## 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
    *
    *     .. 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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