**Purpose**

To reduce the matrices A and B using (and optionally accumulating) state-space and input-space transformations U and V respectively, such that the pair of matrices Ac = U' * A * U, Bc = U' * B * V are in upper "staircase" form. Specifically, [ Acont * ] [ Bcont ] Ac = [ ], Bc = [ ], [ 0 Auncont ] [ 0 ] and [ A11 A12 . . . A1,p-1 A1p ] [ B1 ] [ A21 A22 . . . A2,p-1 A2p ] [ 0 ] [ 0 A32 . . . A3,p-1 A3p ] [ 0 ] Acont = [ . . . . . . . ], Bc = [ . ], [ . . . . . . ] [ . ] [ . . . . . ] [ . ] [ 0 0 . . . Ap,p-1 App ] [ 0 ] where the blocks B1, A21, ..., Ap,p-1 have full row ranks and p is the controllability index of the pair. The size of the block Auncont is equal to the dimension of the uncontrollable subspace of the pair (A, B). The first stage of the reduction, the "forward" stage, accomplishes the reduction to the orthogonal canonical form (see SLICOT library routine AB01ND). The blocks B1, A21, ..., Ap,p-1 are further reduced in a second, "backward" stage to upper triangular form using RQ factorization. Each of these stages is optional.

SUBROUTINE AB01OD( STAGES, JOBU, JOBV, N, M, A, LDA, B, LDB, U, $ LDU, V, LDV, NCONT, INDCON, KSTAIR, TOL, IWORK, $ DWORK, LDWORK, INFO ) C .. Scalar Arguments .. CHARACTER JOBU, JOBV, STAGES INTEGER INDCON, INFO, LDA, LDB, LDU, LDV, LDWORK, M, N, $ NCONT DOUBLE PRECISION TOL C .. Array Arguments .. INTEGER IWORK(*), KSTAIR(*) DOUBLE PRECISION A(LDA,*), B(LDB,*), DWORK(*), U(LDU,*), V(LDV,*)

**Mode Parameters**

STAGES CHARACTER*1 Specifies the reduction stages to be performed as follows: = 'F': Perform the forward stage only; = 'B': Perform the backward stage only; = 'A': Perform both (all) stages. JOBU CHARACTER*1 Indicates whether the user wishes to accumulate in a matrix U the state-space transformations as follows: = 'N': Do not form U; = 'I': U is internally initialized to the unit matrix (if STAGES <> 'B'), or updated (if STAGES = 'B'), and the orthogonal transformation matrix U is returned. JOBV CHARACTER*1 Indicates whether the user wishes to accumulate in a matrix V the input-space transformations as follows: = 'N': Do not form V; = 'I': V is initialized to the unit matrix and the orthogonal transformation matrix V is returned. JOBV is not referenced if STAGES = 'F'.

N (input) INTEGER The actual state dimension, i.e. the order of the matrix A. N >= 0. M (input) INTEGER The actual input dimension. M >= 0. A (input/output) DOUBLE PRECISION array, dimension (LDA,N) On entry, the leading N-by-N part of this array must contain the state transition matrix A to be transformed. If STAGES = 'B', A should be in the orthogonal canonical form, as returned by SLICOT library routine AB01ND. On exit, the leading N-by-N part of this array contains the transformed state transition matrix U' * A * U. The leading NCONT-by-NCONT part contains the upper block Hessenberg state matrix Acont in Ac, given by U' * A * U, of a controllable realization for the original system. The elements below the first block-subdiagonal are set to zero. If STAGES <> 'F', the subdiagonal blocks of A are triangularized by RQ factorization, and the annihilated elements are explicitly zeroed. LDA INTEGER The leading dimension of array A. LDA >= MAX(1,N). B (input/output) DOUBLE PRECISION array, dimension (LDB,M) On entry, the leading N-by-M part of this array must contain the input matrix B to be transformed. If STAGES = 'B', B should be in the orthogonal canonical form, as returned by SLICOT library routine AB01ND. On exit with STAGES = 'F', the leading N-by-M part of this array contains the transformed input matrix U' * B, with all elements but the first block set to zero. On exit with STAGES <> 'F', the leading N-by-M part of this array contains the transformed input matrix U' * B * V, with all elements but the first block set to zero and the first block in upper triangular form. LDB INTEGER The leading dimension of array B. LDB >= MAX(1,N). U (input/output) DOUBLE PRECISION array, dimension (LDU,N) If STAGES <> 'B' or JOBU = 'N', then U need not be set on entry. If STAGES = 'B' and JOBU = 'I', then, on entry, the leading N-by-N part of this array must contain the transformation matrix U that reduced the pair to the orthogonal canonical form. On exit, if JOBU = 'I', the leading N-by-N part of this array contains the transformation matrix U that performed the specified reduction. If JOBU = 'N', the array U is not referenced and can be supplied as a dummy array (i.e. set parameter LDU = 1 and declare this array to be U(1,1) in the calling program). LDU INTEGER The leading dimension of array U. If JOBU = 'I', LDU >= MAX(1,N); if JOBU = 'N', LDU >= 1. V (output) DOUBLE PRECISION array, dimension (LDV,M) If JOBV = 'I', then the leading M-by-M part of this array contains the transformation matrix V. If STAGES = 'F', or JOBV = 'N', the array V is not referenced and can be supplied as a dummy array (i.e. set parameter LDV = 1 and declare this array to be V(1,1) in the calling program). LDV INTEGER The leading dimension of array V. If STAGES <> 'F' and JOBV = 'I', LDV >= MAX(1,M); if STAGES = 'F' or JOBV = 'N', LDV >= 1. NCONT (input/output) INTEGER The order of the controllable state-space representation. NCONT is input only if STAGES = 'B'. INDCON (input/output) INTEGER The number of stairs in the staircase form (also, the controllability index of the controllable part of the system representation). INDCON is input only if STAGES = 'B'. KSTAIR (input/output) INTEGER array, dimension (N) The leading INDCON elements of this array contain the dimensions of the stairs, or, also, the orders of the diagonal blocks of Acont. KSTAIR is input if STAGES = 'B', and output otherwise.

TOL DOUBLE PRECISION The tolerance to be used in rank determination when transforming (A, B). If the user sets TOL > 0, then the given value of TOL is used as a lower bound for the reciprocal condition number (see the description of the argument RCOND in the SLICOT routine MB03OD); a (sub)matrix whose estimated condition number is less than 1/TOL is considered to be of full rank. If the user sets TOL <= 0, then an implicitly computed, default tolerance, defined by TOLDEF = N*N*EPS, is used instead, where EPS is the machine precision (see LAPACK Library routine DLAMCH). TOL is not referenced if STAGES = 'B'.

IWORK INTEGER array, dimension (M) IWORK is not referenced if STAGES = 'B'. DWORK DOUBLE PRECISION array, dimension (LDWORK) On exit, if INFO = 0, DWORK(1) returns the optimal value of LDWORK. LDWORK INTEGER The length of the array DWORK. If STAGES <> 'B', LDWORK >= MAX(1, N + MAX(N,3*M)); If STAGES = 'B', LDWORK >= MAX(1, M + MAX(N,M)). For optimum performance LDWORK should be larger.

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

Staircase reduction of the pencil [B|sI - A] is used. Orthogonal transformations U and V are constructed such that |B |sI-A * . . . * * | | 1| 11 . . . | | | A sI-A . . . | | | 21 22 . . . | | | . . * * | [U'BV|sI - U'AU] = |0 | 0 . . | | | A sI-A * | | | p,p-1 pp | | | | |0 | 0 0 sI-A | | | p+1,p+1| where the i-th diagonal block of U'AU has dimension KSTAIR(i), for i = 1,...,p. The value of p is returned in INDCON. The last block contains the uncontrollable modes of the (A,B)-pair which are also the generalized eigenvalues of the above pencil. The complete reduction is performed in two stages. The first, forward stage accomplishes the reduction to the orthogonal canonical form. The second, backward stage consists in further reduction to triangular form by applying left and right orthogonal transformations.

[1] Van Dooren, P. The generalized eigenvalue problem in linear system theory. IEEE Trans. Auto. Contr., AC-26, pp. 111-129, 1981. [2] Miminis, G. and Paige, C. An algorithm for pole assignment of time-invariant multi-input linear systems. Proc. 21st IEEE CDC, Orlando, Florida, 1, pp. 62-67, 1982.

The algorithm requires O((N + M) x N**2) operations and is backward stable (see [1]).

If the system matrices A and B are badly scaled, it would be useful to scale them with SLICOT routine TB01ID, before calling the routine.

**Program Text**

* AB01OD EXAMPLE PROGRAM TEXT * Copyright (c) 2002-2017 NICONET e.V. * * .. Parameters .. INTEGER NIN, NOUT PARAMETER ( NIN = 5, NOUT = 6 ) INTEGER NMAX, MMAX PARAMETER ( NMAX = 20, MMAX = 20 ) INTEGER LDA, LDB, LDU, LDV PARAMETER ( LDA = NMAX, LDB = NMAX, LDU = NMAX, $ LDV = MMAX ) INTEGER LIWORK PARAMETER ( LIWORK = MMAX ) INTEGER LDWORK PARAMETER ( LDWORK = NMAX + MAX( NMAX, 3*MMAX ) ) * .. Local Scalars .. DOUBLE PRECISION TOL INTEGER I, INDCON, INFO, J, M, N, NCONT CHARACTER*1 JOBU, JOBV, STAGES * .. Local Arrays .. DOUBLE PRECISION A(LDA,NMAX), B(LDB,MMAX), DWORK(LDWORK), $ U(LDU,NMAX), V(LDV,MMAX) INTEGER IWORK(LIWORK), KSTAIR(NMAX) * .. External Subroutines .. EXTERNAL AB01OD * .. Intrinsic Functions .. INTRINSIC MAX * .. Executable Statements .. * WRITE ( NOUT, FMT = 99999 ) * Skip the heading in the data file and read the data. READ ( NIN, FMT = '()' ) READ ( NIN, FMT = * ) N, M, TOL, STAGES, JOBU, JOBV IF ( N.LE.0 .OR. N.GT.NMAX ) THEN WRITE ( NOUT, FMT = 99992 ) N ELSE READ ( NIN, FMT = * ) ( ( A(I,J), I = 1,N ), J = 1,N ) IF ( M.LE.0 .OR. M.GT.MMAX ) THEN WRITE ( NOUT, FMT = 99991 ) M ELSE READ ( NIN, FMT = * ) ( ( B(I,J), J = 1,M ), I = 1,N ) * Reduce the matrices A and B to upper "staircase" form. CALL AB01OD( STAGES, JOBU, JOBV, N, M, A, LDA, B, LDB, U, $ LDU, V, LDV, NCONT, INDCON, KSTAIR, TOL, IWORK, $ 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 = 99995 ) ( A(I,J), J = 1,N ) 20 CONTINUE WRITE ( NOUT, FMT = 99996 ) DO 40 I = 1, N WRITE ( NOUT, FMT = 99995 ) ( B(I,J), J = 1,M ) 40 CONTINUE WRITE ( NOUT, FMT = 99994 ) INDCON WRITE ( NOUT, FMT = 99993 ) ( KSTAIR(I), I = 1,INDCON ) END IF END IF END IF STOP * 99999 FORMAT (' AB01OD EXAMPLE PROGRAM RESULTS',/1X) 99998 FORMAT (' INFO on exit from AB01OD = ',I2) 99997 FORMAT (' The transformed state transition matrix is ') 99996 FORMAT (/' The transformed input matrix is ') 99995 FORMAT (20(1X,F8.4)) 99994 FORMAT (/' The number of stairs in the staircase form = ',I3,/) 99993 FORMAT (' The dimensions of the stairs are ',/(20(I3,2X))) 99992 FORMAT (/' N is out of range.',/' N = ',I5) 99991 FORMAT (/' M is out of range.',/' M = ',I5) END

AB01OD EXAMPLE PROGRAM DATA 5 2 0.0 F N N 17.0 24.0 1.0 8.0 15.0 23.0 5.0 7.0 14.0 16.0 4.0 6.0 13.0 20.0 22.0 10.0 12.0 19.0 21.0 3.0 11.0 18.0 25.0 2.0 9.0 -1.0 -4.0 4.0 9.0 -9.0 -16.0 16.0 25.0 -25.0 -36.0

AB01OD EXAMPLE PROGRAM RESULTS The transformed state transition matrix is 12.8848 3.2345 11.8211 3.3758 -0.8982 4.4741 -12.5544 5.3509 5.9403 1.4360 14.4576 7.6855 23.1452 26.3872 -29.9557 0.0000 1.4805 27.4668 22.6564 -0.0072 0.0000 0.0000 -30.4822 0.6745 18.8680 The transformed input matrix is 31.1199 47.6865 3.2480 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 The number of stairs in the staircase form = 3 The dimensions of the stairs are 2 2 1