Higher-Order Sliding-Mode Control for Linear Time-Varying Systems

In this paper, assuming uniform controllability, a formula to select an output for desired relative degree and zero dynamics in single-input linear time-varying (LTV) systems is developed. The proposed formula allows us to construct a Lyapunov transformation, such that the original LTV system is Lyapunov equivalent to a linear time-invariant one with prescribed zero dynamics. This formula is then suggested to be used as a part of a two-step higher-order sliding-mode control strategy for LTV systems, ensuring finite-time stability in the presence of matched perturbations. It is also shown that if the formula is used to select an output of relative degree equal to the system order, then the formula to select a flat output is recovered. The proposed method is tested on an extended version of a perturbed mass-spring system.