Subspace model identification Part 2. Analysis of the elementary output-error state-space model identification algorithm

Abstract

The elementary MOESP algorithm presented in the first part of this series of papers is analysed in this paper. This is done in three different ways. First, we study the asymptotic properties of the estimated state-space model when only considering zero-mean white noise perturbations on the output sequence. It is shown that, in this case, the MOESPl implementation yields asymptotically unbiased estimates. An important constraint to this result is that the underlying system must have a finite impulse response and subsequently the size of the Hankel matrices, constructed from the input and output data at the beginning of the computations, depends on the number of non-zero Markov parameters. This analysis, however, leads to a second implementation of the elementary MOESP scheme, namely MOESP2. The latter implementation has the same asymptotic properties without the finite impulse response constraint. Secondly, we compare the MOESP2 algorithm with a classical state space model identification scheme. The latter scheme, referred to as the CLASSIC algorithm, is based on the Ho and Kalman realization scheme and estimated Markov parameters. The comparison is done by a sensitivity study, where the effect is studied of the errors on the data on the calculated column space of the shift-invariant subspace. This study demonstrates that the elementary MOESP2 scheme is more robust with respect to the errors considered than the CLASSIC algorithm. In the third part, the model reduction capabilities of the elementary MOESP schemes are analysed when the observations are error-free. We demonstrate in which sense the reduced order model is optimal when acquired with the MOESP schemes. The optimality is expressed by the difference between the 2-norm of the errors on the state (or output) sequence of the reduced-order model and the 2-norm of the matrix containing the rejected singular values being as small as possible. The insights obtained in these three parts are evaluated in a simulation study, and validated in this paper. They lead to the assertion that the MOESP2 implementation allows identification of a compact, low-dimensional, state-space model accurately describing the input -output behaviour of the system to be identified, while making use of ‘perturbed’ input-output data. This can be done efficiently.