标量二阶非线性不确定系统PID镇定的一种较不保守的充分条件
A Less Conservative Sufficient Condition for PID Stabilization of Scalar Second-Order Nonlinear Uncertain Systems
AI总结：
针对标量二阶非线性不确定系统，提出一种端点平衡的二次加积分Lyapunov函数，以降低PID镇定条件的保守性，保证全局渐近调节。
摘要：
本文研究使用经典定增益PID控制器对标量二阶非线性不确定系统进行鲁棒设定点调节。标量二阶模型为非线性机械和机电动力学提供了最小原型，而其速度相关项捕获了物理阻尼和摩擦等不确定性。对于正的速度导数界，现有的Lyapunov充分条件所证明的定增益PID参数区域与从最坏情况线性模型获得的必要条件对应的边界保持分离。为降低这种保守性，本文提出一种端点平衡的二次加积分Lyapunov函数。关键思想是选择二次交叉项系数，使得在提取标量PID不等式之前，混合项惩罚在可容许有效阻尼区间的两个端点处平衡。所得条件保证了整个导数有界不确定性类别的全局渐近调节。当速度导数界为正时，所提条件证明的定增益PID区域严格包含Zhao-Guo和Zhang-Guo所证明的区域。当该界为零时，相应边界与必要边界重合。在Lyapunov分析层面，该构造降低了整个有效阻尼区间上的均匀混合项惩罚。
英文摘要：
This letter studies robust set-point regulation of scalar second-order nonlinear uncertain systems using a classical PID controller with constant gains. The scalar second-order model provides a minimal prototype for nonlinear mechanical and electromechanical dynamics, while its velocity-dependent term captures uncertainties such as physical damping and friction. For a positive velocity-derivative bound, existing Lyapunov sufficient conditions certify fixed-gain PID parameter regions that remain separated from the boundary associated with the necessary condition obtained from the worst-case linear model. To reduce this conservatism, this letter proposes an endpoint-balanced quadratic-plus-integral Lyapunov certificate. The key idea is to choose the quadratic cross-term coefficient so that the mixed-term penalty is balanced at the two endpoints of the admissible effective-damping interval before extracting the scalar PID inequality. The resulting condition guarantees global asymptotic regulation for the full derivative-bounded uncertainty class. When the velocity-derivative bound is positive, the proposed condition certifies a fixed-gain PID region that strictly contains those certified by Zhao--Guo and Zhang--Guo. When this bound is zero, the corresponding boundary coincides with that necessary boundary. At the level of Lyapunov analysis, the construction reduces the uniform mixed-term penalty over the entire effective-damping interval.
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