## IB01QD

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

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

Purpose

      To estimate the initial state and the system matrices  B  and  D
      of a linear time-invariant (LTI) discrete-time system, given the
      matrix pair  (A,C)  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 IB01QD( JOBX0, JOB, N, M, L, NSMP, A, LDA, C, LDC, U,
         $                   LDU, Y, LDY, X0, B, LDB, D, LDD, 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, JOBX0
    C     .. Array Arguments ..
          DOUBLE PRECISION   A(LDA, *), B(LDB, *), C(LDC, *), D(LDD, *),
         $                   DWORK(*),  U(LDU, *), X0(*), Y(LDY, *)
          INTEGER            IWORK(*)

Arguments

Mode Parameters

      JOBX0   CHARACTER*1
              Specifies whether or not the initial state should be
              computed, as follows:
              = 'X':  compute the initial state x(0);
              = 'N':  do not compute the initial state (x(0) is known
                      to be zero).

      JOB     CHARACTER*1
              Specifies which matrices should be computed, as follows:
              = 'B':  compute the matrix B only (D is known to be zero);
              = 'D':  compute the matrices B and D.

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,  t).
              NSMP >= N*M + a + e,  where
              a = 0,  if  JOBX0 = 'N';
              a = N,  if  JOBX0 = 'X';
              e = 0,  if  JOBX0 = 'X'  and  JOB = 'B';
              e = 1,  if  JOBX0 = 'N'  and  JOB = 'B';
              e = M,  if  JOB   = 'D'.

      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).

      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.

      U       (input/output) DOUBLE PRECISION array, dimension (LDU,M)
              On entry, 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.
              On exit, if  JOB = 'D',  the leading NSMP-by-M part of
              this array contains details of the QR factorization of
              the t-by-m matrix  U, possibly computed sequentially
              (see METHOD).
              If  JOB = 'B',  this array is unchanged on exit.
              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)
              If  JOBX0 = 'X',  the estimated initial state of the
              system,  x(0).
              If  JOBX0 = 'N',  x(0)  is set to zero without any
              calculations.

      B       (output) DOUBLE PRECISION array, dimension (LDB,M)
              If  N > 0,  M > 0,  and  INFO = 0,  the leading N-by-M
              part of this array contains the system input matrix  B
              in the coordinates 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.

      D       (output) DOUBLE PRECISION array, dimension (LDD,M)
              If  M > 0,  JOB = 'D',  and  INFO = 0,  the leading
              L-by-M part of this array contains the system input-output
              matrix  D.
              If  M = 0  or  JOB = 'B',  this array is not referenced.

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

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 (LIWORK), where
              LIWORK >= N*M + a,            if  JOB = 'B',
              LIWORK >= max( N*M + a, M ),  if  JOB = 'D',
              with  a = 0,  if  JOBX0 = 'N';
                    a = N,  if  JOBX0 = 'X'.

      DWORK   DOUBLE PRECISION array, dimension (LDWORK)
              On exit, if  INFO = 0,  DWORK(1) returns the optimal value
              of LDWORK;  DWORK(2)  contains the reciprocal condition
              number of the triangular factor of the QR factorization of
              the matrix  W2  (see METHOD); if  M > 0  and  JOB = 'D',
              DWORK(3)  contains the reciprocal condition number of the
              triangular factor of the QR factorization of  U.
              On exit, if  INFO = -23,  DWORK(1)  returns the minimum
              value of LDWORK.

      LDWORK  INTEGER
              The length of the array DWORK.
              LDWORK >= max( LDW1, min( LDW2, LDW3 ) ),  where
              LDW1 = 2,          if  M = 0  or   JOB = 'B',
              LDW1 = 3,          if  M > 0  and  JOB = 'D',
              LDWa = t*L*(r + 1) + max( N + max( d, f ), 6*r ),
              LDW2 = LDWa,       if  M = 0  or  JOB = 'B',
              LDW2 = max( LDWa, t*L*(r + 1) + 2*M*M + 6*M ),
                                 if  M > 0  and JOB = 'D',
              LDWb = (b + r)*(r + 1) +
                      max( q*(r + 1) + N*N*M + c + max( d, f ), 6*r ),
              LDW3 = LDWb,       if  M = 0  or  JOB = 'B',
              LDW3 = max( LDWb, (b + r)*(r + 1) + 2*M*M + 6*M ),
                                 if  M > 0  and JOB = 'D',
                 r = N*M + a,
                 a = 0,                  if  JOBX0 = 'N',
                 a = N,                  if  JOBX0 = 'X';
                 b = 0,                  if  JOB   = 'B',
                 b = L*M,                if  JOB   = 'D';
                 c = 0,                  if  JOBX0 = 'N',
                 c = L*N,                if  JOBX0 = 'X';
                 d = 0,                  if  JOBX0 = 'N',
                 d = 2*N*N + N,          if  JOBX0 = 'X';
                 f = 2*r,                if  JOB   = 'B'   or  M = 0,
                 f = M + max( 2*r, M ),  if  JOB   = 'D'  and  M > 0;
                 q = b + r*L.
              For good performance,  LDWORK  should be larger.
              If  LDWORK >= LDW2  or
                  LDWORK >= t*L*(r + 1) + (b + r)*(r + 1) + N*N*M + c +
                            max( d, f ),
              then standard QR factorizations of the matrices  U  and/or
              W2  (see METHOD) are used.
              Otherwise, the QR factorizations are computed sequentially
              by performing  NCYCLE  cycles, each cycle (except possibly
              the last one) processing  s < t  samples, where  s  is
              chosen from the equation
                LDWORK = s*L*(r + 1) + (b + r)*(r + 1) + N*N*M + c +
                         max( d, f ).
              (s  is at least  N*M+a+e,  the minimum value of  NSMP.)
              The computational effort may increase and the accuracy may
              decrease with the decrease of  s.  Recommended value is
              LDWORK = LDW2,  assuming a large enough cache size, to
              also accommodate  A,  C,  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 [1,2] is used.
      Specifically, denoting

            X = [ vec(D')' vec(B)' x0' ]',

      where  vec(M)  is the vector obtained by stacking the columns of
      the matrix  M,  then  X  is the least squares solution of the
      system  S*X = vec(Y),  with the matrix  S = [ diag(U)  W ],
      defined by

            ( U         |     | ... |     |     | ... |     |         )
            (   U       |  11 | ... |  n1 |  12 | ... |  nm |         )
        S = (     :     | y   | ... | y   | y   | ... | y   | P*Gamma ),
            (       :   |     | ... |     |     | ... |     |         )
            (         U |     | ... |     |     | ... |     |         )
                                                                      ij
      diag(U)  having  L  block rows and columns.  In this formula,  y
      are the outputs of the system for zero initial state computed
      using the following model, for j = 1:m, and for i = 1:n,
             ij          ij                    ij
            x  (k+1) = Ax  (k) + e_i u_j(k),  x  (0) = 0,

             ij          ij
            y  (k)   = Cx  (k),

      where  e_i  is the i-th n-dimensional unit vector,  Gamma  is
      given by

                 (     C     )
                 (    C*A    )
         Gamma = (   C*A^2   ),
                 (     :     )
                 ( C*A^(t-1) )

      and  P  is a permutation matrix that groups together the rows of
      Gamma  depending on the same row of  C,  namely
      [ c_j;  c_j*A;  c_j*A^2; ...  c_j*A^(t-1) ],  for j = 1:L.
      The first block column,  diag(U),  is not explicitly constructed,
      but its structure is exploited. The last block column is evaluated
      using powers of A with exponents 2^k. No interchanges are applied.
      A special QR decomposition of the matrix  S  is computed. Let
      U = q*[ r' 0 ]'  be the QR decomposition of  U,  if  M > 0,  where
      r  is  M-by-M.   Then,  diag(q')  is applied to  W  and  vec(Y).
      The block-rows of  S  and  vec(Y)  are implicitly permuted so that
      matrix  S  becomes

         ( diag(r)  W1 )
         (    0     W2 ),

      where  W1  has L*M rows. Then, the QR decomposition of  W2 is
      computed (sequentially, if  M > 0) and used to obtain  B  and  x0.
      The intermediate results and the QR decomposition of  U  are
      needed to find  D.  If a triangular factor is too ill conditioned,
      then singular value decomposition (SVD) is employed. SVD is not
      generally needed if the input sequence is sufficiently
      persistently exciting and  NSMP  is large enough.
      If the matrix  W  cannot be stored in the workspace (i.e.,
      LDWORK < LDW2),  the QR decompositions of  W2  and  U  are
      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.

      [2] Sima, V., and Varga, A.
          RASP-IDENT : Subspace Model Identification Programs.
          Deutsche Forschungsanstalt fur Luft- und Raumfahrt e. V.,
          Report TR R888-94, DLR Oberpfaffenhofen, Oct. 1994.

Numerical Aspects

      The implemented method is numerically stable.

Further Comments

      The algorithm for computing the system matrices  B  and  D  is
      less efficient than the MOESP or N4SID algorithms implemented in
      SLICOT Library routine IB01PD, because a large least squares
      problem has to be solved, but the accuracy is better, as the
      computed matrices  B  and  D  are fitted to the input and output
      trajectories. However, if matrix  A  is unstable, the computed
      matrices  B  and  D  could be inaccurate.

Example

Program Text

      None

Program Data

      None

Program Results

      None

Return to Supporting Routines index