具有输入延迟的Kuramoto--Sivashinsky方程湍流解的基于预测器的指数跟踪
Predictor-Based Exponential Tracking of Turbulent Solutions for the Kuramoto--Sivashinsky Equation with Input Delay
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中文总结 AI 辅助
针对具有输入延迟的Kuramoto--Sivashinsky方程，提出基于预测器的反推控制，实现湍流解的均匀指数跟踪，数值验证了补偿延迟的有效性。
中文摘要 AI 辅助
针对受恒定输入延迟影响的非线性Kuramoto--Sivashinsky方程的非平稳轨迹，研究了均匀指数跟踪问题。参考轨迹属于全局吸引子中的一族完整轨迹，不必是平稳的、周期的、缓慢变化的或由有限维外系统生成的。延迟输入由一阶输运方程表示，该方程与四阶跟踪误差动力学耦合。预测器-反推变换通过将增广闭环系统映射为由齐次输运子系统的出射迹驱动的标称无延迟误差动力学，补偿了命令生成与执行之间的时间失配。该子系统在一个延迟间隔后消失。均匀吸引子界允许反馈参数和稳定性常数独立于初始时间和参考轨迹进行选择。在增广状态空间中建立了全局适定性和均匀指数稳定性。数值结果表明，对于所考虑的配置，未补偿的延迟反馈会放大跟踪误差，而预测器补偿则恢复持续衰减。预测器一致性、有限时间消失和离散化细化诊断支持了数值实现。
英文摘要
Uniform exponential tracking is addressed for nonstationary trajectories of the nonlinear Kuramoto--Sivashinsky equation subject to a constant input delay. The reference belongs to a family of complete trajectories contained in the global attractor and need not be stationary, periodic, slowly varying, or generated by a finite-dimensional exosystem. The delayed input is represented by a first-order transport equation coupled with the fourth-order tracking-error dynamics. A predictor--backstepping transformation compensates for the temporal mismatch between command generation and actuation by mapping the augmented closed-loop system into the nominal delay-free error dynamics driven by the outgoing trace of a homogeneous transport subsystem. This subsystem vanishes after one delay interval. Uniform attractor bounds permit the feedback parameters and stability constants to be selected independently of the initial time and the reference trajectory. Global well-posedness and uniform exponential stability are established in the augmented state space. Numerical results show that, for the considered configuration, uncompensated delayed feedback amplifies the tracking error, whereas predictor compensation restores sustained decay. Predictor-consistency, finite-time-extinction, and discretization-refinement diagnostics support the numerical implementation.
发表机构
- CRAN, CNRS UMR 7039, University of Lorraine（洛林大学）
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