SB10TD

H2 optimal controller matrices for a discrete-time system

Purpose

  To compute the matrices of the H2 optimal discrete-time controller

           | AK | BK |
       K = |----|----|,
           | CK | DK |

  from the matrices of the controller for the normalized system,
  as determined by the SLICOT Library routine SB10SD.

      SUBROUTINE SB10TD( N, M, NP, NCON, NMEAS, D, LDD, TU, LDTU, TY,
     $                   LDTY, AK, LDAK, BK, LDBK, CK, LDCK, DK, LDDK,
     $                   RCOND, TOL, IWORK, DWORK, LDWORK, INFO )
C     .. Scalar Arguments ..
      INTEGER            INFO, LDAK, LDBK, LDCK, LDD, LDDK, LDTU, LDTY,
     $                   LDWORK, M, N, NCON, NMEAS, NP
      DOUBLE PRECISION   RCOND, TOL
C     .. Array Arguments ..
      INTEGER            IWORK( * )
      DOUBLE PRECISION   AK( LDAK, * ), BK( LDBK, * ), CK( LDCK, * ),
     $                   D( LDD, * ), DK( LDDK, * ), DWORK( * ),
     $                   TU( LDTU, * ), TY( LDTY, * )

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.

  D       (input) DOUBLE PRECISION array, dimension (LDD,M)
          The leading NP-by-M part of this array must contain the
          system input/output matrix D. Only the trailing
          NMEAS-by-NCON submatrix D22 is used.

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

  TU      (input) DOUBLE PRECISION array, dimension (LDTU,M2)
          The leading M2-by-M2 part of this array must contain the
          control transformation matrix TU, as obtained by the
          SLICOT Library routine SB10PD.

  LDTU    INTEGER
          The leading dimension of the array TU.  LDTU >= max(1,M2).

  TY      (input) DOUBLE PRECISION array, dimension (LDTY,NP2)
          The leading NP2-by-NP2 part of this array must contain the
          measurement transformation matrix TY, as obtained by the
          SLICOT Library routine SB10PD.

  LDTY    INTEGER
          The leading dimension of the array TY.
          LDTY >= max(1,NP2).

  AK      (input/output) DOUBLE PRECISION array, dimension (LDAK,N)
          On entry, the leading N-by-N part of this array must
          contain controller state matrix for the normalized system
          as obtained by the SLICOT Library routine SB10SD.
          On exit, the leading N-by-N part of this array contains
          controller state matrix AK.

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

  BK      (input/output) DOUBLE PRECISION array, dimension
          (LDBK,NMEAS)
          On entry, the leading N-by-NMEAS part of this array must
          contain controller input matrix for the normalized system
          as obtained by the SLICOT Library routine SB10SD.
          On exit, the leading N-by-NMEAS part of this array
          contains controller input matrix BK.

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

  CK      (input/output) DOUBLE PRECISION array, dimension (LDCK,N)
          On entry, the leading NCON-by-N part of this array must
          contain controller output matrix for the normalized
          system as obtained by the SLICOT Library routine SB10SD.
          On exit, the leading NCON-by-N part of this array contains
          controller output matrix CK.

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

  DK      (input/output) DOUBLE PRECISION array, dimension
          (LDDK,NMEAS)
          On entry, the leading NCON-by-NMEAS part of this array
          must contain controller matrix DK for the normalized
          system as obtained by the SLICOT Library routine SB10SD.
          On exit, the leading NCON-by-NMEAS part of this array
          contains controller input/output matrix DK.

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

  RCOND   (output) DOUBLE PRECISION
          RCOND contains an estimate of the reciprocal condition
          number of the matrix Im2 + DKHAT*D22 which must be
          inverted in the computation of the controller.

  TOL     DOUBLE PRECISION
          Tolerance used in determining the nonsingularity of the
          matrix which must be inverted. If TOL <= 0, then a default
          value equal to sqrt(EPS) is used, where EPS is the
          relative machine precision.

  IWORK   INTEGER array, dimension (2*M2)

  DWORK   DOUBLE PRECISION array, dimension (LDWORK)

  LDWORK  INTEGER
          The dimension of the array DWORK.
          LDWORK >= max(N*M2,N*NP2,M2*NP2,M2*M2+4*M2).

  INFO    INTEGER
          = 0:  successful exit;
          < 0:  if INFO = -i, the i-th argument had an illegal
                value;
          = 1:  if the matrix Im2 + DKHAT*D22 is singular, or the
                estimated condition number is larger than or equal
                to 1/TOL.

  The routine implements the formulas given in [1].

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

  [2] Petkov, P.Hr., Gu, D.W., and Konstantinov, M.M.
      Fortran 77 routines for Hinf and H2 design of linear
      discrete-time control systems.
      Report 99-8, Department of Engineering, Leicester University,
      April 1999.

  The accuracy of the result depends on the condition numbers of the
  input and output transformations and of the matrix Im2 +
  DKHAT*D22.

  None

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