MIMO system identification: state-space and subspace approximations versus transfer function and instrumental variables

The identification of multi-input multi-output (MIMO) linear systems has previously received a new impetus with the introduction of the state-space (SS) approach based on subspace approximations. This approach has immediately gained popularity, owing to the fact that it avoids the use of canonical forms, requires the determination of only one structural parameter, and has been empirically shown to yield MIMO models with good accuracy in many cases, However, the SS approach suffers from several drawbacks: there is no well-established rule tied to this approach for determining the structural parameter, and, perhaps more important the SS parameter estimates depend on the data in a rather complicated way, which renders almost futile any attempt to analyze and optimize the performance of the estimator. In this paper, we consider a transfer function (TF) approach based on instrumental variables (IV), as an alternative to the SS approach. We use the simplest canonical TF parameterization in which the denominator is equal to a scalar polynomial times the identity matrix. The analysis and optimization of the statistical accuracy of the TF approach is straightforward. Additionally, a simple test tailored to this approach is devised for estimating the single structural parameter needed. A simulation study, in which we compare the performances of the SS and the TF approaches, shows that the latter can provide more accurate models than the former at a lower computational cost.