## AB05QD

### Appending two systems in state-space form

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

Purpose

```  To append two systems G1 and G2 in state-space form together.
If G1 = (A1,B1,C1,D1) and G2 = (A2,B2,C2,D2) are the state-space
models of the given two systems having the transfer-function
matrices G1 and G2, respectively, this subroutine constructs the
state-space model G = (A,B,C,D) which corresponds to the
transfer-function matrix

( G1 0  )
G = (       )
( 0  G2 )

```
Specification
```      SUBROUTINE AB05QD( OVER, N1, M1, P1, N2, M2, P2, A1, LDA1, B1,
\$                   LDB1, C1, LDC1, D1, LDD1, A2, LDA2, B2, LDB2,
\$                   C2, LDC2, D2, LDD2, N, M, P, A, LDA, B, LDB,
\$                   C, LDC, D, LDD, INFO )
C     .. Scalar Arguments ..
CHARACTER         OVER
INTEGER           INFO, LDA, LDA1, LDA2, LDB, LDB1, LDB2, LDC,
\$                  LDC1, LDC2, LDD, LDD1, LDD2, M, M1, M2, N, N1,
\$                  N2, P, P1, P2
C     .. Array Arguments ..
DOUBLE PRECISION  A(LDA,*), A1(LDA1,*), A2(LDA2,*), B(LDB,*),
\$                  B1(LDB1,*), B2(LDB2,*), C(LDC,*), C1(LDC1,*),
\$                  C2(LDC2,*), D(LDD,*), D1(LDD1,*), D2(LDD2,*)

```
Arguments

Mode Parameters

```  OVER    CHARACTER*1
Indicates whether the user wishes to overlap pairs of
arrays, as follows:
= 'N':  Do not overlap;
= 'O':  Overlap pairs of arrays: A1 and A, B1 and B,
C1 and C, and D1 and D, i.e. the same name is
effectively used for each pair (for all pairs)
in the routine call.  In this case, setting
LDA1 = LDA, LDB1 = LDB, LDC1 = LDC, and LDD1 = LDD
will give maximum efficiency.

```
Input/Output Parameters
```  N1      (input) INTEGER
The number of state variables in the first system, i.e.
the order of the matrix A1, the number of rows of B1 and
the number of columns of C1.  N1 >= 0.

M1      (input) INTEGER
The number of input variables in the first system, i.e.
the number of columns of matrices B1 and D1.  M1 >= 0.

P1      (input) INTEGER
The number of output variables in the first system, i.e.
the number of rows of matrices C1 and D1.  P1 >= 0.

N2      (input) INTEGER
The number of state variables in the second system, i.e.
the order of the matrix A2, the number of rows of B2 and
the number of columns of C2.  N2 >= 0.

M2      (input) INTEGER
The number of input variables in the second system, i.e.
the number of columns of matrices B2 and D2.  M2 >= 0.

P2      (input) INTEGER
The number of output variables in the second system, i.e.
the number of rows of matrices C2 and D2.  P2 >= 0.

A1      (input) DOUBLE PRECISION array, dimension (LDA1,N1)
The leading N1-by-N1 part of this array must contain the
state transition matrix A1 for the first system.

LDA1    INTEGER
The leading dimension of array A1.  LDA1 >= MAX(1,N1).

B1      (input) DOUBLE PRECISION array, dimension (LDB1,M1)
The leading N1-by-M1 part of this array must contain the
input/state matrix B1 for the first system.

LDB1    INTEGER
The leading dimension of array B1.  LDB1 >= MAX(1,N1).

C1      (input) DOUBLE PRECISION array, dimension (LDC1,N1)
The leading P1-by-N1 part of this array must contain the
state/output matrix C1 for the first system.

LDC1    INTEGER
The leading dimension of array C1.
LDC1 >= MAX(1,P1) if N1 > 0.
LDC1 >= 1 if N1 = 0.

D1      (input) DOUBLE PRECISION array, dimension (LDD1,M1)
The leading P1-by-M1 part of this array must contain the
input/output matrix D1 for the first system.

LDD1    INTEGER
The leading dimension of array D1.  LDD1 >= MAX(1,P1).

A2      (input) DOUBLE PRECISION array, dimension (LDA2,N2)
The leading N2-by-N2 part of this array must contain the
state transition matrix A2 for the second system.

LDA2    INTEGER
The leading dimension of array A2.  LDA2 >= MAX(1,N2).

B2      (input) DOUBLE PRECISION array, dimension (LDB2,M2)
The leading N2-by-M2 part of this array must contain the
input/state matrix B2 for the second system.

LDB2    INTEGER
The leading dimension of array B2.  LDB2 >= MAX(1,N2).

C2      (input) DOUBLE PRECISION array, dimension (LDC2,N2)
The leading P2-by-N2 part of this array must contain the
state/output matrix C2 for the second system.

LDC2    INTEGER
The leading dimension of array C2.
LDC2 >= MAX(1,P2) if N2 > 0.
LDC2 >= 1 if N2 = 0.

D2      (input) DOUBLE PRECISION array, dimension (LDD2,M2)
The leading P2-by-M2 part of this array must contain the
input/output matrix D2 for the second system.

LDD2    INTEGER
The leading dimension of array D2.  LDD2 >= MAX(1,P2).

N       (output) INTEGER
The number of state variables (N1 + N2) in the resulting
system, i.e. the order of the matrix A, the number of rows
of B and the number of columns of C.

M       (output) INTEGER
The number of input variables (M1 + M2) in the resulting
system, i.e. the number of columns of B and D.

P       (output) INTEGER
The number of output variables (P1 + P2) of the resulting
system, i.e. the number of rows of C and D.

A       (output) DOUBLE PRECISION array, dimension (LDA,N1+N2)
The leading N-by-N part of this array contains the state
transition matrix A for the resulting system.
The array A can overlap A1 if OVER = 'O'.

LDA     INTEGER
The leading dimension of array A.  LDA >= MAX(1,N1+N2).

B       (output) DOUBLE PRECISION array, dimension (LDB,M1+M2)
The leading N-by-M part of this array contains the
input/state matrix B for the resulting system.
The array B can overlap B1 if OVER = 'O'.

LDB     INTEGER
The leading dimension of array B.  LDB >= MAX(1,N1+N2).

C       (output) DOUBLE PRECISION array, dimension (LDC,N1+N2)
The leading P-by-N part of this array contains the
state/output matrix C for the resulting system.
The array C can overlap C1 if OVER = 'O'.

LDC     INTEGER
The leading dimension of array C.
LDC >= MAX(1,P1+P2) if N1+N2 > 0.
LDC >= 1 if N1+N2 = 0.

D       (output) DOUBLE PRECISION array, dimension (LDD,M1+M2)
The leading P-by-M part of this array contains the
input/output matrix D for the resulting system.
The array D can overlap D1 if OVER = 'O'.

LDD     INTEGER
The leading dimension of array D.  LDD >= MAX(1,P1+P2).

```
Error Indicator
```  INFO    INTEGER
= 0:  successful exit;
< 0:  if INFO = -i, the i-th argument had an illegal
value.

```
Method
```  The matrices of the resulting systems are determined as:

( A1   0  )         ( B1  0  )
A = (         ) ,   B = (        ) ,
( 0    A2 )         ( 0   B2 )

( C1   0  )         ( D1  0  )
C = (         ) ,   D = (        ) .
( 0    C2 )         ( 0   D2 )

```
References
```  None

```
```  None
```
Example

Program Text

```*     AB05QD EXAMPLE PROGRAM TEXT
*     Copyright (c) 2002-2017 NICONET e.V.
*
*     .. Parameters ..
INTEGER          NIN, NOUT
PARAMETER        ( NIN = 5, NOUT = 6 )
INTEGER          N1MAX, N2MAX, NMAX, M1MAX, M2MAX, MMAX, P1MAX,
\$                 P2MAX, PMAX
PARAMETER        ( N1MAX = 20, N2MAX = 20, NMAX = N1MAX+N2MAX,
\$                   M1MAX = 20, M2MAX = 20, MMAX = M1MAX+M2MAX,
\$                   P1MAX = 20, P2MAX = 20, PMAX = P1MAX+P2MAX )
INTEGER          LDA, LDA1, LDA2, LDB, LDB1, LDB2, LDC, LDC1,
\$                 LDC2, LDD, LDD1, LDD2
PARAMETER        ( LDA = NMAX, LDA1 = N1MAX, LDA2 = N2MAX,
\$                   LDB = NMAX, LDB1 = N1MAX, LDB2 = N2MAX,
\$                   LDC = PMAX, LDC1 = P1MAX, LDC2 = P1MAX,
\$                   LDD = PMAX, LDD1 = P1MAX, LDD2 = P1MAX )
DOUBLE PRECISION ONE
PARAMETER        ( ONE=1.0D0 )
*     .. Local Scalars ..
CHARACTER*1      OVER
INTEGER          I, INFO, J, M, M1, M2, N, N1, N2, P, P1, P2
*     .. Local Arrays ..
DOUBLE PRECISION A(LDA,NMAX), A1(LDA1,N1MAX), A2(LDA2,N2MAX),
\$                 B(LDB,MMAX), B1(LDB1,M1MAX), B2(LDB2,M2MAX),
\$                 C(LDC,NMAX), C1(LDC1,N1MAX), C2(LDC2,N2MAX),
\$                 D(LDD,MMAX), D1(LDD1,M1MAX), D2(LDD2,M2MAX)
*     .. External Subroutines ..
EXTERNAL         AB05QD
*     .. Executable Statements ..
*
OVER = 'N'
WRITE ( NOUT, FMT = 99999 )
*     Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N1, M1, P1, N2, M2, P2
IF ( N1.LE.0 .OR. N1.GT.N1MAX ) THEN
WRITE ( NOUT, FMT = 99992 ) N1
ELSE
READ ( NIN, FMT = * ) ( ( A1(I,J), J = 1,N1 ), I = 1,N1 )
IF ( M1.LE.0 .OR. M1.GT.M1MAX ) THEN
WRITE ( NOUT, FMT = 99991 ) M1
ELSE
READ ( NIN, FMT = * ) ( ( B1(I,J), I = 1,N1 ), J = 1,M1 )
IF ( P1.LE.0 .OR. P1.GT.P1MAX ) THEN
WRITE ( NOUT, FMT = 99990 ) P1
ELSE
READ ( NIN, FMT = * ) ( ( C1(I,J), J = 1,N1 ), I = 1,P1 )
READ ( NIN, FMT = * ) ( ( D1(I,J), J = 1,M1 ), I = 1,P1 )
IF ( N2.LE.0 .OR. N2.GT.N2MAX ) THEN
WRITE ( NOUT, FMT = 99989 ) N2
ELSE
READ ( NIN, FMT = * )
\$                 ( ( A2(I,J), J = 1,N2 ), I = 1,N2 )
IF ( M2.LE.0 .OR. M2.GT.M2MAX ) THEN
WRITE ( NOUT, FMT = 99988 ) M2
ELSE
READ ( NIN, FMT = * )
\$                    ( ( B2(I,J), I = 1,N2 ), J = 1,M2 )
IF ( P2.LE.0 .OR. P2.GT.P2MAX ) THEN
WRITE ( NOUT, FMT = 99987 ) P2
ELSE
READ ( NIN, FMT = * )
\$                       ( ( C2(I,J), J = 1,N2 ), I = 1,P2 )
READ ( NIN, FMT = * )
\$                       ( ( D2(I,J), J = 1,M2 ), I = 1,P2 )
*                          Find the state-space model (A,B,C,D).
CALL AB05QD( OVER, N1, M1, P1, N2, M2, P2, A1,
\$                               LDA1, B1, LDB1, C1, LDC1, D1, LDD1,
\$                               A2, LDA2, B2, LDB2, C2, LDC2, D2,
\$                               LDD2, N, M, P, A, LDA, B, LDB, C,
\$                               LDC, D, LDD, 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 = 99993 )
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
END IF
END IF
END IF
STOP
*
99999 FORMAT (' AB05QD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from AB05QD = ',I2)
99997 FORMAT (' The state transition matrix of the connected system is')
99996 FORMAT (20(1X,F8.4))
99995 FORMAT (/' The input/state matrix of the connected system is ')
99994 FORMAT (/' The state/output matrix of the connected system is ')
99993 FORMAT (/' The input/output matrix of the connected system is ')
99992 FORMAT (/' N1 is out of range.',/' N1 = ',I5)
99991 FORMAT (/' M1 is out of range.',/' M1 = ',I5)
99990 FORMAT (/' P1 is out of range.',/' P1 = ',I5)
99989 FORMAT (/' N2 is out of range.',/' N2 = ',I5)
99988 FORMAT (/' M2 is out of range.',/' M2 = ',I5)
99987 FORMAT (/' P2 is out of range.',/' P2 = ',I5)
END
```
Program Data
``` AB05QD EXAMPLE PROGRAM DATA
3     2     2     3     2     2
1.0   0.0  -1.0
0.0  -1.0   1.0
1.0   1.0   2.0
1.0   1.0   0.0
2.0   0.0   1.0
3.0  -2.0   1.0
0.0   1.0   0.0
1.0   0.0
0.0   1.0
-3.0   0.0   0.0
1.0   0.0   1.0
0.0  -1.0   2.0
0.0  -1.0   0.0
1.0   0.0   2.0
1.0   1.0   0.0
1.0   1.0  -1.0
1.0   1.0
0.0   1.0
```
Program Results
``` AB05QD EXAMPLE PROGRAM RESULTS

The state transition matrix of the connected system is
1.0000   0.0000  -1.0000   0.0000   0.0000   0.0000
0.0000  -1.0000   1.0000   0.0000   0.0000   0.0000
1.0000   1.0000   2.0000   0.0000   0.0000   0.0000
0.0000   0.0000   0.0000  -3.0000   0.0000   0.0000
0.0000   0.0000   0.0000   1.0000   0.0000   1.0000
0.0000   0.0000   0.0000   0.0000  -1.0000   2.0000

The input/state matrix of the connected system is
1.0000   2.0000   0.0000   0.0000
1.0000   0.0000   0.0000   0.0000
0.0000   1.0000   0.0000   0.0000
0.0000   0.0000   0.0000   1.0000
0.0000   0.0000  -1.0000   0.0000
0.0000   0.0000   0.0000   2.0000

The state/output matrix of the connected system is
3.0000  -2.0000   1.0000   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   1.0000   0.0000
0.0000   0.0000   0.0000   1.0000   1.0000  -1.0000

The input/output matrix of the connected system is
1.0000   0.0000   0.0000   0.0000
0.0000   1.0000   0.0000   0.0000
0.0000   0.0000   1.0000   1.0000
0.0000   0.0000   0.0000   1.0000
```