## IB01PY

### Estimating system matrices B and D using a structure exploiting QR factorization

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

Purpose

      1. To compute the triangular  (QR)  factor of the  p-by-L*s
      structured matrix  Q,

          [ Q_1s  Q_1,s-1  Q_1,s-2  ...  Q_12  Q_11 ]
          [  0      Q_1s   Q_1,s-1  ...  Q_13  Q_12 ]
      Q = [  0       0       Q_1s   ...  Q_14  Q_13 ],
          [  :       :        :           :     :   ]
          [  0       0        0     ...   0    Q_1s ]

      and apply the transformations to the p-by-m matrix  Kexpand,

                [ K_1 ]
                [ K_2 ]
      Kexpand = [ K_3 ],
                [  :  ]
                [ K_s ]

      where, for MOESP approach (METH = 'M'), p = s*(L*s-n), and
      Q_1i = u2(L*(i-1)+1:L*i,:)'  is  (Ls-n)-by-L,  for  i = 1:s,
      u2 = Un(1:L*s,n+1:L*s),  K_i = K(:,(i-1)*m+1:i*m)  (i = 1:s)
      is  (Ls-n)-by-m, and for N4SID approach (METH = 'N'), p = s*(n+L),
      and

                [   -L_1|1    ]          [ M_i-1 - L_1|i ]
         Q_11 = [             ],  Q_1i = [               ],  i = 2:s,
                [ I_L - L_2|1 ]          [     -L_2|i    ]

      are  (n+L)-by-L  matrices, and
      K_i = K(:,(i-1)*m+1:i*m),  i = 1:s,  is  (n+L)-by-m.
      The given matrices are:
      For  METH = 'M',  u2 = Un(1:L*s,n+1:L*s),
                        K(1:Ls-n,1:m*s);

                            [ L_1|1  ...  L_1|s ]
      For  METH = 'N',  L = [                   ],   (n+L)-by-L*s,
                            [ L_2|1  ...  L_2|s ]

                        M = [ M_1  ...  M_s-1 ],  n-by-L*(s-1),  and
                        K,                        (n+L)-by-m*s.
                        Matrix M is the pseudoinverse of the matrix GaL,
                        built from the first  n  relevant singular
                        vectors,  GaL = Un(1:L(s-1),1:n),  and computed
                        by SLICOT Library routine IB01PD for METH = 'N'.

      Matrix  Q  is triangularized  (in  R),  exploiting its structure,
      and the transformations are applied from the left to  Kexpand.

      2. To estimate the matrices B and D of a linear time-invariant
      (LTI) state space model, using the factor  R,  transformed matrix
      Kexpand, and the singular value decomposition information provided
      by other routines.

      IB01PY  routine is intended for speed and efficient use of the
      memory space. It is generally not recommended for  METH = 'N',  as
      IB01PX  routine can produce more accurate results.

Specification

          SUBROUTINE IB01PY( METH, JOB, NOBR, N, M, L, RANKR1, UL, LDUL,
         $                   R1, LDR1, TAU1, PGAL, LDPGAL, K, LDK, R, LDR,
         $                   H, LDH, B, LDB, D, LDD, TOL, IWORK, DWORK,
         $                   LDWORK, IWARN, INFO )
    C     .. Scalar Arguments ..
          DOUBLE PRECISION   TOL
          INTEGER            INFO, IWARN, L, LDB, LDD, LDH, LDK, LDPGAL,
         $                   LDR, LDR1, LDUL, LDWORK, M, N, NOBR, RANKR1
          CHARACTER          JOB, METH
    C     .. Array Arguments ..
          DOUBLE PRECISION   B(LDB, *), D(LDD, *), DWORK(*), H(LDH, *),
         $                   K(LDK, *), PGAL(LDPGAL, *), R(LDR, *),
         $                   R1(LDR1, *), TAU1(*), UL(LDUL, *)
          INTEGER            IWORK( * )

Arguments

Mode Parameters

      METH    CHARACTER*1
              Specifies the subspace identification method to be used,
              as follows:
              = 'M':  MOESP  algorithm with past inputs and outputs;
              = 'N':  N4SID  algorithm.

      JOB     CHARACTER*1
              Specifies whether or not the matrices B and D should be
              computed, as follows:
              = 'B':  compute the matrix B, but not the matrix D;
              = 'D':  compute both matrices B and D;
              = 'N':  do not compute the matrices B and D, but only the
                      R  factor of  Q  and the transformed Kexpand.

Input/Output Parameters

      NOBR    (input) INTEGER
              The number of block rows,  s,  in the input and output
              Hankel matrices processed by other routines.  NOBR > 1.

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

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

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

      RANKR1  (input) INTEGER
              The effective rank of the upper triangular matrix  r1,
              i.e., the triangular QR factor of the matrix  GaL,
              computed by SLICOT Library routine IB01PD. It is also
              the effective rank of the matrix  GaL.  0 <= RANKR1 <= N.
              If  JOB = 'N',  or  M = 0,  or  METH = 'N',  this
              parameter is not used.

      UL      (input/workspace) DOUBLE PRECISION array, dimension
              ( LDUL,L*NOBR )
              On entry, if  METH = 'M',  the leading  L*NOBR-by-L*NOBR
              part of this array must contain the matrix  Un  of
              relevant singular vectors. The first  N  columns of  UN
              need not be specified for this routine.
              On entry, if  METH = 'N',  the leading  (N+L)-by-L*NOBR
              part of this array must contain the given matrix  L.
              On exit, the leading  LDF-by-L*(NOBR-1) part of this array
              is overwritten by the matrix  F  of the algorithm in [4],
              where  LDF = MAX( 1, L*NOBR-N-L ), if  METH = 'M';
                     LDF = N,                    if  METH = 'N'.

      LDUL    INTEGER
              The leading dimension of the array  UL.
              LDUL >= L*NOBR, if  METH = 'M';
              LDUL >= N+L,    if  METH = 'N'.

      R1      (input) DOUBLE PRECISION array, dimension ( LDR1,N )
              If  JOB <> 'N',  M > 0,  METH = 'M',  and  RANKR1 = N,
              the leading  L*(NOBR-1)-by-N  part of this array must
              contain details of the QR factorization of the matrix
              GaL,  as computed by SLICOT Library routine IB01PD.
              Specifically, the leading N-by-N upper triangular part
              must contain the upper triangular factor  r1  of  GaL,
              and the lower  L*(NOBR-1)-by-N  trapezoidal part, together
              with array TAU1, must contain the factored form of the
              orthogonal matrix  Q1  in the QR factorization of  GaL.
              If  JOB = 'N',  or  M = 0,  or  METH = 'N', or  METH = 'M'
              and  RANKR1 < N,  this array is not referenced.

      LDR1    INTEGER
              The leading dimension of the array  R1.
              LDR1 >= L*(NOBR-1), if  JOB <> 'N',  M > 0,  METH = 'M',
                                  and  RANKR1 = N;
              LDR1 >= 1,          otherwise.

      TAU1    (input) DOUBLE PRECISION array, dimension ( N )
              If  JOB <> 'N',  M > 0,  METH = 'M',  and  RANKR1 = N,
              this array must contain the scalar factors of the
              elementary reflectors used in the QR factorization of the
              matrix  GaL,  computed by SLICOT Library routine IB01PD.
              If  JOB = 'N',  or  M = 0,  or  METH = 'N', or  METH = 'M'
              and  RANKR1 < N,  this array is not referenced.

      PGAL    (input) DOUBLE PRECISION array, dimension
              ( LDPGAL,L*(NOBR-1) )
              If  METH = 'N',  or  JOB <> 'N',  M > 0,  METH = 'M'  and
              RANKR1 < N,  the leading  N-by-L*(NOBR-1)  part of this
              array must contain the pseudoinverse of the matrix  GaL,
              as computed by SLICOT Library routine IB01PD.
              If  METH = 'M'  and  JOB = 'N',  or  M = 0,  or
              RANKR1 = N,  this array is not referenced.

      LDPGAL  INTEGER
              The leading dimension of the array  PGAL.
              LDPGAL >= N,  if   METH = 'N',  or  JOB <> 'N',  M > 0,
                            and  METH = 'M'  and RANKR1 < N;
              LDPGAL >= 1,  otherwise.

      K       (input/output) DOUBLE PRECISION array, dimension
              ( LDK,M*NOBR )
              On entry, the leading  (p/s)-by-M*NOBR  part of this array
              must contain the given matrix  K  defined above.
              On exit, the leading  (p/s)-by-M*NOBR  part of this array
              contains the transformed matrix  K.

      LDK     INTEGER
              The leading dimension of the array  K.  LDK >= p/s.

      R       (output) DOUBLE PRECISION array, dimension ( LDR,L*NOBR )
              If  JOB = 'N',  or  M = 0,  or  Q  has full rank, the
              leading  L*NOBR-by-L*NOBR  upper triangular part of this
              array contains the  R  factor of the QR factorization of
              the matrix  Q.
              If  JOB <> 'N',  M > 0,  and  Q  has not a full rank, the
              leading  L*NOBR-by-L*NOBR  upper trapezoidal part of this
              array contains details of the complete orhogonal
              factorization of the matrix  Q,  as constructed by SLICOT
              Library routines MB03OD and MB02QY.

      LDR     INTEGER
              The leading dimension of the array  R.  LDR >= L*NOBR.

      H       (output) DOUBLE PRECISION array, dimension ( LDH,M )
              If  JOB = 'N'  or  M = 0,  the leading  L*NOBR-by-M  part
              of this array contains the updated part of the matrix
              Kexpand  corresponding to the upper triangular factor  R
              in the QR factorization of the matrix  Q.
              If  JOB <> 'N',  M > 0,  and  METH = 'N'  or  METH = 'M'
              and  RANKR1 < N,  the leading  L*NOBR-by-M  part of this
              array contains the minimum norm least squares solution of
              the linear system  Q*X = Kexpand,  from which the matrices
              B  and  D  are found. The first  NOBR-1  row blocks of  X
              appear in the reverse order in  H.
              If  JOB <> 'N',  M > 0,  METH = 'M'  and  RANKR1 = N,  the
              leading  L*(NOBR-1)-by-M  part of this array contains the
              matrix product  Q1'*X,  and the subarray
              L*(NOBR-1)+1:L*NOBR-by-M  contains the  corresponding
              submatrix of  X,  with  X  defined in the phrase above.

      LDH     INTEGER
              The leading dimension of the array  H.  LDH >= L*NOBR.

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

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

      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.
              If  M = 0  or  JOB = 'B'  or  'N',  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' or 'N'.

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;  an m-by-n matrix whose estimated
              condition number is less than  1/TOL  is considered to
              be of full rank.  If the user sets  TOL <= 0,  then an
              implicitly computed, default tolerance, defined by
              TOLDEF = m*n*EPS,  is used instead, where  EPS  is the
              relative machine precision (see LAPACK Library routine
              DLAMCH).
              This parameter is not used if  M = 0  or  JOB = 'N'.

Workspace

      IWORK   INTEGER array, dimension ( LIWORK )
              where  LIWORK >= 0,       if  JOB =  'N',  or   M = 0;
                     LIWORK >= L*NOBR,  if  JOB <> 'N',  and  M > 0.

      DWORK   DOUBLE PRECISION array, dimension ( LDWORK )
              On exit, if  INFO = 0,  DWORK(1) returns the optimal value
              of  LDWORK,  and, if  JOB <> 'N',  and  M > 0,  DWORK(2)
              contains the reciprocal condition number of the triangular
              factor of the matrix  R.
              On exit, if  INFO = -28,  DWORK(1)  returns the minimum
              value of LDWORK.

      LDWORK  INTEGER
              The length of the array DWORK.
              LDWORK >= MAX( 2*L, L*NOBR, L+M*NOBR ),
                                          if  JOB = 'N',  or  M = 0;
              LDWORK >= MAX( L+M*NOBR, L*NOBR + MAX( 3*L*NOBR+1, M ) ),
                                          if  JOB <> 'N',  and  M > 0.
              For good performance,  LDWORK  should be larger.

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;
              = 3:  a singular upper triangular matrix was found.

Method

      The QR factorization is computed exploiting the structure,
      as described in [4].
      The matrices  B  and  D  are then obtained by solving certain
      linear systems in a least squares sense.

References

      [1] Verhaegen M., and Dewilde, P.
          Subspace Model Identification. Part 1: The output-error
          state-space model identification class of algorithms.
          Int. J. Control, 56, pp. 1187-1210, 1992.

      [2] Van Overschee, P., and De Moor, B.
          N4SID: Two Subspace Algorithms for the Identification
          of Combined Deterministic-Stochastic Systems.
          Automatica, Vol.30, No.1, pp. 75-93, 1994.

      [3] Van Overschee, P.
          Subspace Identification : Theory - Implementation -
          Applications.
          Ph. D. Thesis, Department of Electrical Engineering,
          Katholieke Universiteit Leuven, Belgium, Feb. 1995.

      [4] Sima, V.
          Subspace-based Algorithms for Multivariable System
          Identification.
          Studies in Informatics and Control, 5, pp. 335-344, 1996.

Numerical Aspects

      The implemented method for computing the triangular factor and
      updating Kexpand is numerically stable.

Further Comments

      The computed matrices B and D are not the least squares solutions
      delivered by either MOESP or N4SID algorithms, except for the
      special case n = s - 1, L = 1. However, the computed B and D are
      frequently good enough estimates, especially for  METH = 'M'.
      Better estimates could be obtained by calling SLICOT Library
      routine IB01PX, but it is less efficient, and requires much more
      workspace.

Example

Program Text

      None

Program Data

      None

Program Results

      None

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