**Purpose**

To estimate the matrices A, C, B, and D of a linear time-invariant (LTI) state space model, using the singular value decomposition information provided by other routines. Optionally, the system and noise covariance matrices, needed for the Kalman gain, are also determined.

SUBROUTINE IB01PD( METH, JOB, JOBCV, NOBR, N, M, L, NSMPL, R, $ LDR, A, LDA, C, LDC, B, LDB, D, LDD, Q, LDQ, $ RY, LDRY, S, LDS, O, LDO, TOL, IWORK, DWORK, $ LDWORK, IWARN, INFO ) C .. Scalar Arguments .. DOUBLE PRECISION TOL INTEGER INFO, IWARN, L, LDA, LDB, LDC, LDD, LDO, LDQ, $ LDR, LDRY, LDS, LDWORK, M, N, NOBR, NSMPL CHARACTER JOB, JOBCV, METH C .. Array Arguments .. DOUBLE PRECISION A(LDA, *), B(LDB, *), C(LDC, *), D(LDD, *), $ DWORK(*), O(LDO, *), Q(LDQ, *), R(LDR, *), $ RY(LDRY, *), S(LDS, *) INTEGER IWORK( * )

**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 which matrices should be computed, as follows: = 'A': compute all system matrices, A, B, C, and D; = 'C': compute the matrices A and C only; = 'B': compute the matrix B only; = 'D': compute the matrices B and D only. JOBCV CHARACTER*1 Specifies whether or not the covariance matrices are to be computed, as follows: = 'C': the covariance matrices should be computed; = 'N': the covariance matrices should not be computed.

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. NSMPL (input) INTEGER If JOBCV = 'C', the total number of samples used for calculating the covariance matrices. NSMPL >= 2*(M+L)*NOBR. This parameter is not meaningful if JOBCV = 'N'. R (input/workspace) DOUBLE PRECISION array, dimension ( LDR,2*(M+L)*NOBR ) On entry, the leading 2*(M+L)*NOBR-by-2*(M+L)*NOBR part of this array must contain the relevant data for the MOESP or N4SID algorithms, as constructed by SLICOT Library routines IB01AD or IB01ND. Let R_ij, i,j = 1:4, be the ij submatrix of R (denoted S in IB01AD and IB01ND), partitioned by M*NOBR, L*NOBR, M*NOBR, and L*NOBR rows and columns. The submatrix R_22 contains the matrix of left singular vectors used. Also needed, for METH = 'N' or JOBCV = 'C', are the submatrices R_11, R_14 : R_44, and, for METH = 'M' and JOB <> 'C', the submatrices R_31 and R_12, containing the processed matrices R_1c and R_2c, respectively, as returned by SLICOT Library routines IB01AD or IB01ND. Moreover, if METH = 'N' and JOB = 'A' or 'C', the block-row R_41 : R_43 must contain the transpose of the block-column R_14 : R_34 as returned by SLICOT Library routines IB01AD or IB01ND. The remaining part of R is used as workspace. On exit, part of this array is overwritten. Specifically, if METH = 'M', R_22 and R_31 are overwritten if JOB = 'B' or 'D', and R_12, R_22, R_14 : R_34, and possibly R_11 are overwritten if JOBCV = 'C'; if METH = 'N', all needed submatrices are overwritten. LDR INTEGER The leading dimension of the array R. LDR >= 2*(M+L)*NOBR. A (input or output) DOUBLE PRECISION array, dimension (LDA,N) On entry, if METH = 'N' and JOB = 'B' or 'D', the leading N-by-N part of this array must contain the system state matrix. If METH = 'M' or (METH = 'N' and JOB = 'A' or 'C'), this array need not be set on input. On exit, if JOB = 'A' or 'C' and INFO = 0, the leading N-by-N part of this array contains the system state matrix. LDA INTEGER The leading dimension of the array A. LDA >= N, if JOB = 'A' or 'C', or METH = 'N' and JOB = 'B' or 'D'; LDA >= 1, otherwise. C (input or output) DOUBLE PRECISION array, dimension (LDC,N) On entry, if METH = 'N' and JOB = 'B' or 'D', the leading L-by-N part of this array must contain the system output matrix. If METH = 'M' or (METH = 'N' and JOB = 'A' or 'C'), this array need not be set on input. On exit, if JOB = 'A' or 'C' and INFO = 0, or INFO = 3 (or INFO >= 0, for METH = 'M'), the leading L-by-N part of this array contains the system output matrix. LDC INTEGER The leading dimension of the array C. LDC >= L, if JOB = 'A' or 'C', or METH = 'N' and JOB = 'B' or 'D'; LDC >= 1, otherwise. B (output) DOUBLE PRECISION array, dimension (LDB,M) If M > 0, JOB = 'A', '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 = 'C', this array is not referenced. LDB INTEGER The leading dimension of the array B. LDB >= N, if M > 0 and JOB = 'A', 'B', or 'D'; LDB >= 1, if M = 0 or JOB = 'C'. D (output) DOUBLE PRECISION array, dimension (LDD,M) If M > 0, JOB = 'A' or 'D' and INFO = 0, the leading L-by-M part of this array contains the system input-output matrix. If M = 0 or JOB = 'C' or 'B', this array is not referenced. LDD INTEGER The leading dimension of the array D. LDD >= L, if M > 0 and JOB = 'A' or 'D'; LDD >= 1, if M = 0 or JOB = 'C' or 'B'. Q (output) DOUBLE PRECISION array, dimension (LDQ,N) If JOBCV = 'C', the leading N-by-N part of this array contains the positive semidefinite state covariance matrix to be used as state weighting matrix when computing the Kalman gain. This parameter is not referenced if JOBCV = 'N'. LDQ INTEGER The leading dimension of the array Q. LDQ >= N, if JOBCV = 'C'; LDQ >= 1, if JOBCV = 'N'. RY (output) DOUBLE PRECISION array, dimension (LDRY,L) If JOBCV = 'C', the leading L-by-L part of this array contains the positive (semi)definite output covariance matrix to be used as output weighting matrix when computing the Kalman gain. This parameter is not referenced if JOBCV = 'N'. LDRY INTEGER The leading dimension of the array RY. LDRY >= L, if JOBCV = 'C'; LDRY >= 1, if JOBCV = 'N'. S (output) DOUBLE PRECISION array, dimension (LDS,L) If JOBCV = 'C', the leading N-by-L part of this array contains the state-output cross-covariance matrix to be used as cross-weighting matrix when computing the Kalman gain. This parameter is not referenced if JOBCV = 'N'. LDS INTEGER The leading dimension of the array S. LDS >= N, if JOBCV = 'C'; LDS >= 1, if JOBCV = 'N'. O (output) DOUBLE PRECISION array, dimension ( LDO,N ) If METH = 'M' and JOBCV = 'C', or METH = 'N', the leading L*NOBR-by-N part of this array contains the estimated extended observability matrix, i.e., the first N columns of the relevant singular vectors. If METH = 'M' and JOBCV = 'N', this array is not referenced. LDO INTEGER The leading dimension of the array O. LDO >= L*NOBR, if JOBCV = 'C' or METH = 'N'; LDO >= 1, otherwise.

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).

IWORK INTEGER array, dimension (LIWORK) LIWORK = N, if METH = 'M' and M = 0 or JOB = 'C' and JOBCV = 'N'; LIWORK = M*NOBR+N, if METH = 'M', JOB = 'C', and JOBCV = 'C'; LIWORK = max(L*NOBR,M*NOBR), if METH = 'M', JOB <> 'C', and JOBCV = 'N'; LIWORK = max(L*NOBR,M*NOBR+N), if METH = 'M', JOB <> 'C', and JOBCV = 'C'; LIWORK = max(M*NOBR+N,M*(N+L)), if METH = 'N'. DWORK DOUBLE PRECISION array, dimension (LDWORK) On exit, if INFO = 0, DWORK(1) returns the optimal value of LDWORK, and DWORK(2), DWORK(3), DWORK(4), and DWORK(5) contain the reciprocal condition numbers of the triangular factors of the matrices, defined in the code, GaL (GaL = Un(1:(s-1)*L,1:n)), R_1c (if METH = 'M'), M (if JOBCV = 'C' or METH = 'N'), and Q or T (see SLICOT Library routines IB01PY or IB01PX), respectively. If METH = 'N', DWORK(3) is set to one without any calculations. Similarly, if METH = 'M' and JOBCV = 'N', DWORK(4) is set to one. If M = 0 or JOB = 'C', DWORK(3) and DWORK(5) are set to one. On exit, if INFO = -30, DWORK(1) returns the minimum value of LDWORK. LDWORK INTEGER The length of the array DWORK. LDWORK >= max( LDW1,LDW2 ), where, if METH = 'M', LDW1 >= max( 2*(L*NOBR-L)*N+2*N, (L*NOBR-L)*N+N*N+7*N ), if JOB = 'C' or JOB = 'A' and M = 0; LDW1 >= max( 2*(L*NOBR-L)*N+N*N+7*N, (L*NOBR-L)*N+N+6*M*NOBR, (L*NOBR-L)*N+N+ max( L+M*NOBR, L*NOBR + max( 3*L*NOBR+1, M ) ) ) if M > 0 and JOB = 'A', 'B', or 'D'; LDW2 >= 0, if JOBCV = 'N'; LDW2 >= max( (L*NOBR-L)*N+Aw+2*N+max(5*N,(2*M+L)*NOBR+L), 4*(M*NOBR+N)+1, M*NOBR+2*N+L ), if JOBCV = 'C', where Aw = N+N*N, if M = 0 or JOB = 'C'; Aw = 0, otherwise; and, if METH = 'N', LDW1 >= max( (L*NOBR-L)*N+2*N+(2*M+L)*NOBR+L, 2*(L*NOBR-L)*N+N*N+8*N, N+4*(M*NOBR+N)+1, M*NOBR+3*N+L ); LDW2 >= 0, if M = 0 or JOB = 'C'; LDW2 >= M*NOBR*(N+L)*(M*(N+L)+1)+ max( (N+L)**2, 4*M*(N+L)+1 ), if M > 0 and JOB = 'A', 'B', or 'D'. For good performance, LDWORK should be larger.

IWARN INTEGER = 0: no warning; = 4: a least squares problem to be solved has a rank-deficient coefficient matrix; = 5: the computed covariance matrices are too small. The problem seems to be a deterministic one.

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

In the MOESP approach, the matrices A and C are first computed from an estimated extended observability matrix [1], and then, the matrices B and D are obtained by solving an extended linear system in a least squares sense. In the N4SID approach, besides the estimated extended observability matrix, the solutions of two least squares problems are used to build another least squares problem, whose solution is needed to compute the system matrices A, C, B, and D. The solutions of the two least squares problems are also optionally used by both approaches to find the covariance matrices.

[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.

The implemented method is numerically stable.

In some applications, it is useful to compute the system matrices using two calls to this routine, the first one with JOB = 'C', and the second one with JOB = 'B' or 'D'. This is slightly less efficient than using a single call with JOB = 'A', because some calculations are repeated. If METH = 'N', all the calculations at the first call are performed again at the second call; moreover, it is required to save the needed submatrices of R before the first call and restore them before the second call. If the covariance matrices are desired, JOBCV should be set to 'C' at the second call. If B and D are both needed, they should be computed at once. It is possible to compute the matrices A and C using the MOESP algorithm (METH = 'M'), and the matrices B and D using the N4SID algorithm (METH = 'N'). This combination could be slightly more efficient than N4SID algorithm alone and it could be more accurate than MOESP algorithm. No saving/restoring is needed in such a combination, provided JOBCV is set to 'N' at the first call. Recommended usage: either one call with JOB = 'A', or first call with METH = 'M', JOB = 'C', JOBCV = 'N', second call with METH = 'M', JOB = 'D', JOBCV = 'C', or first call with METH = 'M', JOB = 'C', JOBCV = 'N', second call with METH = 'N', JOB = 'D', JOBCV = 'C'.

**Program Text**

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