SUBROUTINE SB08DD( DICO, N, M, P, A, LDA, B, LDB, C, LDC, D, LDD, $ NQ, NR, CR, LDCR, DR, LDDR, TOL, DWORK, LDWORK, $ IWARN, INFO ) C C SLICOT RELEASE 5.0. C C Copyright (c) 2002-2009 NICONET e.V. C C This program is free software: you can redistribute it and/or C modify it under the terms of the GNU General Public License as C published by the Free Software Foundation, either version 2 of C the License, or (at your option) any later version. C C This program is distributed in the hope that it will be useful, C but WITHOUT ANY WARRANTY; without even the implied warranty of C MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the C GNU General Public License for more details. C C You should have received a copy of the GNU General Public License C along with this program. If not, see C . C C PURPOSE C C To construct, for a given system G = (A,B,C,D), a feedback matrix C F, an orthogonal transformation matrix Z, and a gain matrix V, C such that the systems C C Q = (Z'*(A+B*F)*Z, Z'*B*V, (C+D*F)*Z, D*V) C and C R = (Z'*(A+B*F)*Z, Z'*B*V, F*Z, V) C C provide a stable right coprime factorization of G in the form C -1 C G = Q * R , C C where G, Q and R are the corresponding transfer-function matrices C and the denominator R is inner, that is, R'(-s)*R(s) = I in the C continuous-time case, or R'(1/z)*R(z) = I in the discrete-time C case. The Z matrix is not explicitly computed. C C Note: G must have no controllable poles on the imaginary axis C for a continuous-time system, or on the unit circle for a C discrete-time system. If the given state-space representation C is not stabilizable, the unstabilizable part of the original C system is automatically deflated and the order of the systems C Q and R is accordingly reduced. C C ARGUMENTS C C Mode Parameters C C DICO CHARACTER*1 C Specifies the type of the original system as follows: C = 'C': continuous-time system; C = 'D': discrete-time system. C C Input/Output Parameters C C N (input) INTEGER C The dimension of the state vector, i.e. the order of the C matrix A, and also the number of rows of the matrix B and C the number of columns of the matrices C and CR. N >= 0. C C M (input) INTEGER C The dimension of input vector, i.e. the number of columns C of the matrices B, D and DR and the number of rows of the C matrices CR and DR. M >= 0. C C P (input) INTEGER C The dimension of output vector, i.e. the number of rows C of the matrices C and D. P >= 0. C C A (input/output) DOUBLE PRECISION array, dimension (LDA,N) C On entry, the leading N-by-N part of this array must C contain the state dynamics matrix A. The matrix A must not C have controllable eigenvalues on the imaginary axis, if C DICO = 'C', or on the unit circle, if DICO = 'D'. C On exit, the leading NQ-by-NQ part of this array contains C the leading NQ-by-NQ part of the matrix Z'*(A+B*F)*Z, the C state dynamics matrix of the numerator factor Q, in a C real Schur form. The trailing NR-by-NR part of this matrix C represents the state dynamics matrix of a minimal C realization of the denominator factor R. C C LDA INTEGER C The leading dimension of array A. LDA >= MAX(1,N). C C B (input/output) DOUBLE PRECISION array, dimension (LDB,M) C On entry, the leading N-by-M part of this array must C contain the input/state matrix. C On exit, the leading NQ-by-M part of this array contains C the leading NQ-by-M part of the matrix Z'*B*V, the C input/state matrix of the numerator factor Q. The last C NR rows of this matrix form the input/state matrix of C a minimal realization of the denominator factor R. C C LDB INTEGER C The leading dimension of array B. LDB >= MAX(1,N). C C C (input/output) DOUBLE PRECISION array, dimension (LDC,N) C On entry, the leading P-by-N part of this array must C contain the state/output matrix C. C On exit, the leading P-by-NQ part of this array contains C the leading P-by-NQ part of the matrix (C+D*F)*Z, C the state/output matrix of the numerator factor Q. C C LDC INTEGER C The leading dimension of array C. LDC >= MAX(1,P). C C D (input/output) DOUBLE PRECISION array, dimension (LDD,M) C On entry, the leading P-by-M part of this array must C contain the input/output matrix. C On exit, the leading P-by-M part of this array contains C the matrix D*V representing the input/output matrix C of the numerator factor Q. C C LDD INTEGER C The leading dimension of array D. LDD >= MAX(1,P). C C NQ (output) INTEGER C The order of the resulting factors Q and R. C Generally, NQ = N - NS, where NS is the number of C uncontrollable eigenvalues outside the stability region. C C NR (output) INTEGER C The order of the minimal realization of the factor R. C Generally, NR is the number of controllable eigenvalues C of A outside the stability region (the number of modified C eigenvalues). C C CR (output) DOUBLE PRECISION array, dimension (LDCR,N) C The leading M-by-NQ part of this array contains the C leading M-by-NQ part of the feedback matrix F*Z, which C reflects the eigenvalues of A lying outside the stable C region to values which are symmetric with respect to the C imaginary axis (if DICO = 'C') or the unit circle (if C DICO = 'D'). The last NR columns of this matrix form the C state/output matrix of a minimal realization of the C denominator factor R. C C LDCR INTEGER C The leading dimension of array CR. LDCR >= MAX(1,M). C C DR (output) DOUBLE PRECISION array, dimension (LDDR,M) C The leading M-by-M part of this array contains the upper C triangular matrix V of order M representing the C input/output matrix of the denominator factor R. C C LDDR INTEGER C The leading dimension of array DR. LDDR >= MAX(1,M). C C Tolerances C C TOL DOUBLE PRECISION C The absolute tolerance level below which the elements of C B are considered zero (used for controllability tests). C If the user sets TOL <= 0, then an implicitly computed, C default tolerance, defined by TOLDEF = N*EPS*NORM(B), C is used instead, where EPS is the machine precision C (see LAPACK Library routine DLAMCH) and NORM(B) denotes C the 1-norm of B. C C Workspace C C DWORK DOUBLE PRECISION array, dimension (LDWORK) C On exit, if INFO = 0, DWORK(1) returns the optimal value C of LDWORK. C C LDWORK INTEGER C The dimension of working array DWORK. C LDWORK >= MAX( 1, N*(N+5), M*(M+2), 4*M, 4*P ). C For optimum performance LDWORK should be larger. C C Warning Indicator C C IWARN INTEGER C = 0: no warning; C = K: K violations of the numerical stability condition C NORM(F) <= 10*NORM(A)/NORM(B) occured during the C assignment of eigenvalues. C C Error Indicator C C INFO INTEGER C = 0: successful exit; C < 0: if INFO = -i, the i-th argument had an illegal C value; C = 1: the reduction of A to a real Schur form failed; C = 2: a failure was detected during the ordering of the C real Schur form of A, or in the iterative process C for reordering the eigenvalues of Z'*(A + B*F)*Z C along the diagonal; C = 3: if DICO = 'C' and the matrix A has a controllable C eigenvalue on the imaginary axis, or DICO = 'D' C and A has a controllable eigenvalue on the unit C circle. C C METHOD C C The subroutine is based on the factorization algorithm of [1]. C C REFERENCES C C [1] Varga A. C A Schur method for computing coprime factorizations with inner C denominators and applications in model reduction. C Proc. ACC'93, San Francisco, CA, pp. 2130-2131, 1993. C C NUMERICAL ASPECTS C 3 C The algorithm requires no more than 14N floating point C operations. C C CONTRIBUTOR C C A. Varga, German Aerospace Center, C DLR Oberpfaffenhofen, July 1998. C Based on the RASP routine RCFID. C C REVISIONS C C Nov. 1998, V. Sima, Research Institute for Informatics, Bucharest. C Dec. 1998, V. Sima, Katholieke Univ. Leuven, Leuven. C Feb. 1999, May 2003, A. Varga, DLR Oberpfaffenhofen. C C KEYWORDS C C Coprime factorization, eigenvalue, eigenvalue assignment, C feedback control, pole placement, state-space model. C C ****************************************************************** C C .. Parameters .. DOUBLE PRECISION ONE, TEN, ZERO PARAMETER ( ONE = 1.0D0, TEN = 1.0D1, ZERO = 0.0D0 ) C .. Scalar Arguments .. CHARACTER DICO INTEGER INFO, IWARN, LDA, LDB, LDC, LDCR, LDD, LDDR, $ LDWORK, M, N, NQ, NR, P DOUBLE PRECISION TOL C .. Array Arguments .. DOUBLE PRECISION A(LDA,*), B(LDB,*), C(LDC,*), CR(LDCR,*), $ D(LDD,*), DR(LDDR,*), DWORK(*) C .. Local Scalars .. LOGICAL DISCR INTEGER I, IB, IB1, J, K, KFI, KV, KW, KWI, KWR, KZ, L, $ L1, NB, NCUR, NFP, NLOW, NSUP DOUBLE PRECISION ALPHA, BNORM, CS, PR, RMAX, SM, SN, TOLER, $ WRKOPT, X, Y C .. Local Arrays .. DOUBLE PRECISION Z(4,4) C .. External Functions .. DOUBLE PRECISION DLAMCH, DLANGE LOGICAL LSAME EXTERNAL DLAMCH, DLANGE, LSAME C .. External Subroutines .. EXTERNAL DGEMM, DLACPY, DLAEXC, DLANV2, DLASET, DROT, $ DTRMM, DTRMV, SB01FY, TB01LD, XERBLA C .. Intrinsic Functions .. INTRINSIC DBLE, MAX, MIN C .. Executable Statements .. C DISCR = LSAME( DICO, 'D' ) IWARN = 0 INFO = 0 C C Check the scalar input parameters. C IF( .NOT. ( LSAME( DICO, 'C' ) .OR. DISCR ) ) THEN INFO = -1 ELSE IF( N.LT.0 ) THEN INFO = -2 ELSE IF( M.LT.0 ) THEN INFO = -3 ELSE IF( P.LT.0 ) THEN INFO = -4 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN INFO = -6 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN INFO = -8 ELSE IF( LDC.LT.MAX( 1, P ) ) THEN INFO = -10 ELSE IF( LDD.LT.MAX( 1, P ) ) THEN INFO = -12 ELSE IF( LDCR.LT.MAX( 1, M ) ) THEN INFO = -16 ELSE IF( LDDR.LT.MAX( 1, M ) ) THEN INFO = -18 ELSE IF( LDWORK.LT.MAX( 1, N*(N+5), M*(M+2), 4*M, 4*P ) ) THEN INFO = -21 END IF IF( INFO.NE.0 )THEN C C Error return. C CALL XERBLA( 'SB08DD', -INFO ) RETURN END IF C C Set DR = I and quick return if possible. C NR = 0 IF( MIN( M, P ).GT.0 ) $ CALL DLASET( 'Full', M, M, ZERO, ONE, DR, LDDR ) IF( MIN( N, M ).EQ.0 ) THEN NQ = 0 DWORK(1) = ONE RETURN END IF C C Set F = 0 in the array CR. C CALL DLASET( 'Full', M, N, ZERO, ZERO, CR, LDCR ) C C Compute the norm of B and set the default tolerance if necessary. C BNORM = DLANGE( '1-norm', N, M, B, LDB, DWORK ) TOLER = TOL IF( TOLER.LE.ZERO ) $ TOLER = DBLE( N ) * BNORM * DLAMCH( 'Epsilon' ) IF( BNORM.LE.TOLER ) THEN NQ = 0 DWORK(1) = ONE RETURN END IF C C Compute the bound for the numerical stability condition. C RMAX = TEN * DLANGE( '1-norm', N, N, A, LDA, DWORK ) / BNORM C C Allocate working storage. C KZ = 1 KWR = KZ + N*N KWI = KWR + N KW = KWI + N C C Reduce A to an ordered real Schur form using an orthogonal C similarity transformation A <- Z'*A*Z and accumulate the C transformations in Z. The separation of spectrum of A is C performed such that the leading NFP-by-NFP submatrix of A C corresponds to the "stable" eigenvalues which will be not C modified. The bottom (N-NFP)-by-(N-NFP) diagonal block of A C corresponds to the "unstable" eigenvalues to be modified. C Apply the transformation to B and C: B <- Z'*B and C <- C*Z. C C Workspace needed: N*(N+2); C Additional workspace: need 3*N; C prefer larger. C IF( DISCR ) THEN ALPHA = ONE ELSE ALPHA = ZERO END IF CALL TB01LD( DICO, 'Stable', 'General', N, M, P, ALPHA, A, LDA, $ B, LDB, C, LDC, NFP, DWORK(KZ), N, DWORK(KWR), $ DWORK(KWI), DWORK(KW), LDWORK-KW+1, INFO ) IF( INFO.NE.0 ) $ RETURN C WRKOPT = DWORK(KW) + DBLE( KW-1 ) C C Perform the pole assignment if there exist "unstable" eigenvalues. C NQ = N IF( NFP.LT.N ) THEN KV = 1 KFI = KV + M*M KW = KFI + 2*M C C Set the limits for the bottom diagonal block. C NLOW = NFP + 1 NSUP = N C C WHILE (NLOW <= NSUP) DO 10 IF( NLOW.LE.NSUP ) THEN C C Main loop for assigning one or two poles. C C Determine the dimension of the last block. C IB = 1 IF( NLOW.LT.NSUP ) THEN IF( A(NSUP,NSUP-1).NE.ZERO ) IB = 2 END IF L = NSUP - IB + 1 C C Check the controllability of the last block. C IF( DLANGE( '1-norm', IB, M, B(L,1), LDB, DWORK(KW) ) $ .LE.TOLER ) THEN C C Deflate the uncontrollable block and resume the main C loop. C NSUP = NSUP - IB ELSE C C Determine the M-by-IB feedback matrix FI which assigns C the selected IB poles for the pair C ( A(L:L+IB-1,L:L+IB-1), B(L:L+IB-1,1:M) ). C C Workspace needed: M*(M+2). C CALL SB01FY( DISCR, IB, M, A(L,L), LDA, B(L,1), LDB, $ DWORK(KFI), M, DWORK(KV), M, INFO ) IF( INFO.EQ.2 ) THEN INFO = 3 RETURN END IF C C Check for possible numerical instability. C IF( DLANGE( '1-norm', M, IB, DWORK(KFI), M, DWORK(KW) ) $ .GT.RMAX ) IWARN = IWARN + 1 C C Update the state matrix A <-- A + B*[0 FI]. C CALL DGEMM( 'NoTranspose', 'NoTranspose', NSUP, IB, M, $ ONE, B, LDB, DWORK(KFI), M, ONE, A(1,L), $ LDA ) C C Update the feedback matrix F <-- F + V*[0 FI] in CR. C IF( DISCR ) $ CALL DTRMM( 'Left', 'Upper', 'NoTranspose', 'NonUnit', $ M, IB, ONE, DR, LDDR, DWORK(KFI), M ) K = KFI DO 30 J = L, L + IB - 1 DO 20 I = 1, M CR(I,J) = CR(I,J) + DWORK(K) K = K + 1 20 CONTINUE 30 CONTINUE C IF( DISCR ) THEN C C Update the input matrix B <-- B*V. C CALL DTRMM( 'Right', 'Upper', 'NoTranspose', $ 'NonUnit', N, M, ONE, DWORK(KV), M, B, $ LDB ) C C Update the feedthrough matrix DR <-- DR*V. C K = KV DO 40 I = 1, M CALL DTRMV( 'Upper', 'Transpose', 'NonUnit', $ M-I+1, DWORK(K), M, DR(I,I), LDDR ) K = K + M + 1 40 CONTINUE END IF C IF( IB.EQ.2 ) THEN C C Put the 2x2 block in a standard form. C L1 = L + 1 CALL DLANV2( A(L,L), A(L,L1), A(L1,L), A(L1,L1), $ X, Y, PR, SM, CS, SN ) C C Apply the transformation to A, B, C and F. C IF( L1.LT.NSUP ) $ CALL DROT( NSUP-L1, A(L,L1+1), LDA, A(L1,L1+1), $ LDA, CS, SN ) CALL DROT( L-1, A(1,L), 1, A(1,L1), 1, CS, SN ) CALL DROT( M, B(L,1), LDB, B(L1,1), LDB, CS, SN ) IF( P.GT.0 ) $ CALL DROT( P, C(1,L), 1, C(1,L1), 1, CS, SN ) CALL DROT( M, CR(1,L), 1, CR(1,L1), 1, CS, SN ) END IF IF( NLOW+IB.LE.NSUP ) THEN C C Move the last block(s) to the leading position(s) of C the bottom block. C C Workspace: need MAX(4*N, 4*M, 4*P). C NCUR = NSUP - IB C WHILE (NCUR >= NLOW) DO 50 IF( NCUR.GE.NLOW ) THEN C C Loop for positioning of the last block. C C Determine the dimension of the current block. C IB1 = 1 IF( NCUR.GT.NLOW ) THEN IF( A(NCUR,NCUR-1).NE.ZERO ) IB1 = 2 END IF NB = IB1 + IB C C Initialize the local transformation matrix Z. C CALL DLASET( 'Full', NB, NB, ZERO, ONE, Z, 4 ) L = NCUR - IB1 + 1 C C Exchange two adjacent blocks and accumulate the C transformations in Z. C CALL DLAEXC( .TRUE., NB, A(L,L), LDA, Z, 4, 1, IB1, $ IB, DWORK, INFO ) IF( INFO.NE.0 ) THEN INFO = 2 RETURN END IF C C Apply the transformation to the rest of A. C L1 = L + NB IF( L1.LE.NSUP ) THEN CALL DGEMM( 'Transpose', 'NoTranspose', NB, $ NSUP-L1+1, NB, ONE, Z, 4, A(L,L1), $ LDA, ZERO, DWORK, NB ) CALL DLACPY( 'Full', NB, NSUP-L1+1, DWORK, NB, $ A(L,L1), LDA ) END IF CALL DGEMM( 'NoTranspose', 'NoTranspose', L-1, NB, $ NB, ONE, A(1,L), LDA, Z, 4, ZERO, $ DWORK, N ) CALL DLACPY( 'Full', L-1, NB, DWORK, N, A(1,L), $ LDA ) C C Apply the transformation to B, C and F. C CALL DGEMM( 'Transpose', 'NoTranspose', NB, M, NB, $ ONE, Z, 4, B(L,1), LDB, ZERO, DWORK, $ NB ) CALL DLACPY( 'Full', NB, M, DWORK, NB, B(L,1), $ LDB ) C IF( P.GT.0 ) THEN CALL DGEMM( 'NoTranspose', 'NoTranspose', P, NB, $ NB, ONE, C(1,L), LDC, Z, 4, ZERO, $ DWORK, P ) CALL DLACPY( 'Full', P, NB, DWORK, P, $ C(1,L), LDC ) END IF C CALL DGEMM( 'NoTranspose', 'NoTranspose', M, NB, $ NB, ONE, CR(1,L), LDCR, Z, 4, ZERO, $ DWORK, M ) CALL DLACPY( 'Full', M, NB, DWORK, M, CR(1,L), $ LDCR ) C NCUR = NCUR - IB1 GO TO 50 END IF C END WHILE 50 C END IF NLOW = NLOW + IB END IF GO TO 10 END IF C END WHILE 10 C NQ = NSUP NR = NSUP - NFP C C Annihilate the elements below the first subdiagonal of A. C IF( NQ.GT.2 ) $ CALL DLASET( 'Lower', NQ-2, NQ-2, ZERO, ZERO, A(3,1), LDA ) END IF C C Compute C <-- CQ = C + D*F and D <-- DQ = D*DR. C CALL DGEMM( 'NoTranspose', 'NoTranspose', P, NQ, M, ONE, D, LDD, $ CR, LDCR, ONE, C, LDC ) IF( DISCR ) $ CALL DTRMM( 'Right', 'Upper', 'NoTranspose', 'NonUnit', P, M, $ ONE, DR, LDDR, D, LDD ) C DWORK(1) = MAX( WRKOPT, DBLE( MAX( M*(M+2), 4*M, 4*P ) ) ) C RETURN C *** Last line of SB08DD *** END