## SB10AD

### H-infinity optimal controller using modified Glover's and Doyle's formulas (continuous-time)

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

Purpose

      To compute the matrices of an H-infinity optimal n-state
      controller

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

      using modified 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 the estimated minimal possible value of gamma with respect
      to GTOL, 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, and then to compute the matrices
      of the closed-loop system

               | AC | BC |
           G = |----|----|,
               | CC | DC |

      if the stabilizing controller exists.

      It is assumed that

      (A1) (A,B2) is stabilizable and (C2,A) is detectable,

      (A2) D12 is full column rank and D21 is full row rank,

      (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 |

Specification

          SUBROUTINE SB10AD( JOB, N, M, NP, NCON, NMEAS, GAMMA, A, LDA,
         $                   B, LDB, C, LDC, D, LDD, AK, LDAK, BK, LDBK, CK,
         $                   LDCK, DK, LDDK, AC, LDAC, BC, LDBC, CC, LDCC,
         $                   DC, LDDC, RCOND, GTOL, ACTOL, IWORK, LIWORK,
         $                   DWORK, LDWORK, BWORK, LBWORK, INFO )
    C     .. Scalar Arguments ..
          INTEGER            INFO, JOB, LBWORK, LDA, LDAC, LDAK, LDB, LDBC,
         $                   LDBK, LDC, LDCC, LDCK, LDD, LDDC, LDDK, LDWORK,
         $                   LIWORK, M, N, NCON, NMEAS, NP
          DOUBLE PRECISION   ACTOL, GAMMA, GTOL
    C     .. Array Arguments ..
          LOGICAL            BWORK( * )
          INTEGER            IWORK( * )
          DOUBLE PRECISION   A( LDA, * ), AC( LDAC, * ), AK( LDAK, * ),
         $                   B( LDB, * ), BC( LDBC, * ), BK( LDBK, * ),
         $                   C( LDC, * ), CC( LDCC, * ), CK( LDCK, * ),
         $                   D( LDD, * ), DC( LDDC, * ), DK( LDDK, * ),
         $                   DWORK( * ), RCOND( 4 )

Arguments

Input/Output Parameters

      JOB     (input) INTEGER
              Indicates the strategy for reducing the GAMMA value, as
              follows:
              = 1: Use bisection method for decreasing GAMMA from GAMMA
                   to GAMMAMIN until the closed-loop system leaves
                   stability.
              = 2: Scan from GAMMA to 0 trying to find the minimal GAMMA
                   for which the closed-loop system retains stability.
              = 3: First bisection, then scanning.
              = 4: Find suboptimal controller only.

      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/output) DOUBLE PRECISION
              The initial value of gamma on input. It is assumed that
              gamma is sufficiently large so that the controller is
              admissible. GAMMA >= 0.
              On output it contains the minimal estimated gamma.

      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).

      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).

      AC      (output) DOUBLE PRECISION array, dimension (LDAC,2*N)
              The leading 2*N-by-2*N part of this array contains the
              closed-loop system state matrix AC.

      LDAC    INTEGER
              The leading dimension of the array AC.
              LDAC >= max(1,2*N).

      BC      (output) DOUBLE PRECISION array, dimension (LDBC,M-NCON)
              The leading 2*N-by-(M-NCON) part of this array contains
              the closed-loop system input matrix BC.

      LDBC    INTEGER
              The leading dimension of the array BC.
              LDBC >= max(1,2*N).

      CC      (output) DOUBLE PRECISION array, dimension (LDCC,2*N)
              The leading (NP-NMEAS)-by-2*N part of this array contains
              the closed-loop system output matrix CC.

      LDCC    INTEGER
              The leading dimension of the array CC.
              LDCC >= max(1,NP-NMEAS).

      DC      (output) DOUBLE PRECISION array, dimension (LDDC,M-NCON)
              The leading (NP-NMEAS)-by-(M-NCON) part of this array
              contains the closed-loop system input/output matrix DC.

      LDDC    INTEGER
              The leading dimension of the array DC.
              LDDC >= max(1,NP-NMEAS).

      RCOND   (output) DOUBLE PRECISION array, dimension (4)
                       For the last successful step:
              RCOND(1) contains the reciprocal condition number of the
                       control transformation matrix;
              RCOND(2) contains the reciprocal condition number of the
                       measurement transformation matrix;
              RCOND(3) contains an estimate of the reciprocal condition
                       number of the X-Riccati equation;
              RCOND(4) contains an estimate of the reciprocal condition
                       number of the Y-Riccati equation.

Tolerances

      GTOL    DOUBLE PRECISION
              Tolerance used for controlling the accuracy of GAMMA
              and its distance to the estimated minimal possible
              value of GAMMA.
              If GTOL <= 0, then a default value equal to sqrt(EPS)
              is used, where EPS is the relative machine precision.

      ACTOL   DOUBLE PRECISION
              Upper bound for the poles of the closed-loop system
              used for determining if it is stable.
              ACTOL <= 0 for stable systems.

Workspace

      IWORK   INTEGER array, dimension (LIWORK)

      LIWORK  INTEGER
              The dimension of the array IWORK.
              LIWORK >= max(2*max(N,M-NCON,NP-NMEAS,NCON,NMEAS),N*N)

      DWORK   DOUBLE PRECISION array, dimension (LDWORK)
              On exit, if INFO = 0, DWORK(1) contains the optimal
              value of LDWORK.

      LDWORK  INTEGER
              The dimension of the array DWORK.
              LDWORK >= LW1 + max(1,LW2,LW3,LW4,LW5 + MAX(LW6,LW7)),
              where
              LW1 = N*M + NP*N + NP*M + M2*M2 + NP2*NP2;
              LW2 = max( ( N + NP1 + 1 )*( N + M2 ) +
                           max( 3*( N + M2 ) + N + NP1, 5*( N + M2 ) ),
                         ( N + NP2 )*( N + M1 + 1 ) +
                           max( 3*( N + NP2 ) + N + M1, 5*( N + NP2 ) ),
                         M2 + NP1*NP1 + max( NP1*max( N, M1 ),
                                             3*M2 + NP1, 5*M2 ),
                         NP2 + M1*M1 +  max( max( N, NP1 )*M1,
                                             3*NP2 + M1, 5*NP2 ) );
              LW3 = max( ND1*M1 + max( 4*min( ND1, M1 ) + max( ND1,M1 ),
                                       6*min( ND1, M1 ) ),
                         NP1*ND2 + max( 4*min( NP1, ND2 ) +
                                                         max( NP1,ND2 ),
                                        6*min( NP1, ND2 ) ) );
              LW4 = 2*M*M + NP*NP + 2*M*N + M*NP + 2*N*NP;
              LW5 = 2*N*N + M*N + N*NP;
              LW6 = max( 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 ) ));
              LW7 = M2*NP2 + NP2*NP2 + M2*M2 +
                    max( ND1*ND1 + max( 2*ND1, ( ND1 + ND2 )*NP2 ),
                         ND2*ND2 + max( 2*ND2, ND2*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 ) ) ) ) );
              M1  = M   - M2, NP1 = NP - NP2,
              ND1 = NP1 - M2, ND2 = M1 - NP2.
              For good performance, LDWORK must generally be larger.

      BWORK   LOGICAL array, dimension (LBWORK)

      LBWORK  INTEGER
              The dimension of the array BWORK.  LBWORK >= 2*N.

Error Indicator

      INFO    INTEGER
              = 0:  successful exit;
              < 0:  if INFO = -i, the i-th argument had an illegal
                    value;
              = 1:  if the matrix | A-j*omega*I  B2  | had not full
                                  |    C1        D12 |
                    column rank in respect to the tolerance EPS;
              = 2:  if the matrix | A-j*omega*I  B1  |  had not full row
                                  |    C2        D21 |
                    rank in respect to the tolerance EPS;
              = 3:  if the matrix D12 had not full column rank in
                    respect to the tolerance SQRT(EPS);
              = 4:  if the matrix D21 had not full row rank in respect
                    to the tolerance SQRT(EPS);
              = 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|
              = 6:  if the controller is not admissible (too small value
                    of gamma);
              = 7:  if the X-Riccati equation was not solved
                    successfully (the controller is not admissible or
                    there are numerical difficulties);
              = 8:  if the Y-Riccati equation was not solved
                    successfully (the controller is not admissible or
                    there are numerical difficulties);
              = 9:  if the determinant of Im2 + Tu*D11HAT*Ty*D22 is
                    zero [3];
              = 10: if there are numerical problems when estimating
                    singular values of D1111, D1112, D1111', D1121';
              = 11: if the matrices Inp2 - D22*DK or Im2 - DK*D22
                    are singular to working precision;
              = 12: if a stabilizing controller cannot be found.

Method

      The routine implements the Glover's and Doyle's 1988 formulas [1],
      [2], modified to improve the efficiency as described in [3].

      JOB = 1: It tries with a decreasing value of GAMMA, starting with
      the given, and with the newly obtained controller estimates of the
      closed-loop system. If it is stable, (i.e., max(eig(AC)) < ACTOL)
      the iterations can be continued until the given tolerance between
      GAMMA and the estimated GAMMAMIN is reached. Otherwise, in the
      next step GAMMA is increased. The step in the all next iterations
      is step = step/2. The closed-loop system is obtained by the
      formulas given in [2].

      JOB = 2: The same as for JOB = 1, but with non-varying step till
      GAMMA = 0, step = max(0.1, GTOL).

      JOB = 3: Combines the JOB = 1 and JOB = 2 cases for a quicker
      procedure.

      JOB = 4: Suboptimal controller for current GAMMA only.

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, MA, 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.

Numerical Aspects

      The accuracy of the result depends on the condition numbers of the
      input and output transformations and on the condition numbers of
      the two Riccati equations, as given by the values of RCOND(1),
      RCOND(2), RCOND(3) and RCOND(4), respectively.
      This approach by estimating the closed-loop system and checking
      its poles seems to be reliable.

Further Comments

      None

Example

Program Text

      None

Program Data

      None

Program Results

      None

Return to index