**Purpose**

To construct an upper triangular factor R of the concatenated block Hankel matrices using input-output data. The input-output data can, optionally, be processed sequentially.

SUBROUTINE IB01MD( METH, ALG, BATCH, CONCT, NOBR, M, L, NSMP, U, $ LDU, Y, LDY, R, LDR, IWORK, DWORK, LDWORK, $ IWARN, INFO ) C .. Scalar Arguments .. INTEGER INFO, IWARN, L, LDR, LDU, LDWORK, LDY, M, NOBR, $ NSMP CHARACTER ALG, BATCH, CONCT, METH C .. Array Arguments .. INTEGER IWORK(*) DOUBLE PRECISION DWORK(*), R(LDR, *), U(LDU, *), Y(LDY, *)

**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. ALG CHARACTER*1 Specifies the algorithm for computing the triangular factor R, as follows: = 'C': Cholesky algorithm applied to the correlation matrix of the input-output data; = 'F': Fast QR algorithm; = 'Q': QR algorithm applied to the concatenated block Hankel matrices. BATCH CHARACTER*1 Specifies whether or not sequential data processing is to be used, and, for sequential processing, whether or not the current data block is the first block, an intermediate block, or the last block, as follows: = 'F': the first block in sequential data processing; = 'I': an intermediate block in sequential data processing; = 'L': the last block in sequential data processing; = 'O': one block only (non-sequential data processing). NOTE that when 100 cycles of sequential data processing are completed for BATCH = 'I', a warning is issued, to prevent for an infinite loop. CONCT CHARACTER*1 Specifies whether or not the successive data blocks in sequential data processing belong to a single experiment, as follows: = 'C': the current data block is a continuation of the previous data block and/or it will be continued by the next data block; = 'N': there is no connection between the current data block and the previous and/or the next ones. This parameter is not used if BATCH = 'O'.

NOBR (input) INTEGER The number of block rows, s, in the input and output block Hankel matrices to be processed. NOBR > 0. (In the MOESP theory, NOBR should be larger than n, the estimated dimension of state vector.) M (input) INTEGER The number of system inputs. M >= 0. When M = 0, no system inputs are processed. 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). (When sequential data processing is used, NSMP is the number of samples of the current data block.) NSMP >= 2*(M+L+1)*NOBR - 1, for non-sequential processing; NSMP >= 2*NOBR, for sequential processing. The total number of samples when calling the routine with BATCH = 'L' should be at least 2*(M+L+1)*NOBR - 1. The NSMP argument may vary from a cycle to another in sequential data processing, but NOBR, M, and L should be kept constant. For efficiency, it is advisable to use NSMP as large as possible. U (input) DOUBLE PRECISION array, dimension (LDU,M) 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 >= 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 >= NSMP. R (output or input/output) DOUBLE PRECISION array, dimension ( LDR,2*(M+L)*NOBR ) On exit, if INFO = 0 and ALG = 'Q', or (ALG = 'C' or 'F', and BATCH = 'L' or 'O'), the leading 2*(M+L)*NOBR-by-2*(M+L)*NOBR upper triangular part of this array contains the (current) upper triangular factor R from the QR factorization of the concatenated block Hankel matrices. The diagonal elements of R are positive when the Cholesky algorithm was successfully used. On exit, if ALG = 'C' and BATCH = 'F' or 'I', the leading 2*(M+L)*NOBR-by-2*(M+L)*NOBR upper triangular part of this array contains the current upper triangular part of the correlation matrix in sequential data processing. If ALG = 'F' and BATCH = 'F' or 'I', the array R is not referenced. On entry, if ALG = 'C', or ALG = 'Q', and BATCH = 'I' or 'L', the leading 2*(M+L)*NOBR-by-2*(M+L)*NOBR upper triangular part of this array must contain the upper triangular matrix R computed at the previous call of this routine in sequential data processing. The array R need not be set on entry if ALG = 'F' or if BATCH = 'F' or 'O'. LDR INTEGER The leading dimension of the array R. LDR >= 2*(M+L)*NOBR.

IWORK INTEGER array, dimension (LIWORK) LIWORK >= M+L, if ALG = 'F'; LIWORK >= 0, if ALG = 'C' or 'Q'. DWORK DOUBLE PRECISION array, dimension (LDWORK) On exit, if INFO = 0, DWORK(1) returns the optimal value of LDWORK. On exit, if INFO = -17, DWORK(1) returns the minimum value of LDWORK. Let k = 0, if CONCT = 'N' and ALG = 'C' or 'Q'; k = 2*NOBR-1, if CONCT = 'C' and ALG = 'C' or 'Q'; k = 2*NOBR*(M+L+1), if CONCT = 'N' and ALG = 'F'; k = 2*NOBR*(M+L+2), if CONCT = 'C' and ALG = 'F'. The first (M+L)*k elements of DWORK should be preserved during successive calls of the routine with BATCH = 'F' or 'I', till the final call with BATCH = 'L'. LDWORK INTEGER The length of the array DWORK. LDWORK >= (4*NOBR-2)*(M+L), if ALG = 'C', BATCH <> 'O' and CONCT = 'C'; LDWORK >= 1, if ALG = 'C', BATCH = 'O' or CONCT = 'N'; LDWORK >= (M+L)*2*NOBR*(M+L+3), if ALG = 'F', BATCH <> 'O' and CONCT = 'C'; LDWORK >= (M+L)*2*NOBR*(M+L+1), if ALG = 'F', BATCH = 'F', 'I' and CONCT = 'N'; LDWORK >= (M+L)*4*NOBR*(M+L+1)+(M+L)*2*NOBR, if ALG = 'F', BATCH = 'L' and CONCT = 'N', or BATCH = 'O'; LDWORK >= 4*(M+L)*NOBR, if ALG = 'Q', BATCH = 'F' or 'O', and LDR >= NS = NSMP - 2*NOBR + 1; LDWORK >= 6*(M+L)*NOBR, if ALG = 'Q', BATCH = 'F' or 'O', and LDR < NS, or BATCH = 'I' or 'L' and CONCT = 'N'; LDWORK >= 4*(NOBR+1)*(M+L)*NOBR, if ALG = 'Q', BATCH = 'I' or 'L' and CONCT = 'C'. The workspace used for ALG = 'Q' is LDRWRK*2*(M+L)*NOBR + 4*(M+L)*NOBR, where LDRWRK = LDWORK/(2*(M+L)*NOBR) - 2; recommended value LDRWRK = NS, assuming a large enough cache size. For good performance, LDWORK should be larger. If LDWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the DWORK array, returns this value as the first entry of the DWORK array, and no error message related to LDWORK is issued by XERBLA.

IWARN INTEGER = 0: no warning; = 1: the number of 100 cycles in sequential data processing has been exhausted without signaling that the last block of data was get; the cycle counter was reinitialized; = 2: a fast algorithm was requested (ALG = 'C' or 'F'), but it failed, and the QR algorithm was then used (non-sequential data processing).

INFO INTEGER = 0: successful exit; < 0: if INFO = -i, the i-th argument had an illegal value; = 1: a fast algorithm was requested (ALG = 'C', or 'F') in sequential data processing, but it failed. The routine can be repeatedly called again using the standard QR algorithm.

1) For non-sequential data processing using QR algorithm, a t x 2(m+l)s matrix H is constructed, where H = [ Uf' Up' Y' ], for METH = 'M', s+1,2s,t 1,s,t 1,2s,t H = [ U' Y' ], for METH = 'N', 1,2s,t 1,2s,t and Up , Uf , U , and Y are block Hankel 1,s,t s+1,2s,t 1,2s,t 1,2s,t matrices defined in terms of the input and output data [3]. A QR factorization is used to compress the data. The fast QR algorithm uses a QR factorization which exploits the block-Hankel structure. Actually, the Cholesky factor of H'*H is computed. 2) For sequential data processing using QR algorithm, the QR decomposition is done sequentially, by updating the upper triangular factor R. This is also performed internally if the workspace is not large enough to accommodate an entire batch. 3) For non-sequential or sequential data processing using Cholesky algorithm, the correlation matrix of input-output data is computed (sequentially, if requested), taking advantage of the block Hankel structure [7]. Then, the Cholesky factor of the correlation matrix is found, if possible.

[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] Verhaegen M. Subspace Model Identification. Part 3: Analysis of the ordinary output-error state-space model identification algorithm. Int. J. Control, 58, pp. 555-586, 1993. [3] Verhaegen M. Identification of the deterministic part of MIMO state space models given in innovations form from input-output data. Automatica, Vol.30, No.1, pp.61-74, 1994. [4] Van Overschee, P., and De Moor, B. N4SID: Subspace Algorithms for the Identification of Combined Deterministic-Stochastic Systems. Automatica, Vol.30, No.1, pp. 75-93, 1994. [5] Peternell, K., Scherrer, W. and Deistler, M. Statistical Analysis of Novel Subspace Identification Methods. Signal Processing, 52, pp. 161-177, 1996. [6] Sima, V. Subspace-based Algorithms for Multivariable System Identification. Studies in Informatics and Control, 5, pp. 335-344, 1996. [7] Sima, V. Cholesky or QR Factorization for Data Compression in Subspace-based Identification ? Proceedings of the Second NICONET Workshop on ``Numerical Control Software: SLICOT, a Useful Tool in Industry'', December 3, 1999, INRIA Rocquencourt, France, pp. 75-80, 1999.

The implemented method is numerically stable (when QR algorithm is used), reliable and efficient. The fast Cholesky or QR algorithms are more efficient, but the accuracy could diminish by forming the correlation matrix. 2 The QR algorithm needs 0(t(2(m+l)s) ) floating point operations. 2 3 The Cholesky algorithm needs 0(2t(m+l) s)+0((2(m+l)s) ) floating point operations. 2 3 2 The fast QR algorithm needs 0(2t(m+l) s)+0(4(m+l) s ) floating point operations.

For ALG = 'Q', BATCH = 'O' and LDR < NS, or BATCH <> 'O', the calculations could be rather inefficient if only minimal workspace (see argument LDWORK) is provided. It is advisable to provide as much workspace as possible. Almost optimal efficiency can be obtained for LDWORK = (NS+2)*(2*(M+L)*NOBR), assuming that the cache size is large enough to accommodate R, U, Y, and DWORK.

**Program Text**

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