SB10PD

Normalization of a system for H-infinity controller design

Purpose

  To reduce the matrices D12 and D21 of the linear time-invariant
  system

                | A  | B1  B2  |   | A | B |
            P = |----|---------| = |---|---|
                | C1 | D11 D12 |   | C | D |
                | C2 | D21 D22 |

  to unit diagonal form, to transform the matrices B, C, and D11 to
  satisfy the formulas in the computation of an H2 and H-infinity
  (sub)optimal controllers and to check the rank conditions.

      SUBROUTINE SB10PD( N, M, NP, NCON, NMEAS, A, LDA, B, LDB, C, LDC,
     $                   D, LDD, TU, LDTU, TY, LDTY, RCOND, TOL, DWORK,
     $                   LDWORK, INFO )
C     .. Scalar Arguments ..
      INTEGER            INFO, LDA, LDB, LDC, LDD, LDTU, LDTY, LDWORK,
     $                   M, N, NCON, NMEAS, NP
      DOUBLE PRECISION   TOL
C     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), B( LDB, * ), C( LDC, * ),
     $                   D( LDD, * ), DWORK( * ), RCOND( 2 ),
     $                   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.

  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/output) DOUBLE PRECISION array, dimension (LDB,M)
          On entry, the leading N-by-M part of this array must
          contain the system input matrix B.
          On exit, the leading N-by-M part of this array contains
          the transformed system input matrix B.

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

  C       (input/output) DOUBLE PRECISION array, dimension (LDC,N)
          On entry, the leading NP-by-N part of this array must
          contain the system output matrix C.
          On exit, the leading NP-by-N part of this array contains
          the transformed system output matrix C.

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

  D       (input/output) DOUBLE PRECISION array, dimension (LDD,M)
          On entry, the leading NP-by-M part of this array must
          contain the system input/output matrix D. The
          NMEAS-by-NCON trailing submatrix D22 is not referenced.
          On exit, the leading (NP-NMEAS)-by-(M-NCON) part of this
          array contains the transformed submatrix D11.
          The transformed submatrices D12 = [ 0  Im2 ]' and
          D21 = [ 0  Inp2 ] are not stored. The corresponding part
          of this array contains no useful information.

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

  TU      (output) DOUBLE PRECISION array, dimension (LDTU,M2)
          The leading M2-by-M2 part of this array contains the
          control transformation matrix TU.

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

  TY      (output) DOUBLE PRECISION array, dimension (LDTY,NP2)
          The leading NP2-by-NP2 part of this array contains the
          measurement transformation matrix TY.

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

  RCOND   (output) DOUBLE PRECISION array, dimension (2)
          RCOND(1) contains the reciprocal condition number of the
                   control transformation matrix TU;
          RCOND(2) contains the reciprocal condition number of the
                   measurement transformation matrix TY.
          RCOND is set even if INFO = 3 or INFO = 4; if INFO = 3,
          then RCOND(2) was not computed, but it is set to 0.

  TOL     DOUBLE PRECISION
          Tolerance used for controlling the accuracy of the applied
          transformations. Transformation matrices TU and TY whose
          reciprocal condition numbers are less than TOL are not
          allowed. If TOL <= 0, then a default value equal to
          sqrt(EPS) is used, where EPS is the relative machine
          precision.

  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 >= MAX(1,LW1,LW2,LW3,LW4), where
          LW1 = (N+NP1+1)*(N+M2) + MAX(3*(N+M2)+N+NP1,5*(N+M2)),
          LW2 = (N+NP2)*(N+M1+1) + MAX(3*(N+NP2)+N+M1,5*(N+NP2)),
          LW3 = M2 + NP1*NP1 + MAX(NP1*MAX(N,M1),3*M2+NP1,5*M2),
          LW4 = NP2 + M1*M1 + MAX(MAX(N,NP1)*M1,3*NP2+M1,5*NP2),
          with M1 = M - M2 and NP1 = NP - NP2.
          For good performance, LDWORK must generally be larger.
          Denoting Q = MAX(M1,M2,NP1,NP2), an upper bound is
          MAX(1,(N+Q)*(N+Q+6),Q*(Q+MAX(N,Q,5)+1).

  INFO    INTEGER
          = 0:  successful exit;
          < 0:  if INFO = -i, the i-th argument had an illegal
                value;
          = 1:  if the matrix | A   B2  | had not full column rank
                              | C1  D12 |
                in respect to the tolerance EPS;
          = 2:  if the matrix | A   B1  | had not full row rank in
                              | C2  D21 |
                respect to the tolerance EPS;
          = 3:  if the matrix D12 had not full column rank in
                respect to the tolerance TOL;
          = 4:  if the matrix D21 had not full row rank in respect
                to the tolerance TOL;
          = 5:  if the singular value decomposition (SVD) algorithm
                did not converge (when computing the SVD of one of
                the matrices |A   B2 |, |A   B1 |, D12 or D21).
                             |C1  D12|  |C2  D21|

  The routine performs the transformations described in [2].

  [1] Glover, K. and Doyle, J.C.
      State-space formulae for all stabilizing controllers that
      satisfy an Hinf norm bound and relations to risk sensitivity.
      Systems and Control Letters, vol. 11, pp. 167-172, 1988.

  [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.

  The precision of the transformations can be controlled by the
  condition numbers of the matrices TU and TY as given by the
  values of RCOND(1) and RCOND(2), respectively. An error return
  with INFO = 3 or INFO = 4 will be obtained if the condition
  number of TU or TY, respectively, would exceed 1/TOL.

  None

Program Text

  None

  None

  None