An improved subspace identification method for bilinear systems

Several subspace algorithms for the identification of bilinear systems have been proposed. A key practical problem with all of these is the very large size of the data-based matrices which must be constructed in order to `linearise´ the problem and allow parameter estimation essentially by regression. Favoreel et al. (1997) proposed an algorithm which gave unbiased results only if the measured input signal was white. Favoreel and De Moor (1998) suggested an alternative algorithm for general input signals, but which gave biased estimates. Chen and Maciejowski proposed algorithms for the deterministic (2000) and combined deterministic-stochastic (2000) cases which give asymptotically unbiased estimates with general inputs, and for which the rate of reduction of bias can be estimated. The computational complexity of these algorithms was also significantly lower than the earlier ones, both because the matrix dimensions were smaller, and because convergence to correct estimates (with sample size) appears to be much faster. In this paper, we reduce the matrix dimensions further, by making different choices of subspaces for the decomposition of the input-output data. In fact we propose two algorithms: an unbiased one for the case of l⩾n, (where l: number of outputs, n: number of states), and an asymptotically unbiased one for the case l<n. In each case, the matrix dimensions are smaller than in earlier algorithms. Even with these improvements, the dimensions remain large, so that the algorithms are currently practical only for low values of n