Model-free adaptive control based on grey number measure for non-Newtonian mechanical systems with bounded disturbances

In the literature, it is presumed that the pattern class variable and its measured values are equivalent for the non-Newtonian mechanical systems. This presumption ignores the existence of uncertainty in the cross mapping process within a bounded range. In this work, a new measurement of uncertainties based on grey numbers, consisting of time-varying bounded disturbances, is proposed to represent the pattern class variable. On this basis, an equivalent data model with disturbance compensation is established through the dynamic linearization technique. Motivated by measurable and unmeasurable disturbance scenarios, two modified model-free adaptive control (MFAC) methods are proposed. In particular, the grey number measure based MFAC (GM-MFAC) scheme is put forward using the disturbance compensation parameter and the quantified classification algorithm. Moreover, an improved Kalman filter based MFAC (HOKF-MFAC) scheme is designed to handle measurement noise, which adopts both high-order pseudo partial derivatives (HOPPD) and the optimal estimation output. This paper delves into the convergence properties of parameter estimation and tracking errors, and further conducts a statistical analysis to evaluate robustness against measurement disturbances. Two numerical examples are carried out to validate the effectiveness of the proposed methods.