Recursive Estimation of ARX Systems Using Binary Sensors with Adjustable Thresholds

We consider the identification of ARX systems which are observed via a binary sensor. Previous solutions typically assumed the complete knowledge of the noise distribution and that the inputs can be chosen by the user. Here, we only make mild assumptions on the noise and we make no assumptions that we can control the inputs. However, we assume that we can choose the threshold of the binary sensor or, equivalently, we can apply a dither signal to the system. We propose two recursive algorithms for this problem. The first one is based on an FIR approximation of the ARX system and requires post-processing. We prove that it provides a strongly consistent estimator. The other algorithm estimates the parameters directly, without post-processing, by simultaneously estimating the parameters and the outputs of the system. Numerical experiments which show that both algorithms work effectively are also presented.