SUBROUTINE SB16CY( DICO, JOBCF, N, M, P, A, LDA, B, LDB, C, LDC,
$ F, LDF, G, LDG, SCALEC, SCALEO, S, LDS, R, LDR,
$ DWORK, LDWORK, INFO )
C
C PURPOSE
C
C To compute, for a given open-loop model (A,B,C,0), and for
C given state feedback gain F and full observer gain G,
C such that A+B*F and A+G*C are stable, the Cholesky factors
C Su and Ru of a controllability Grammian P = Su*Su' and of
C an observability Grammian Q = Ru'*Ru corresponding to a
C frequency-weighted model reduction of the left or right coprime
C factors of the state-feedback controller.
C
C ARGUMENTS
C
C Mode Parameters
C
C DICO CHARACTER*1
C Specifies the type of the open-loop system as follows:
C = 'C': continuous-time system;
C = 'D': discrete-time system.
C
C JOBCF CHARACTER*1
C Specifies whether a left or right coprime factorization
C of the state-feedback controller is to be used as follows:
C = 'L': use a left coprime factorization;
C = 'R': use a right coprime factorization.
C
C Input/Output Parameters
C
C N (input) INTEGER
C The order of the open-loop state-space representation,
C i.e., 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 A (input) DOUBLE PRECISION array, dimension (LDA,N)
C The leading N-by-N part of this array must contain the
C state matrix A of the open-loop system.
C
C LDA INTEGER
C The leading dimension of array A. LDA >= MAX(1,N).
C
C B (input) DOUBLE PRECISION array, dimension (LDB,M)
C The leading N-by-M part of this array must contain the
C input/state matrix B of the open-loop system.
C
C LDB INTEGER
C The leading dimension of array B. LDB >= MAX(1,N).
C
C C (input) DOUBLE PRECISION array, dimension (LDC,N)
C The leading P-by-N part of this array must contain the
C state/output matrix C of the open-loop system.
C
C LDC INTEGER
C The leading dimension of array C. LDC >= MAX(1,P).
C
C F (input) DOUBLE PRECISION array, dimension (LDF,N)
C The leading M-by-N part of this array must contain a
C stabilizing state feedback matrix.
C
C LDF INTEGER
C The leading dimension of array F. LDF >= MAX(1,M).
C
C G (input) DOUBLE PRECISION array, dimension (LDG,P)
C The leading N-by-P part of this array must contain a
C stabilizing observer gain matrix.
C
C LDG INTEGER
C The leading dimension of array G. LDG >= MAX(1,N).
C
C SCALEC (output) DOUBLE PRECISION
C Scaling factor for the controllability Grammian.
C See METHOD.
C
C SCALEO (output) DOUBLE PRECISION
C Scaling factor for the observability Grammian.
C See METHOD.
C
C S (output) DOUBLE PRECISION array, dimension (LDS,N)
C The leading N-by-N upper triangular part of this array
C contains the Cholesky factor Su of frequency-weighted
C cotrollability Grammian P = Su*Su'. See METHOD.
C
C LDS INTEGER
C The leading dimension of the array S. LDS >= MAX(1,N).
C
C R (output) DOUBLE PRECISION array, dimension (LDR,N)
C The leading N-by-N upper triangular part of this array
C contains the Cholesky factor Ru of the frequency-weighted
C observability Grammian Q = Ru'*Ru. See METHOD.
C
C LDR INTEGER
C The leading dimension of the array R. LDR >= MAX(1,N).
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 length of the array DWORK.
C LDWORK >= MAX(1, N*(N + MAX(N,M) + MIN(N,M) + 6)),
C if JOBCF = 'L';
C LDWORK >= MAX(1, N*(N + MAX(N,P) + MIN(N,P) + 6)),
C if JOBCF = 'R'.
C For optimum performance LDWORK should be larger.
C An upper bound for both cases is
C LDWORK >= MAX(1, N*(N + MAX(N,M,P) + 7)).
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: eigenvalue computation failure;
C = 2: the matrix A+G*C is not stable;
C = 3: the matrix A+B*F is not stable;
C = 4: the Lyapunov equation for computing the
C observability Grammian is (nearly) singular;
C = 5: the Lyapunov equation for computing the
C controllability Grammian is (nearly) singular.
C
C METHOD
C
C In accordance with the type of the coprime factorization
C of the controller (left or right), the Cholesky factors Su and Ru
C of the frequency-weighted controllability Grammian P = Su*Su' and
C of the frequency-weighted observability Grammian Q = Ru'*Ru are
C computed by solving appropriate Lyapunov or Stein equations [1].
C
C If JOBCF = 'L' and DICO = 'C', P and Q are computed as the
C solutions of the following Lyapunov equations:
C
C (A+B*F)*P + P*(A+B*F)' + scalec^2*B*B' = 0, (1)
C
C (A+G*C)'*Q + Q*(A+G*C) + scaleo^2*F'*F = 0. (2)
C
C If JOBCF = 'L' and DICO = 'D', P and Q are computed as the
C solutions of the following Stein equations:
C
C (A+B*F)*P*(A+B*F)' - P + scalec^2*B*B' = 0, (3)
C
C (A+G*C)'*Q*(A+G*C) - Q + scaleo^2*F'*F = 0. (4)
C
C If JOBCF = 'R' and DICO = 'C', P and Q are computed as the
C solutions of the following Lyapunov equations:
C
C (A+B*F)*P + P*(A+B*F)' + scalec^2*G*G' = 0, (5)
C
C (A+G*C)'*Q + Q*(A+G*C) + scaleo^2*C'*C = 0. (6)
C
C If JOBCF = 'R' and DICO = 'D', P and Q are computed as the
C solutions of the following Stein equations:
C
C (A+B*F)*P*(A+B*F)' - P + scalec^2*G*G' = 0, (7)
C
C (A+G*C)'*Q*(A+G*C) - Q + scaleo^2*C'*C = 0. (8)
C
C REFERENCES
C
C [1] Liu, Y., Anderson, B.D.O. and Ly, O.L.
C Coprime factorization controller reduction with Bezout
C identity induced frequency weighting.
C Automatica, vol. 26, pp. 233-249, 1990.
C
C CONTRIBUTORS
C
C A. Varga, German Aerospace Center, Oberpfaffenhofen, October 2000.
C D. Sima, University of Bucharest, October 2000.
C V. Sima, Research Institute for Informatics, Bucharest, Oct. 2000.
C
C REVISIONS
C
C A. Varga, Australian National University, Canberra, November 2000.
C
C KEYWORDS
C
C Controller reduction, frequency weighting, multivariable system,
C state-space model, state-space representation.
C
C ******************************************************************
C
C .. Parameters ..
DOUBLE PRECISION ONE
PARAMETER ( ONE = 1.0D0 )
C .. Scalar Arguments ..
CHARACTER DICO, JOBCF
INTEGER INFO, LDA, LDB, LDC, LDF, LDG, LDR, LDS, LDWORK,
$ M, N, P
DOUBLE PRECISION SCALEC, SCALEO
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), B(LDB,*), C(LDC,*), DWORK(*),
$ F(LDF,*), G(LDG,*), R(LDR,*), S(LDS,*)
C .. Local Scalars ..
LOGICAL DISCR, LEFTW
INTEGER IERR, KAW, KU, KW, KWI, KWR, LDU, LW, ME, MP,
$ WRKOPT
C .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
C .. External Subroutines ..
EXTERNAL DGEMM, DLACPY, SB03OD, XERBLA
C .. Intrinsic Functions ..
INTRINSIC INT, MAX, MIN
C .. Executable Statements ..
C
DISCR = LSAME( DICO, 'D' )
LEFTW = LSAME( JOBCF, 'L' )
C
INFO = 0
IF( LEFTW ) THEN
MP = M
ELSE
MP = P
END IF
LW = N*( N + MAX( N, MP ) + MIN( N, MP ) + 6 )
C
IF( .NOT. ( LSAME( DICO, 'C' ) .OR. DISCR ) ) THEN
INFO = -1
ELSE IF( .NOT.( LEFTW .OR. LSAME( JOBCF, 'R' ) ) ) THEN
INFO = -2
ELSE IF( N.LT.0 ) THEN
INFO = -3
ELSE IF( M.LT.0 ) THEN
INFO = -4
ELSE IF( P.LT.0 ) THEN
INFO = -5
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -7
ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
INFO = -9
ELSE IF( LDC.LT.MAX( 1, P ) ) THEN
INFO = -11
ELSE IF( LDF.LT.MAX( 1, M ) ) THEN
INFO = -13
ELSE IF( LDG.LT.MAX( 1, N ) ) THEN
INFO = -15
ELSE IF( LDS.LT.MAX( 1, N ) ) THEN
INFO = -19
ELSE IF( LDR.LT.MAX( 1, N ) ) THEN
INFO = -21
ELSE IF( LDWORK.LT.MAX( 1, LW ) ) THEN
INFO = -23
END IF
C
IF( INFO.NE.0 ) THEN
C
C Error return.
C
CALL XERBLA( 'SB16CY', -INFO )
RETURN
END IF
C
C Quick return if possible.
C
IF( MIN( N, M, P ).EQ.0 ) THEN
SCALEC = ONE
SCALEO = ONE
DWORK(1) = ONE
RETURN
END IF
C
C Allocate storage for work arrays.
C
KAW = 1
KU = KAW + N*N
KWR = KU + N*MAX( N, MP )
KWI = KWR + N
KW = KWI + N
C
C Form A+G*C.
C
CALL DLACPY( 'Full', N, N, A, LDA, DWORK(KAW), N )
CALL DGEMM( 'No-transpose', 'No-transpose', N, N, P, ONE,
$ G, LDG, C, LDC, ONE, DWORK(KAW), N )
C
C Form the factor H of the free term.
C
IF( LEFTW ) THEN
C
C H = F.
C
LDU = MAX( N, M )
ME = M
CALL DLACPY( 'Full', M, N, F, LDF, DWORK(KU), LDU )
ELSE
C
C H = C.
C
LDU = MAX( N, P )
ME = P
CALL DLACPY( 'Full', P, N, C, LDC, DWORK(KU), LDU )
END IF
C
C Solve for the Cholesky factor Ru of Q, Q = Ru'*Ru,
C the continuous-time Lyapunov equation (if DICO = 'C')
C
C (A+G*C)'*Q + Q*(A+G*C) + scaleo^2*H'*H = 0,
C
C or the discrete-time Lyapunov equation (if DICO = 'D')
C
C (A+G*C)'*Q*(A+G*C) - Q + scaleo^2*H'*H = 0.
C
C Workspace: need N*(N + MAX(N,M) + MIN(N,M) + 6) if JOBCF = 'L';
C N*(N + MAX(N,P) + MIN(N,P) + 6) if JOBCF = 'R'.
C prefer larger.
C
CALL SB03OD( DICO, 'NoFact', 'NoTransp', N, ME, DWORK(KAW), N,
$ R, LDR, DWORK(KU), LDU, SCALEO, DWORK(KWR),
$ DWORK(KWI), DWORK(KW), LDWORK-KW+1, IERR )
IF( IERR.NE.0 ) THEN
IF( IERR.EQ.2 ) THEN
INFO = 2
ELSE IF( IERR.EQ.1 ) THEN
INFO = 4
ELSE IF( IERR.EQ.6 ) THEN
INFO = 1
END IF
RETURN
END IF
C
WRKOPT = INT( DWORK(KW) ) + KW - 1
CALL DLACPY( 'Upper', N, N, DWORK(KU), LDU, R, LDR )
C
C Form A+B*F.
C
CALL DLACPY( 'Full', N, N, A, LDA, DWORK(KAW), N )
CALL DGEMM( 'No-transpose', 'No-transpose', N, N, M, ONE,
$ B, LDB, F, LDF, ONE, DWORK(KAW), N )
C
C Form the factor K of the free term.
C
LDU = N
IF( LEFTW ) THEN
C
C K = B.
C
ME = M
CALL DLACPY( 'Full', N, M, B, LDB, DWORK(KU), LDU )
ELSE
C
C K = G.
C
ME = P
CALL DLACPY( 'Full', N, P, G, LDG, DWORK(KU), LDU )
END IF
C
C Solve for the Cholesky factor Su of P, P = Su*Su',
C the continuous-time Lyapunov equation (if DICO = 'C')
C
C (A+B*F)*P + P*(A+B*F)' + scalec^2*K*K' = 0,
C
C or the discrete-time Lyapunov equation (if DICO = 'D')
C
C (A+B*F)*P*(A+B*F)' - P + scalec^2*K*K' = 0.
C
C Workspace: need N*(N + MAX(N,M) + MIN(N,M) + 6) if JOBCF = 'L';
C N*(N + MAX(N,P) + MIN(N,P) + 6) if JOBCF = 'R'.
C prefer larger.
C
CALL SB03OD( DICO, 'NoFact', 'Transp', N, ME, DWORK(KAW), N,
$ S, LDS, DWORK(KU), LDU, SCALEC, DWORK(KWR),
$ DWORK(KWI), DWORK(KW), LDWORK-KW+1, IERR )
IF( IERR.NE.0 ) THEN
IF( IERR.EQ.2 ) THEN
INFO = 3
ELSE IF( IERR.EQ.1 ) THEN
INFO = 5
ELSE IF( IERR.EQ.6 ) THEN
INFO = 1
END IF
RETURN
END IF
WRKOPT = MAX( WRKOPT, INT( DWORK(KW) ) + KW - 1 )
CALL DLACPY( 'Upper', N, N, DWORK(KU), LDU, S, LDS )
C
C Save the optimal workspace.
C
DWORK(1) = WRKOPT
C
RETURN
C *** Last line of SB16CY ***
END