Performance analysis of second-order sliding-mode control systems with fast actuators

The robustness of the second-order sliding-mode control (2-SMC) algorithm known as the "Generalized Sub-optimal" algorithm is analyzed with respect to the cascade introduction of a linear fast actuator. It is shown that if the actuator dynamics are sufficiently fast then the system trajectories converge to an invariant set that includes the second-order sliding domain s = ṡ =0. It is also shown that the size of the invariant set has quadratic dependence with respect to the actuator "small parameter" μ for the sliding variable s, and linear dependence for its derivative ṡ. This means that in the steady state the system trajectories converge to an invariant domain described by the following conditions: s ≤ O(μ²) and ṡ ≤ O(μ). A simulation example is given to confirm the proposed analysis.