SUBROUTINE AB09AD( DICO, JOB, EQUIL, ORDSEL, N, M, P, NR, A, LDA, $ B, LDB, C, LDC, HSV, TOL, IWORK, 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 compute a reduced order model (Ar,Br,Cr) for a stable original C state-space representation (A,B,C) by using either the square-root C or the balancing-free square-root Balance & Truncate (B & T) C model reduction method. 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 JOB CHARACTER*1 C Specifies the model reduction approach to be used C as follows: C = 'B': use the square-root Balance & Truncate method; C = 'N': use the balancing-free square-root C Balance & Truncate method. C C EQUIL CHARACTER*1 C Specifies whether the user wishes to preliminarily C equilibrate the triplet (A,B,C) as follows: C = 'S': perform equilibration (scaling); C = 'N': do not perform equilibration. C C ORDSEL CHARACTER*1 C Specifies the order selection method as follows: C = 'F': the resulting order NR is fixed; C = 'A': the resulting order NR is automatically determined C on basis of the given tolerance TOL. C C Input/Output Parameters C C N (input) INTEGER C The order of the original state-space representation, i.e. C the order of the matrix A. N >= 0. C C M (input) INTEGER C The number of system inputs. M >= 0. C C P (input) INTEGER C The number of system outputs. P >= 0. C C NR (input/output) INTEGER C On entry with ORDSEL = 'F', NR is the desired order of the C resulting reduced order system. 0 <= NR <= N. C On exit, if INFO = 0, NR is the order of the resulting C reduced order model. NR is set as follows: C if ORDSEL = 'F', NR is equal to MIN(NR,NMIN), where NR C is the desired order on entry and NMIN is the order of a C minimal realization of the given system; NMIN is C determined as the number of Hankel singular values greater C than N*EPS*HNORM(A,B,C), where EPS is the machine C precision (see LAPACK Library Routine DLAMCH) and C HNORM(A,B,C) is the Hankel norm of the system (computed C in HSV(1)); C if ORDSEL = 'A', NR is equal to the number of Hankel C singular values greater than MAX(TOL,N*EPS*HNORM(A,B,C)). 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. C On exit, if INFO = 0, the leading NR-by-NR part of this C array contains the state dynamics matrix Ar of the reduced C order system. 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 original input/state matrix B. C On exit, if INFO = 0, the leading NR-by-M part of this C array contains the input/state matrix Br of the reduced C order system. 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 original state/output matrix C. C On exit, if INFO = 0, the leading P-by-NR part of this C array contains the state/output matrix Cr of the reduced C order system. C C LDC INTEGER C The leading dimension of array C. LDC >= MAX(1,P). C C HSV (output) DOUBLE PRECISION array, dimension (N) C If INFO = 0, it contains the Hankel singular values of C the original system ordered decreasingly. HSV(1) is the C Hankel norm of the system. C C Tolerances C C TOL DOUBLE PRECISION C If ORDSEL = 'A', TOL contains the tolerance for C determining the order of reduced system. C For model reduction, the recommended value is C TOL = c*HNORM(A,B,C), where c is a constant in the C interval [0.00001,0.001], and HNORM(A,B,C) is the C Hankel-norm of the given system (computed in HSV(1)). C For computing a minimal realization, the recommended C value is TOL = N*EPS*HNORM(A,B,C), where EPS is the C machine precision (see LAPACK Library Routine DLAMCH). C This value is used by default if TOL <= 0 on entry. C If ORDSEL = 'F', the value of TOL is ignored. C C Workspace C C IWORK INTEGER array, dimension (LIWORK) C LIWORK = 0, if JOB = 'B'; C LIWORK = N, if JOB = 'N'. 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 length of the array DWORK. C LDWORK >= MAX(1,N*(2*N+MAX(N,M,P)+5)+N*(N+1)/2). C For optimum performance LDWORK should be larger. C C Warning Indicator C C IWARN INTEGER C = 0: no warning; C = 1: with ORDSEL = 'F', the selected order NR is greater C than the order of a minimal realization of the C given system. In this case, the resulting NR is C set automatically to a value corresponding to the C order of a minimal realization of the system. 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 the real Schur form failed; C = 2: the state matrix A is not stable (if DICO = 'C') C or not convergent (if DICO = 'D'); C = 3: the computation of Hankel singular values failed. C C METHOD C C Let be the stable linear system C C d[x(t)] = Ax(t) + Bu(t) C y(t) = Cx(t) (1) C C where d[x(t)] is dx(t)/dt for a continuous-time system and x(t+1) C for a discrete-time system. The subroutine AB09AD determines for C the given system (1), the matrices of a reduced order system C C d[z(t)] = Ar*z(t) + Br*u(t) C yr(t) = Cr*z(t) (2) C C such that C C HSV(NR) <= INFNORM(G-Gr) <= 2*[HSV(NR+1) + ... + HSV(N)], C C where G and Gr are transfer-function matrices of the systems C (A,B,C) and (Ar,Br,Cr), respectively, and INFNORM(G) is the C infinity-norm of G. C C If JOB = 'B', the square-root Balance & Truncate method of [1] C is used and, for DICO = 'C', the resulting model is balanced. C By setting TOL <= 0, the routine can be used to compute balanced C minimal state-space realizations of stable systems. C C If JOB = 'N', the balancing-free square-root version of the C Balance & Truncate method [2] is used. C By setting TOL <= 0, the routine can be used to compute minimal C state-space realizations of stable systems. C C REFERENCES C C [1] Tombs M.S. and Postlethwaite I. C Truncated balanced realization of stable, non-minimal C state-space systems. C Int. J. Control, Vol. 46, pp. 1319-1330, 1987. C C [2] Varga A. C Efficient minimal realization procedure based on balancing. C Proc. of IMACS/IFAC Symp. MCTS, Lille, France, May 1991, C A. El Moudui, P. Borne, S. G. Tzafestas (Eds.), C Vol. 2, pp. 42-46. C C NUMERICAL ASPECTS C C The implemented methods rely on accuracy enhancing square-root or C balancing-free square-root techniques. C 3 C The algorithms require less than 30N floating point operations. C C CONTRIBUTOR C C C. Oara and A. Varga, German Aerospace Center, C DLR Oberpfaffenhofen, March 1998. C Based on the RASP routines SRBT and SRBFT. C C REVISIONS C C May 2, 1998. C November 11, 1998, V. Sima, Research Institute for Informatics, C Bucharest. C C KEYWORDS C C Balancing, minimal state-space representation, model reduction, C multivariable system, state-space model. C C ****************************************************************** C C .. Parameters .. DOUBLE PRECISION ONE, C100 PARAMETER ( ONE = 1.0D0, C100 = 100.0D0 ) C .. Scalar Arguments .. CHARACTER DICO, EQUIL, JOB, ORDSEL INTEGER INFO, IWARN, LDA, LDB, LDC, LDWORK, M, N, NR, P DOUBLE PRECISION TOL C .. Array Arguments .. INTEGER IWORK(*) DOUBLE PRECISION A(LDA,*), B(LDB,*), C(LDC,*), DWORK(*), HSV(*) C .. Local Scalars .. LOGICAL FIXORD INTEGER IERR, KI, KR, KT, KTI, KW, NN DOUBLE PRECISION MAXRED, WRKOPT C .. External Functions .. LOGICAL LSAME EXTERNAL LSAME C .. External Subroutines .. EXTERNAL AB09AX, TB01ID, TB01WD, XERBLA C .. Intrinsic Functions .. INTRINSIC DBLE, MAX, MIN C .. Executable Statements .. C INFO = 0 IWARN = 0 FIXORD = LSAME( ORDSEL, 'F' ) C C Test the input scalar arguments. C IF( .NOT. ( LSAME( DICO, 'C' ) .OR. LSAME( DICO, 'D' ) ) ) THEN INFO = -1 ELSE IF( .NOT. ( LSAME( JOB, 'B' ) .OR. LSAME( JOB, 'N' ) ) ) THEN INFO = -2 ELSE IF( .NOT. ( LSAME( EQUIL, 'S' ) .OR. $ LSAME( EQUIL, 'N' ) ) ) THEN INFO = -3 ELSE IF( .NOT. ( FIXORD .OR. LSAME( ORDSEL, 'A' ) ) ) THEN INFO = -4 ELSE IF( N.LT.0 ) THEN INFO = -5 ELSE IF( M.LT.0 ) THEN INFO = -6 ELSE IF( P.LT.0 ) THEN INFO = -7 ELSE IF( FIXORD .AND. ( NR.LT.0 .OR. NR.GT.N ) ) THEN INFO = -8 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN INFO = -10 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN INFO = -12 ELSE IF( LDC.LT.MAX( 1, P ) ) THEN INFO = -14 ELSE IF( LDWORK.LT.MAX( 1, N*( 2*N + MAX( N, M, P ) + 5 ) + $ ( N*( N + 1 ) )/2 ) ) THEN INFO = -19 END IF C IF( INFO.NE.0 ) THEN C C Error return. C CALL XERBLA( 'AB09AD', -INFO ) RETURN END IF C C Quick return if possible. C IF( MIN( N, M, P ).EQ.0 .OR. ( FIXORD .AND. NR.EQ.0 ) ) THEN NR = 0 DWORK(1) = ONE RETURN END IF C C Allocate working storage. C NN = N*N KT = 1 KR = KT + NN KI = KR + N KW = KI + N C IF( LSAME( EQUIL, 'S' ) ) THEN C C Scale simultaneously the matrices A, B and C: C A <- inv(D)*A*D, B <- inv(D)*B and C <- C*D, where D is a C diagonal matrix. C MAXRED = C100 CALL TB01ID( 'A', N, M, P, MAXRED, A, LDA, B, LDB, C, LDC, $ DWORK, INFO ) END IF C C Reduce A to the real Schur form using an orthogonal similarity C transformation A <- T'*A*T and apply the transformation to C B and C: B <- T'*B and C <- C*T. C CALL TB01WD( N, M, P, A, LDA, B, LDB, C, LDC, DWORK(KT), N, $ DWORK(KR), DWORK(KI), DWORK(KW), LDWORK-KW+1, IERR ) IF( IERR.NE.0 ) THEN INFO = 1 RETURN END IF C WRKOPT = DWORK(KW) + DBLE( KW-1 ) KTI = KT + NN KW = KTI + NN C CALL AB09AX( DICO, JOB, ORDSEL, N, M, P, NR, A, LDA, B, LDB, C, $ LDC, HSV, DWORK(KT), N, DWORK(KTI), N, TOL, IWORK, $ DWORK(KW), LDWORK-KW+1, IWARN, IERR ) C IF( IERR.NE.0 ) THEN INFO = IERR + 1 RETURN END IF C DWORK(1) = MAX( WRKOPT, DWORK(KW) + DBLE( KW-1 ) ) C RETURN C *** Last line of AB09AD *** END