**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.

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(*)

**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.

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).

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'.

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

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.

None

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

None

**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

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

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