470 lines
17 KiB
Fortran
470 lines
17 KiB
Fortran
SUBROUTINE SB10FD( N, M, NP, NCON, NMEAS, GAMMA, A, LDA, B, LDB,
|
|
$ C, LDC, D, LDD, AK, LDAK, BK, LDBK, CK, LDCK,
|
|
$ DK, LDDK, RCOND, TOL, IWORK, DWORK, LDWORK,
|
|
$ BWORK, 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 <http://www.gnu.org/licenses/>.
|
|
C
|
|
C PURPOSE
|
|
C
|
|
C To compute the matrices of an H-infinity (sub)optimal n-state
|
|
C controller
|
|
C
|
|
C | AK | BK |
|
|
C K = |----|----|,
|
|
C | CK | DK |
|
|
C
|
|
C using modified Glover's and Doyle's 1988 formulas, for the system
|
|
C
|
|
C | A | B1 B2 | | A | B |
|
|
C P = |----|---------| = |---|---|
|
|
C | C1 | D11 D12 | | C | D |
|
|
C | C2 | D21 D22 |
|
|
C
|
|
C and for a given value of gamma, where B2 has as column size the
|
|
C number of control inputs (NCON) and C2 has as row size the number
|
|
C of measurements (NMEAS) being provided to the controller.
|
|
C
|
|
C It is assumed that
|
|
C
|
|
C (A1) (A,B2) is stabilizable and (C2,A) is detectable,
|
|
C
|
|
C (A2) D12 is full column rank and D21 is full row rank,
|
|
C
|
|
C (A3) | A-j*omega*I B2 | has full column rank for all omega,
|
|
C | C1 D12 |
|
|
C
|
|
C (A4) | A-j*omega*I B1 | has full row rank for all omega.
|
|
C | C2 D21 |
|
|
C
|
|
C ARGUMENTS
|
|
C
|
|
C Input/Output Parameters
|
|
C
|
|
C N (input) INTEGER
|
|
C The order of the system. N >= 0.
|
|
C
|
|
C M (input) INTEGER
|
|
C The column size of the matrix B. M >= 0.
|
|
C
|
|
C NP (input) INTEGER
|
|
C The row size of the matrix C. NP >= 0.
|
|
C
|
|
C NCON (input) INTEGER
|
|
C The number of control inputs (M2). M >= NCON >= 0,
|
|
C NP-NMEAS >= NCON.
|
|
C
|
|
C NMEAS (input) INTEGER
|
|
C The number of measurements (NP2). NP >= NMEAS >= 0,
|
|
C M-NCON >= NMEAS.
|
|
C
|
|
C GAMMA (input) DOUBLE PRECISION
|
|
C The value of gamma. It is assumed that gamma is
|
|
C sufficiently large so that the controller is admissible.
|
|
C GAMMA >= 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 system state matrix A.
|
|
C
|
|
C LDA INTEGER
|
|
C The leading dimension of the 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 system input matrix B.
|
|
C
|
|
C LDB INTEGER
|
|
C The leading dimension of the array B. LDB >= max(1,N).
|
|
C
|
|
C C (input) DOUBLE PRECISION array, dimension (LDC,N)
|
|
C The leading NP-by-N part of this array must contain the
|
|
C system output matrix C.
|
|
C
|
|
C LDC INTEGER
|
|
C The leading dimension of the array C. LDC >= max(1,NP).
|
|
C
|
|
C D (input) DOUBLE PRECISION array, dimension (LDD,M)
|
|
C The leading NP-by-M part of this array must contain the
|
|
C system input/output matrix D.
|
|
C
|
|
C LDD INTEGER
|
|
C The leading dimension of the array D. LDD >= max(1,NP).
|
|
C
|
|
C AK (output) DOUBLE PRECISION array, dimension (LDAK,N)
|
|
C The leading N-by-N part of this array contains the
|
|
C controller state matrix AK.
|
|
C
|
|
C LDAK INTEGER
|
|
C The leading dimension of the array AK. LDAK >= max(1,N).
|
|
C
|
|
C BK (output) DOUBLE PRECISION array, dimension (LDBK,NMEAS)
|
|
C The leading N-by-NMEAS part of this array contains the
|
|
C controller input matrix BK.
|
|
C
|
|
C LDBK INTEGER
|
|
C The leading dimension of the array BK. LDBK >= max(1,N).
|
|
C
|
|
C CK (output) DOUBLE PRECISION array, dimension (LDCK,N)
|
|
C The leading NCON-by-N part of this array contains the
|
|
C controller output matrix CK.
|
|
C
|
|
C LDCK INTEGER
|
|
C The leading dimension of the array CK.
|
|
C LDCK >= max(1,NCON).
|
|
C
|
|
C DK (output) DOUBLE PRECISION array, dimension (LDDK,NMEAS)
|
|
C The leading NCON-by-NMEAS part of this array contains the
|
|
C controller input/output matrix DK.
|
|
C
|
|
C LDDK INTEGER
|
|
C The leading dimension of the array DK.
|
|
C LDDK >= max(1,NCON).
|
|
C
|
|
C RCOND (output) DOUBLE PRECISION array, dimension (4)
|
|
C RCOND(1) contains the reciprocal condition number of the
|
|
C control transformation matrix;
|
|
C RCOND(2) contains the reciprocal condition number of the
|
|
C measurement transformation matrix;
|
|
C RCOND(3) contains an estimate of the reciprocal condition
|
|
C number of the X-Riccati equation;
|
|
C RCOND(4) contains an estimate of the reciprocal condition
|
|
C number of the Y-Riccati equation.
|
|
C
|
|
C Tolerances
|
|
C
|
|
C TOL DOUBLE PRECISION
|
|
C Tolerance used for controlling the accuracy of the applied
|
|
C transformations for computing the normalized form in
|
|
C SLICOT Library routine SB10PD. Transformation matrices
|
|
C whose reciprocal condition numbers are less than TOL are
|
|
C not allowed. If TOL <= 0, then a default value equal to
|
|
C sqrt(EPS) is used, where EPS is the relative machine
|
|
C precision.
|
|
C
|
|
C Workspace
|
|
C
|
|
C IWORK INTEGER array, dimension (LIWORK), where
|
|
C LIWORK = max(2*max(N,M-NCON,NP-NMEAS,NCON),N*N)
|
|
C
|
|
C DWORK DOUBLE PRECISION array, dimension (LDWORK)
|
|
C On exit, if INFO = 0, DWORK(1) contains the optimal
|
|
C LDWORK.
|
|
C
|
|
C LDWORK INTEGER
|
|
C The dimension of the array DWORK.
|
|
C LDWORK >= N*M + NP*(N+M) + M2*M2 + NP2*NP2 +
|
|
C max(1,LW1,LW2,LW3,LW4,LW5,LW6), where
|
|
C LW1 = (N+NP1+1)*(N+M2) + max(3*(N+M2)+N+NP1,5*(N+M2)),
|
|
C LW2 = (N+NP2)*(N+M1+1) + max(3*(N+NP2)+N+M1,5*(N+NP2)),
|
|
C LW3 = M2 + NP1*NP1 + max(NP1*max(N,M1),3*M2+NP1,5*M2),
|
|
C LW4 = NP2 + M1*M1 + max(max(N,NP1)*M1,3*NP2+M1,5*NP2),
|
|
C LW5 = 2*N*N + N*(M+NP) +
|
|
C max(1,M*M + max(2*M1,3*N*N+max(N*M,10*N*N+12*N+5)),
|
|
C NP*NP + max(2*NP1,3*N*N +
|
|
C max(N*NP,10*N*N+12*N+5))),
|
|
C LW6 = 2*N*N + N*(M+NP) +
|
|
C max(1, M2*NP2 + NP2*NP2 + M2*M2 +
|
|
C max(D1*D1 + max(2*D1, (D1+D2)*NP2),
|
|
C D2*D2 + max(2*D2, D2*M2), 3*N,
|
|
C N*(2*NP2 + M2) +
|
|
C max(2*N*M2, M2*NP2 +
|
|
C max(M2*M2+3*M2, NP2*(2*NP2+
|
|
C M2+max(NP2,N)))))),
|
|
C with D1 = NP1 - M2, D2 = M1 - NP2,
|
|
C NP1 = NP - NP2, M1 = M - M2.
|
|
C For good performance, LDWORK must generally be larger.
|
|
C Denoting Q = max(M1,M2,NP1,NP2), an upper bound is
|
|
C 2*Q*(3*Q+2*N)+max(1,(N+Q)*(N+Q+6),Q*(Q+max(N,Q,5)+1),
|
|
C 2*N*(N+2*Q)+max(1,4*Q*Q+
|
|
C max(2*Q,3*N*N+max(2*N*Q,10*N*N+12*N+5)),
|
|
C Q*(3*N+3*Q+max(2*N,4*Q+max(N,Q))))).
|
|
C
|
|
C BWORK LOGICAL array, dimension (2*N)
|
|
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 | A-j*omega*I B2 | had not full
|
|
C | C1 D12 |
|
|
C column rank in respect to the tolerance EPS;
|
|
C = 2: if the matrix | A-j*omega*I B1 | had not full row
|
|
C | C2 D21 |
|
|
C rank in respect to the tolerance EPS;
|
|
C = 3: if the matrix D12 had not full column rank in
|
|
C respect to the tolerance TOL;
|
|
C = 4: if the matrix D21 had not full row rank in respect
|
|
C to the tolerance TOL;
|
|
C = 5: if the singular value decomposition (SVD) algorithm
|
|
C did not converge (when computing the SVD of one of
|
|
C the matrices |A B2 |, |A B1 |, D12 or D21).
|
|
C |C1 D12| |C2 D21|
|
|
C = 6: if the controller is not admissible (too small value
|
|
C of gamma);
|
|
C = 7: if the X-Riccati equation was not solved
|
|
C successfully (the controller is not admissible or
|
|
C there are numerical difficulties);
|
|
C = 8: if the Y-Riccati equation was not solved
|
|
C successfully (the controller is not admissible or
|
|
C there are numerical difficulties);
|
|
C = 9: if the determinant of Im2 + Tu*D11HAT*Ty*D22 is
|
|
C zero [3].
|
|
C
|
|
C METHOD
|
|
C
|
|
C The routine implements the Glover's and Doyle's 1988 formulas [1],
|
|
C [2] modified to improve the efficiency as described in [3].
|
|
C
|
|
C REFERENCES
|
|
C
|
|
C [1] Glover, K. and Doyle, J.C.
|
|
C State-space formulae for all stabilizing controllers that
|
|
C satisfy an Hinf norm bound and relations to risk sensitivity.
|
|
C Systems and Control Letters, vol. 11, pp. 167-172, 1988.
|
|
C
|
|
C [2] Balas, G.J., Doyle, J.C., Glover, K., Packard, A., and
|
|
C Smith, R.
|
|
C mu-Analysis and Synthesis Toolbox.
|
|
C The MathWorks Inc., Natick, Mass., 1995.
|
|
C
|
|
C [3] Petkov, P.Hr., Gu, D.W., and Konstantinov, M.M.
|
|
C Fortran 77 routines for Hinf and H2 design of continuous-time
|
|
C linear control systems.
|
|
C Rep. 98-14, Department of Engineering, Leicester University,
|
|
C Leicester, U.K., 1998.
|
|
C
|
|
C NUMERICAL ASPECTS
|
|
C
|
|
C The accuracy of the result depends on the condition numbers of the
|
|
C input and output transformations and on the condition numbers of
|
|
C the two Riccati equations, as given by the values of RCOND(1),
|
|
C RCOND(2), RCOND(3) and RCOND(4), respectively.
|
|
C
|
|
C CONTRIBUTORS
|
|
C
|
|
C P.Hr. Petkov, D.W. Gu and M.M. Konstantinov, October 1998.
|
|
C
|
|
C REVISIONS
|
|
C
|
|
C V. Sima, Research Institute for Informatics, Bucharest, May 1999,
|
|
C Sept. 1999, Feb. 2000.
|
|
C
|
|
C KEYWORDS
|
|
C
|
|
C Algebraic Riccati equation, H-infinity optimal control, robust
|
|
C control.
|
|
C
|
|
C ******************************************************************
|
|
C
|
|
C .. Parameters ..
|
|
DOUBLE PRECISION ZERO, ONE
|
|
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
|
|
C ..
|
|
C .. Scalar Arguments ..
|
|
INTEGER INFO, LDA, LDAK, LDB, LDBK, LDC, LDCK, LDD,
|
|
$ LDDK, LDWORK, M, N, NCON, NMEAS, NP
|
|
DOUBLE PRECISION GAMMA, TOL
|
|
C ..
|
|
C .. Array Arguments ..
|
|
LOGICAL BWORK( * )
|
|
INTEGER IWORK( * )
|
|
DOUBLE PRECISION A( LDA, * ), AK( LDAK, * ), B( LDB, * ),
|
|
$ BK( LDBK, * ), C( LDC, * ), CK( LDCK, * ),
|
|
$ D( LDD, * ), DK( LDDK, * ), DWORK( * ),
|
|
$ RCOND( 4 )
|
|
C ..
|
|
C .. Local Scalars ..
|
|
INTEGER INFO2, IWC, IWD, IWF, IWH, IWRK, IWTU, IWTY,
|
|
$ IWX, IWY, LW1, LW2, LW3, LW4, LW5, LW6,
|
|
$ LWAMAX, M1, M2, MINWRK, ND1, ND2, NP1, NP2
|
|
DOUBLE PRECISION TOLL
|
|
C ..
|
|
C .. External Functions ..
|
|
DOUBLE PRECISION DLAMCH
|
|
EXTERNAL DLAMCH
|
|
C ..
|
|
C .. External Subroutines ..
|
|
EXTERNAL DLACPY, SB10PD, SB10QD, SB10RD, XERBLA
|
|
C ..
|
|
C .. Intrinsic Functions ..
|
|
INTRINSIC DBLE, INT, MAX, SQRT
|
|
C ..
|
|
C .. Executable Statements ..
|
|
C
|
|
C Decode and Test input parameters.
|
|
C
|
|
M1 = M - NCON
|
|
M2 = NCON
|
|
NP1 = NP - NMEAS
|
|
NP2 = NMEAS
|
|
C
|
|
INFO = 0
|
|
IF( N.LT.0 ) THEN
|
|
INFO = -1
|
|
ELSE IF( M.LT.0 ) THEN
|
|
INFO = -2
|
|
ELSE IF( NP.LT.0 ) THEN
|
|
INFO = -3
|
|
ELSE IF( NCON.LT.0 .OR. M1.LT.0 .OR. M2.GT.NP1 ) THEN
|
|
INFO = -4
|
|
ELSE IF( NMEAS.LT.0 .OR. NP1.LT.0 .OR. NP2.GT.M1 ) THEN
|
|
INFO = -5
|
|
ELSE IF( GAMMA.LT.ZERO ) THEN
|
|
INFO = -6
|
|
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
|
|
INFO = -8
|
|
ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
|
|
INFO = -10
|
|
ELSE IF( LDC.LT.MAX( 1, NP ) ) THEN
|
|
INFO = -12
|
|
ELSE IF( LDD.LT.MAX( 1, NP ) ) THEN
|
|
INFO = -14
|
|
ELSE IF( LDAK.LT.MAX( 1, N ) ) THEN
|
|
INFO = -16
|
|
ELSE IF( LDBK.LT.MAX( 1, N ) ) THEN
|
|
INFO = -18
|
|
ELSE IF( LDCK.LT.MAX( 1, M2 ) ) THEN
|
|
INFO = -20
|
|
ELSE IF( LDDK.LT.MAX( 1, M2 ) ) THEN
|
|
INFO = -22
|
|
ELSE
|
|
C
|
|
C Compute workspace.
|
|
C
|
|
ND1 = NP1 - M2
|
|
ND2 = M1 - NP2
|
|
LW1 = ( N + NP1 + 1 )*( N + M2 ) + MAX( 3*( N + M2 ) + N + NP1,
|
|
$ 5*( N + M2 ) )
|
|
LW2 = ( N + NP2 )*( N + M1 + 1 ) + MAX( 3*( N + NP2 ) + N +
|
|
$ M1, 5*( N + NP2 ) )
|
|
LW3 = M2 + NP1*NP1 + MAX( NP1*MAX( N, M1 ), 3*M2 + NP1, 5*M2 )
|
|
LW4 = NP2 + M1*M1 + MAX( MAX( N, NP1 )*M1, 3*NP2 + M1, 5*NP2 )
|
|
LW5 = 2*N*N + N*( M + NP ) +
|
|
$ MAX( 1, M*M + MAX( 2*M1, 3*N*N +
|
|
$ MAX( N*M, 10*N*N + 12*N + 5 ) ),
|
|
$ NP*NP + MAX( 2*NP1, 3*N*N +
|
|
$ MAX( N*NP, 10*N*N + 12*N + 5 ) ) )
|
|
LW6 = 2*N*N + N*( M + NP ) +
|
|
$ MAX( 1, M2*NP2 + NP2*NP2 + M2*M2 +
|
|
$ MAX( ND1*ND1 + MAX( 2*ND1, ( ND1 + ND2 )*NP2 ),
|
|
$ ND2*ND2 + MAX( 2*ND2, ND2*M2 ), 3*N,
|
|
$ N*( 2*NP2 + M2 ) +
|
|
$ MAX( 2*N*M2, M2*NP2 +
|
|
$ MAX( M2*M2 + 3*M2, NP2*( 2*NP2 +
|
|
$ M2 + MAX( NP2, N ) ) ) ) ) )
|
|
MINWRK = N*M + NP*( N + M ) + M2*M2 + NP2*NP2 +
|
|
$ MAX( 1, LW1, LW2, LW3, LW4, LW5, LW6 )
|
|
IF( LDWORK.LT.MINWRK )
|
|
$ INFO = -27
|
|
END IF
|
|
IF( INFO.NE.0 ) THEN
|
|
CALL XERBLA( 'SB10FD', -INFO )
|
|
RETURN
|
|
END IF
|
|
C
|
|
C Quick return if possible.
|
|
C
|
|
IF( N.EQ.0 .OR. M.EQ.0 .OR. NP.EQ.0 .OR. M1.EQ.0 .OR. M2.EQ.0
|
|
$ .OR. NP1.EQ.0 .OR. NP2.EQ.0 ) THEN
|
|
RCOND( 1 ) = ONE
|
|
RCOND( 2 ) = ONE
|
|
RCOND( 3 ) = ONE
|
|
RCOND( 4 ) = ONE
|
|
DWORK( 1 ) = ONE
|
|
RETURN
|
|
END IF
|
|
C
|
|
TOLL = TOL
|
|
IF( TOLL.LE.ZERO ) THEN
|
|
C
|
|
C Set the default value of the tolerance.
|
|
C
|
|
TOLL = SQRT( DLAMCH( 'Epsilon' ) )
|
|
END IF
|
|
C
|
|
C Workspace usage.
|
|
C
|
|
IWC = 1 + N*M
|
|
IWD = IWC + NP*N
|
|
IWTU = IWD + NP*M
|
|
IWTY = IWTU + M2*M2
|
|
IWRK = IWTY + NP2*NP2
|
|
C
|
|
CALL DLACPY( 'Full', N, M, B, LDB, DWORK, N )
|
|
CALL DLACPY( 'Full', NP, N, C, LDC, DWORK( IWC ), NP )
|
|
CALL DLACPY( 'Full', NP, M, D, LDD, DWORK( IWD ), NP )
|
|
C
|
|
C Transform the system so that D12 and D21 satisfy the formulas
|
|
C in the computation of the Hinf (sub)optimal controller.
|
|
C
|
|
CALL SB10PD( N, M, NP, NCON, NMEAS, A, LDA, DWORK, N,
|
|
$ DWORK( IWC ), NP, DWORK( IWD ), NP, DWORK( IWTU ),
|
|
$ M2, DWORK( IWTY ), NP2, RCOND, TOLL, DWORK( IWRK ),
|
|
$ LDWORK-IWRK+1, INFO2 )
|
|
IF( INFO2.GT.0 ) THEN
|
|
INFO = INFO2
|
|
RETURN
|
|
END IF
|
|
LWAMAX = INT( DWORK( IWRK ) ) + IWRK - 1
|
|
C
|
|
IWX = IWRK
|
|
IWY = IWX + N*N
|
|
IWF = IWY + N*N
|
|
IWH = IWF + M*N
|
|
IWRK = IWH + N*NP
|
|
C
|
|
C Compute the (sub)optimal state feedback and output injection
|
|
C matrices.
|
|
C
|
|
CALL SB10QD( N, M, NP, NCON, NMEAS, GAMMA, A, LDA, DWORK, N,
|
|
$ DWORK( IWC ), NP, DWORK( IWD ), NP, DWORK( IWF ),
|
|
$ M, DWORK( IWH ), N, DWORK( IWX ), N, DWORK( IWY ),
|
|
$ N, RCOND(3), IWORK, DWORK( IWRK ), LDWORK-IWRK+1,
|
|
$ BWORK, INFO2 )
|
|
IF( INFO2.GT.0 ) THEN
|
|
INFO = INFO2 + 5
|
|
RETURN
|
|
END IF
|
|
LWAMAX = MAX( INT( DWORK( IWRK ) ) + IWRK - 1, LWAMAX )
|
|
C
|
|
C Compute the Hinf (sub)optimal controller.
|
|
C
|
|
CALL SB10RD( N, M, NP, NCON, NMEAS, GAMMA, A, LDA, DWORK, N,
|
|
$ DWORK( IWC ), NP, DWORK( IWD ), NP, DWORK( IWF ),
|
|
$ M, DWORK( IWH ), N, DWORK( IWTU ), M2, DWORK( IWTY ),
|
|
$ NP2, DWORK( IWX ), N, DWORK( IWY ), N, AK, LDAK, BK,
|
|
$ LDBK, CK, LDCK, DK, LDDK, IWORK, DWORK( IWRK ),
|
|
$ LDWORK-IWRK+1, INFO2 )
|
|
IF( INFO2.EQ.1 ) THEN
|
|
INFO = 6
|
|
RETURN
|
|
ELSE IF( INFO2.EQ.2 ) THEN
|
|
INFO = 9
|
|
RETURN
|
|
END IF
|
|
LWAMAX = MAX( INT( DWORK( IWRK ) ) + IWRK - 1, LWAMAX )
|
|
C
|
|
DWORK( 1 ) = DBLE( LWAMAX )
|
|
RETURN
|
|
C *** Last line of SB10FD ***
|
|
END
|