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<HTML>
<HEAD><TITLE>AB05MD - SLICOT Library Routine Documentation</TITLE>
</HEAD>
<BODY>
<H2><A Name="AB05MD">AB05MD</A></H2>
<H3>
Cascade inter-connection of two systems in state-space form
</H3>
<A HREF ="#Specification"><B>[Specification]</B></A>
<A HREF ="#Arguments"><B>[Arguments]</B></A>
<A HREF ="#Method"><B>[Method]</B></A>
<A HREF ="#References"><B>[References]</B></A>
<A HREF ="#Comments"><B>[Comments]</B></A>
<A HREF ="#Example"><B>[Example]</B></A>
<P>
<B><FONT SIZE="+1">Purpose</FONT></B>
<PRE>
To obtain the state-space model (A,B,C,D) for the cascaded
inter-connection of two systems, each given in state-space form.
</PRE>
<A name="Specification"><B><FONT SIZE="+1">Specification</FONT></B></A>
<PRE>
SUBROUTINE AB05MD( UPLO, OVER, N1, M1, P1, N2, P2, A1, LDA1, B1,
$ LDB1, C1, LDC1, D1, LDD1, A2, LDA2, B2, LDB2,
$ C2, LDC2, D2, LDD2, N, A, LDA, B, LDB, C, LDC,
$ D, LDD, DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER OVER, UPLO
INTEGER INFO, LDA, LDA1, LDA2, LDB, LDB1, LDB2, LDC,
$ LDC1, LDC2, LDD, LDD1, LDD2, LDWORK, M1, N, N1,
$ N2, 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,*),
$ DWORK(*)
</PRE>
<A name="Arguments"><B><FONT SIZE="+1">Arguments</FONT></B></A>
<P>
<B>Mode Parameters</B>
<PRE>
UPLO CHARACTER*1
Indicates whether the user wishes to obtain the matrix A
in the upper or lower block diagonal form, as follows:
= 'U': Obtain A in the upper block diagonal form;
= 'L': Obtain A in the lower block diagonal form.
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 (for UPLO = 'L'), or A2
and A, B2 and B, C2 and C, and D2 and D (for
UPLO = 'U'), 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, or
LDA2 = LDA, LDB2 = LDB, LDC2 = LDC, and LDD2 = LDD
will give maximum efficiency.
</PRE>
<B>Input/Output Parameters</B>
<PRE>
N1 (input) INTEGER
The number of state variables in the first system, i.e.
the order of the matrix A1. N1 >= 0.
M1 (input) INTEGER
The number of input variables for the first system.
M1 >= 0.
P1 (input) INTEGER
The number of output variables from the first system and
the number of input variables for the second system.
P1 >= 0.
N2 (input) INTEGER
The number of state variables in the second system, i.e.
the order of the matrix A2. N2 >= 0.
P2 (input) INTEGER
The number of output variables from the second system.
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,P1)
The leading N2-by-P1 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,P1)
The leading P2-by-P1 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.
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 cascaded system.
If OVER = 'O', the array A can overlap A1, if UPLO = 'L',
or A2, if UPLO = 'U'.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,N1+N2).
B (output) DOUBLE PRECISION array, dimension (LDB,M1)
The leading N-by-M1 part of this array contains the
input/state matrix B for the cascaded system.
If OVER = 'O', the array B can overlap B1, if UPLO = 'L',
or B2, if UPLO = 'U'.
LDB INTEGER
The leading dimension of array B. LDB >= MAX(1,N1+N2).
C (output) DOUBLE PRECISION array, dimension (LDC,N1+N2)
The leading P2-by-N part of this array contains the
state/output matrix C for the cascaded system.
If OVER = 'O', the array C can overlap C1, if UPLO = 'L',
or C2, if UPLO = 'U'.
LDC INTEGER
The leading dimension of array C.
LDC >= MAX(1,P2) if N1+N2 > 0.
LDC >= 1 if N1+N2 = 0.
D (output) DOUBLE PRECISION array, dimension (LDD,M1)
The leading P2-by-M1 part of this array contains the
input/output matrix D for the cascaded system.
If OVER = 'O', the array D can overlap D1, if UPLO = 'L',
or D2, if UPLO = 'U'.
LDD INTEGER
The leading dimension of array D. LDD >= MAX(1,P2).
</PRE>
<B>Workspace</B>
<PRE>
DWORK DOUBLE PRECISION array, dimension (LDWORK)
The array DWORK is not referenced if OVER = 'N'.
LDWORK INTEGER
The length of the array DWORK.
LDWORK >= MAX( 1, P1*MAX(N1, M1, N2, P2) ) if OVER = 'O'.
LDWORK >= 1 if OVER = 'N'.
</PRE>
<B>Error Indicator</B>
<PRE>
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
</PRE>
<A name="Method"><B><FONT SIZE="+1">Method</FONT></B></A>
<PRE>
After cascaded inter-connection of the two systems
X1' = A1*X1 + B1*U
V = C1*X1 + D1*U
X2' = A2*X2 + B2*V
Y = C2*X2 + D2*V
(where ' denotes differentiation with respect to time)
the following state-space model will be obtained:
X' = A*X + B*U
Y = C*X + D*U
where matrix A has the form ( A1 0 ),
( B2*C1 A2)
matrix B has the form ( B1 ),
( B2*D1 )
matrix C has the form ( D2*C1 C2 ) and
matrix D has the form ( D2*D1 ).
This form is returned by the routine when UPLO = 'L'. Note that
when A1 and A2 are block lower triangular, the resulting state
matrix is also block lower triangular.
By applying a similarity transformation to the system above,
using the matrix ( 0 I ), where I is the identity matrix of
( J 0 )
order N2, and J is the identity matrix of order N1, the
system matrices become
A = ( A2 B2*C1 ),
( 0 A1 )
B = ( B2*D1 ),
( B1 )
C = ( C2 D2*C1 ) and
D = ( D2*D1 ).
This form is returned by the routine when UPLO = 'U'. Note that
when A1 and A2 are block upper triangular (for instance, in the
real Schur form), the resulting state matrix is also block upper
triangular.
</PRE>
<A name="References"><B><FONT SIZE="+1">References</FONT></B></A>
<PRE>
None
</PRE>
<A name="Numerical Aspects"><B><FONT SIZE="+1">Numerical Aspects</FONT></B></A>
<PRE>
The algorithm requires P1*(N1+M1)*(N2+P2) operations.
</PRE>
<A name="Comments"><B><FONT SIZE="+1">Further Comments</FONT></B></A>
<PRE>
None
</PRE>
<A name="Example"><B><FONT SIZE="+1">Example</FONT></B></A>
<P>
<B>Program Text</B>
<PRE>
* AB05MD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER N1MAX, N2MAX, NMAX, M1MAX, P1MAX, P2MAX
PARAMETER ( N1MAX = 20, N2MAX = 20, NMAX = N1MAX+N2MAX,
$ M1MAX = 20, P1MAX = 20, P2MAX = 20 )
INTEGER LDA, LDA1, LDA2, LDB, LDB1, LDB2, LDC, LDC1,
$ LDC2, LDD, LDD1, LDD2, LDWORK
PARAMETER ( LDA = NMAX, LDA1 = N1MAX, LDA2 = N2MAX,
$ LDB = NMAX,LDB1 = N1MAX, LDB2 = N2MAX,
$ LDC = P2MAX, LDC1 = P1MAX, LDC2 = P2MAX,
$ LDD = P2MAX, LDD1 = P1MAX, LDD2 = P2MAX,
$ LDWORK = P1MAX*N1MAX )
* .. Local Scalars ..
CHARACTER*1 OVER, UPLO
INTEGER I, INFO, J, M1, N, N1, N2, P1, P2
* .. Local Arrays ..
DOUBLE PRECISION A(LDA,NMAX), A1(LDA1,N1MAX), A2(LDA2,N2MAX),
$ B(LDB,M1MAX), B1(LDB1,M1MAX), B2(LDB2,P1MAX),
$ C(LDC,NMAX), C1(LDC1,N1MAX), C2(LDC2,N2MAX),
$ D(LDD,M1MAX), D1(LDD1,M1MAX), D2(LDD2,P1MAX),
$ DWORK(LDWORK)
* .. External Subroutines ..
EXTERNAL AB05MD
* .. Executable Statements ..
*
UPLO = 'Lower'
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, 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 )
READ ( NIN, FMT = * )
$ ( ( B2(I,J), I = 1,N2 ), J = 1,P1 )
IF ( P2.LE.0 .OR. P2.GT.P2MAX ) THEN
WRITE ( NOUT, FMT = 99988 ) P2
ELSE
READ ( NIN, FMT = * )
$ ( ( C2(I,J), J = 1,N2 ), I = 1,P2 )
READ ( NIN, FMT = * )
$ ( ( D2(I,J), J = 1,P1 ), I = 1,P2 )
* Find the state-space model (A,B,C,D).
CALL AB05MD( UPLO, OVER, N1, M1, P1, N2, P2, A1,
$ LDA1, B1, LDB1, C1, LDC1, D1, LDD1,
$ A2, LDA2, B2, LDB2, C2, LDC2, D2,
$ LDD2, N, A, LDA, B, LDB, C, LDC, D,
$ LDD, 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,M1 )
40 CONTINUE
WRITE ( NOUT, FMT = 99994 )
DO 60 I = 1, P2
WRITE ( NOUT, FMT = 99996 )
$ ( C(I,J), J = 1,N )
60 CONTINUE
WRITE ( NOUT, FMT = 99993 )
DO 80 I = 1, P2
WRITE ( NOUT, FMT = 99996 )
$ ( D(I,J), J = 1,M1 )
80 CONTINUE
END IF
END IF
END IF
END IF
END IF
END IF
STOP
*
99999 FORMAT (' AB05MD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from AB05MD = ',I2)
99997 FORMAT (' The state transition matrix of the cascaded system is ')
99996 FORMAT (20(1X,F8.4))
99995 FORMAT (/' The input/state matrix of the cascaded system is ')
99994 FORMAT (/' The state/output matrix of the cascaded system is ')
99993 FORMAT (/' The input/output matrix of the cascaded 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 (/' P2 is out of range.',/' P2 = ',I5)
END
</PRE>
<B>Program Data</B>
<PRE>
AB05MD EXAMPLE PROGRAM DATA
3 2 2 3 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
</PRE>
<B>Program Results</B>
<PRE>
AB05MD EXAMPLE PROGRAM RESULTS
The state transition matrix of the cascaded 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 1.0000 0.0000 -3.0000 0.0000 0.0000
-3.0000 2.0000 -1.0000 1.0000 0.0000 1.0000
0.0000 2.0000 0.0000 0.0000 -1.0000 2.0000
The input/state matrix of the cascaded system is
1.0000 2.0000
1.0000 0.0000
0.0000 1.0000
0.0000 1.0000
-1.0000 0.0000
0.0000 2.0000
The state/output matrix of the cascaded system is
3.0000 -1.0000 1.0000 1.0000 1.0000 0.0000
0.0000 1.0000 0.0000 1.0000 1.0000 -1.0000
The input/output matrix of the cascaded system is
1.0000 1.0000
0.0000 1.0000
</PRE>
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