SUBROUTINE CKBF(NR,N,M,L,A,G,C,R,D,P,F,FCL,GDG,WXYZ,AT,
1 CT,E,RI,RS,S,U,WK,ALFI,ALFR,BETA,
2 CPERM,CSCALE,RSD,RTOL,MAXIT,
3 RSTRUC,IBAL,IND,IERR,ERRMSG)
C
C FUNCTION:
CF
CF The subroutine CKBF designs a steady state Kalman-Bucy
CF Filter of the form
CF
CF Xhat'(t) = A*Xhat(t) + F*(Y(t) - Xhat(t))
CF
CF for the continous-time linear system:
CF
CF X'(t) = A*X(t) + Gd(t)
CF Y(t) = C*X(t) + r(t)
CF
CF where
CF A is the N by N state matrix.
CF C is the L by N output matrix.
CF G is an N by M matrix, usually the identity.
CF d(t) is the input noise process with
CF M by M input noise covariance matrix
CF D. The input noise is assumed to
CF be white Gaussian.
CF r(t) is the observation noise process
CF with L by L observation noise co-
CF variance matrix R, also assumed
CF to be white Gaussian.
CF
CF The filter gain matrix F is of the form
CF
CF T -1
CF F = P * C * R
CF
CF where P is the solution to the continuous time
CF Algebraic Riccati Equation (ARE):
CF
CF T T -1 T
CF A*P + P*A - P*C *R *C*P + G*D*G = 0
CF
CF The closed loop gain matrix FCL of the system is then given
CF by:
CF FCL = A - F*C.
CF
C USAGE:
CU
CU The subroutine CKBF can be used to design a Kalman filter for
CU a given system. The subroutine is invoked by the call
CU
CU SUBROUTINE CKBF(NR,N,M,L,A,G,C,R,D,P,F,FCL,GDG,WXYZ,AT,
CU 1 CT,E,RI,RS,S,U,WK,ALFI,ALFR,BETA,
CU 2 CPERM,CSCALE,RSD,RTOL,MAXIT,
CU 3 RSTRUC,IBAL,IND,IERR,ERRMSG)
CU
CU where the variables are described below. When the subroutine
CU is called, it is only necessary to have NR, N, M, L, A, G, C,
CU R, D, RTOL, MAXIT, and RSTRUC initialized.
CU
C INPUTS:
CI
CI NR - INTEGER variable which is the row dimension of
CI all the arrays as declared in the calling program.
CI
CI N - INTEGER variable; the order the system.
CI
CI M - INTEGER variable; the number of system inputs.
CI
CI L - INTEGER variable; the number of system outputs.
CI
CI A - DP variable; the N by N `A' (state) matrix of the system.
CI
CI C - DP variable; the L by N `C' (output) matrix of the system.
CI
CI R - DP variable; the L by L control weighting matrix.
CI
CI WXYZ - DP variable; a scratch array.
CI
CI G - DP variable; the N by M input noise matrix.
CI
CI D - DP variable; the M by M input noise covariance matrix.
CI
CI AT - DP variable; a scratch array.
CI
CI CT - DP variable; a scratch array.
CI
CI GDG - DP variable; a scratch array.
CI
CI E - DP variable; a scratch array.
CI
CI S - DP variable; a scratch array.
CI
CI U - DP variable; a scratch array.
CI
CI WK - DP variable; a scratch array.
CI
CI RTOL - DP variable; set this to the worst condition R can be
CI and still be considered invertible. The condition RTOL
CI is compared to is 1/RCOND where RCOND is the condition
CI estimate used in LINPACK. If RTOL=0, then a default
CI value of 1.0D+6 is used.
CI
CI MAXIT - the maximum number of iterations to use for improving
CI the accuracy of the solution by Newton's method.
CI
CI RSTRUC - INTEGER variable indicating the structure of the
CI R matrix:
CI RSTRUC = 0 if R is the identity;
CI RSTRUC = 1 if R is diagonal but
CI not the identity;
CI RSTRUC = 2 otherwise.
CI
CI IBAL - INTEGER variable determining if the equation is
CI balanced:
CI IBAL = 1 for WARD (pre-QZ) balancing.
CI IBAL = 0 for no balancing.
CI
C OUTPUTS:
CO
CO P - DP variable; P contains the N by N solution to the ARE.
CO
CO F - DP variable; F contains the N by L optimal gain matrix.
CO
CO FCL - DP variable; FCL contains the N by N closed loop
CO matrix for the system, which is A-F*C.
CO
CO RI - DP variable; contains the L by L matrix R-inverse;
CO however, the program may not have have used
CO R-inverse in the calculation of the ARE solution.
CO
CO AT - DP variable; contains the N by N matrix A-transpose.
CO
CO CT - DP variable; contains the L by N matrix C-transpose.
CO
CO RS - DP variable; contains the N by N residual matrix of
CO of the ARE with F as the solution.
CO
CO RSD - DP variable; the 1-Norm of the residual matrix RS
CO divided by the 1-Norm of the solution matrix F.
CO
CO MAXIT - INTEGER variable; on output, MAXIT contains the
CO number Newton iterations actually performed by
CO NEWT. A number less than the original MAXIT is
CO not necessarily bad since the solution may have
CO converged.
CO
CO GDG - The matrix contains the product G*D*(G-transpose).
CO
CO ALFR - DP variable; this matrix contains the real parts of
CO the eigenvalues of FCL.
CO
CO ALFI - DP variable; this matrix contains the imaginary parts
CO of the eigenvalues of FCL.
CO
CO IERR - An INTEGER variable indicating whether an error or
CO warning has occured during execution.
CO IERR = 0 for a normal return.
CO IERR < 0 is a warning, indicating abnormal
CO but nonfatal conditions. NOTE that if
CO a warning occurs, then it is the last
CO warning condition which occured, and
CO therefore may be misleading, since an
CO earlier warning may have caused other
CO warnings and then been masked.
CO IERR > 0 indicates a fatal error.
CO
CO ERRMSG = a CHARACTER variable which is only meaningful if
CO IERR .NE. 0, it contains a description of the
CO error or warning.
CO
CO
C ALGORITHM:
CA
CA The algorithm and much of the code is adapted (copied where
CA possible) from the interactive driver program to RICPACK.
CA See:
CA 1. Arnold, W. F., and A. J. Laub, "Generalized
CA eigenproblem algorithms and software for
CA algebraic Riccati equations," Proceedings of
CA the IEEE, vol. 72, no. 12, December 1984,
CA pp. 1746-1754.
CA
CA 2. The RICPACK code.
CA
CA Note that this program currently only solves the ARE for
CA the case where S=0 and E=I.
CA
C MACHINE DEPENDENCIES:
CM
CM None.
CM
C HISTORY:
CH
CH written by: Bobby Bodenheimer
CH date: September 1986
CH current version: 1.2
CH modifications: zero'ed out cross term S - jdb - 5/10/88
CH defined undefined variables - jdb - 5/13/88
CH
C ROUTINES CALLED:
CC
CC RINV, RICSOL, CMPRS, SAVE, MADD, MULB, TRNATB,
CC MLINEQ, FBGAIN, NEWT, RESID, MSUB, ICNVRT.
CC
C----------------------------------------------------------------------
C written for: The CASCADE Project
C Oak Ridge National Laboratory
C U.S. Department of Energy
C contract number DE-AC05-840R21400
C subcontract number 37B-07685C S13
C organization: The University of Tennessee
C----------------------------------------------------------------------
C THIS SOFTWARE IS IN THE PUBLIC DOMAIN
C NO RESTRICTIONS ON ITS USE ARE IMPLIED
C----------------------------------------------------------------------
C
C Global Variables.
C
INTEGER NR
INTEGER RSTRUC
INTEGER IERR
INTEGER IND(NR)
INTEGER IBAL
INTEGER INFO
INTEGER L
INTEGER M
INTEGER MAXIT
INTEGER N
C
CHARACTER*(*) ERRMSG
C
DOUBLE PRECISION A(NR,NR)
DOUBLE PRECISION C(NR,NR)
DOUBLE PRECISION D(NR,NR)
DOUBLE PRECISION E(NR,NR)
DOUBLE PRECISION P(NR,NR)
DOUBLE PRECISION G(NR,NR)
DOUBLE PRECISION GDG(NR,NR)
DOUBLE PRECISION R(NR,NR)
DOUBLE PRECISION RI(NR,NR)
DOUBLE PRECISION RS(NR,NR)
DOUBLE PRECISION S(NR,NR)
DOUBLE PRECISION U(NR,NR)
DOUBLE PRECISION WK(NR,NR)
DOUBLE PRECISION F(NR,NR)
DOUBLE PRECISION WXYZ(NR,NR)
DOUBLE PRECISION FCL(NR,NR)
DOUBLE PRECISION AT(NR,NR)
DOUBLE PRECISION CT(NR,NR)
C
DOUBLE PRECISION ALFI(NR)
DOUBLE PRECISION ALFR(NR)
DOUBLE PRECISION BETA(NR)
DOUBLE PRECISION CPERM(NR)
DOUBLE PRECISION CSCALE(NR)
C
DOUBLE PRECISION RSD
DOUBLE PRECISION RTOL
C
C Local Variables.
C
INTEGER I
INTEGER J
INTEGER LENGTH
INTEGER MERR
INTEGER NN
INTEGER NNPL
INTEGER NOUT
C
DOUBLE PRECISION SEP
DOUBLE PRECISION COND
C
PARAMETER(NOUT=6)
C
CHARACTER RDFLG
CHARACTER RFLAG
CHARACTER*30 STRING
C
C Begin
C
IERR = 0
IF (RTOL.EQ.0.0D0) RTOL = 1.0D+6
NN = 2*N
NNPL = NN + L
C
C Check system size.
C
IF (NNPL.GT.NR) THEN
CALL ICNVRT(0,NNPL,STRING,LENGTH,MERR)
IERR = 1
ERRMSG = 'CKBF: Fatal Error: System too large.'//
1 ' For this system, NR should be at '//
2 ' at least '//STRING(1:LENGTH)
RETURN
END IF
C
C Set structure flags for R.
C
IF (RSTRUC.EQ.0) THEN
RFLAG = 'N'
RDFLG = 'Y'
ELSE IF (RSTRUC.EQ.1) THEN
RFLAG = 'Y'
RDFLG = 'Y'
ELSE IF (RSTRUC.EQ.2) THEN
RFLAG = 'Y'
RDFLG = 'N'
ELSE
IERR = 2
ERRMSG = ' CKBF: Fatal Error: RSTRUC must be '//
1 '0, 1, or 2.'
RETURN
END IF
C
C Zero out the cross term S
C
DO 4 J = 1, L
DO 4 I = 1, N
S(I,J) = 0.D0
4 CONTINUE
C
C Make E the identity.
C
DO 6 J=1,N
DO 5 I=1,N
E(I,J) = 0.0D0
5 CONTINUE
E(J,J) = 1.0D0
6 CONTINUE
C
C Compute A-transpose, store it in AT; compute
C C-transpose and store it in CT.
C
CALL TRNATB(NR,NR,N,N,A,AT)
CALL TRNATB(NR,NR,L,N,C,CT)
C
C Compute the matrix product G*D*(G-transpose) and store it
C in GDG.
C
CALL TRNATB(NR,NR,N,M,G,GDG)
CALL MULB(NR,NR,M,M,N,D,GDG,CSCALE)
CALL MULB(NR,NR,N,M,N,G,GDG,CSCALE)
C
IF (RSTRUC.EQ.0) THEN
C
C R = Identity.
C
CALL SAVE(NR,NR,L,L,R,RI)
CALL RINV(NR,NR,N,NN,L,
1 E,AT,CT,GDG,RI,WXYZ,P,RS,CPERM,
2 RDFLG,RFLAG,'N',.TRUE.)
C
C R is diagonal.
C
ELSE IF (RSTRUC.EQ.1) THEN
DO 10 I=1,L
RI(I,I) = 1.0D0/R(I,I)
10 CONTINUE
CALL RINV(NR,NR,N,NN,L,
1 E,AT,CT,GDG,RI,WXYZ,P,RS,CPERM,
2 RDFLG,RFLAG,'N',.TRUE.)
C
ELSE IF (RSTRUC.EQ.2) THEN
C
C No special structure in R to exploit.
C
DO 30 J=1,L
DO 20 I=1,L
RI(I,J) = 0.0D0
20 CONTINUE
RI(J,J) = 1.0D0
30 CONTINUE
C
C Check the condition of R with respect to inversion.
C RI contains R-inverse, but if R is ill-conditioned,
C we won't use it.
C
CALL MLINEQ(NR,NR,L,L,R,RI,
1 COND,IND,CPERM)
C
C R ill-conditioned.
C
IF (COND.GE.RTOL) THEN
CALL CMPRS(NR,NR,NR,N,NN,
1 NNPL,L,E,AT,CT,GDG,R,
2 S,WXYZ,P,U,WK,CPERM,CSCALE,
3 BETA,'N','N',INFO)
IF (INFO.NE.0) THEN
IERR = 100 + INFO
ERRMSG = 'CKBF: Fatal Error: Attempt to '//
1 'compress R failed. IERR = 100 +'//
2 'INFO, INFO returned from DSVDC.'
RETURN
END IF
ELSE
C
C R not ill-conditioned.
C
CALL RINV(NR,NR,N,NN,L,
1 E,AT,CT,GDG,RI,WXYZ,P,RS,CPERM,
2 RDFLG,RFLAG,'N',.TRUE.)
END IF
END IF
C
C Compute the Riccati solution.
C
CALL RICSOL(NR,NR,NN,N,WXYZ,P,E,F,
1 ALFR,ALFI,BETA,CPERM,CSCALE,
2 IND,-1,IBAL,.TRUE.,'N')
C
C Compute the generalized eigenvalues in case the routine
C bombs later.
C
DO 40, I=1,N
ALFR(I) = ALFR(I)/BETA(I)
ALFI(I) = ALFI(I)/BETA(I)
40 CONTINUE
C
C Check the RICSOL computation for errors.
C
IF (IND(1).NE.0) THEN
IERR = 200 + IND(1)
ERRMSG = 'CKBF: Fatal Error: More than 50 '//
1 'iterations required by QZITW. '//
2 'IERR = 200 + IND(1), where IND(1) '//
3 ' is the ierr from QZITW.'
RETURN
END IF
IF (CPERM(1).EQ.1.0D+20) THEN
CALL SAVE(NR,NR,N,N,F,P)
IERR = 300
ERRMSG = ' CKBF: Fatal Error: Schur Vector '//
1 'singular to working precision, '//
2 'probably because the original system '//
3 'is not stabilizable. Hence solution '//
4 'by this method is not possible. The '//
5 'Schur matrix is returned in P.'
RETURN
END IF
IF (IND(2).NE.0) THEN
IERR = 400
ERRMSG = 'CKBF: Fatal Error: Convergence '//
1 ' failure in ordering routine '//
2 'ORDER. '
RETURN
END IF
C
C Attempt to improve the solution by iterating
C with Newton's method.
C
CALL SAVE(NR,NR,N,N,P,U)
CALL NEWT(NR,NR,NR,NR,N,L,E,AT,CT,GDG,R,S,RI,RS,U,
1 WXYZ,P,F,ALFR,ALFI,BETA,IND,RTOL,'N',RFLAG,
2 RDFLG,'N',SEP,.TRUE.,MAXIT,INFO,NOUT)
IF (INFO.NE.0) THEN
IERR = -200 - INFO
ERRMSG = 'CKBF: Warning: Newton iteration '//
1 'failed. Solution after last '//
2 'iteration given. IERR = -200 - INFO '//
3 'where INFO is returned from NEWT.'
END IF
C
C Compute the Residual of the refined ARE solution.
C
CALL RESID(NR,NR,NR,NR,N,L,E,AT,CT,GDG,R,S,RI,RS,
1 U,WXYZ,F,CPERM,IND,RTOL,'N',RFLAG,RDFLG,
2 'N',RSD,.TRUE.,NOUT)
IF (IND(1).NE.0) THEN
IERR = -300
ERRMSG = 'CKBF: Warning: Residual estimation '//
1 'may be invalid. IPVT(1) from RESID '//
2 'is non-zero.'
END IF
CALL SAVE(NR,NR,N,N,U,P)
C
C T -1
C Compute the filter gain matrix F = P*C *R.
C
IF (RSTRUC.EQ.0.OR.RSTRUC.EQ.1) THEN
DO 70 J=1,L
DO 60 I=1,L
F(I,J) = 0.0D0
60 CONTINUE
F(J,J) = RI(J,J)
70 CONTINUE
ELSE IF(RSTRUC.EQ.2) THEN
CALL SAVE(NR,NR,L,L,RI,F)
END IF
CALL MULB(NR,NR,N,L,L,CT,F,CSCALE)
CALL MULB(NR,NR,N,N,L,P,F,CSCALE)
C
C Calculate the Closed Loop Filter Matrix for the system.
C FCL = A - F*C
C
CALL MMUL(NR,NR,NR,N,N,L,F,C,WXYZ)
CALL MSUB(NR,NR,NR,N,N,A,WXYZ,FCL)
C
C Last Lines of CKBF.
C
RETURN
END