Sliding Mode Control with Time-Varying Sliding Surface for Linear Uncertain Impulsive Systems

This paper presents a new sliding mode control (SMC) design method for linear uncertain impulsive systems, where the sliding function is designed to be linear with a time-varying projection matrix. When the time-varying projection matrix satisfies a so-called continuity condition, the sliding function is continuous along the state trajectories, which will facilitate the analysis and design of SMC laws. To obtain a tractable design algorithm on the time-varying projection matrix, a regular form is introduced for the representation of the considered impulsive system. Within this framework, the projection matrix is represented as a piecewise time-varying form based on a partition on the impulse intervals, which can be determined by finite constant gain matrices. Then, a piecewise Lyapunov function associated with the partition is constructed to analyze the stability of the reduced-order sliding dynamics. By this means, the solvability condition for the desired sliding function is obtained by solving a convex optimization problem. Finally, a numerical example which considers four types of impulses is provided to show the effectiveness of the proposed design scheme.