SUBROUTINE AB05RD( FBTYPE, JOBD, N, M, P, MV, PZ, ALPHA, BETA, A,
$ LDA, B, LDB, C, LDC, D, LDD, F, LDF, K, LDK,
$ G, LDG, H, LDH, RCOND, BC, LDBC, CC, LDCC,
$ DC, LDDC, 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 mixed output and
C state feedback control law
C
C u = alpha*F*y + beta*K*x + G*v
C z = H*y.
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
C in 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 dimension of state vector x, 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 dimension of input vector u, i.e. the number of
C columns of matrices B and D, and the number of rows of F.
C M >= 0.
C
C P (input) INTEGER
C The dimension of output vector y, i.e. the number of rows
C of matrices C and D, and the number of columns of F.
C P >= 0 and P = M if FBTYPE = 'I'.
C
C MV (input) INTEGER
C The dimension of the new input vector v, i.e. the number
C of columns of matrix G. MV >= 0.
C
C PZ (input) INTEGER.
C The dimension of the new output vector z, i.e. the number
C of rows of matrix H. PZ >= 0.
C
C ALPHA (input) DOUBLE PRECISION
C The coefficient alpha in the output feedback law.
C
C BETA (input) DOUBLE PRECISION.
C The coefficient beta in the state 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 intermediary input matrix B1 (see METHOD).
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 intermediary output matrix C1+BETA*D1*K (see METHOD).
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, if JOBD = 'D', the leading P-by-M part of this
C array must contain the system direct input/output
C transmission matrix D.
C On exit, the leading P-by-M part of this array contains
C the intermediary direct input/output transmission matrix
C D1 (see METHOD).
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 K (input) DOUBLE PRECISION array, dimension (LDK,N)
C The leading M-by-N part of this array must contain the
C state feedback matrix K.
C The array K is not referenced if BETA = 0.
C
C LDK INTEGER
C The leading dimension of the array K.
C LDK >= MAX(1,M) if BETA <> 0.
C LDK >= 1 if BETA = 0.
C
C G (input) DOUBLE PRECISION array, dimension (LDG,MV)
C The leading M-by-MV part of this array must contain the
C system input scaling matrix G.
C
C LDG INTEGER
C The leading dimension of the array G. LDG >= MAX(1,M).
C
C H (input) DOUBLE PRECISION array, dimension (LDH,P)
C The leading PZ-by-P part of this array must contain the
C system output scaling matrix H.
C
C LDH INTEGER
C The leading dimension of the array H. LDH >= MAX(1,PZ).
C
C RCOND (output) DOUBLE PRECISION
C The reciprocal condition number of the matrix
C I - alpha*D*F.
C
C BC (output) DOUBLE PRECISION array, dimension (LDBC,MV)
C The leading N-by-MV part of this array contains the input
C matrix Bc of the closed-loop system.
C
C LDBC INTEGER
C The leading dimension of array BC. LDBC >= MAX(1,N).
C
C CC (output) DOUBLE PRECISION array, dimension (LDCC,N)
C The leading PZ-by-N part of this array contains the
C system output matrix Cc of the closed-loop system.
C
C LDCC INTEGER
C The leading dimension of array CC.
C LDCC >= MAX(1,PZ) if N > 0.
C LDCC >= 1 if N = 0.
C
C DC (output) DOUBLE PRECISION array, dimension (LDDC,MV)
C If JOBD = 'D', the leading PZ-by-MV part of this array
C contains the direct input/output transmission matrix Dc
C of the closed-loop system.
C The array DC is not referenced if JOBD = 'Z'.
C
C LDDC INTEGER
C The leading dimension of array DC.
C LDDC >= MAX(1,PZ) if JOBD = 'D'.
C LDDC >= 1 if JOBD = 'Z'.
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*MV, 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 = A1 + beta*B1*K, Bc = B1*G,
C Cc = H*(C1 + beta*D1*K), Dc = H*D1*G,
C
C where
C
C A1 = A + alpha*B*F*E*C, B1 = B + alpha*B*F*E*D,
C C1 = E*C, D1 = E*D,
C
C with 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, February 18, 1998.
C V. Sima, Research Institute for Informatics, Bucharest, July 2003,
C Jan. 2005.
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, LDBC, LDC, LDCC, LDD, LDDC,
$ LDF, LDG, LDH, LDK, LDWORK, M, MV, N, P, PZ
DOUBLE PRECISION ALPHA, BETA, RCOND
C .. Array Arguments ..
INTEGER IWORK(*)
DOUBLE PRECISION A(LDA,*), B(LDB,*), BC(LDBC,*), C(LDC,*),
$ CC(LDCC,*), D(LDD,*), DC(LDDC,*), DWORK(*),
$ F(LDF,*), G(LDG,*), H(LDH,*), K(LDK,*)
C .. Local Scalars ..
LOGICAL LJOBD, OUTPF, UNITF
INTEGER LDWP
C .. External functions ..
LOGICAL LSAME
EXTERNAL LSAME
C .. External subroutines ..
EXTERNAL AB05SD, DGEMM, 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' )
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( MV.LT.0 ) THEN
INFO = -6
ELSE IF( PZ.LT.0 ) THEN
INFO = -7
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -11
ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
INFO = -13
ELSE IF( ( N.GT.0 .AND. LDC.LT.MAX( 1, P ) ) .OR.
$ ( N.EQ.0 .AND. LDC.LT.1 ) ) THEN
INFO = -15
ELSE IF( ( LJOBD .AND. LDD.LT.MAX( 1, P ) ) .OR.
$ ( .NOT.LJOBD .AND. LDD.LT.1 ) ) THEN
INFO = -17
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 = -19
ELSE IF( ( BETA.NE.ZERO .AND. LDK.LT.MAX( 1, M ) ) .OR.
$ ( BETA.EQ.ZERO .AND. LDK.LT.1 ) ) THEN
INFO = -21
ELSE IF( LDG.LT.MAX( 1, M ) ) THEN
INFO = -23
ELSE IF( LDH.LT.MAX( 1, PZ ) ) THEN
INFO = -25
ELSE IF( LDBC.LT.MAX( 1, N ) ) THEN
INFO = -28
ELSE IF( ( N.GT.0 .AND. LDCC.LT.MAX( 1, PZ ) ) .OR.
$ ( N.EQ.0 .AND. LDCC.LT.1 ) ) THEN
INFO = -30
ELSE IF( ( ( LJOBD .AND. LDDC.LT.MAX( 1, PZ ) ) .OR.
$ ( .NOT.LJOBD .AND. LDDC.LT.1 ) ) ) THEN
INFO = -32
ELSE IF( ( LJOBD .AND. LDWORK.LT.MAX( 1, M, P*MV, P*P + 4*P ) )
$ .OR. ( .NOT.LJOBD .AND. LDWORK.LT.MAX( 1, M ) ) ) THEN
INFO = -35
END IF
C
IF ( INFO.NE.0 ) THEN
C
C Error return.
C
CALL XERBLA( 'AB05RD', -INFO )
RETURN
END IF
C
C Quick return if possible.
C
IF ( MAX( N, MIN( M, P ), MIN( MV, PZ ) ).EQ.0 ) THEN
RCOND = ONE
RETURN
END IF
C
C Apply the partial output feedback u = alpha*F*y + v1
C
CALL AB05SD( FBTYPE, JOBD, N, M, P, ALPHA, A, LDA, B, LDB, C,
$ LDC, D, LDD, F, LDF, RCOND, IWORK, DWORK, LDWORK,
$ INFO )
IF ( INFO.NE.0 ) RETURN
C
C Apply the partial state feedback v1 = beta*K*x + v2.
C
C Compute Ac = A1 + beta*B1*K and C1 <- C1 + beta*D1*K.
C
IF( BETA.NE.ZERO .AND. N.GT.0 ) THEN
CALL DGEMM( 'N', 'N', N, N, M, BETA, B, LDB, K, LDK, ONE, A,
$ LDA )
IF( LJOBD )
$ CALL DGEMM( 'N', 'N', P, N, M, BETA, D, LDD, K, LDK, ONE,
$ C, LDC )
END IF
C
C Apply the input and output conversions v2 = G*v, z = H*y.
C
C Compute Bc = B1*G.
C
CALL DGEMM( 'N', 'N', N, MV, M, ONE, B, LDB, G, LDG, ZERO, BC,
$ LDBC )
C
C Compute Cc = H*C1.
C
IF( N.GT.0 )
$ CALL DGEMM( 'N', 'N', PZ, N, P, ONE, H, LDH, C, LDC, ZERO, CC,
$ LDCC )
C
C Compute Dc = H*D1*G.
C
IF( LJOBD ) THEN
LDWP = MAX( 1, P )
CALL DGEMM( 'N', 'N', P, MV, M, ONE, D, LDD, G, LDG, ZERO,
$ DWORK, LDWP )
CALL DGEMM( 'N', 'N', PZ, MV, P, ONE, H, LDH, DWORK, LDWP,
$ ZERO, DC, LDDC )
END IF
C
RETURN
C *** Last line of AB05RD ***
END