IB01RD

Estimating initial state, given A, B, C, D, and input-output trajectories

Purpose

  To estimate the initial state of a linear time-invariant (LTI)
  discrete-time system, given the system matrices  (A,B,C,D)  and
  the input and output trajectories of the system. The model
  structure is :

        x(k+1) = Ax(k) + Bu(k),   k >= 0,
        y(k)   = Cx(k) + Du(k),

  where  x(k)  is the  n-dimensional state vector (at time k),
         u(k)  is the  m-dimensional input vector,
         y(k)  is the  l-dimensional output vector,
  and  A, B, C, and D  are real matrices of appropriate dimensions.
  Matrix  A  is assumed to be in a real Schur form.

      SUBROUTINE IB01RD( JOB, N, M, L, NSMP, A, LDA, B, LDB, C, LDC, D,
     $                   LDD, U, LDU, Y, LDY, X0, TOL, IWORK, DWORK,
     $                   LDWORK, IWARN, INFO )
C     .. Scalar Arguments ..
      DOUBLE PRECISION   TOL
      INTEGER            INFO, IWARN, L, LDA, LDB, LDC, LDD, LDU,
     $                   LDWORK, LDY, M, N, NSMP
      CHARACTER          JOB
C     .. Array Arguments ..
      DOUBLE PRECISION   A(LDA, *), B(LDB, *), C(LDC, *), D(LDD, *),
     $                   DWORK(*),  U(LDU, *), X0(*), Y(LDY, *)
      INTEGER            IWORK(*)

Mode Parameters

  JOB     CHARACTER*1
          Specifies whether or not the matrix D is zero, as follows:
          = 'Z':  the matrix  D  is zero;
          = 'N':  the matrix  D  is not zero.

  N       (input) INTEGER
          The order of the system.  N >= 0.

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

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

  NSMP    (input) INTEGER
          The number of rows of matrices  U  and  Y  (number of
          samples used,  t).  NSMP >= N.

  A       (input) DOUBLE PRECISION array, dimension (LDA,N)
          The leading N-by-N part of this array must contain the
          system state matrix  A  in a real Schur form.

  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  (corresponding to the real Schur
          form of  A).
          If  N = 0  or  M = 0,  this array is not referenced.

  LDB     INTEGER
          The leading dimension of the array B.
          LDB >= N,  if  N > 0  and  M > 0;
          LDB >= 1,  if  N = 0  or   M = 0.

  C       (input) DOUBLE PRECISION array, dimension (LDC,N)
          The leading L-by-N part of this array must contain the
          system output matrix  C  (corresponding to the real Schur
          form of  A).

  LDC     INTEGER
          The leading dimension of the array C.  LDC >= L.

  D       (input) DOUBLE PRECISION array, dimension (LDD,M)
          The leading L-by-M part of this array must contain the
          system input-output matrix.
          If  M = 0  or  JOB = 'Z',  this array is not referenced.

  LDD     INTEGER
          The leading dimension of the array D.
          LDD >= L,  if  M > 0  and  JOB = 'N';
          LDD >= 1,  if  M = 0  or   JOB = 'Z'.

  U       (input) DOUBLE PRECISION array, dimension (LDU,M)
          If  M > 0,  the leading NSMP-by-M part of this array must
          contain the t-by-m input-data sequence matrix  U,
          U = [u_1 u_2 ... u_m].  Column  j  of  U  contains the
          NSMP  values of the j-th input component for consecutive
          time increments.
          If M = 0, this array is not referenced.

  LDU     INTEGER
          The leading dimension of the array U.
          LDU >= MAX(1,NSMP),  if M > 0;
          LDU >= 1,            if M = 0.

  Y       (input) DOUBLE PRECISION array, dimension (LDY,L)
          The leading NSMP-by-L part of this array must contain the
          t-by-l output-data sequence matrix  Y,
          Y = [y_1 y_2 ... y_l].  Column  j  of  Y  contains the
          NSMP  values of the j-th output component for consecutive
          time increments.

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

  X0      (output) DOUBLE PRECISION array, dimension (N)
          The estimated initial state of the system,  x(0).

  TOL     DOUBLE PRECISION
          The tolerance to be used for estimating the rank of
          matrices. If the user sets  TOL > 0,  then the given value
          of  TOL  is used as a lower bound for the reciprocal
          condition number;  a matrix whose estimated condition
          number is less than  1/TOL  is considered to be of full
          rank.  If the user sets  TOL <= 0,  then  EPS  is used
          instead, where  EPS  is the relative machine precision
          (see LAPACK Library routine DLAMCH).  TOL <= 1.

  IWORK   INTEGER array, dimension (N)

  DWORK   DOUBLE PRECISION array, dimension (LDWORK)
          On exit, if  INFO = 0,  DWORK(1) returns the optimal value
          of LDWORK and  DWORK(2)  contains the reciprocal condition
          number of the triangular factor of the QR factorization of
          the matrix  Gamma  (see METHOD).
          On exit, if  INFO = -22,  DWORK(1)  returns the minimum
          value of LDWORK.

  LDWORK  INTEGER
          The length of the array DWORK.
          LDWORK >= max( 2, min( LDW1, LDW2 ) ),  where
          LDW1 = t*L*(N + 1) + 2*N + max( 2*N*N, 4*N ),
          LDW2 =   N*(N + 1) + 2*N +
                   max( q*(N + 1) + 2*N*N + L*N, 4*N ),
             q = N*L.
          For good performance,  LDWORK  should be larger.
          If  LDWORK >= LDW1,  then standard QR factorization of
          the matrix  Gamma  (see METHOD) is used. Otherwise, the
          QR factorization is computed sequentially by performing
          NCYCLE  cycles, each cycle (except possibly the last one)
          processing  s  samples, where  s  is chosen by equating
          LDWORK  to  LDW2,  for  q  replaced by  s*L.
          The computational effort may increase and the accuracy may
          decrease with the decrease of  s.  Recommended value is
          LDRWRK = LDW1,  assuming a large enough cache size, to
          also accommodate  A, B, C, D, U,  and  Y.

  IWARN   INTEGER
          = 0:  no warning;
          = 4:  the least squares problem to be solved has a
                rank-deficient coefficient matrix.

  INFO    INTEGER
          = 0:  successful exit;
          < 0:  if INFO = -i, the i-th argument had an illegal
                value;
          = 2:  the singular value decomposition (SVD) algorithm did
                not converge.

  An extension and refinement of the method in [1] is used.
  Specifically, the output y0(k) of the system for zero initial
  state is computed for k = 0, 1, ...,  t-1 using the given model.
  Then the following least squares problem is solved for x(0)

                      (     C     )            (   y(0) - y0(0)   )
                      (    C*A    )            (   y(1) - y0(1)   )
     Gamma * x(0)  =  (     :     ) * x(0)  =  (        :         ).
                      (     :     )            (        :         )
                      ( C*A^(t-1) )            ( y(t-1) - y0(t-1) )

  The coefficient matrix  Gamma  is evaluated using powers of A with
  exponents 2^k. The QR decomposition of this matrix is computed.
  If its triangular factor  R  is too ill conditioned, then singular
  value decomposition of  R  is used.

  If the coefficient matrix cannot be stored in the workspace (i.e.,
  LDWORK < LDW1),  the QR decomposition is computed sequentially.

  [1] Verhaegen M., and Varga, A.
      Some Experience with the MOESP Class of Subspace Model
      Identification Methods in Identifying the BO105 Helicopter.
      Report TR R165-94, DLR Oberpfaffenhofen, 1994.

  The implemented method is numerically stable.

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Program Text

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