## SB08GD

### State-space representation of a left coprime factorization

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

Purpose

```  To construct the state-space representation for the system
G = (A,B,C,D) from the factors Q = (AQR,BQ,CQR,DQ) and
R = (AQR,BR,CQR,DR) of its left coprime factorization
-1
G = R  * Q,

where G, Q and R are the corresponding transfer-function matrices.

```
Specification
```      SUBROUTINE SB08GD( N, M, P, A, LDA, B, LDB, C, LDC, D, LDD, BR,
\$                   LDBR, DR, LDDR, IWORK, DWORK, INFO )
C     .. Scalar Arguments ..
INTEGER           INFO, LDA, LDB, LDBR, LDC, LDD, LDDR, M, N, P
C     .. Array Arguments ..
DOUBLE PRECISION  A(LDA,*), B(LDB,*), BR(LDBR,*), C(LDC,*),
\$                  D(LDD,*), DR(LDDR,*), DWORK(*)
INTEGER           IWORK(*)

```
Arguments

Input/Output Parameters

```  N       (input) INTEGER
The order of the matrix A. Also the number of rows of the
matrices B and BR and the number of columns of the matrix
C. N represents the order of the systems Q and R.  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 AQR of the systems
Q and R.
On exit, the leading N-by-N part of this array contains
the state dynamics matrix of the system G.

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 BQ of the system Q.
On exit, the leading N-by-M part of this array contains
the input/state matrix of the system G.

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 CQR of the systems
Q and R.
On exit, the leading P-by-N part of this array contains
the state/output matrix of the system G.

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 DQ of the system Q.
On exit, the leading P-by-M part of this array contains
the input/output matrix of the system G.

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

BR      (input) DOUBLE PRECISION array, dimension (LDBR,P)
The leading N-by-P part of this array must contain the
input/state matrix BR of the system R.

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

DR      (input/output) DOUBLE PRECISION array, dimension (LDDR,P)
On entry, the leading P-by-P part of this array must
contain the input/output matrix DR of the system R.
On exit, the leading P-by-P part of this array contains
the LU factorization of the matrix DR, as computed by
LAPACK Library routine DGETRF.

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

```
Workspace
```  IWORK   INTEGER array, dimension (P)

DWORK   DOUBLE PRECISION array, dimension (MAX(1,4*P))
On exit, DWORK(1) contains an estimate of the reciprocal
condition number of the matrix DR.

```
Error Indicator
```  INFO    INTEGER
= 0:  successful exit;
< 0:  if INFO = -i, the i-th argument had an illegal
value;
= 1:  the matrix DR is singular;
= 2:  the matrix DR is numerically singular (warning);
the calculations continued.

```
Method
```  The subroutine computes the matrices of the state-space
representation G = (A,B,C,D) by using the formulas:

-1              -1
A = AQR - BR * DR  * CQR,  C = DR  * CQR,
-1              -1
B = BQ  - BR * DR  * DQ,   D = DR  * DQ.

```
References
```   Varga A.
Coprime factors model reduction method based on
square-root balancing-free techniques.
System Analysis, Modelling and Simulation,
vol. 11, pp. 303-311, 1993.

```
```  None
```
Example

Program Text

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Program Data
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Program Results
```  None
```