System identification with multi-threshold quantized observations and bounded persistent excitations

This paper takes the gain system identification with multi-threshold quantization observations as an example to explore how to select the optimal thresholds and quantized values. A projection recursive algorithm is proposed to estimate the parameter and proved to be both mean-square and almost surely convergent under a class of persistently exciting inputs. The upper bound of the convergence rate is also obtained, which has the same order as the one of optimal estimation in the case where the system output is accurately measured and not quantized. Then, the asymptotic property is analyzed and the optimal scheme of quantization values and thresholds is given by use of the multi-linear transformation. A numerical example is used to demonstrate the effectiveness of the algorithms and the main results obtained.