离散时间 LTI 系统的精确不稳定半径
Exact Instability Radius of Discrete-Time LTI Systems
AI总结：
本文提出离散时间 LTI 系统的精确鲁棒不稳定半径，通过峰值增益频率处的相位变化率最大化与小增益条件证明一阶全通摄动最优，并以磁悬浮控制和 FitzHugh–Nagumo 神经尖峰模型验证其实用性。
摘要：
受稳定摄动影响的不稳定被控对象的鲁棒不稳定性具有重要意义，并出现在非线性系统持续振荡现象的研究中。本文通过鲁棒不稳定半径（RIR）这一度量不稳定性的概念，分析线性离散时间系统在稳定摄动下的鲁棒不稳定性。我们利用小增益型条件，将问题表述为在唯一峰值增益频率处受适当约束的相位变化率最大化问题，从而确定了某些不稳定系统的精确 RIR；其中稳定的一阶全通函数被证明是最优的。文中给出了两个实际应用——磁悬浮系统的最小努力采样数据控制，以及受摄动的 FitzHugh–Nagumo 模型中的神经尖峰生成——以说明所得结果的效用。
英文摘要：
The robust instability of an unstable plant subject to stable perturbations is of significant importance and arises in the study of sustained oscillatory phenomena in nonlinear systems. This paper analyzes the robust instability of linear discrete-time systems against stable perturbations via the notion of robust instability radius (RIR) as a measure of instability. We determine the exact RIR for certain unstable systems using small-gain type conditions by formulating the problem in terms of a phase change rate maximization subject to appropriate constraints at unique peak-gain frequencies, for which stable first-order all-pass functions are shown to be optimal. Two real-world applications -- minimum-effort sampled-data control of magnetic levitation systems and neural spike generations in the FitzHugh--Nagumo model subject to perturbations -- are provided to illustrate the utility of our results.
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