SUBROUTINE AB05SD( FBTYPE, JOBD, N, M, P, ALPHA, A, LDA, B, LDB,
$ C, LDC, D, LDD, F, LDF, RCOND, IWORK, DWORK,
$ LDWORK, INFO)
C
C PURPOSE
C
C To construct for a given state space system (A,B,C,D) the closed-
C loop system (Ac,Bc,Cc,Dc) corresponding to the output feedback
C control law
C
C u = alpha*F*y + v.
C
C ARGUMENTS
C
C Mode Parameters
C
C FBTYPE CHARACTER*1
C Specifies the type of the feedback law as follows:
C = 'I': Unitary output feedback (F = I);
C = 'O': General output feedback.
C
C JOBD CHARACTER*1
C Specifies whether or not a non-zero matrix D appears in
C the given state space model:
C = 'D': D is present;
C = 'Z': D is assumed a zero matrix.
C
C Input/Output Parameters
C
C N (input) INTEGER
C The number of state variables, i.e. the order of the
C matrix A, the number of rows of B and the number of
C columns of C. N >= 0.
C
C M (input) INTEGER
C The number of input variables, i.e. the number of columns
C of matrices B and D, and the number of rows of F. M >= 0.
C
C P (input) INTEGER
C The number of output variables, i.e. the number of rows of
C matrices C and D, and the number of columns of F. P >= 0
C and P = M if FBTYPE = 'I'.
C
C ALPHA (input) DOUBLE PRECISION
C The coefficient alpha in the output feedback law.
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 system state transition matrix A.
C On exit, the leading N-by-N part of this array contains
C the state matrix Ac of the closed-loop 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 system input matrix B.
C On exit, the leading N-by-M part of this array contains
C the input matrix Bc of the closed-loop 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 system output matrix C.
C On exit, the leading P-by-N part of this array contains
C the output matrix Cc of the closed-loop system.
C
C LDC INTEGER
C The leading dimension of array C.
C LDC >= MAX(1,P) if N > 0.
C LDC >= 1 if N = 0.
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 system direct input/output transmission
C matrix D.
C On exit, if JOBD = 'D', the leading P-by-M part of this
C array contains the direct input/output transmission
C matrix Dc of the closed-loop system.
C The array D is not referenced if JOBD = 'Z'.
C
C LDD INTEGER
C The leading dimension of array D.
C LDD >= MAX(1,P) if JOBD = 'D'.
C LDD >= 1 if JOBD = 'Z'.
C
C F (input) DOUBLE PRECISION array, dimension (LDF,P)
C If FBTYPE = 'O', the leading M-by-P part of this array
C must contain the output feedback matrix F.
C If FBTYPE = 'I', then the feedback matrix is assumed to be
C an M x M order identity matrix.
C The array F is not referenced if FBTYPE = 'I' or
C ALPHA = 0.
C
C LDF INTEGER
C The leading dimension of array F.
C LDF >= MAX(1,M) if FBTYPE = 'O' and ALPHA <> 0.
C LDF >= 1 if FBTYPE = 'I' or ALPHA = 0.
C
C RCOND (output) DOUBLE PRECISION
C The reciprocal condition number of the matrix
C I - alpha*D*F.
C
C Workspace
C
C IWORK INTEGER array, dimension (LIWORK)
C LIWORK >= MAX(1,2*P) if JOBD = 'D'.
C LIWORK >= 1 if JOBD = 'Z'.
C IWORK is not referenced if JOBD = 'Z'.
C
C DWORK DOUBLE PRECISION array, dimension (LDWORK)
C
C LDWORK INTEGER
C The length of the array DWORK.
C LDWORK >= wspace, where
C wspace = MAX( 1, M, P*P + 4*P ) if JOBD = 'D',
C wspace = MAX( 1, M ) if JOBD = 'Z'.
C For best performance, LDWORK >= MAX( wspace, N*M, N*P ).
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: if the matrix I - alpha*D*F is numerically singular.
C
C METHOD
C
C The matrices of the closed-loop system have the expressions:
C
C Ac = A + alpha*B*F*E*C, Bc = B + alpha*B*F*E*D,
C Cc = E*C, Dc = E*D,
C
C where E = (I - alpha*D*F)**-1.
C
C NUMERICAL ASPECTS
C
C The accuracy of computations basically depends on the conditioning
C of the matrix I - alpha*D*F. If RCOND is very small, it is likely
C that the computed results are inaccurate.
C
C CONTRIBUTORS
C
C A. Varga, German Aerospace Research Establishment,
C Oberpfaffenhofen, Germany, and V. Sima, Katholieke Univ. Leuven,
C Belgium, Nov. 1996.
C
C REVISIONS
C
C January 14, 1997.
C V. Sima, Research Institute for Informatics, Bucharest, July 2003.
C
C KEYWORDS
C
C Multivariable system, state-space model, state-space
C representation.
C
C ******************************************************************
C
C .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
C .. Scalar Arguments ..
CHARACTER FBTYPE, JOBD
INTEGER INFO, LDA, LDB, LDC, LDD, LDF, LDWORK, M, N, P
DOUBLE PRECISION ALPHA, RCOND
C .. Array Arguments ..
INTEGER IWORK(*)
DOUBLE PRECISION A(LDA,*), B(LDB,*), C(LDC,*), D(LDD,*),
$ DWORK(*), F(LDF,*)
C .. Local Scalars ..
LOGICAL LJOBD, OUTPF, UNITF
INTEGER I, IW, LDWN, LDWP
DOUBLE PRECISION ENORM
C .. Local Arrays ..
DOUBLE PRECISION DUMMY(1)
C .. External functions ..
LOGICAL LSAME
DOUBLE PRECISION DLAMCH, DLANGE
EXTERNAL DLAMCH, DLANGE, LSAME
C .. External subroutines ..
EXTERNAL DAXPY, DCOPY, DGECON, DGEMM, DGEMV, DGETRF,
$ DGETRS, DLACPY, DLASCL, XERBLA
C .. Intrinsic Functions ..
INTRINSIC MAX, MIN
C
C .. Executable Statements ..
C
C Check the input scalar arguments.
C
UNITF = LSAME( FBTYPE, 'I' )
OUTPF = LSAME( FBTYPE, 'O' )
LJOBD = LSAME( JOBD, 'D' )
LDWN = MAX( 1, N )
LDWP = MAX( 1, P )
C
INFO = 0
C
IF( .NOT.UNITF .AND. .NOT.OUTPF ) THEN
INFO = -1
ELSE IF( .NOT.LJOBD .AND. .NOT.LSAME( JOBD, 'Z' ) ) THEN
INFO = -2
ELSE IF( N.LT.0 ) THEN
INFO = -3
ELSE IF( M.LT.0 ) THEN
INFO = -4
ELSE IF( P.LT.0 .OR. UNITF.AND.P.NE.M ) THEN
INFO = -5
ELSE IF( LDA.LT.LDWN ) THEN
INFO = -7
ELSE IF( LDB.LT.LDWN ) THEN
INFO = -9
ELSE IF( ( N.GT.0 .AND. LDC.LT.LDWP ) .OR.
$ ( N.EQ.0 .AND. LDC.LT.1 ) ) THEN
INFO = -11
ELSE IF( ( LJOBD .AND. LDD.LT.LDWP ) .OR.
$ ( .NOT.LJOBD .AND. LDD.LT.1 ) ) THEN
INFO = -13
ELSE IF( ( OUTPF .AND. ALPHA.NE.ZERO .AND. LDF.LT.MAX( 1, M ) )
$ .OR. ( ( UNITF .OR. ALPHA.EQ.ZERO ) .AND. LDF.LT.1 ) ) THEN
INFO = -16
ELSE IF( ( LJOBD .AND. LDWORK.LT.MAX( 1, M, P*P + 4*P ) ) .OR.
$ ( .NOT.LJOBD .AND. LDWORK.LT.MAX( 1, M ) ) ) THEN
INFO = -20
END IF
C
IF ( INFO.NE.0 ) THEN
C
C Error return.
C
CALL XERBLA( 'AB05SD', -INFO )
RETURN
END IF
C
C Quick return if possible.
C
RCOND = ONE
IF ( MAX( N, MIN( M, P ) ).EQ.0 .OR. ALPHA.EQ.ZERO )
$ RETURN
C
IF (LJOBD) THEN
IW = P*P + 1
C
C Compute I - alpha*D*F.
C
IF( UNITF) THEN
CALL DLACPY( 'F', P, P, D, LDD, DWORK, LDWP )
IF ( ALPHA.NE.-ONE )
$ CALL DLASCL( 'G', 0, 0, ONE, -ALPHA, P, P, DWORK, LDWP,
$ INFO )
ELSE
CALL DGEMM( 'N', 'N', P, P, M, -ALPHA, D, LDD, F, LDF, ZERO,
$ DWORK, LDWP )
END IF
C
DUMMY(1) = ONE
CALL DAXPY( P, ONE, DUMMY, 0, DWORK, P+1 )
C
C Compute Cc = E*C, Dc = E*D, where E = (I - alpha*D*F)**-1.
C
ENORM = DLANGE( '1', P, P, DWORK, LDWP, DWORK(IW) )
CALL DGETRF( P, P, DWORK, LDWP, IWORK, INFO )
IF( INFO.GT.0 ) THEN
C
C Error return.
C
RCOND = ZERO
INFO = 1
RETURN
END IF
CALL DGECON( '1', P, DWORK, LDWP, ENORM, RCOND, DWORK(IW),
$ IWORK(P+1), INFO )
IF( RCOND.LE.DLAMCH('E') ) THEN
C
C Error return.
C
INFO = 1
RETURN
END IF
C
IF( N.GT.0 )
$ CALL DGETRS( 'N', P, N, DWORK, LDWP, IWORK, C, LDC, INFO )
CALL DGETRS( 'N', P, M, DWORK, LDWP, IWORK, D, LDD, INFO )
END IF
C
IF ( N.EQ.0 )
$ RETURN
C
C Compute Ac = A + alpha*B*F*Cc and Bc = B + alpha*B*F*Dc.
C
IF( UNITF ) THEN
CALL DGEMM( 'N', 'N', N, N, M, ALPHA, B, LDB, C, LDC, ONE, A,
$ LDA )
IF( LJOBD ) THEN
C
IF( LDWORK.LT.N*M ) THEN
C
C Not enough working space for using DGEMM.
C
DO 10 I = 1, N
CALL DCOPY( P, B(I,1), LDB, DWORK, 1 )
CALL DGEMV( 'T', P, P, ALPHA, D, LDD, DWORK, 1, ONE,
$ B(I,1), LDB )
10 CONTINUE
C
ELSE
CALL DLACPY( 'F', N, M, B, LDB, DWORK, LDWN )
CALL DGEMM( 'N', 'N', N, P, M, ALPHA, DWORK, LDWN, D,
$ LDD, ONE, B, LDB )
END IF
END IF
ELSE
C
IF( LDWORK.LT.N*P ) THEN
C
C Not enough working space for using DGEMM.
C
DO 20 I = 1, N
CALL DGEMV( 'N', M, P, ALPHA, F, LDF, C(1,I), 1, ZERO,
$ DWORK, 1 )
CALL DGEMV( 'N', N, M, ONE, B, LDB, DWORK, 1, ONE,
$ A(1,I), 1 )
20 CONTINUE
C
IF( LJOBD ) THEN
C
DO 30 I = 1, N
CALL DGEMV( 'T', M, P, ALPHA, F, LDF, B(I,1), LDB,
$ ZERO, DWORK, 1 )
CALL DGEMV( 'T', P, M, ONE, D, LDD, DWORK, 1, ONE,
$ B(I,1), LDB )
30 CONTINUE
C
END IF
ELSE
C
CALL DGEMM( 'N', 'N', N, P, M, ALPHA, B, LDB, F, LDF,
$ ZERO, DWORK, LDWN )
CALL DGEMM( 'N', 'N', N, N, P, ONE, DWORK, LDWN, C, LDC,
$ ONE, A, LDA )
IF( LJOBD )
$ CALL DGEMM( 'N', 'N', N, M, P, ONE, DWORK, LDWN, D, LDD,
$ ONE, B, LDB )
END IF
END IF
C
RETURN
C *** Last line of AB05SD ***
END