不确定二阶MIMO非线性系统全局镇定的必要和充分PID增益区域
Necessary and Sufficient PID Gain Regions for Global Stabilization of Uncertain Second-Order MIMO Nonlinear Systems
AI总结：
研究不确定二阶MIMO非线性系统全局镇定的PID增益区域，通过构造充分区域保证全局稳定性和渐近跟踪，推导必要区域揭示保守性，在特定假设下给出精确充要表征，区域以封闭形式给出，仅取决于相关界和输入增益下界。
摘要：
众所周知，经典PID控制在工业过程中无处不在，但针对非线性不确定MIMO二阶系统的严格且明确的设计理论仍不完善。本文考虑一类具有不确定动力学和未知但严格正输入增益的此类系统，其中非线性不确定性由关于状态变量的雅可比矩阵的界来表征。我们明确构造了一个三维PID增益区域，足以保证满足这些雅可比矩阵界的所有非线性系统的全局稳定性和对恒定参考的渐近跟踪。然后我们推导了相应的必要区域，从而揭示了应对最坏情况不确定性所需的固有保守性。此外，在对非线性的额外结构假设下，这些充分和必要区域重合，给出了所有全局镇定PID增益的精确充要表征。所有这些区域都以封闭形式给出，并且仅取决于规定的雅可比矩阵界和输入增益的已知下界，这与文献中的许多定性调谐方法形成对比。
英文摘要：
As is well known, classical PID control is ubiquitous in industrial processes, yet a rigorous and explicit design theory for nonlinear uncertain MIMO second-order systems remains underdeveloped. In this paper we consider a class of such systems with both uncertain dynamics and an unknown but strictly positive input gain, where the nonlinear uncertainty is characterized by bounds on the Jacobian with respect to the state variables. We explicitly construct a three-dimensional region for the PID gains that is sufficient to guarantee global stability and asymptotic tracking of constant references for all nonlinearities satisfying these Jacobian bounds. We then derive a corresponding necessary region, thereby revealing the inherent conservatism required to cope with worst-case uncertainties. Moreover, under additional structural assumptions on the nonlinearities, these sufficient and necessary regions coincide, yielding a precise necessary-and-sufficient characterization of all globally stabilizing PID gains. All these regions are given in closed form and depend only on the prescribed Jacobian bounds and the known lower bound of the input gain, in contrast to many qualitative tuning methods in the literature.
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