AB05OD

Rowwise concatenation 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 rowwise
  concatenation (parallel inter-connection on outputs, with separate
  inputs) of two systems, each given in state-space form.

Specification
      SUBROUTINE AB05OD( OVER, N1, M1, P1, N2, M2, ALPHA, A1, LDA1, B1,
     $                   LDB1, C1, LDC1, D1, LDD1, A2, LDA2, B2, LDB2,
     $                   C2, LDC2, D2, LDD2, N, M, 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, P1
      DOUBLE PRECISION  ALPHA
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.  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 each 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.

  M2      (input) INTEGER
          The number of input variables for the second system.
          M2 >= 0.

  ALPHA   (input) DOUBLE PRECISION
          A coefficient multiplying the transfer-function matrix
          (or the output equation) of the second system.

  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 P1-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,P1) if N2 > 0.
          LDC2 >= 1 if N2 = 0.

  D2      (input) DOUBLE PRECISION array, dimension (LDD2,M2)
          The leading P1-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,P1).

  N       (output) INTEGER
          The number of state variables (N1 + N2) in the connected
          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) for the connected
          system, i.e. the number of columns of B 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 connected 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 connected 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 P1-by-N part of this array contains the
          state/output matrix C for the connected system.
          The array C can overlap C1 if OVER = 'O'.

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

  D       (output) DOUBLE PRECISION array, dimension (LDD,M1+M2)
          The leading P1-by-M part of this array contains the
          input/output matrix D for the connected system.
          The array D can overlap D1 if OVER = 'O'.

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

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

Method
  After rowwise concatenation (parallel inter-connection with
  separate inputs) of the two systems,

  X1'     = A1*X1 + B1*U
  Y1      = C1*X1 + D1*U

  X2'     = A2*X2 + B2*V
  Y2      = C2*X2 + D2*V

  (where  '  denotes differentiation with respect to time),

  with the output equation for the second system multiplied by a
  scalar alpha, the following state-space model will be obtained:

  X'      = A*X + B*(U)
                    (V)

  Y       = C*X + D*(U)
                    (V)

  where matrix  A  has the form    ( A1   0  ),
                                   ( 0    A2 )

        matrix  B  has the form    ( B1   0  ),
                                   ( 0    B2 )

        matrix  C  has the form    ( C1   alpha*C2 ) and

        matrix  D  has the form    ( D1   alpha*D2 ).

References
  None

Numerical Aspects
  None

Further Comments
  None
Example

Program Text

*     AB05OD 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
      PARAMETER        ( N1MAX = 20, N2MAX = 20, NMAX = N1MAX+N2MAX,
     $                   M1MAX = 20, M2MAX = 20, MMAX = M1MAX+M2MAX,
     $                   P1MAX = 20 )
      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 = P1MAX, LDC1 = P1MAX, LDC2 = P1MAX,
     $                   LDD = P1MAX, 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, P1
      DOUBLE PRECISION ALPHA
*     .. 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         AB05OD
*     .. Executable Statements ..
*
      OVER = 'N'
      ALPHA = ONE
      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
      IF ( N1.LE.0 .OR. N1.GT.N1MAX ) THEN
         WRITE ( NOUT, FMT = 99993 ) 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 = 99992 ) 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 = 99991 ) 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 = 99990 ) 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 = 99989 ) M2
                  ELSE
                     READ ( NIN, FMT = * )
     $                    ( ( B2(I,J), I = 1,N2 ), J = 1,M2 )
                     READ ( NIN, FMT = * )
     $                    ( ( C2(I,J), J = 1,N2 ), I = 1,P1 )
                     READ ( NIN, FMT = * )
     $                    ( ( D2(I,J), J = 1,M2 ), I = 1,P1 )
*                       Find the state-space model (A,B,C,D).
                     CALL AB05OD( OVER, N1, M1, P1, N2, M2, ALPHA, A1,
     $                            LDA1, B1, LDB1, C1, LDC1, D1, LDD1,
     $                            A2, LDA2, B2, LDB2, C2, LDC2, D2,
     $                            LDD2, N, M, 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, P1
                           WRITE ( NOUT, FMT = 99996 )
     $                           ( C(I,J), J = 1,N )
   60                   CONTINUE
                        WRITE ( NOUT, FMT = 99993 )
                        DO 80 I = 1, P1
                           WRITE ( NOUT, FMT = 99996 )
     $                           ( D(I,J), J = 1,M )
   80                   CONTINUE
                     END IF
                  END IF
               END IF
            END IF
         END IF
      END IF
      STOP
*
99999 FORMAT (' AB05OD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from AB05OD = ',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)
      END
Program Data
 AB05OD 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
 AB05OD 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   1.0000   1.0000   0.0000
   0.0000   1.0000   0.0000   1.0000   1.0000  -1.0000

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

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