On- and off-line identification of linear state-space models

A geometrically inspired matrix algorithm is derived for the identification of statespace models for multivariable linear time-invariant systems using (possibly noisy) input-output (I/O) measurements only. As opposed to other (mostly stochastic) identification schemes, no variance-covariance information whatever is involved, and only a limited number of I/O-data are required for the determination of the system matrices. Hence, the algorithm can be best described and understood in the matrix formalism, and consists of the following two steps. First, a state vector sequence is realized as the intersection of the row spaces of two block Hankel matrices, constructed with I/O-data. Then the system matrices are obtained at once from the least-squares solution of a set of linear equations. When dealing with noisy data, this algorithm draws its excellent performance from repeated use of the numerically stable and accurate singular value decomposition. Also, the algorithm is easily applied to slowly time-varying systems using windowing or exponential weighting. These results are illustrated by examples, including the identification of an industrial plant.