SB16AY

Cholesky factors of the frequency-weighted controllability and observability Grammians for controller reduction

Purpose

  To compute for given state-space representations (A,B,C,D) and
  (Ac,Bc,Cc,Dc) of the transfer-function matrices of the
  open-loop system G and feedback controller K, respectively,
  the Cholesky factors of the frequency-weighted
  controllability and observability Grammians corresponding
  to a frequency-weighted model reduction problem.
  The controller must stabilize the closed-loop system.
  The state matrix Ac must be in a block-diagonal real Schur form
  Ac = diag(Ac1,Ac2), where Ac1 contains the unstable eigenvalues
  of Ac and Ac2 contains the stable eigenvalues of Ac.

      SUBROUTINE SB16AY( DICO, JOBC, JOBO, WEIGHT, N, M, P, NC, NCS,
     $                   A, LDA, B, LDB, C, LDC, D, LDD,
     $                   AC, LDAC, BC, LDBC, CC, LDCC, DC, LDDC,
     $                   SCALEC, SCALEO, S, LDS, R, LDR,
     $                   IWORK, DWORK, LDWORK, INFO )
C     .. Scalar Arguments ..
      CHARACTER        DICO, JOBC, JOBO, WEIGHT
      INTEGER          INFO, LDA, LDAC, LDB, LDBC, LDC, LDCC, LDD, LDDC,
     $                 LDR, LDS, LDWORK, M, N, NC, NCS, P
      DOUBLE PRECISION SCALEC, SCALEO
C     .. Array Arguments ..
      INTEGER          IWORK(*)
      DOUBLE PRECISION A(LDA,*), AC(LDAC,*), B(LDB,*), BC(LDBC,*),
     $                 C(LDC,*), CC(LDCC,*), D(LDD,*), DC(LDDC,*),
     $                 DWORK(*), R(LDR,*),   S(LDS,*)

Mode Parameters

  DICO    CHARACTER*1
          Specifies the type of the systems as follows:
          = 'C':  G and K are continuous-time systems;
          = 'D':  G and K are discrete-time systems.

  JOBC    CHARACTER*1
          Specifies the choice of frequency-weighted controllability
          Grammian as follows:
          = 'S': choice corresponding to standard Enns' method [1];
          = 'E': choice corresponding to the stability enhanced
                 modified Enns' method of [2].

  JOBO    CHARACTER*1
          Specifies the choice of frequency-weighted observability
          Grammian as follows:
          = 'S': choice corresponding to standard Enns' method [1];
          = 'E': choice corresponding to the stability enhanced
                 modified combination method of [2].

  WEIGHT  CHARACTER*1
          Specifies the type of frequency-weighting, as follows:
          = 'N':  no weightings are used (V = I, W = I);
          = 'O':  stability enforcing left (output) weighting
                            -1
                  V = (I-G*K) *G is used (W = I);
          = 'I':  stability enforcing right (input) weighting
                            -1
                  W = (I-G*K) *G is used (V = I);
          = 'P':  stability and performance enforcing weightings
                            -1                -1
                  V = (I-G*K) *G ,  W = (I-G*K)  are used.

  N       (input) INTEGER
          The order of the open-loop system state-space
          representation, i.e., the order of the matrix A.  N >= 0.

  M       (input) INTEGER
          The number of system inputs.  M >= 0.

  P       (input) INTEGER
          The number of system outputs.  P >= 0.

  NC      (input) INTEGER
          The order of the controller state-space representation,
          i.e., the order of the matrix AC.  NC >= 0.

  NCS     (input) INTEGER
          The dimension of the stable part of the controller, i.e.,
          the order of matrix Ac2.  NC >= NCS >= 0.

  A       (input) DOUBLE PRECISION array, dimension (LDA,N)
          The leading N-by-N part of this array must contain the
          state matrix A of the system with the transfer-function
          matrix G.

  LDA     INTEGER
          The leading dimension of 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
          input/state matrix B.

  LDB     INTEGER
          The leading dimension of array B.  LDB >= MAX(1,N).

  C       (input) DOUBLE PRECISION array, dimension (LDC,N)
          The leading P-by-N part of this array must contain the
          state/output matrix C.

  LDC     INTEGER
          The leading dimension of array C.  LDC >= MAX(1,P).

  D       (input) DOUBLE PRECISION array, dimension (LDD,M)
          The leading P-by-M part of this array must contain the
          input/output matrix D of the open-loop system.

  LDD     INTEGER
          The leading dimension of array D.  LDD >= MAX(1,P).

  AC      (input) DOUBLE PRECISION array, dimension (LDAC,NC)
          The leading NC-by-NC part of this array must contain
          the state dynamics matrix Ac of the controller in a
          block diagonal real Schur form Ac = diag(Ac1,Ac2), where
          Ac1 is (NC-NCS)-by-(NC-NCS) and contains the unstable
          eigenvalues of Ac, and Ac2 is NCS-by-NCS and contains
          the stable eigenvalues of Ac.

  LDAC    INTEGER
          The leading dimension of array AC.  LDAC >= MAX(1,NC).

  BC      (input) DOUBLE PRECISION array, dimension (LDBC,P)
          The leading NC-by-P part of this array must contain
          the input/state matrix Bc of the controller.

  LDBC    INTEGER
          The leading dimension of array BC.  LDBC >= MAX(1,NC).

  CC      (input) DOUBLE PRECISION array, dimension (LDCC,NC)
          The leading M-by-NC part of this array must contain
          the state/output matrix Cc of the controller.

  LDCC    INTEGER
          The leading dimension of array CC.  LDCC >= MAX(1,M).

  DC      (input) DOUBLE PRECISION array, dimension (LDDC,P)
          The leading M-by-P part of this array must contain
          the input/output matrix Dc of the controller.

  LDDC    INTEGER
          The leading dimension of array DC.  LDDC >= MAX(1,M).

  SCALEC  (output) DOUBLE PRECISION
          Scaling factor for the controllability Grammian.
          See METHOD.

  SCALEO  (output) DOUBLE PRECISION
          Scaling factor for the observability Grammian. See METHOD.

  S       (output) DOUBLE PRECISION array, dimension (LDS,NCS)
          The leading NCS-by-NCS upper triangular part of this array
          contains the Cholesky factor S of the frequency-weighted
          controllability Grammian P = S*S'. See METHOD.

  LDS     INTEGER
          The leading dimension of array S.  LDS >= MAX(1,NCS).

  R       (output) DOUBLE PRECISION array, dimension (LDR,NCS)
          The leading NCS-by-NCS upper triangular part of this array
          contains the Cholesky factor R of the frequency-weighted
          observability Grammian Q = R'*R. See METHOD.

  LDR     INTEGER
          The leading dimension of array R.  LDR >= MAX(1,NCS).

  IWORK   INTEGER array, dimension (LIWRK)
          LIWRK = 0,       if WEIGHT = 'N';
          LIWRK = 2(M+P),  if WEIGHT = 'O', 'I', or 'P'.

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

  LDWORK  INTEGER
          The length of the array DWORK.
          LDWORK >= MAX( 1, LFREQ ),
          where
          LFREQ = (N+NC)*(N+NC+2*M+2*P)+
                  MAX((N+NC)*(N+NC+MAX(N+NC,M,P)+7), (M+P)*(M+P+4))
                                   if WEIGHT = 'I' or 'O' or 'P';
          LFREQ  = NCS*(MAX(M,P)+5) if WEIGHT = 'N'.
          For optimum performance LDWORK should be larger.

  INFO    INTEGER
          = 0:  successful exit;
          < 0:  if INFO = -i, the i-th argument had an illegal
                value;
          = 1:  the closed-loop system is not well-posed;
                its feedthrough matrix is (numerically) singular;
          = 2:  the computation of the real Schur form of the
                closed-loop state matrix failed;
          = 3:  the closed-loop state matrix is not stable;
          = 4:  the solution of a symmetric eigenproblem failed;
          = 5:  the NCS-by-NCS trailing part Ac2 of the state
                matrix Ac is not stable or not in a real Schur form.

  If JOBC = 'S', the controllability Grammian P is determined as
  follows:

  - if WEIGHT = 'O' or 'N', P satisfies for a continuous-time
    controller the Lyapunov equation

         Ac2*P + P*Ac2' +  scalec^2*Bc*Bc' = 0

    and for a discrete-time controller

         Ac2*P*Ac2' - P +  scalec^2*Bc*Bc' = 0;

  - if WEIGHT = 'I' or 'P', let Pi be the solution of the
    continuous-time Lyapunov equation

         Ai*Pi + Pi*Ai' +  scalec^2*Bi*Bi' = 0

    or of the discrete-time Lyapunov equation

         Ai*Pi*Ai' - Pi +  scalec^2*Bi*Bi' = 0,

    where Ai and Bi are the state and input matrices of a special
    state-space realization of the input frequency weight (see [2]);
    P results as the trailing NCS-by-NCS part of Pi partitioned as

        Pi = ( *  * ).
             ( *  P )

  If JOBC = 'E', a modified controllability Grammian P1 >= P is
  determined to guarantee stability for a modified Enns' method [2].

  If JOBO = 'S', the observability Grammian Q is determined as
  follows:

  - if WEIGHT = 'I' or 'N', Q satisfies for a continuous-time
    controller the Lyapunov equation

         Ac2'*Q + Q*Ac2 +  scaleo^2*Cc'*Cc = 0

    and for a discrete-time controller

         Ac2'*Q*Ac2 - Q +  scaleo^2*Cc'*Cc = 0;

  - if WEIGHT = 'O' or 'P', let Qo be the solution of the
    continuous-time Lyapunov equation

         Ao'*Qo + Qo*Ao +  scaleo^2*Co'*Co = 0

    or of the discrete-time Lyapunov equation

         Ao'*Qo*Ao - Qo +  scaleo^2*Co'*Co = 0,

    where Ao and Co are the state and output matrices of a
    special state-space realization of the output frequency weight
    (see [2]); if WEIGHT = 'O', Q results as the leading NCS-by-NCS
    part of Qo partitioned as

        Qo = ( Q  * )
             ( *  * )

    while if WEIGHT = 'P', Q results as the trailing NCS-by-NCS
    part of Qo partitioned as

        Qo = ( *  * ).
             ( *  Q )

  If JOBO = 'E', a modified observability Grammian Q1 >= Q is
  determined to guarantee stability for a modified Enns' method [2].

  The routine computes directly the Cholesky factors S and R
  such that P = S*S' and Q = R'*R according to formulas
  developed in [2].

  [1] Enns, D.
      Model reduction with balanced realizations: An error bound
      and a frequency weighted generalization.
      Proc. CDC, Las Vegas, pp. 127-132, 1984.

  [2] Varga, A. and Anderson, B.D.O.
      Frequency-weighted balancing related controller reduction.
      Proceedings of the 15th IFAC World Congress, July 21-26, 2002,
      Barcelona, Spain, Vol.15, Part 1, 2002-07-21.

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