## SB10PD

### Normalization of a system for H-infinity controller design

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

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.

Specification

          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, * )

Arguments

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.

Tolerances

      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.

Workspace

      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).

Error Indicator

      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|

Method

      The routine performs the transformations described in [2].

References

      [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.

Numerical Aspects

      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.

Further Comments

      None

Example

Program Text

      None

Program Data

      None

Program Results

      None

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