A Subspace Approach for Identifying Bilinear Systems with Deterministic Inputs

In this paper we introduce an identification algorithm for MIMO bilinear systems subject to deterministic inputs. The new algorithm is based on an expanding dimensions concept, leading to a rectangular, dimension varying, linear system. In this framework the observability, controllability, and Markov parameters are similar to those of a time-varying system. The fact that the system is time invariant, leads to an equaivaleet linear deterministic subspace algorithm. Provided a rank condition is satisfied, the algorithm will produce unbiased parameter estimates. This rank condition can be guaranteed to hold if the ratio of the number of outputs to the number of inputs is larger than the system order. This is due to the typical exponential blow-out in the dimensions of the Hankel data matrices of bilinear systems, in particular for deterministic inputs since part of the input subspace cannot be projected out. Other algorithms in the literature, based on Walsh functions, require that the number of outputs is at least equal to the system order. For ease of notation and clarification, the algorithm is presented as an intersection based subspace algorithm. Numerical results show that the algorithm reproduces the system parameters very well, provided the rank condition is satisfied. When the rank condition is not satisfied, the algorithm will return biased parameter estimates, which is a typical bottleneck of bilinear system identification algorithms for deterministic inputs.