SB10QD

State feedback and output injection matrices for an H-infinity (sub)optimal state controller (continuous-time)

Purpose

  To compute the state feedback and the output injection
  matrices for an H-infinity (sub)optimal n-state controller,
  using Glover's and Doyle's 1988 formulas, for the system

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

  and for a given value of gamma, 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
       subroutine SB10PD,

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

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

      SUBROUTINE SB10QD( N, M, NP, NCON, NMEAS, GAMMA, A, LDA, B, LDB,
     $                   C, LDC, D, LDD, F, LDF, H, LDH, X, LDX, Y, LDY,
     $                   XYCOND, IWORK, DWORK, LDWORK, BWORK, INFO )
C     .. Scalar Arguments ..
      INTEGER            INFO, LDA, LDB, LDC, LDD, LDF, LDH, LDWORK,
     $                   LDX, LDY, M, N, NCON, NMEAS, NP
      DOUBLE PRECISION   GAMMA
C     .. Array Arguments ..
      INTEGER            IWORK( * )
      DOUBLE PRECISION   A( LDA, * ), B( LDB, * ), C( LDC, * ),
     $                   D( LDD, * ), DWORK( * ),  F( LDF, * ),
     $                   H( LDH, * ), X( LDX, * ), XYCOND( 2 ),
     $                   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.

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

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

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

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

  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).

  XYCOND  (output) DOUBLE PRECISION array, dimension (2)
          XYCOND(1) contains an estimate of the reciprocal condition
                    number of the X-Riccati equation;
          XYCOND(2) contains an estimate of the reciprocal condition
                    number of the Y-Riccati equation.

  IWORK   INTEGER array, dimension
          (max(2*max(N,M-NCON,NP-NMEAS),N*N))

  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,M*M + max(2*M1,3*N*N +
                                    max(N*M,10*N*N+12*N+5)),
                        NP*NP + max(2*NP1,3*N*N +
                                    max(N*NP,10*N*N+12*N+5))),
          where 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(1,4*Q*Q+max(2*Q,3*N*N + max(2*N*Q,10*N*N+12*N+5))).

  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 controller is not admissible (too small value
                of gamma);
          = 2:  if the X-Riccati equation was not solved
                successfully (the controller is not admissible or
                there are numerical difficulties);
          = 3:  if the Y-Riccati equation was not solved
                successfully (the controller is not admissible or
                there are numerical difficulties).

  The routine implements the Glover's and Doyle's formulas [1],[2]
  modified as described in [3]. The X- and Y-Riccati equations
  are solved with condition and accuracy estimates [4].

  [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.

  [3] Petkov, P.Hr., Gu, D.W., and Konstantinov, M.M.
      Fortran 77 routines for Hinf and H2 design of continuous-time
      linear control systems.
      Rep. 98-14, Department of Engineering, Leicester University,
      Leicester, U.K., 1998.

  [4] Petkov, P.Hr., Konstantinov, M.M., and Mehrmann, V.
      DGRSVX and DMSRIC: Fortan 77 subroutines for solving
      continuous-time matrix algebraic Riccati equations with
      condition and accuracy estimates.
      Preprint SFB393/98-16, Fak. f. Mathematik, Tech. Univ.
      Chemnitz, May 1998.

  The precision of the solution of the matrix Riccati equations
  can be controlled by the values of the condition numbers
  XYCOND(1) and XYCOND(2) of these equations.

  The Riccati equations are solved by the Schur approach
  implementing condition and accuracy estimates.

Program Text

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