Purpose
To solve for X either the continuous-time algebraic Riccati
equation
-1
Q + A'*X + X*A - X*B*R B'*X = 0 (1)
or the discrete-time algebraic Riccati equation
-1
X = A'*X*A - A'*X*B*(R + B'*X*B) B'*X*A + Q (2)
where A, B, Q and R are N-by-N, N-by-M, N-by-N and M-by-M matrices
respectively, with Q symmetric and R symmetric nonsingular; X is
an N-by-N symmetric matrix.
-1
The matrix G = B*R B' must be provided on input, instead of B and
R, that is, for instance, the continuous-time equation
Q + A'*X + X*A - X*G*X = 0 (3)
is solved, where G is an N-by-N symmetric matrix. SLICOT Library
routine SB02MT should be used to compute G, given B and R. SB02MT
also enables to solve Riccati equations corresponding to optimal
problems with coupling terms.
The routine also returns the computed values of the closed-loop
spectrum of the optimal system, i.e., the stable eigenvalues
lambda(1),...,lambda(N) of the corresponding Hamiltonian or
symplectic matrix associated to the optimal problem.
Specification
SUBROUTINE SB02MD( DICO, HINV, UPLO, SCAL, SORT, N, A, LDA, G,
$ LDG, Q, LDQ, RCOND, WR, WI, S, LDS, U, LDU,
$ IWORK, DWORK, LDWORK, BWORK, INFO )
C .. Scalar Arguments ..
CHARACTER DICO, HINV, SCAL, SORT, UPLO
INTEGER INFO, LDA, LDG, LDQ, LDS, LDU, LDWORK, N
DOUBLE PRECISION RCOND
C .. Array Arguments ..
LOGICAL BWORK(*)
INTEGER IWORK(*)
DOUBLE PRECISION A(LDA,*), DWORK(*), G(LDG,*), Q(LDQ,*),
$ S(LDS,*), U(LDU,*), WR(*), WI(*)
Arguments
Mode Parameters
DICO CHARACTER*1
Specifies the type of Riccati equation to be solved as
follows:
= 'C': Equation (3), continuous-time case;
= 'D': Equation (2), discrete-time case.
HINV CHARACTER*1
If DICO = 'D', specifies which symplectic matrix is to be
constructed, as follows:
= 'D': The matrix H in (5) (see METHOD) is constructed;
= 'I': The inverse of the matrix H in (5) is constructed.
HINV is not used if DICO = 'C'.
UPLO CHARACTER*1
Specifies which triangle of the matrices G and Q is
stored, as follows:
= 'U': Upper triangle is stored;
= 'L': Lower triangle is stored.
SCAL CHARACTER*1
Specifies whether or not a scaling strategy should be
used, as follows:
= 'G': General scaling should be used;
= 'N': No scaling should be used.
SORT CHARACTER*1
Specifies which eigenvalues should be obtained in the top
of the Schur form, as follows:
= 'S': Stable eigenvalues come first;
= 'U': Unstable eigenvalues come first.
Input/Output Parameters
N (input) INTEGER
The order of the matrices A, Q, G and X. N >= 0.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading N-by-N part of this array must
contain the coefficient matrix A of the equation.
On exit, if DICO = 'D', and INFO = 0 or INFO > 1, the
-1
leading N-by-N part of this array contains the matrix A .
Otherwise, the array A is unchanged on exit.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,N).
G (input) DOUBLE PRECISION array, dimension (LDG,N)
The leading N-by-N 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 matrix G. The stricly
lower triangular part (if UPLO = 'U') or stricly upper
triangular part (if UPLO = 'L') is not referenced.
LDG INTEGER
The leading dimension of array G. LDG >= MAX(1,N).
Q (input/output) DOUBLE PRECISION array, dimension (LDQ,N)
On entry, the leading N-by-N 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 matrix Q.
The stricly lower triangular part (if UPLO = 'U') or
stricly upper triangular part (if UPLO = 'L') is not used.
On exit, if INFO = 0, the leading N-by-N part of this
array contains the solution matrix X of the problem.
LDQ INTEGER
The leading dimension of array N. LDQ >= MAX(1,N).
RCOND (output) DOUBLE PRECISION
An estimate of the reciprocal of the condition number (in
the 1-norm) of the N-th order system of algebraic
equations from which the solution matrix X is obtained.
WR (output) DOUBLE PRECISION array, dimension (2*N)
WI (output) DOUBLE PRECISION array, dimension (2*N)
If INFO = 0 or INFO = 5, these arrays contain the real and
imaginary parts, respectively, of the eigenvalues of the
2N-by-2N matrix S, ordered as specified by SORT (except
for the case HINV = 'D', when the order is opposite to
that specified by SORT). The leading N elements of these
arrays contain the closed-loop spectrum of the system
-1
matrix A - B*R *B'*X, if DICO = 'C', or of the matrix
-1
A - B*(R + B'*X*B) B'*X*A, if DICO = 'D'. Specifically,
lambda(k) = WR(k) + j*WI(k), for k = 1,2,...,N.
S (output) DOUBLE PRECISION array, dimension (LDS,2*N)
If INFO = 0 or INFO = 5, the leading 2N-by-2N part of this
array contains the ordered real Schur form S of the
Hamiltonian or symplectic matrix H. That is,
(S S )
( 11 12)
S = ( ),
(0 S )
( 22)
where S , S and S are N-by-N matrices.
11 12 22
LDS INTEGER
The leading dimension of array S. LDS >= MAX(1,2*N).
U (output) DOUBLE PRECISION array, dimension (LDU,2*N)
If INFO = 0 or INFO = 5, the leading 2N-by-2N part of this
array contains the transformation matrix U which reduces
the Hamiltonian or symplectic matrix H to the ordered real
Schur form S. That is,
(U U )
( 11 12)
U = ( ),
(U U )
( 21 22)
where U , U , U and U are N-by-N matrices.
11 12 21 22
LDU INTEGER
The leading dimension of array U. LDU >= MAX(1,2*N).
Workspace
IWORK INTEGER array, dimension (2*N)
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK and DWORK(2) returns the scaling factor used
(set to 1 if SCAL = 'N'), also set if INFO = 5;
if DICO = 'D', DWORK(3) returns the reciprocal condition
number of the given matrix A.
LDWORK INTEGER
The length of the array DWORK.
LDWORK >= MAX(2,6*N) if DICO = 'C';
LDWORK >= MAX(3,6*N) if DICO = 'D'.
For optimum performance LDWORK should be larger.
BWORK LOGICAL array, dimension (2*N)
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: if matrix A is (numerically) singular in discrete-
time case;
= 2: if the Hamiltonian or symplectic matrix H cannot be
reduced to real Schur form;
= 3: if the real Schur form of the Hamiltonian or
symplectic matrix H cannot be appropriately ordered;
= 4: if the Hamiltonian or symplectic matrix H has less
than N stable eigenvalues;
= 5: if the N-th order system of linear algebraic
equations, from which the solution matrix X would
be obtained, is singular to working precision.
Method
The method used is the Schur vector approach proposed by Laub.
It is assumed that [A,B] is a stabilizable pair (where for (3) B
is any matrix such that B*B' = G with rank(B) = rank(G)), and
[E,A] is a detectable pair, where E is any matrix such that
E*E' = Q with rank(E) = rank(Q). Under these assumptions, any of
the algebraic Riccati equations (1)-(3) is known to have a unique
non-negative definite solution. See [2].
Now consider the 2N-by-2N Hamiltonian or symplectic matrix
( A -G )
H = ( ), (4)
(-Q -A'),
for continuous-time equation, and
-1 -1
( A A *G )
H = ( -1 -1 ), (5)
(Q*A A' + Q*A *G)
-1
for discrete-time equation, respectively, where G = B*R *B'.
The assumptions guarantee that H in (4) has no pure imaginary
eigenvalues, and H in (5) has no eigenvalues on the unit circle.
If Y is an N-by-N matrix then there exists an orthogonal matrix U
such that U'*Y*U is an upper quasi-triangular matrix. Moreover, U
can be chosen so that the 2-by-2 and 1-by-1 diagonal blocks
(corresponding to the complex conjugate eigenvalues and real
eigenvalues respectively) appear in any desired order. This is the
ordered real Schur form. Thus, we can find an orthogonal
similarity transformation U which puts (4) or (5) in ordered real
Schur form
U'*H*U = S = (S(1,1) S(1,2))
( 0 S(2,2))
where S(i,j) is an N-by-N matrix and the eigenvalues of S(1,1)
have negative real parts in case of (4), or moduli greater than
one in case of (5). If U is conformably partitioned into four
N-by-N blocks
U = (U(1,1) U(1,2))
(U(2,1) U(2,2))
with respect to the assumptions we then have
(a) U(1,1) is invertible and X = U(2,1)*inv(U(1,1)) solves (1),
(2), or (3) with X = X' and non-negative definite;
(b) the eigenvalues of S(1,1) (if DICO = 'C') or S(2,2) (if
DICO = 'D') are equal to the eigenvalues of optimal system
(the 'closed-loop' spectrum).
[A,B] is stabilizable if there exists a matrix F such that (A-BF)
is stable. [E,A] is detectable if [A',E'] is stabilizable.
References
[1] Laub, A.J.
A Schur Method for Solving Algebraic Riccati equations.
IEEE Trans. Auto. Contr., AC-24, pp. 913-921, 1979.
[2] Wonham, W.M.
On a matrix Riccati equation of stochastic control.
SIAM J. Contr., 6, pp. 681-697, 1968.
[3] Sima, V.
Algorithms for Linear-Quadratic Optimization.
Pure and Applied Mathematics: A Series of Monographs and
Textbooks, vol. 200, Marcel Dekker, Inc., New York, 1996.
Numerical Aspects
3 The algorithm requires 0(N ) operations.Further Comments
To obtain a stabilizing solution of the algebraic Riccati
equation for DICO = 'D', set SORT = 'U', if HINV = 'D', or set
SORT = 'S', if HINV = 'I'.
The routine can also compute the anti-stabilizing solutions of
the algebraic Riccati equations, by specifying
SORT = 'U' if DICO = 'D' and HINV = 'I', or DICO = 'C', or
SORT = 'S' if DICO = 'D' and HINV = 'D'.
Usually, the combinations HINV = 'D' and SORT = 'U', or HINV = 'I'
and SORT = 'U', will be faster then the other combinations [3].
Example
Program Text
* SB02MD EXAMPLE PROGRAM TEXT.
* Copyright (c) 2002-2017 NICONET e.V.
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX
PARAMETER ( NMAX = 20 )
INTEGER LDA, LDG, LDQ, LDS, LDU
PARAMETER ( LDA = NMAX, LDG = NMAX, LDQ = NMAX,
$ LDS = 2*NMAX, LDU = 2*NMAX )
INTEGER LIWORK
PARAMETER ( LIWORK = 2*NMAX )
INTEGER LDWORK
PARAMETER ( LDWORK = 6*NMAX )
* .. Local Scalars ..
DOUBLE PRECISION RCOND
INTEGER I, INFO, J, N
CHARACTER DICO, HINV, SCAL, SORT, UPLO
* .. Local Arrays ..
DOUBLE PRECISION A(LDA,NMAX), DWORK(LDWORK), G(LDG,NMAX),
$ Q(LDQ,NMAX), S(LDS,2*NMAX), U(LDU,2*NMAX),
$ WI(2*NMAX), WR(2*NMAX)
INTEGER IWORK(LIWORK)
LOGICAL BWORK(LIWORK)
* .. External Subroutines ..
EXTERNAL SB02MD
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, DICO, HINV, UPLO, SCAL, SORT
IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99995 ) N
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
READ ( NIN, FMT = * ) ( ( Q(I,J), J = 1,N ), I = 1,N )
READ ( NIN, FMT = * ) ( ( G(I,J), J = 1,N ), I = 1,N )
* Find the solution matrix X.
CALL SB02MD( DICO, HINV, UPLO, SCAL, SORT, N, A, LDA, G, LDG,
$ Q, LDQ, RCOND, WR, WI, S, LDS, U, LDU, IWORK,
$ DWORK, LDWORK, BWORK, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99997 ) RCOND
DO 20 I = 1, N
WRITE ( NOUT, FMT = 99996 ) ( Q(I,J), J = 1,N )
20 CONTINUE
END IF
END IF
STOP
*
99999 FORMAT (' SB02MD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from SB02MD = ',I2)
99997 FORMAT (' RCOND = ',F4.2,//' The solution matrix X is ')
99996 FORMAT (20(1X,F8.4))
99995 FORMAT (/' N is out of range.',/' N = ',I5)
END
Program Data
SB02MD EXAMPLE PROGRAM DATA 2 C D U N S 0.0 1.0 0.0 0.0 1.0 0.0 0.0 2.0 0.0 0.0 0.0 1.0Program Results
SB02MD EXAMPLE PROGRAM RESULTS RCOND = 0.31 The solution matrix X is 2.0000 1.0000 1.0000 2.0000