## SB08CD

### Left coprime factorization with inner denominator

[Specification] [Arguments] [Method] [References] [Comments] [Example]

Purpose

```  To construct, for a given system G = (A,B,C,D), an output
injection matrix H, an orthogonal transformation matrix Z, and a
gain matrix V, such that the systems

Q = (Z'*(A+H*C)*Z, Z'*(B+H*D), V*C*Z, V*D)
and
R = (Z'*(A+H*C)*Z, Z'*H, V*C*Z, V)

provide a stable left coprime factorization of G in the form
-1
G = R  * Q,

where G, Q and R are the corresponding transfer-function matrices
and the denominator R is co-inner, that is, R(s)*R'(-s) = I in
the continuous-time case, or R(z)*R'(1/z) = I in the discrete-time
case. The Z matrix is not explicitly computed.

Note: G must have no observable poles on the imaginary axis
for a continuous-time system, or on the unit circle for a
discrete-time system. If the given state-space representation
is not detectable, the undetectable part of the original
system is automatically deflated and the order of the systems
Q and R is accordingly reduced.

```
Specification
```      SUBROUTINE SB08CD( DICO, N, M, P, A, LDA, B, LDB, C, LDC, D, LDD,
\$                   NQ, NR, BR, LDBR, DR, LDDR, TOL, DWORK, LDWORK,
\$                   IWARN, INFO )
C     .. Scalar Arguments ..
CHARACTER         DICO
INTEGER           INFO, IWARN, LDA, LDB, LDBR, LDC, LDD, LDDR,
\$                  LDWORK, M, N, NQ, NR, P
DOUBLE PRECISION  TOL
C     .. Array Arguments ..
DOUBLE PRECISION  A(LDA,*), B(LDB,*), BR(LDBR,*), C(LDC,*),
\$                  D(LDD,*), DR(LDDR,*), DWORK(*)

```
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, and also the number of rows of the matrices B
and BR, 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 the matrices B and D.  M >= 0.

P       (input) INTEGER
The dimension of output vector, i.e. the number of rows
of the matrices C, D and DR, and the number of columns
of the matrices BR and DR.  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 A. The matrix A must not
have observable eigenvalues on the imaginary axis, if
DICO = 'C', or on the unit circle, if DICO = 'D'.
On exit, the leading NQ-by-NQ part of this array contains
the leading NQ-by-NQ part of the matrix Z'*(A+H*C)*Z, the
state dynamics matrix of the numerator factor Q, in a
real Schur form. The leading NR-by-NR part of this matrix
represents the state dynamics matrix of a minimal
realization of the denominator factor R.

LDA     INTEGER
The leading dimension of array A.  LDA >= MAX(1,N).

B       (input/output) DOUBLE PRECISION array, dimension
(LDB,MAX(M,P))
On entry, the leading N-by-M part of this array must
contain the input/state matrix.
On exit, the leading NQ-by-M part of this array contains
the leading NQ-by-M part of the matrix Z'*(B+H*D), the
input/state matrix of the numerator factor Q.
The remaining part of this array is needed as workspace.

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 C.
On exit, the leading P-by-NQ part of this array contains
the leading P-by-NQ part of the matrix V*C*Z, the
state/output matrix of the numerator factor Q.
The first NR columns of this array represent the
state/output matrix of a minimal realization of the
denominator factor R.
The remaining part of this array is needed as workspace.

LDC     INTEGER
The leading dimension of array C.
LDC >= MAX(1,M,P), if N > 0.
LDC >= 1,          if N = 0.

D       (input/output) DOUBLE PRECISION array, dimension
(LDD,MAX(M,P))
On entry, the leading P-by-M part of this array must
contain the input/output matrix.
On exit, the leading P-by-M part of this array contains
the matrix V*D representing the input/output matrix
of the numerator factor Q.
The remaining part of this array is needed as workspace.

LDD     INTEGER
The leading dimension of array D.  LDD >= MAX(1,M,P).

NQ      (output) INTEGER
The order of the resulting factors Q and R.
Generally, NQ = N - NS, where NS is the number of
unobservable eigenvalues outside the stability region.

NR      (output) INTEGER
The order of the minimal realization of the factor R.
Generally, NR is the number of observable eigenvalues
of A outside the stability region (the number of modified
eigenvalues).

BR      (output) DOUBLE PRECISION array, dimension (LDBR,P)
The leading NQ-by-P part of this array contains the
leading NQ-by-P part of the output injection matrix
Z'*H, which reflects the eigenvalues of A lying outside
the stable region to values which are symmetric with
respect to the imaginary axis (if DICO = 'C') or the unit
circle (if DICO = 'D'). The first NR rows of this matrix
form the input/state matrix of a minimal realization of
the denominator factor R.

LDBR    INTEGER
The leading dimension of array BR.  LDBR >= MAX(1,N).

DR      (output) DOUBLE PRECISION array, dimension (LDDR,P)
The leading P-by-P part of this array contains the lower
triangular matrix V representing the input/output matrix
of the denominator factor R.

LDDR    INTEGER
The leading dimension of array DR.  LDDR >= MAX(1,P).

```
Tolerances
```  TOL     DOUBLE PRECISION
The absolute tolerance level below which the elements of
C are considered zero (used for observability tests).
If the user sets TOL <= 0, then an implicitly computed,
default tolerance, defined by  TOLDEF = N*EPS*NORM(C),
is used instead, where EPS is the machine precision
(see LAPACK Library routine DLAMCH) and NORM(C) denotes
the infinity-norm of C.

```
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, P*N + MAX( N*(N+5),P*(P+2),4*P,4*M ) ).
For optimum performance LDWORK should be larger.

```
Warning Indicator
```  IWARN   INTEGER
= 0:  no warning;
= K:  K violations of the numerical stability condition
NORM(H) <= 10*NORM(A)/NORM(C) occured during the
assignment of eigenvalues.

```
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 ordering of the
real Schur form of A, or in the iterative process
for reordering the eigenvalues of Z'*(A + H*C)*Z
along the diagonal;
= 3:  if DICO = 'C' and the matrix A has an observable
eigenvalue on the imaginary axis, or DICO = 'D' and
A has an observable eigenvalue on the unit circle.

```
Method
```  The subroutine uses the right coprime factorization algorithm with
inner denominator of  applied to G'.

```
References
```   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.

```
```  None
```
Example

Program Text

```*     SB08CD 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          MPMAX
PARAMETER        ( MPMAX = MAX( MMAX, PMAX ) )
INTEGER          LDA, LDB, LDBR, LDC, LDD, LDDR
PARAMETER        ( LDA = NMAX, LDB = NMAX, LDBR = NMAX,
\$                   LDC = MPMAX, LDD = MPMAX, LDDR = PMAX )
INTEGER          LDWORK
PARAMETER        ( LDWORK = NMAX*PMAX + MAX( NMAX*( NMAX + 5 ),
\$                            PMAX*( PMAX + 2 ), 4*PMAX, 4*MMAX ) )
*     .. Local Scalars ..
DOUBLE PRECISION TOL
INTEGER          I, INFO, IWARN, J, M, N, NQ, NR, P
CHARACTER*1      DICO
*     .. Local Arrays ..
DOUBLE PRECISION A(LDA,NMAX), B(LDB,MPMAX), BR(LDBR,PMAX),
\$                 C(LDC,NMAX), D(LDD,MPMAX), DR(LDDR,PMAX),
\$                 DWORK(LDWORK)
*     .. External Subroutines ..
EXTERNAL         SB08CD
*     .. 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, P, TOL, 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 RCFID for (A,B,C,D).
CALL SB08CD( DICO, N, M, P, A, LDA, B, LDB, C, LDC,
\$                      D, LDD, NQ, NR, BR, LDBR, DR, LDDR, TOL,
\$                      DWORK, LDWORK, IWARN, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
IF( NQ.GT.0 ) WRITE ( NOUT, FMT = 99996 )
DO 20 I = 1, NQ
WRITE ( NOUT, FMT = 99995 ) ( A(I,J), J = 1, NQ )
20             CONTINUE
IF( NQ.GT.0 ) WRITE ( NOUT, FMT = 99993 )
DO 40 I = 1, NQ
WRITE ( NOUT, FMT = 99995 ) ( B(I,J), J = 1, M )
40             CONTINUE
IF( NQ.GT.0 ) WRITE ( NOUT, FMT = 99992 )
DO 60 I = 1, P
WRITE ( NOUT, FMT = 99995 ) ( C(I,J), J = 1, NQ )
60             CONTINUE
WRITE ( NOUT, FMT = 99991 )
DO 70 I = 1, P
WRITE ( NOUT, FMT = 99995 ) ( D(I,J), J = 1, M )
70             CONTINUE
IF( NR.GT.0 ) WRITE ( NOUT, FMT = 99986 )
DO 80 I = 1, NR
WRITE ( NOUT, FMT = 99995 )
\$                     ( A(I,J), J = 1, NR )
80             CONTINUE
IF( NR.GT.0 ) WRITE ( NOUT, FMT = 99985 )
DO 90 I = 1, NR
WRITE ( NOUT, FMT = 99995 ) ( BR(I,J), J = 1, P )
90             CONTINUE
IF( NR.GT.0 ) WRITE ( NOUT, FMT = 99984 )
DO 100 I = 1, P
WRITE ( NOUT, FMT = 99995 )
\$                     ( C(I,J), J = 1, NR )
100             CONTINUE
WRITE ( NOUT, FMT = 99983 )
DO 110 I = 1, P
WRITE ( NOUT, FMT = 99995 ) ( DR(I,J), J = 1, P )
110             CONTINUE
END IF
END IF
END IF
END IF
STOP
*
99999 FORMAT (' SB08CD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from SB08CD = ',I2)
99996 FORMAT (/' The numerator state dynamics matrix AQ is ')
99995 FORMAT (20(1X,F8.4))
99993 FORMAT (/' The numerator input/state matrix BQ is ')
99992 FORMAT (/' The numerator state/output matrix CQ is ')
99991 FORMAT (/' The numerator input/output matrix DQ 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)
99986 FORMAT (/' The denominator state dynamics matrix AR is ')
99985 FORMAT (/' The denominator input/state matrix BR is ')
99984 FORMAT (/' The denominator state/output matrix CR is ')
99983 FORMAT (/' The denominator input/output matrix DR is ')
END
```
Program Data
``` SB08CD EXAMPLE PROGRAM DATA (Continuous system)
7  2  3   1.E-10 C
-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   0.0000
```
Program Results
``` SB08CD EXAMPLE PROGRAM RESULTS

The numerator state dynamics matrix AQ is
-0.1605   0.0523   0.9423   2.0193   0.4166   0.2518   1.6140
-0.4489  -0.1605   1.7955   3.8719  -0.2394   0.0491  -0.8740
0.0000   0.0000 -12.4245   3.5463  -0.0057   0.0254  -0.0053
0.0000   0.0000   0.0000  -3.5957  -0.0153  -0.0290  -0.0616
0.0000   0.0000   0.0000   0.0000 -13.1627  -1.9835  -3.6182
0.0000   0.0000   0.0000   0.0000   0.0000  -1.4178   5.6218
0.0000   0.0000   0.0000   0.0000   0.0000  -0.8374  -1.4178

The numerator input/state matrix BQ is
-1.0157   0.2554
0.5523   0.4443
0.0056 -11.6989
0.0490   4.3728
11.7198  -0.0038
-2.8173   0.0308
3.1018  -0.0009

The numerator state/output matrix CQ is
0.1975  -0.1063  -0.0006  -0.0083   0.1279   0.8797   0.3994
0.8541  -0.4513  -0.0007  -0.0041   0.0305  -0.2562   0.0122
0.4668   0.8826   0.0248  -0.0506   0.0000   0.0022  -0.0017

The numerator input/output matrix DQ is
0.0000   0.0000
0.0000   0.0000
0.0000   0.0000

The denominator state dynamics matrix AR is
-0.1605   0.0523
-0.4489  -0.1605

The denominator input/state matrix BR is
-0.0158  -0.0692  -0.1688
0.0306   0.1281  -0.4984

The denominator state/output matrix CR is
0.1975  -0.1063
0.8541  -0.4513
0.4668   0.8826

The denominator input/output matrix DR is
1.0000   0.0000   0.0000
0.0000   1.0000   0.0000
0.0000   0.0000   1.0000
```