System identification under saturated precise or set-valued measurements

This paper considers the system identification problem based on saturated precise or set-valued measurements, which is widely used in various fields and has essential difficulties. Since existing methodologies cannot make full use of the mixed data, this paper is aiming to fill the gap and build a unified framework in dealing with such problems rigorously and comprehensively. New algorithms are introduced and their properties are established. Most significantly, the Cramér-Rao (CR) lower bound based on the measurements is established, which consists of two parts with respect to the precise data and set-valued data, respectively. This prompts the idea of designing an estimation algorithm by grouping and combining the estimations under two classifications of data. As a result, a CR lower bound-based algorithm (CRBA) is constructed. The convergence properties are theoretically analyzed in terms of consistency and asymptotic efficiency under periodic inputs. For general inputs, an algorithm based on the CRBA that combines the expectation maximization (EM) algorithm for set-valued subsystems and the gradient descent algorithm for precise subsystems is proposed. Numerical simulations validate the superiority of the proposed algorithms.