SB10RD - SLICOT Library Routine Documentation

## SB10RD

### H-infinity (sub)optimal controller matrices using state feedback and output injection matrices (continuous-time)

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

Purpose

      To compute the matrices of an H-infinity (sub)optimal controller

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

      from the state feedback matrix F and output injection matrix H as
      determined by the SLICOT Library routine SB10QD.

Specification

          SUBROUTINE SB10RD( N, M, NP, NCON, NMEAS, GAMMA, A, LDA, B, LDB,
         $                   C, LDC, D, LDD, F, LDF, H, LDH, TU, LDTU, TY,
         $                   LDTY, X, LDX, Y, LDY, AK, LDAK, BK, LDBK, CK,
         $                   LDCK, DK, LDDK, IWORK, DWORK, LDWORK, INFO )
    C     .. Scalar Arguments ..
          INTEGER            INFO, LDA, LDAK, LDB, LDBK, LDC, LDCK, LDD,
         $                   LDDK, LDF, LDH, LDTU, LDTY, LDWORK, LDX, LDY,
         $                   M, N, NCON, NMEAS, NP
          DOUBLE PRECISION   GAMMA
    C     .. Array Arguments ..
          INTEGER            IWORK( * )
          DOUBLE PRECISION   A( LDA, * ), AK( LDAK, * ), B( LDB, * ),
         $                   BK( LDBK, * ), C( LDC, * ), CK( LDCK, * ),
         $                   D( LDD, * ), DK( LDDK, * ), DWORK( * ),
         $                   F( LDF, * ), H( LDH, * ), TU( LDTU, * ),
         $                   TY( LDTY, * ), X( LDX, * ), Y( LDY, * )

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.

      GAMMA   (input) DOUBLE PRECISION
              The value of gamma. It is assumed that gamma is
              sufficiently large so that the controller is admissible.
              GAMMA >= 0.

      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).

      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.

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

      F       (input) DOUBLE PRECISION array, dimension (LDF,N)
              The leading M-by-N part of this array must contain the
              state feedback matrix F.

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

      H       (input) DOUBLE PRECISION array, dimension (LDH,NP)
              The leading N-by-NP part of this array must contain the
              output injection matrix H.

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

      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).

      X       (input) DOUBLE PRECISION array, dimension (LDX,N)
              The leading N-by-N part of this array must contain the
              matrix X, solution of the X-Riccati equation, as obtained
              by the SLICOT Library routine SB10QD.

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

      Y       (input) DOUBLE PRECISION array, dimension (LDY,N)
              The leading N-by-N part of this array must contain the
              matrix Y, solution of the Y-Riccati equation, as obtained
              by the SLICOT Library routine SB10QD.

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

      AK      (output) DOUBLE PRECISION array, dimension (LDAK,N)
              The leading N-by-N part of this array contains the
              controller state matrix AK.

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

      BK      (output) DOUBLE PRECISION array, dimension (LDBK,NMEAS)
              The leading N-by-NMEAS part of this array contains the
              controller input matrix BK.

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

      CK      (output) DOUBLE PRECISION array, dimension (LDCK,N)
              The leading NCON-by-N part of this array contains the
              controller output matrix CK.

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

      DK      (output) DOUBLE PRECISION array, dimension (LDDK,NMEAS)
              The leading NCON-by-NMEAS part of this array contains the
              controller input/output matrix DK.

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

Workspace

      IWORK   INTEGER array, dimension (LIWORK), where
              LIWORK = max(2*(max(NP,M)-M2-NP2,M2,N),NP2)

      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, M2*NP2 + NP2*NP2 + M2*M2 +
                            max(D1*D1 + max(2*D1, (D1+D2)*NP2),
                                D2*D2 + max(2*D2, D2*M2), 3*N,
                                N*(2*NP2 + M2) +
                                max(2*N*M2, M2*NP2 +
                                            max(M2*M2+3*M2, NP2*(2*NP2+
                                                   M2+max(NP2,N))))))
              where D1 = NP1 - M2, D2 = M1 - NP2,
                   NP1 = NP - NP2, M1 = M - M2.
              For good performance, LDWORK must generally be larger.
              Denoting Q = max(M1,M2,NP1,NP2), an upper bound is
              max( 1, Q*(3*Q + 3*N + max(2*N, 4*Q + max(Q, N)))).

Error Indicator

      INFO    INTEGER
              = 0:  successful exit;
              < 0:  if INFO = -i, the i-th argument had an illegal
                    value;
              = 1:  if the controller is not admissible (too small value
                    of gamma);
              = 2:  if the determinant of Im2 + Tu*D11HAT*Ty*D22 is zero.

Method

      The routine implements the Glover's and Doyle's formulas [1],[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 accuracy of the result depends on the condition numbers of the
      input and output transformations.

Further Comments

      None

Example

Program Text

      None

Program Data

      None

Program Results

      None

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