## IB01RD

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

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

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.

Specification

          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(*)

Arguments

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.

Input/Output Parameters

      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).

Tolerances

      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.

Workspace

      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.

Warning Indicator

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

Error Indicator

      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.

Method

      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.

References

      [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.

Numerical Aspects

      The implemented method is numerically stable.

Further Comments

      None

Example

Program Text

      None

Program Data

      None

Program Results

      None

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