SB10SD

H2 optimal controller matrices for a normalized discrete-time system

Purpose

  To compute the matrices of the H2 optimal controller

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

  for the normalized discrete-time system

                | A  | B1  B2  |   | A | B |
            P = |----|---------| = |---|---|
                | C1 | D11 D12 |   | C | D |
                | C2 | D21  0  |

  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 SB10PD,

            j*Theta
  (A3) | A-e       *I  B2  | has full column rank for all
       |    C1         D12 |

       0 <= Theta < 2*Pi ,

            j*Theta
  (A4) | A-e       *I  B1  | has full row rank for all
       |    C2         D21 |

       0 <= Theta < 2*Pi .

      SUBROUTINE SB10SD( N, M, NP, NCON, NMEAS, A, LDA, B, LDB, C, LDC,
     $                   D, LDD, AK, LDAK, BK, LDBK, CK, LDCK, DK, LDDK,
     $                   X, LDX, Y, LDY, RCOND, TOL, IWORK, DWORK,
     $                   LDWORK, BWORK, INFO )
C     .. Scalar Arguments ..
      INTEGER            INFO, LDA, LDAK, LDB, LDBK, LDC, LDCK, LDD,
     $                   LDDK, LDWORK, LDX, LDY, M, N, NCON, NMEAS, NP
      DOUBLE PRECISION   TOL
C     .. Array Arguments ..
      INTEGER            IWORK( * )
      DOUBLE PRECISION   A( LDA, * ), AK( LDAK, * ), B( LDB, * ),
     $                   BK( LDBK, * ), C( LDC, * ), CK( LDCK, * ),
     $                   D( LDD, * ), DK( LDDK, * ), DWORK( * ),
     $                   RCOND( * ), X( LDX, * ), Y( LDY, * )
      LOGICAL            BWORK( * )

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).

  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 leading
          (NP-NP2)-by-(M-M2) submatrix D11 is used.

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

  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).

  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).

  RCOND   (output) DOUBLE PRECISION array, dimension (4)
          RCOND contains estimates of the reciprocal condition
          numbers of the matrices which are to be inverted and the
          reciprocal condition numbers of the Riccati equations
          which have to be solved during the computation of the
          controller. (See the description of the algorithm in [2].)
          RCOND(1) contains the reciprocal condition number of the
                   matrix Im2 + B2'*X2*B2;
          RCOND(2) contains the reciprocal condition number of the
                   matrix Ip2 + C2*Y2*C2';
          RCOND(3) contains the reciprocal condition number of the
                   X-Riccati equation;
          RCOND(4) contains the reciprocal condition number of the
                   Y-Riccati equation.

  TOL     DOUBLE PRECISION
          Tolerance used in determining the nonsingularity of the
          matrices 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 (max(M2,2*N,N*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, 14*N*N+6*N+max(14*N+23,16*N),
                           M2*(N+M2+max(3,M1)), NP2*(N+NP2+3)),
          where M1 = M - M2.
          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 matrix Im2 + B2'*X2*B2 is not positive
                definite, or it is numerically singular (with
                respect to the tolerance TOL);
          = 3:  if the Y-Riccati equation was not solved
                successfully;
          = 4:  if the matrix Ip2 + C2*Y2*C2' is not positive
                definite, or it is numerically singular (with
                respect to the tolerance TOL).

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

  [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
  matrices which are to be inverted and on the condition numbers of
  the matrix Riccati equations which are to be solved in the
  computation of the controller. (The corresponding reciprocal
  condition numbers are given in the output array RCOND.)

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