A Modified Lifting for Aperiodic Sampled‐Data Systems and L1$$ {L}_1 $$‐Induced Norm Analysis

ABSTRACT This paper is concerned with sampled‐data systems with uncertain sampling intervals by introducing a modified version of the existing lifting technique, which is tailored only for periodic sampled‐data systems. The modified lifting technique is equipped with an aperiodic truncation and a norm‐preserving normalization for functions, and converts an aperiodic sampled‐data system to an operator‐based linear time‐varying discrete‐time system. This allows us to derive a tractable expression for the ‐induced norm of aperiodic sampled‐data systems. For a sophisticated analysis of the ‐induced norm, a gridding‐based approach is applied to the set of admissible sampling sequences, by which an approximate but asymptotically exact value of this induced norm is obtained. More precisely, the ‐induced norm is reduced to the maximum of the 1‐norms of finite‐dimensional matrices over a gridding‐based finite‐dimensional set, ensuring that the associated error converges to 0 as the gridding parameter becomes larger. Finally, a numerical study is conducted to verify the effectiveness of the proposed method.