SB10VD

State feedback and output injection matrices for an H2 optimal state controller (continuous-time)

Purpose

  To compute the state feedback and the output injection
  matrices for an H2 optimal n-state controller for the system

                | A  | B1  B2  |   | A | B |
            P = |----|---------| = |---|---|
                | C1 |  0  D12 |   | C | D |
                | C2 | D21 D22 |

  where B2 has as column size the number of control inputs (NCON)
  and C2 has as row size the number of measurements (NMEAS) being
  provided to the controller.

  It is assumed that

  (A1) (A,B2) is stabilizable and (C2,A) is detectable,

  (A2) D12 is full column rank with D12 = | 0 | and D21 is
                                          | I |
       full row rank with D21 = | 0 I | as obtained by the
       SLICOT Library routine SB10UD. Matrix D is not used
       explicitly.

      SUBROUTINE SB10VD( N, M, NP, NCON, NMEAS, A, LDA, B, LDB, C, LDC,
     $                   F, LDF, H, LDH, X, LDX, Y, LDY, XYCOND, IWORK,
     $                   DWORK, LDWORK, BWORK, INFO )
C     .. Scalar Arguments ..
      INTEGER            INFO, LDA, LDB, LDC, LDF, LDH, LDWORK, LDX,
     $                   LDY, M, N, NCON, NMEAS, NP
C     .. Array Arguments ..
      LOGICAL            BWORK( * )
      INTEGER            IWORK( * )
      DOUBLE PRECISION   A( LDA, * ), B( LDB, * ), C( LDC, * ),
     $                   DWORK( * ),  F( LDF, * ), H( LDH, * ),
     $                   X( LDX, * ), XYCOND( 2 ), Y( LDY, * )

Input/Output Parameters

  N       (input) INTEGER
          The order of the system.  N >= 0.

  M       (input) INTEGER
          The column size of the matrix B.  M >= 0.

  NP      (input) INTEGER
          The row size of the matrix C.  NP >= 0.

  NCON    (input) INTEGER
          The number of control inputs (M2).  M >= NCON >= 0,
          NP-NMEAS >= NCON.

  NMEAS   (input) INTEGER
          The number of measurements (NP2).  NP >= NMEAS >= 0,
          M-NCON >= NMEAS.

  A       (input) DOUBLE PRECISION array, dimension (LDA,N)
          The leading N-by-N part of this array must contain the
          system state matrix A.

  LDA     INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).

  B       (input) DOUBLE PRECISION array, dimension (LDB,M)
          The leading N-by-M part of this array must contain the
          system input matrix B.

  LDB     INTEGER
          The leading dimension of the array B.  LDB >= max(1,N).

  C       (input) DOUBLE PRECISION array, dimension (LDC,N)
          The leading NP-by-N part of this array must contain the
          system output matrix C.

  LDC     INTEGER
          The leading dimension of the array C.  LDC >= max(1,NP).

  F       (output) DOUBLE PRECISION array, dimension (LDF,N)
          The leading NCON-by-N part of this array contains the
          state feedback matrix F.

  LDF     INTEGER
          The leading dimension of the array F.  LDF >= max(1,NCON).

  H       (output) DOUBLE PRECISION array, dimension (LDH,NMEAS)
          The leading N-by-NMEAS part of this array contains the
          output injection matrix H.

  LDH     INTEGER
          The leading dimension of the array H.  LDH >= max(1,N).

  X       (output) DOUBLE PRECISION array, dimension (LDX,N)
          The leading N-by-N part of this array contains the matrix
          X, solution of the X-Riccati equation.

  LDX     INTEGER
          The leading dimension of the array X.  LDX >= max(1,N).

  Y       (output) DOUBLE PRECISION array, dimension (LDY,N)
          The leading N-by-N part of this array contains the matrix
          Y, solution of the Y-Riccati equation.

  LDY     INTEGER
          The leading dimension of the array Y.  LDY >= max(1,N).

  XYCOND  (output) DOUBLE PRECISION array, dimension (2)
          XYCOND(1) contains an estimate of the reciprocal condition
                    number of the X-Riccati equation;
          XYCOND(2) contains an estimate of the reciprocal condition
                    number of the Y-Riccati equation.

  IWORK   INTEGER array, dimension (max(2*N,N*N))

  DWORK   DOUBLE PRECISION array, dimension (LDWORK)
          On exit, if INFO = 0, DWORK(1) contains the optimal
          LDWORK.

  LDWORK  INTEGER
          The dimension of the array DWORK.
          LDWORK >= 13*N*N + 12*N + 5.
          For good performance, LDWORK must generally be larger.

  BWORK   LOGICAL array, dimension (2*N)

  INFO    INTEGER
          = 0:  successful exit;
          < 0:  if INFO = -i, the i-th argument had an illegal
                value;
          = 1:  if the X-Riccati equation was not solved
                successfully;
          = 2:  if the Y-Riccati equation was not solved
                successfully.

  The routine implements the formulas given in [1], [2]. The X-
  and Y-Riccati equations are solved with condition and accuracy
  estimates [3].

  [1] Zhou, K., Doyle, J.C., and Glover, K.
      Robust and Optimal Control.
      Prentice-Hall, Upper Saddle River, NJ, 1996.

  [2] Balas, G.J., Doyle, J.C., Glover, K., Packard, A., and
      Smith, R.
      mu-Analysis and Synthesis Toolbox.
      The MathWorks Inc., Natick, Mass., 1995.

  [3] Petkov, P.Hr., Konstantinov, M.M., and Mehrmann, V.
      DGRSVX and DMSRIC: Fortan 77 subroutines for solving
      continuous-time matrix algebraic Riccati equations with
      condition and accuracy estimates.
      Preprint SFB393/98-16, Fak. f. Mathematik, Tech. Univ.
      Chemnitz, May 1998.

  The precision of the solution of the matrix Riccati equations
  can be controlled by the values of the condition numbers
  XYCOND(1) and XYCOND(2) of these equations.

  The Riccati equations are solved by the Schur approach
  implementing condition and accuracy estimates.

Program Text

  None

  None

  None