm of B.
Workspace
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK.
LDWORK INTEGER
The dimension of working array DWORK.
LDWORK >= MAX( 1, M*(N+M) + MAX( N*(N+5), M*(M+2), 4*P ),
N*( MAX( N, P ) + 4 ) + MIN( N, P ) ).
For optimum performance LDWORK should be larger.
Warning Indicator
IWARN INTEGER
= 0: no warning;
= K: K violations of the numerical stability condition
occured during the assignment of eigenvalues in
computing the right coprime factorization with inner
denominator of G (see the SLICOT subroutine SB08DD).
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: the reduction of A to a real Schur form failed;
= 2: a failure was detected during the reordering of the
real Schur form of A, or in the iterative process
for reordering the eigenvalues of Z'*(A + B*F)*Z
along the diagonal (see SLICOT routine SB08DD);
= 3: if DICO = 'C' and the matrix A has a controllable
eigenvalue on the imaginary axis, or DICO = 'D'
and A has a controllable eigenvalue on the unit
circle;
= 4: the solution of Lyapunov equation failed because
the equation is singular;
= 5: if DICO = 'C' and D is a nonzero matrix;
= 6: if JOBN = 'H' and the system is unstable.
Method
The subroutine is based on the algorithms proposed in [1] and [2].
If the given transfer-function matrix G is unstable, then a right
coprime factorization with inner denominator of G is first
computed
-1
G = Q*R ,
where Q and R are stable transfer-function matrices and R is
inner. If G is stable, then Q = G and R = I.
Let (AQ,BQ,CQ,DQ) be the state-space representation of Q.
If DICO = 'C', then the L2-norm of G is computed as
NORM2(G) = NORM2(Q) = SQRT(TRACE(BQ'*X*BQ)),
where X satisfies the continuous-time Lyapunov equation
AQ'*X + X*AQ + CQ'*CQ = 0.
If DICO = 'D', then the l2-norm of G is computed as
NORM2(G) = NORM2(Q) = SQRT(TRACE(BQ'*X*BQ+DQ'*DQ)),
where X satisfies the discrete-time Lyapunov equation
AQ'*X*AQ - X + CQ'*CQ = 0.
References
[1] Varga A.
On computing 2-norms of transfer-function matrices.
Proc. 1992 ACC, Chicago, June 1992.
[2] Varga A.
A Schur method for computing coprime factorizations with
inner denominators and applications in model reduction.
Proc. ACC'93, San Francisco, CA, pp. 2130-2131, 1993.
Numerical Aspects
3 The algorithm requires no more than 14N floating point operations.Further Comments
NoneExample
Program Text
* AB13BD 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 = 20, MMAX = 20, PMAX = 20 )
INTEGER LDA, LDB, LDC, LDD
PARAMETER ( LDA = NMAX, LDB = NMAX, LDC = PMAX,
$ LDD = PMAX )
INTEGER LDWORK
PARAMETER ( LDWORK = MAX( MMAX*( NMAX + MMAX ) +
$ MAX( NMAX*( NMAX + 5 ),
$ MMAX*( MMAX + 2 ), 4*PMAX ),
$ NMAX*( MAX( NMAX, PMAX ) + 4 ) +
$ MIN( NMAX, PMAX ) ) )
* .. Local Scalars ..
DOUBLE PRECISION S2NORM, TOL
INTEGER I, INFO, IWARN, J, M, N, NQ, P
CHARACTER*1 DICO, JOBN
* .. Local Arrays ..
DOUBLE PRECISION A(LDA,NMAX), B(LDB,MMAX), C(LDC,NMAX),
$ D(LDD,MMAX), DWORK(LDWORK)
* .. External Functions ..
LOGICAL LSAME
DOUBLE PRECISION AB13BD
EXTERNAL AB13BD, LSAME
* .. 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, TOL, DICO, JOBN
IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99990 ) N
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1, N ), I = 1,N )
IF ( M.LT.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99989 ) M
ELSE
READ ( NIN, FMT = * ) ( ( B(I,J), J = 1, M ), I = 1, N )
IF ( P.LT.0 .OR. P.GT.PMAX ) THEN
WRITE ( NOUT, FMT = 99988 ) P
ELSE
READ ( NIN, FMT = * ) ( ( C(I,J), J = 1, N ), I = 1, P )
READ ( NIN, FMT = * ) ( ( D(I,J), J = 1, M ), I = 1, P )
* Compute the H2 or L2 norm of (A,B,C,D).
S2NORM = AB13BD( DICO, JOBN, N, M, P, A, LDA, B, LDB,
* C, LDC, D, LDD, NQ, TOL, DWORK, LDWORK,
* IWARN, INFO)
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
IF( LSAME( JOBN, 'H' ) ) THEN
WRITE ( NOUT, FMT = 99997 ) S2NORM
ELSE
WRITE ( NOUT, FMT = 99996 ) S2NORM
END IF
END IF
END IF
END IF
END IF
STOP
*
99999 FORMAT (' AB13BD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from AB13BD = ',I2)
99997 FORMAT (' The H2-norm of the system = ',1PD14.5)
99996 FORMAT (' The L2-norm of the system = ',1PD14.5)
99990 FORMAT (/' N is out of range.',/' N = ',I5)
99989 FORMAT (/' M is out of range.',/' M = ',I5)
99988 FORMAT (/' P is out of range.',/' P = ',I5)
END
Program Data
AB13BD EXAMPLE PROGRAM DATA (Continuous system) 7 2 3 1.E-10 C L -0.04165 0.0000 4.9200 0.4920 0.0000 0.0000 0.0000 -5.2100 -12.500 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 3.3300 -3.3300 0.0000 0.0000 0.0000 0.0000 0.5450 0.0000 0.0000 0.0000 0.0545 0.0000 0.0000 0.0000 0.0000 0.0000 -0.49200 0.004165 0.0000 4.9200 0.0000 0.0000 0.0000 0.0000 0.5210 -12.500 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 3.3300 -3.3300 0.0000 0.0000 12.500 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 12.500 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000Program Results
AB13BD EXAMPLE PROGRAM RESULTS The L2-norm of the system = 7.93948D+00