Purpose
To compute a reduced order model by using singular perturbation approximation formulas.Specification
SUBROUTINE AB09DD( DICO, N, M, P, NR, A, LDA, B, LDB, C, LDC,
$ D, LDD, RCOND, IWORK, DWORK, INFO )
C .. Scalar Arguments ..
CHARACTER DICO
INTEGER INFO, LDA, LDB, LDC, LDD, M, N, NR, P
DOUBLE PRECISION RCOND
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), B(LDB,*), C(LDC,*), D(LDD,*), DWORK(*)
INTEGER IWORK(*)
Arguments
Mode Parameters
DICO CHARACTER*1
Specifies the type of the original system as follows:
= 'C': continuous-time system;
= 'D': discrete-time system.
Input/Output Parameters
N (input) INTEGER
The dimension of the state vector, i.e. the order of the
matrix A; also the number of rows of matrix B and the
number of columns of the matrix C. N >= 0.
M (input) INTEGER
The dimension of input vector, i.e. the number of columns
of matrices B and D. M >= 0.
P (input) INTEGER
The dimension of output vector, i.e. the number of rows of
matrices C and D. P >= 0.
NR (input) INTEGER
The order of the reduced order system. N >= NR >= 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 original system.
On exit, the leading NR-by-NR part of this array contains
the state dynamics matrix Ar of the reduced order system.
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 original system.
On exit, the leading NR-by-M part of this array contains
the input/state matrix Br of the reduced order system.
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 original system.
On exit, the leading P-by-NR part of this array contains
the state/output matrix Cr of the reduced order system.
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 original system.
On exit, the leading P-by-M part of this array contains
the input/output matrix Dr of the reduced order system.
If NR = 0 and the given system is stable, then D contains
the steady state gain of the system.
LDD INTEGER
The leading dimension of array D. LDD >= MAX(1,P).
RCOND (output) DOUBLE PRECISION
The reciprocal condition number of the matrix A22-g*I
(see METHOD).
Workspace
IWORK INTEGER array, dimension (2*(N-NR)) DWORK DOUBLE PRECISION array, dimension (4*(N-NR))Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: if the matrix A22-g*I (see METHOD) is numerically
singular.
Method
Given the system (A,B,C,D), partition the system matrices as
( A11 A12 ) ( B1 )
A = ( ) , B = ( ) , C = ( C1 C2 ),
( A21 A22 ) ( B2 )
where A11 is NR-by-NR, B1 is NR-by-M, C1 is P-by-NR, and the other
submatrices have appropriate dimensions.
The matrices of the reduced order system (Ar,Br,Cr,Dr) are
computed according to the following residualization formulas:
-1 -1
Ar = A11 + A12*(g*I-A22) *A21 , Br = B1 + A12*(g*I-A22) *B2
-1 -1
Cr = C1 + C2*(g*I-A22) *A21 , Dr = D + C2*(g*I-A22) *B2
where g = 0 if DICO = 'C' and g = 1 if DICO = 'D'.
Further Comments
NoneExample
Program Text
* AB09DD 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 LIWORK
PARAMETER ( LIWORK = 2*NMAX )
INTEGER LDWORK
PARAMETER ( LDWORK = 4*NMAX )
* .. Local Scalars ..
DOUBLE PRECISION RCOND
INTEGER I, INFO, J, M, N, NR, P
CHARACTER*1 DICO
* .. Local Arrays ..
DOUBLE PRECISION A(LDA,NMAX), B(LDB,MMAX), C(LDC,NMAX),
$ D(LDD,MMAX), DWORK(LDWORK)
INTEGER IWORK(LIWORK)
* .. External Subroutines ..
EXTERNAL AB09DD
* .. 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, NR, DICO
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 )
* Find a reduced ssr for (A,B,C).
CALL AB09DD( DICO, N, M, P, NR, A, LDA, B, LDB, C, LDC,
$ D, LDD, RCOND, IWORK, DWORK, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99997 ) RCOND
IF( NR.GT.0 ) WRITE ( NOUT, FMT = 99996 )
DO 20 I = 1, NR
WRITE ( NOUT, FMT = 99995 ) ( A(I,J), J = 1,NR )
20 CONTINUE
IF( NR.GT.0 ) WRITE ( NOUT, FMT = 99993 )
DO 40 I = 1, NR
WRITE ( NOUT, FMT = 99995 ) ( B(I,J), J = 1,M )
40 CONTINUE
IF( NR.GT.0 ) WRITE ( NOUT, FMT = 99992 )
DO 60 I = 1, P
WRITE ( NOUT, FMT = 99995 ) ( C(I,J), J = 1,NR )
60 CONTINUE
WRITE ( NOUT, FMT = 99991 )
DO 70 I = 1, P
WRITE ( NOUT, FMT = 99995 ) ( D(I,J), J = 1,M )
70 CONTINUE
END IF
END IF
END IF
END IF
STOP
*
99999 FORMAT (' AB09DD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from AB09DD = ',I2)
99997 FORMAT (' The computed reciprocal condition number = ',1PD12.5)
99996 FORMAT (/' The reduced state dynamics matrix Ar is ')
99995 FORMAT (20(1X,F8.4))
99993 FORMAT (/' The reduced input/state matrix Br is ')
99992 FORMAT (/' The reduced state/output matrix Cr is ')
99991 FORMAT (/' The reduced input/output matrix Dr is ')
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
AB09DD EXAMPLE PROGRAM DATA (Continuous system)
7 2 3 5 C
-0.04165 4.9200 -4.9200 0 0 0 0
0 -3.3300 0 0 0 3.3300 0
0.5450 0 0 -0.5450 0 0 0
0 0 4.9200 -0.04165 4.9200 0 0
0 0 0 0 -3.3300 0 3.3300
-5.2100 0 0 0 0 -12.5000 0
0 0 0 -5.2100 0 0 -12.5000
0 0
0 0
0 0
0 0
0 0
12.5000 0
0 12.5000
1 0 0 0 0 0 0
0 0 1 0 0 0 0
0 0 0 1 0 0 0
0 0
0 0
0 0
Program Results
AB09DD EXAMPLE PROGRAM RESULTS The computed reciprocal condition number = 1.00000D+00 The reduced state dynamics matrix Ar is -0.0416 4.9200 -4.9200 0.0000 0.0000 -1.3879 -3.3300 0.0000 0.0000 0.0000 0.5450 0.0000 0.0000 -0.5450 0.0000 0.0000 0.0000 4.9200 -0.0416 4.9200 0.0000 0.0000 0.0000 -1.3879 -3.3300 The reduced input/state matrix Br is 0.0000 0.0000 3.3300 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 3.3300 The reduced state/output matrix Cr is 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 The reduced input/output matrix Dr is 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000