Refined instrumental variable methods for unstable continuous-time systems in closed-loop

In continuous-time system identification, refined instrumental variable methods are widely used in open and closed-loop settings. Although their robustness and performance are well documented for stable systems, these estimators are not reliable for estimating unstable continuous-time models. The main difficulty we encounter in modeling unstable systems with refined instrumental variables is that the filtered regressor and instrument vectors, as well as the filtered output, become severely ill-conditioned if the model is unstable during the iterative process. In this work, we propose a solution to this problem by including a tailor-made all-pass filter in the prefiltering step. This approach is used for obtaining an extension of the least-squares state-variable filter method, as well as extensions for the refined instrumental variable method for continuous-time systems (RIVC) and its simplified embodiment (SRIVC), that admit the identification of unstable systems and are shown to minimise the prediction error upon convergence and as the sample size goes to infinity. In addition, several implementations of these methods are proposed depending on the intersample behaviour of the input (zero and first-order hold, multisine and arbitrary). The particular case when the plant has integral action is explicitly considered in this work. In an indirect system identification setting, an extension of the closed-loop version of the SRIVC method is also proposed and discussed in detail. Monte Carlo simulations are used to assess the performance of our methods.