Purpose
To perform a transformation on the parameters (A,B,C,D) of a system, which is equivalent to a bilinear transformation of the corresponding transfer function matrix.Specification
SUBROUTINE AB04MD( TYPE, N, M, P, ALPHA, BETA, A, LDA, B, LDB, C,
$ LDC, D, LDD, IWORK, DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER TYPE
INTEGER INFO, LDA, LDB, LDC, LDD, LDWORK, M, N, P
DOUBLE PRECISION ALPHA, BETA
C .. Array Arguments ..
INTEGER IWORK(*)
DOUBLE PRECISION A(LDA,*), B(LDB,*), C(LDC,*), D(LDD,*), DWORK(*)
Arguments
Mode Parameters
TYPE CHARACTER*1
Indicates the type of the original system and the
transformation to be performed as follows:
= 'D': discrete-time -> continuous-time;
= 'C': continuous-time -> discrete-time.
Input/Output Parameters
N (input) INTEGER
The order of the state matrix A. N >= 0.
M (input) INTEGER
The number of system inputs. M >= 0.
P (input) INTEGER
The number of system outputs. P >= 0.
ALPHA, (input) DOUBLE PRECISION
BETA Parameters specifying the bilinear transformation.
Recommended values for stable systems: ALPHA = 1,
BETA = 1. ALPHA <> 0, BETA <> 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 matrix A of the original system.
On exit, the leading N-by-N part of this array contains
_
the state matrix A of the transformed 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 matrix B of the original system.
On exit, the leading N-by-M part of this array contains
_
the input matrix B of the transformed 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 output matrix C of the original system.
On exit, the leading P-by-N part of this array contains
_
the output matrix C of the transformed 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 D for the original system.
On exit, the leading P-by-M part of this array contains
_
the input/output matrix D of the transformed system.
LDD INTEGER
The leading dimension of array D. LDD >= MAX(1,P).
Workspace
IWORK INTEGER array, dimension (N)
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK.
LDWORK INTEGER
The length of the array DWORK. LDWORK >= MAX(1,N).
For optimum performance LDWORK >= MAX(1,N*NB), where NB
is the optimal blocksize.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: if the matrix (ALPHA*I + A) is exactly singular;
= 2: if the matrix (BETA*I - A) is exactly singular.
Method
The parameters of the discrete-time system are transformed into
the parameters of the continuous-time system (TYPE = 'D'), or
vice-versa (TYPE = 'C') by the transformation:
1. Discrete -> continuous
_ -1
A = beta*(alpha*I + A) * (A - alpha*I)
_ -1
B = sqrt(2*alpha*beta) * (alpha*I + A) * B
_ -1
C = sqrt(2*alpha*beta) * C * (alpha*I + A)
_ -1
D = D - C * (alpha*I + A) * B
which is equivalent to the bilinear transformation
z - alpha
z -> s = beta --------- .
z + alpha
of one transfer matrix onto the other.
2. Continuous -> discrete
_ -1
A = alpha*(beta*I - A) * (beta*I + A)
_ -1
B = sqrt(2*alpha*beta) * (beta*I - A) * B
_ -1
C = sqrt(2*alpha*beta) * C * (beta*I - A)
_ -1
D = D + C * (beta*I - A) * B
which is equivalent to the bilinear transformation
beta + s
s -> z = alpha -------- .
beta - s
of one transfer matrix onto the other.
References
[1] Al-Saggaf, U.M. and Franklin, G.F.
Model reduction via balanced realizations: a extension and
frequency weighting techniques.
IEEE Trans. Autom. Contr., AC-33, pp. 687-692, 1988.
Numerical Aspects
3 The time taken is approximately proportional to N . The accuracy depends mainly on the condition number of the matrix to be inverted.Further Comments
NoneExample
Program Text
* AB04MD 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 = NMAX )
* .. Local Scalars ..
DOUBLE PRECISION ALPHA, BETA
INTEGER I, INFO, J, M, N, P
CHARACTER*1 TYPE
* .. Local Arrays ..
DOUBLE PRECISION A(LDA,NMAX), B(LDB,MMAX), C(LDC,NMAX),
$ D(LDD,MMAX), DWORK(LDWORK)
INTEGER IWORK(NMAX)
* .. External Subroutines ..
EXTERNAL AB04MD
* .. 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, TYPE, ALPHA, BETA
IF ( N.LE.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99993 ) N
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), I = 1,N ), J = 1,N )
IF ( M.LE.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99992 ) M
ELSE
READ ( NIN, FMT = * ) ( ( B(I,J), I = 1,N ), J = 1,M )
IF ( P.LE.0 .OR. P.GT.PMAX ) THEN
WRITE ( NOUT, FMT = 99991 ) P
ELSE
READ ( NIN, FMT = * ) ( ( C(I,J), I = 1,P ), J = 1,N )
READ ( NIN, FMT = * ) ( ( D(I,J), I = 1,P ), J = 1,M )
* Transform the parameters (A,B,C,D).
CALL AB04MD( TYPE, N, M, P, ALPHA, BETA, A, LDA, B, LDB,
$ C, LDC, D, LDD, IWORK, DWORK, LDWORK, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99997 )
DO 20 I = 1, N
WRITE ( NOUT, FMT = 99996 ) ( A(I,J), J = 1,N )
20 CONTINUE
WRITE ( NOUT, FMT = 99995 )
DO 40 I = 1, N
WRITE ( NOUT, FMT = 99996 ) ( B(I,J), J = 1,M )
40 CONTINUE
WRITE ( NOUT, FMT = 99994 )
DO 60 I = 1, P
WRITE ( NOUT, FMT = 99996 ) ( C(I,J), J = 1,N )
60 CONTINUE
WRITE ( NOUT, FMT = 99990 )
DO 80 I = 1, P
WRITE ( NOUT, FMT = 99996 ) ( D(I,J), J = 1,M )
80 CONTINUE
END IF
END IF
END IF
END IF
STOP
*
99999 FORMAT (' AB04MD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from AB04MD = ',I2)
99997 FORMAT (' The transformed state matrix is ')
99996 FORMAT (20(1X,F8.4))
99995 FORMAT (/' The transformed input matrix is ')
99994 FORMAT (/' The transformed output matrix is ')
99993 FORMAT (/' N is out of range.',/' N = ',I5)
99992 FORMAT (/' M is out of range.',/' M = ',I5)
99991 FORMAT (/' P is out of range.',/' P = ',I5)
99990 FORMAT (/' The transformed input/output matrix is ')
END
Program Data
AB04MD EXAMPLE PROGRAM DATA 2 2 2 C 1.0D0 1.0D0 1.0 0.5 0.5 1.0 0.0 -1.0 1.0 0.0 -1.0 0.0 0.0 1.0 1.0 0.0 0.0 -1.0Program Results
AB04MD EXAMPLE PROGRAM RESULTS The transformed state matrix is -1.0000 -4.0000 -4.0000 -1.0000 The transformed input matrix is 2.8284 0.0000 0.0000 -2.8284 The transformed output matrix is 0.0000 2.8284 -2.8284 0.0000 The transformed input/output matrix is -1.0000 0.0000 0.0000 -3.0000