On Parametric Uncertainty in Dynamically Perturbed Sliding Mode Controlled Systems

The unmodeled dynamics inside an SMC control loop such as actuators, sensors, time-delays, etc., dynamically perturb their close-loop response, inducing chattering. Dynamically perturbed SMC systems have been widely analyzed in the frequency domain via the Describing Function (DF), Tzypkin method, Locus of a Perturbed Relay System (LPRS), and others, that require a linear representation of the plant (usually given as a transfer function) to later estimate the resulting chattering parameters. However, if parametric variation/uncertainty is present, a unique value of the chattering parameters cannot be guaranteed. In this paper, a method to analyze dynamically perturbed SMC with parametric uncertainty is presented. Parametric uncertainty is addressed as a family of interval second-order transfer functions, formed by cascading a first-order actuator with a plant with relative-degree of one. The proposed method identifies (in closed-form) the member system among the interval, corresponding with the marginal chattering parameters. Hence, leading to the worst-case condition for the whole systems' family and enabling direct design criteria. Analytic and simulated examples to validate the proposed methods are presented.