Invariant Sets in Sliding Mode Control Theory with Application to Servo Actuator System with Friction

The mismatched perturbations and system chattering are the two main challenging problems in sliding mode control. This paper tries to solve these problems by deriving the invariant sets created by the sliding mode controller where the present work is devoted to a second order nonlinear affine system. If the state started in these sets it will not leave it for all future time. The first invariant set is found function to the initial condition only. Accordingly, the state bound is estimated and used when determining the gain of the sliding mode controller. This step overcomes an arithmetic difficulty that consists of calculating suitable controller gain value that ensures the attractiveness of the switching manifold with lower chattering behavior. Moreover to eliminate system chattering and to attenuate the effects of the mismatched perturbations, the signum function is replaced by an approximate form which yields a differentiable sliding mode controller. Therefore, the state will converge to a second positively invariant set rather than the origin. The size of this set, as derived here, is function to the parameters that can be chosen by the designer. This result enables us to control the size of the steady state error which means also that the effect of mismatched perturbation is attenuated. The sliding mode controller is then applied to the servo actuator system with friction based on the derived invariant sets. The friction model is represented by the major friction components; Coulomb friction, the Stiction friction, and the viscous friction. The simulation results demonstrate the rightness of the derived sets and the ability of the differentiable sliding mode controller to attenuate the friction effect and regulate the state to the positively invariant set with a prescribed steady state error. Key-Words: -Positively Invariant Set, Sliding Mode Control, Servo Actuator, Friction Model.