不确定离散Delta系统的有限时间稳定性分析

Finite-Time Stability Analysis for Uncertain Discrete-Time Delta Systems

摘要: 针对参数不确定和外部扰动下的离散Delta系统，研究其有限时间稳定性问题。通过引入Delta算子描述离散系统，避免传统移位算子在高速采样下的数值病态问题。利用Lyapunov稳定性理论、young不等式等数学方法，建立系统有限时间稳定的充分条件。通过构造二次型Lyapunov函数，结合参数不确定性的范数有界条件和外部扰动的能量有界假设，推导出以LMI形式表示的稳定性判据。数值算例结果表明，所提方法不仅能有效保证系统有限时间稳定，更通过多采样周期对比仿真，凸显了Delta算子在高速采样下相较于传统方法的独特优势，即优异的数值一致性与稳定性，为高速采样场景下离散控制系统的分析与设计提供了更可靠的理论工具。

Abstract: This paper addresses the finite-time stability problem for discrete-time Delta systems subject to parameter uncertainties and external disturbances. The Delta operator is introduced to describe the discrete system, effectively avoiding the numerical ill-conditioning issues associated with traditional shift operators under high-speed sampling conditions. Utilizing Lyapunov stability theory, Young’s inequality, and other mathematical tools, sufficient conditions for finite-time stability are established. By constructing a quadratic Lyapunov function, combined with the norm-bounded condition for parametric uncertainties and the energy-bounded assumption for external disturbances, a stability criterion expressed in the form of Linear Matrix Inequalities (LMIs) is derived. Numerical results demonstrate that the proposed method not only effectively ensures finite-time stability but also, through multi-rate sampling comparison simulations, highlights the unique advantages of the Delta operator in high-speed sampling scenarios—namely, superior numerical consistency and stability. This work provides a more reliable theoretical tool for the analysis and design of discrete control systems in high-speed sampling applications.