不确定非仿射MIMO系统的精确PID和PI增益区域
Exact PID and PI Gain Regions for Uncertain Non-Affine MIMO Systems
- State Key Laboratory of Mathematical Sciences, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院数学科学重点实验室）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文针对两类非仿射MIMO系统，通过公共二次内区域与线性子类外区域重合及边界族无损约简，精确刻画了PID和PI增益区域，并严格扩大了现有充分区域。
AI中文摘要：
刻画所有保证不确定非线性系统一致指数调节的PID增益仍然是一个基本问题。现有的大多数结果仅提供由特定Lyapunov构造导出的充分条件。本文针对具有$m\ge2$个受控坐标和输入通道的两类非仿射MIMO系统，给出了由整个不确定性类上的一致调节直接定义的精确增益区域。对于PID控制下的二阶系统，我们独立地刻画了一个公共二次内区域和一个基于线性子类的外区域。这两个区域重合，因此在系统层面上精确刻画了PID增益区域。对于PI控制下的一阶系统，在不确定性边界上无损约简为线性族，结合公共二次证书，得到了精确的PI区域。这两个区域都严格扩大了现有的充分区域，并且PID结果还提供了顺序整定规则。一个具有一个不确定参数的紧凑线性族表明，公共二次认证通常不是无损的。这些结果确立了端点重合和边界族等价性作为获得精确增益区域的两种途径，并引发了更广泛的问题：哪些非线性不确定性类允许这样的刻画。
英文摘要：
Characterizing all PID gains that guarantee uniform exponential regulation of uncertain nonlinear systems remains a fundamental problem. Most existing results provide only sufficient conditions derived from a particular Lyapunov construction. This paper gives exact gain regions, defined directly by uniform regulation over the entire uncertainty class, for two classes of nonaffine MIMO systems with $m\ge2$ controlled coordinates and input channels. For systems of second order under PID control, we independently characterize a common quadratic inner region and an outer region based on a linear subclass. The two regions coincide and therefore characterize the PID gain region exactly at the system level. For systems of first order under PI control, a lossless reduction to a linear family on the uncertainty boundary, combined with a common quadratic certificate, yields the exact PI region. Both regions strictly enlarge existing sufficient regions, and the PID result also provides a sequential tuning rule. A compact linear family with one uncertain parameter shows that common quadratic certification is not generally lossless. These results establish endpoint coincidence and boundary family equivalence as two routes to exact gain regions and motivate the broader question of which nonlinear uncertainty classes admit such characterizations.
