## SB02MD

### Solution of continuous- or discrete-time algebraic Riccati equations (Schur vectors method)

[Specification] [Arguments] [Method] [References] [Comments] [Example]

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 strictly lower triangular part (if UPLO = 'U') or
              strictly 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 strictly lower triangular part (if UPLO = 'U') or
              strictly 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.
    *
    *     .. 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   1.0

Program Results

     SB02MD EXAMPLE PROGRAM RESULTS

     RCOND = 0.31

     The solution matrix X is
       2.0000   1.0000
       1.0000   2.0000

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