AB05MD

Cascade inter-connection of two systems in state-space form

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

Purpose

  To obtain the state-space model (A,B,C,D) for the cascaded
  inter-connection of two systems, each given in state-space form.

Specification
      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(*)

Arguments

Mode Parameters

  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.

Input/Output Parameters
  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).

Workspace
  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'.

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

Method
  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.

References
  None

Numerical Aspects
  The algorithm requires P1*(N1+M1)*(N2+P2) operations.

Further Comments
  None
Example

Program Text

*     AB05MD EXAMPLE PROGRAM TEXT
*     Copyright (c) 2002-2017 NICONET e.V.
*
*     .. 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
Program Data
 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
Program Results
 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

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