Purpose
To compute the optimal gain matrix K for the problem of optimal
control given by
-1
K = (R + B'XB) (B'Xop(A) + L') (1)
in the discrete-time case and
-1
K = R (B'Xop(E) + L') (2)
in the continuous-time case, where A, E, B and L are N-by-N,
N-by-N, N-by-M, and N-by-M matrices, respectively; R and X are
M-by-M and N-by-N symmetric matrices, respectively, and op(W) is
either W or W'. Matrix op(K) defines the feedback gain matrix, if
op(W) = W, and the estimator matrix, if op(W) = W'. The formulas
above are also useful in Newton's algorithms for solving algebraic
Riccati equations, when X is the current iterate.
Optionally, matrix R may be specified in a factored form, and L
may be zero. If R or R + B'XB (for DICO = 'C', or DICO = 'D',
respectively), is positive definite, let C be its Cholesky factor
(denoted, e.g., C = chol(R), for DICO = 'C'). Optionally, the
matrix H, defined by
H = op(E)'XB + L, if DICO = 'C', or
H = op(A)'XB + L, if DICO = 'D',
is returned on exit, besides K; if C exists, the matrix F, defined
by FC = H may be optionally returned, instead of K and H. The
matrix F or the pair of matrices H and K may be used for computing
the residual matrix for an (approximate) solution of an algebraic
Riccati equation (see SLICOT Library routine SG02CW).
Specification
SUBROUTINE SG02ND( DICO, JOBE, JOB, JOBX, FACT, UPLO, JOBL,
$ TRANS, N, M, P, A, LDA, E, LDE, B, LDB, R, LDR,
$ IPIV, L, LDL, X, LDX, RNORM, K, LDK, H, LDH,
$ XE, LDXE, OUFACT, IWORK, DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER DICO, FACT, JOB, JOBE, JOBL, JOBX, TRANS, UPLO
INTEGER INFO, LDA, LDB, LDE, LDH, LDK, LDL, LDR, LDWORK,
$ LDX, LDXE, M, N, P
DOUBLE PRECISION RNORM
C .. Array Arguments ..
INTEGER IPIV(*), IWORK(*), OUFACT(2)
DOUBLE PRECISION A(LDA,*), B(LDB,*), DWORK(*), E(LDE,*),
$ H(LDH,*), K(LDK,*), L(LDL,*), R(LDR,*),
$ X(LDX,*), XE(LDXE,*)
Arguments
Mode Parameters
DICO CHARACTER*1
Specifies the equation from which K is to be determined,
as follows:
= 'D': Equation (1), discrete-time case;
= 'C': Equation (2), continuous-time case.
JOBE CHARACTER*1
Specifies whether E is a general or an identity matrix,
as follows:
= 'G': The matrix E is general and is given;
= 'I': The matrix E is assumed identity and is not given.
This parameter is not relevant for DICO = 'D'.
JOB CHARACTER*1
Specifies what should be computed, as follows:
= 'K': Compute and return the matrix K only;
= 'H': Compute and return both matrices H and K;
= 'F': Compute the matrix F, if possible; otherwise,
compute and return H and K;
= 'D': Compute and return both matrices H and K, when
B and L have previously been transformed using
SLICOT Library routines SB02MT or SB02MX, which
returned OUFACT = 1. This is useful for computing
K in (2), since then K is the solution of CK = H'.
In this case, FACT should be set to 'C', and the
array R must contain the Cholesky factor of
R + B'XB, if DICO = 'D';
= 'C': Compute and return the matrix F, when B and L have
previously been transformed using SB02MT or
SB02MX, which returned OUFACT = 1. In this case,
FACT should be set to 'C', and the array R must
contain the Cholesky factor of R + B'XB, if
DICO = 'D'.
JOB should not be set to 'F' if FACT = 'U'.
JOBX CHARACTER*1
Specifies whether the matrix op(Xop(E)), if DICO = 'C', or
op(Xop(A)), if DICO = 'D', must be computed, as follows:
= 'C': Compute and return the coresponding matrix;
= 'N': Do not compute that matrix.
This parameter is not relevant for DICO = 'C' and
JOBE = 'I'.
FACT CHARACTER*1
Specifies how the matrix R is given (factored or not), as
follows:
= 'N': Array R contains the matrix R;
= 'D': Array R contains a P-by-M matrix D, where R = D'D;
= 'C': Array R contains the Cholesky factor of R;
= 'U': Array R contains the symmetric indefinite UdU' or
LdL' factorization of R. This option is not
available for DICO = 'D'.
UPLO CHARACTER*1
Specifies which triangle of the possibly factored matrix R
(or R + B'XB, on exit) is or should be stored, as follows:
= 'U': Upper triangle is stored;
= 'L': Lower triangle is stored.
JOBL CHARACTER*1
Specifies whether or not the matrix L is zero, as follows:
= 'Z': L is zero;
= 'N': L is nonzero.
TRANS CHARACTER*1
Specifies the form of op(W) to be used in the matrix
multiplication, as follows:
= 'N': op(W) = W;
= 'T': op(W) = W';
= 'C': op(W) = W'.
Input/Output Parameters
N (input) INTEGER
The order of the matrices A and X. N >= 0.
No computations are performed if MIN(N,M) = 0.
M (input) INTEGER
The order of the matrix R and the number of columns of the
matrices B and L. M >= 0.
P (input) INTEGER
The number of rows of the matrix D.
P >= M for DICO = 'C';
P >= 0 for DICO = 'D'.
This parameter is relevant only for FACT = 'D'.
A (input) DOUBLE PRECISION array, dimension (LDA,N)
If DICO = 'D', the leading N-by-N part of this array must
contain the state matrix A of the system.
If DICO = 'C', this array is not referenced.
LDA INTEGER
The leading dimension of array A.
LDA >= MAX(1,N) if DICO = 'D';
LDA >= 1 if DICO = 'C'.
E (input) DOUBLE PRECISION array, dimension (LDE,*)
If JOBE = 'G' and DICO = 'C', the leading N-by-N part of
this array must contain the matrix E.
If JOBE = 'I' or DICO = 'D', this array is not referenced.
LDE INTEGER
The leading dimension of array E.
LDE >= MAX(1,N), if JOBE = 'G' and DICO = 'C';
LDE >= 1, if JOBE = 'I' or DICO = 'D'.
B (input/worksp.) DOUBLE PRECISION array, dimension (LDB,M)
The leading N-by-M part of this array must contain the
input matrix B of the system, transformed by SB02MT or
SB02MX, if JOB = 'D' or JOB = 'C'.
If DICO = 'D' and FACT = 'D' or 'C', the contents of this
array is destroyed. Specifically, if, on exit,
OUFACT(2) = 1, this array contains chol(X)*B, and if
OUFACT(2) = 2 and INFO < M+2, but INFO >= 0, its trailing
part (in the first N rows) contains the submatrix of
sqrt(V)*U'B corresponding to the non-negligible, positive
eigenvalues of X, where V and U are the matrices with the
eigenvalues and eigenvectors of X.
Otherwise, B is unchanged on exit.
LDB INTEGER
The leading dimension of array B. LDB >= MAX(1,N).
R (input/output) DOUBLE PRECISION array, dimension (LDR,M)
On entry, if FACT = 'N', the leading M-by-M upper
triangular part (if UPLO = 'U') or lower triangular part
(if UPLO = 'L') of this array must contain the upper
triangular part or lower triangular part, respectively,
of the symmetric input weighting matrix R.
On entry, if FACT = 'D', the leading P-by-M part of this
array must contain the direct transmission matrix D of the
system.
On entry, if FACT = 'C', the leading M-by-M upper
triangular part (if UPLO = 'U') or lower triangular part
(if UPLO = 'L') of this array must contain the Cholesky
factor of the positive definite input weighting matrix R
(as produced by LAPACK routine DPOTRF).
On entry, if DICO = 'C' and FACT = 'U', the leading M-by-M
upper triangular part (if UPLO = 'U') or lower triangular
part (if UPLO = 'L') of this array must contain the
factors of the UdU' or LdL' factorization, respectively,
of the symmetric indefinite input weighting matrix R (as
produced by LAPACK routine DSYTRF).
The strictly lower triangular part (if UPLO = 'U') or
strictly upper triangular part (if UPLO = 'L') of this
array is used as workspace (filled in by symmetry with the
other strictly triangular part of R, of R+B'XB, or of the
result, if DICO = 'C', DICO = 'D' (if FACT = 'N', in both
cases), or (DICO = 'D' and (FACT = 'D' or FACT = 'C') and
UPLO = 'L'), respectively.
On exit, if OUFACT(1) = 1, and INFO = 0 (or INFO = M+1),
the leading M-by-M upper triangular part (if UPLO = 'U')
or lower triangular part (if UPLO = 'L') of this array
contains the Cholesky factor of the given input weighting
matrix R (for DICO = 'C'), or that of the matrix R + B'XB
(for DICO = 'D').
On exit, if OUFACT(1) = 2, and INFO = 0 (or INFO = M+1),
the leading M-by-M upper triangular part (if UPLO = 'U')
or lower triangular part (if UPLO = 'L') of this array
contains the factors of the UdU' or LdL' factorization,
respectively, of the given input weighting matrix
(for DICO = 'C'), or that of the matrix R + B'XB
(for DICO = 'D' and FACT = 'N').
On exit R is unchanged if FACT = 'U' or N = 0.
LDR INTEGER.
The leading dimension of the array R.
LDR >= MAX(1,M) if FACT <> 'D';
LDR >= MAX(1,M,P) if FACT = 'D'.
IPIV (input/output) INTEGER array, dimension (M)
On entry, if FACT = 'U', this array must contain details
of the interchanges performed and the block structure of
the d factor in the UdU' or LdL' factorization of matrix R
(as produced by LAPACK routine DSYTRF).
On exit, if OUFACT(1) = 2, this array contains details of
the interchanges performed and the block structure of the
d factor in the UdU' or LdL' factorization of matrix R or
R + B'XB, as produced by LAPACK routine DSYTRF.
This array is not referenced if FACT = 'D', or FACT = 'C',
or N = 0.
L (input) DOUBLE PRECISION array, dimension (LDL,M)
If JOBL = 'N', the leading N-by-M part of this array must
contain the cross weighting matrix L, transformed by
SB02MT or SB02MX, if JOB = 'D' or JOB = 'C'.
If JOBL = 'Z', this array is not referenced.
LDL INTEGER
The leading dimension of array L.
LDL >= MAX(1,N) if JOBL = 'N';
LDL >= 1 if JOBL = 'Z'.
X (input/output) DOUBLE PRECISION array, dimension (LDX,N)
On entry, the leading N-by-N part of this array must
contain the (approximate) solution matrix X of the
algebraic Riccati equation as produced by SLICOT Library
routines SB02MD or SB02OD (or SG02CD). Matrix X is assumed
non-negative definite if DICO = 'D' and (FACT = 'D' or
FACT = 'C').
The full matrix X must be given on input if LDWORK < N*M
and ((JOBX = 'N' and N >= M) or (JOBX = 'C', DICO = 'D',
FACT = 'N' and N < M)).
On exit, if DICO = 'D', FACT = 'D' or FACT = 'C', and
OUFACT(2) = 1, the N-by-N upper triangular part
(if UPLO = 'U') or lower triangular part (if UPLO = 'L')
of this array contains the Cholesky factor of the given
matrix X, which is found to be positive definite.
On exit, if DICO = 'D', FACT = 'D' or 'C', OUFACT(2) = 2,
and INFO <> M+2 (but INFO >= 0), the leading N-by-N part
of this array contains the matrix of orthonormal
eigenvectors of X.
On exit X is unchanged if DICO = 'C' or FACT = 'N'.
LDX INTEGER
The leading dimension of array X. LDX >= MAX(1,N).
RNORM (input) DOUBLE PRECISION
If FACT = 'U', this parameter must contain the 1-norm of
the original matrix R (before factoring it).
Otherwise, this parameter is not used.
K (output) DOUBLE PRECISION array, dimension (LDK,N)
If JOB = 'K' or JOB = 'H' or JOB = 'D' or OUFACT(1) = 2,
the leading M-by-N part of this array contains the gain
matrix K.
LDK INTEGER
The leading dimension of array K. LDK >= MAX(1,M).
H (output) DOUBLE PRECISION array, dimension (LDH,*)
If JOB = 'H' or ((JOB = 'F' or JOB = 'D' or JOB = 'C') and
OUFACT(1) = 2), the leading N-by-M part of this array
contains the matrix H.
If (JOB = 'F' or JOB = 'C') and OUFACT(1) = 1, the leading
N-by-M part of this array contains the matrix F.
If JOB = 'K', this array is not referenced.
LDH INTEGER
The leading dimension of array H.
LDH >= MAX(1,N), if JOB <> 'K';
LDH >= 1, if JOB = 'K'.
XE (output) DOUBLE PRECISION array, dimension (LDXE,*)
If JOBX = 'C', DICO = 'C', and JOBE = 'G', the leading
N-by-N part of this array contains the matrix product X*E,
if TRANS = 'N', or E*X, if TRANS = 'T' or TRANS = 'C'.
If JOBX = 'C' and DICO = 'D', the leading N-by-N part of
this array contains the matrix product X*A, if
TRANS = 'N', or A*X, if TRANS = 'T' or TRANS = 'C'.
These matrix products may be needed for computing the
residual matrix for an (approximate) solution of a Riccati
equation (see SLICOT Library routine SG02CW).
If JOBX = 'N' or (DICO = 'C' and JOBE = 'I'), this array
is not referenced.
LDXE INTEGER
The leading dimension of array XE.
LDXE >= MAX(1,N), if JOBX = 'C', and either DICO = 'C' and
JOBE = 'G', or DICO = 'D';
LDXE >= 1, if JOBX = 'N' or (DICO = 'C' and
JOBE = 'I').
OUFACT (output) INTEGER array, dimension (2)
Information about the factorization finally used.
OUFACT(1) = 1: Cholesky factorization of R (or R + B'XB)
has been used;
OUFACT(1) = 2: UdU' (if UPLO = 'U') or LdL' (if UPLO =
'L') factorization of R (or R + B'XB)
has been used;
OUFACT(2) = 1: Cholesky factorization of X has been used;
OUFACT(2) = 2: Spectral factorization of X has been used.
The value of OUFACT(2) is not set for DICO = 'C' or for
DICO = 'D' and FACT = 'N'.
This array is not set if N = 0 or M = 0.
Workspace
IWORK INTEGER array, dimension (M)
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0 or LDWORK = -1, DWORK(1) returns the
optimal value of LDWORK, and for LDWORK set as specified
below, DWORK(2) contains the reciprocal condition number
of the matrix R (for DICO = 'C') or of R + B'XB (for
DICO = 'D'), if FACT = 'N' or FACT = 'U' or OUFACT(1) = 2,
or of its Cholesky factor, if FACT = 'C' or FACT = 'D' and
OUFACT(1) = 1; DWORK(2) is set to 1 if N = 0.
On exit, if LDWORK = -2 on input or INFO = -35, then
DWORK(1) returns the minimal value of LDWORK.
If on exit INFO = 0, and OUFACT(2) = 2, then DWORK(3),...,
DWORK(N+2) contain the eigenvalues of X, in ascending
order.
LDWORK INTEGER
Dimension of working array DWORK.
Let a = N, if JOBX = 'N' and (DICO = 'D' or JOBE = 'G');
a = 0, otherwise. Then
LDWORK >= max(2,2*M,a) if FACT = 'U';
LDWORK >= max(2,3*M,a) if FACT <> 'U', DICO = 'C',
or FACT = 'N' or
(FACT = 'C' and (JOB = 'C'
or JOB = 'D')), DICO = 'D';
LDWORK >= max(N+3*M+2,4*N+1) if FACT = 'D' or
(FACT = 'C' and JOB <> 'C'
and JOB <> 'D'), DICO = 'D'.
For optimum performance LDWORK should be larger.
If LDWORK = -1, an optimal workspace query is assumed; the
routine only calculates the optimal size of the DWORK
array, returns this value as the first entry of the DWORK
array, and no error message is issued by XERBLA.
If LDWORK = -2, a minimal workspace query is assumed; the
routine only calculates the minimal size of the DWORK
array, returns this value as the first entry of the DWORK
array, and no error message is issued by XERBLA.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= i: if the i-th element of the d factor is exactly zero;
the UdU' (or LdL') factorization has been completed,
but the block diagonal matrix d is exactly singular;
= M+1: if the matrix R (if DICO = 'C'), or R + B'XB
(if DICO = 'D') is numerically singular (to working
precision);
= M+2: if one or more of the eigenvalues of X has not
converged;
= M+3: if the matrix X is indefinite and updating the
triangular factorization failed.
If INFO > M+1, call the routine again with an appropriate,
unfactored matrix R.
Method
The (optimal) gain matrix K is obtained as the solution to the
system of linear equations
(R + B'XB) * K = B'Xop(A) + L'
in the discrete-time case and
R * K = B'Xop(E) + L'
in the continuous-time case, with R replaced by D'D if FACT = 'D'.
If FACT = 'N', Cholesky factorization is tried first, but
if the coefficient matrix is not positive definite, then UdU' (or
LdL') factorization is used. If FACT <> 'N', the factored form
of R is taken into account. The discrete-time case then involves
updating of a triangular factorization of R (or D'D); Cholesky or
symmetric spectral factorization of X is employed to avoid
squaring of the condition number of the matrix. When D is given,
its QR factorization is determined, and the triangular factor is
used as described above.
Numerical Aspects
The algorithm consists of numerically stable steps.
3 2
For DICO = 'C' and JOBE = 'I', it requires O(m + mn ) floating
2
point operations if FACT = 'N' and O(mn ) floating point
operations, otherwise.
For DICO = 'D' or JOBE = 'G', the operation counts are similar,
3
but additional O(n ) floating point operations may be needed in
the worst case.
These estimates assume that M <= N.
Further Comments
NoneExample
Program Text
* SG02ND EXAMPLE PROGRAM TEXT
* Copyright (c) 2002-2017 NICONET e.V.
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX, MMAX, PMAX
PARAMETER ( NMAX = 20, MMAX = 20, PMAX = 20 )
INTEGER NMAX2
PARAMETER ( NMAX2 = 2*NMAX )
INTEGER LDA, LDB, LDC, LDF, LDH, LDL, LDR, LDS, LDT, LDU,
$ LDX, LDXE
PARAMETER ( LDA = NMAX, LDB = NMAX, LDC = PMAX, LDL = NMAX,
$ LDR = MAX(MMAX,PMAX), LDS = NMAX2+MMAX,
$ LDT = NMAX2+MMAX, LDU = NMAX2, LDX = NMAX,
$ LDF = MMAX, LDH = NMAX, LDXE = NMAX )
INTEGER LIWORK
PARAMETER ( LIWORK = MMAX )
INTEGER LDWORK
PARAMETER ( LDWORK = MAX( NMAX+3*MMAX+2, 14*NMAX+23,
$ 16*NMAX ) )
* .. Local Scalars ..
DOUBLE PRECISION TOL, RCOND, RNORM
INTEGER I, INFO1, INFO2, J, M, N, P
CHARACTER*1 DICO, FACT, JOB, JOBB, JOBE, JOBL, JOBX, SORT,
$ TRANS, UPLO
* .. Local Arrays ..
DOUBLE PRECISION AE(LDA,NMAX), ALFAI(2*NMAX), ALFAR(2*NMAX),
$ B(LDB,MMAX), BETA(2*NMAX), C(LDC,NMAX),
$ DWORK(LDWORK), F(LDF,NMAX), H(LDH,MMAX),
$ L(LDL,MMAX), R(LDR,MMAX), S(LDS,NMAX2+MMAX),
$ T(LDT,NMAX2), U(LDU,NMAX2), X(LDX,NMAX),
$ XE(LDXE,NMAX)
INTEGER IPIV(LIWORK), IWORK(LIWORK), OUFACT(2)
LOGICAL BWORK(NMAX2)
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* .. External Subroutines ..
EXTERNAL SG02ND, SB02OD
* .. Intrinsic Functions ..
INTRINSIC MAX
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, M, P, TOL, DICO, JOBE, JOB, JOBX, FACT,
$ JOBL, UPLO, TRANS
IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99993 ) N
ELSE
READ ( NIN, FMT = * ) ( ( AE(I,J), J = 1,N ), I = 1,N )
IF ( M.LT.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99992 ) M
ELSE
READ ( NIN, FMT = * ) ( ( B(I,J), J = 1,M ), I = 1,N )
IF ( P.LT.0 .OR. P.GT.PMAX ) THEN
WRITE ( NOUT, FMT = 99991 ) P
ELSE
READ ( NIN, FMT = * ) ( ( C(I,J), J = 1,N ), I = 1,P )
IF ( LSAME( FACT, 'D' ) ) THEN
READ ( NIN, FMT = * ) ( ( R(I,J), J = 1,M ), I = 1,P )
ELSE
READ ( NIN, FMT = * ) ( ( R(I,J), J = 1,M ), I = 1,M )
END IF
IF ( LSAME( JOBL, 'N' ) )
$ READ ( NIN, FMT = * ) ( ( L(I,J), J = 1,M ), I = 1,N )
* Find the solution matrix X.
JOBB = 'B'
SORT = 'S'
CALL SB02OD( DICO, JOBB, 'Both', UPLO, JOBL, SORT, N, M,
$ P, AE, LDA, B, LDB, C, LDC, R, LDR, L, LDL,
$ RCOND, X, LDX, ALFAR, ALFAI, BETA, S, LDS,
$ T, LDT, U, LDU, TOL, IWORK, DWORK, LDWORK,
$ BWORK, INFO1 )
*
IF ( INFO1.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO1
ELSE
WRITE ( NOUT, FMT = 99996 )
DO 20 I = 1, N
WRITE ( NOUT, FMT = 99994 ) ( X(I,J), J = 1,N )
20 CONTINUE
* Compute the optimal feedback matrix F.
CALL SG02ND( DICO, JOBE, JOB, JOBX, FACT, UPLO, JOBL,
$ TRANS, N, M, P, AE, LDA, AE, LDA, B, LDB,
$ R, LDR, IPIV, L, LDL, X, LDX, RNORM, F,
$ LDF, H, LDH, XE, LDXE, OUFACT, IWORK,
$ DWORK, LDWORK, INFO2 )
*
IF ( INFO2.NE.0 ) THEN
WRITE ( NOUT, FMT = 99997 ) INFO2
ELSE
WRITE ( NOUT, FMT = 99995 )
DO 40 I = 1, M
WRITE ( NOUT, FMT = 99994 ) ( F(I,J), J = 1,N )
40 CONTINUE
END IF
END IF
END IF
END IF
END IF
STOP
*
99999 FORMAT (' SG02ND EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from SB02OD = ',I2)
99997 FORMAT (' INFO on exit from SG02ND = ',I2)
99996 FORMAT (' The solution matrix X is ')
99995 FORMAT (/' The optimal feedback matrix F is ')
99994 FORMAT (20(1X,F8.4))
99993 FORMAT (/' N is out of range.',/' N = ',I5)
99992 FORMAT (/' M is out of range.',/' M = ',I5)
99991 FORMAT (/' P is out of range.',/' P = ',I5)
END
Program Data
SG02ND EXAMPLE PROGRAM DATA 2 1 3 0.0 D I K N N Z U N 2.0 -1.0 1.0 0.0 1.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0 0.0 0.0 0.0Program Results
SG02ND EXAMPLE PROGRAM RESULTS The solution matrix X is 1.0000 0.0000 0.0000 1.0000 The optimal feedback matrix F is 2.0000 -1.0000